U.U.D.M. Project Report 2011:12

A Comparison of Local Volatility and Implied Volatility

Hui Ye

Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011

Department of Mathematics Uppsala University

Preface

Published in 1973, the Black-Scholes model has undoubtfully become one of the most frequently used models in the financial derivative pricing field during the past few decades. As the option pricing benchmark, the Black-Scholes model has shown an extensive application range, however, some assumptions originally made in order to derive this model have been found unlikely to hold in the reality. One of such controversial assumptions is that the volatility of underlying asset remains constant throughout the entire option life. This has been criticized by more and more researchers and practitioners as it is not in line with the research results on volatilities and the observations accumulated from the real market data. Because of this shortcoming of Black-Scholes model, people are eager to find such a model that incorporates the variability of the implied volatility in the estabilishment of the model and still has similar volatility numerical values as opposed to those implied by the Black-Scholes model. Our paper here takes a step in that direction by investigating the replicating ability of a local volatility model. This study is featured by focusing on the relationships between the implied volatility inferred by the Black-Scholes model, the local volatility specified by the local volatility model and the volatility given by the Dupire's formula for implied volatility.

1 Abstract

This paper mainly studies the relationships between the implied volatilities inferred by the Black- Scholes model and the volatilities derived by the local volatility model. By studying the difference between other volatilities and the implied volatilities, we can search for models that have similar volatilities to those of Black-Scholes models, and yet still process more realistic and plausible price processes that do not depend on a constant volatility, unlike the ones of the Black-Scholes models. Our search for such models are illustrated by the local volatility model.

2 Acknowledgement

My gratitude and appreciation goes out to both of my supervisors. I would like thank Prof. Johan Tysk for all the valuable advice during this study. Also I would like to thank Senior Lecturer Erik Eström for the inspiration of the topic of this study. I would not be able to complete this study and finish this paper without their constructive advice, kind help and sincere critiques. I personally benefited a lot from this learning experience. I am really grateful for their help and this studying opportunity. I am looking forward to working with them again someday.

3 Contents

Chapter 1 Introduction------5

1.1 Motivation------5

1.2 Objectives------5

1.3 Chapter Review------6

Chapter 2. Background------8

2.1 The Local Volatility Models------8

2.2 The Dupire Model(Method)------10

Chapter 3 Implied Volatility Models------17

3.1 The Local Volatility Model------17

3.1.1 Option Pricing------17

3.1.2 Implied Volatilities and The Local Volatilities------22

3.1.3 Implied Volatilities and The Dupire Volatilities------41

3.1.4 Summary of Three Types of Volatilities------55

Chapter 4 Conclusions and Future Studies------58

Notation------60

Appendix A------61

Appendix B------89

Bibliography------130

4 Chapter 1. Introduction

1.1 Motivation

In general, volatility is a measure for variation of price of a certain financial instrument over time in finance. There are many types of volatilities categorized by different standards. For example, historical volatility is a type of volatility derived from time series based on the past market prices; a constant volatility is an assumption of the nature of volatility that we usually make in deriving the Black-Scholes formula for option prices. An implied volatility, however, is a type of volatility derived from the market-quoted data of a market traded derivative, such as an option. One of the most frequently used models, the Black-Scholes model which assumes a constant volatility is used to derive the corresponding implied volatility for each quoted market price for options. Indeed, the Black-Scholes model has been a great contribution to option pricing area Nevertheless, there are still some facts that contradict the key assumptions in Black-Scholes model, especially the constant volatility assumption. The evidence to this contradiction is a long-observed pattern of implied volatilities, in which at-the money options tend to have lower implied volatilities than in- or out-of-the-money options. This pattern is called "the volatility smile"(sometimes referred to as "volatility skew") which was starting to show in American markets after the huge stock market crash in 1987. One explanation for this phenomenon is that in reality the volatility of an underlying asset is not really a constant value throughout the lifespan of the derivative. That is why the volatility curve plotted by the using of the values of implied volatility inferred by Black-Scholes model does not appear to be horizontal, but displays a "volatility smile" in the plots. This, however, motivates us to wonder whether such a model can be found, that gives a series of values of volatility close enough to the volatility values in the volatility smile, i.e. , the implied volatility; or more specifically what the difference between the volatility given by this alternative model and the corresponding implied volatility inferred by Black-Scholes model is, if any. Having this thought in mind, we can also apply this scheme of searching for suitable models to testing among different types of models. Our demonstration in this paper uses the local volatility model.

1.2 Objectives

In this paper, the alternative model type for the Black-Scholes model we use is the local volatility model.

5 Our objective here is to set up the pricing model for options using the stock price processes and other conditions specified by the local volatility model, solve the option values for this model, calculate the corresponding implied volatilities for this model, thus to achieve our goal of comparing these two volatilities, the implied volatilities and the local volatilities. Besides the local volatility given by the local volatility model, we also want to compare the implied volatilities to another local volatility, the dupire volatility. The Dupire volatility is a way of calculating volatility under the Dupire model, which treats the strike price K and the maturity time T instead of the stock price S and current time point t as variables in the option value function V (K,T;S,t) . We will introduce this Dupire model and Dupire volatility in detail in Chapter 2. This additional analysis would give us some additional points of views to this local volatility model here.

1.3 Chapter Review

Chapter 1, Introduction, mainly talks about the theoretical and practical reasons that motivate us to write about this topic on implied volatility models in this paper, and sets straight the objectives of our research as well. Chapter 2, Background, introduces us to two types of models that will be focused on in the later chapters of this paper. They are the the local volatility models and the Dupire local volatility model. This chapter familiarize us to the basic knowledges of these models, and we will discuss these models in detail including pricing option values and evaluating volatility values in Chapter 3. Chapter 3, The Implied Volatility Models, concentrates on the difference between the implied volatilities that are inferred by Black-Scholes model and the volatility factors that are specified by the local volatility models, with all parameters of these two models staying the same. The former is obtained by solving the volatility implied by the Black-Scholes formula for options reversely with known option values. The latter in our paper here is the local volatility specified by the local volatility model's price processes with known stock prices and time. And also, we are curiously that what the difference between the implied volatility inferred by the Black-Scholes model evaluated by a reverse calculation and the Dupire volatility computed by the Dupire method of calculating volatility(which is called Dupire volatility) is, since both of the option values used in these two models are essentially based on the Black-Scholes model. Hence, in short, between implied volatility and local volatility(of the local volatility model), the implied volatility and the Dupire volatility, we do two sets of cross-references by evaluating the distances between them to find their inner connections. This is also the main idea of searching for a more realistic alternative asset price processes for the Black-Scholes model(each different asset price processes correspond to a specific

6 option ricing model), which has the implied volatility close to that of an asset price process which follows a Black-Scholes model. We finish this chapter by analysing our numerical results and plots. Chapter 4, Conclusions, which sums up the conclusions for our research and the results in this paper.

7 Chapter 2. Background

In this chapter, we briefly introduce the models we use in this paper.

2.1 The Local Volatility Models

In the 1970s, when Black-Scholes formula was initially derived, most people were convinced that the volatility of a certain asset given the current circumstance was a constant number. Then, later on,after the economic crash in 1987, people were starting to doubt the constant volatility assumption. Especially after more and more evidence of volatility smile was collected, people tend to believe that the implied volatilities can not remain constant during the whole time. They probably have some dependent relationships with some other factors in the option pricing model as well. One of such guesses is that, the implied volatility could be depending on the stock price S(t) and time t . And if we study a model of price processes with a volatility that depends on the stock price S(t) and time t , we can try to explore the inner connection between the implied volatility σ imp , and the local volatility σ (S(t),t) . The volatility in such models depends on the stock price S(t) and time t . This is why we call these types of models the local volatility models, whose volatilities are determined locally. Hence, we take one example out of this category, and consider a case where the volatility is decreasing with respect to the stock prices. Given the local volatility model under an EMM(equivalent martingale measure, we use the same acronym in the following) Q as following,

dS = σ (S,t) ⋅ S ⋅ dWt + r ⋅ dt , (2.1) where we assume, 1 σ (St ) = , (2.2) St

r = 0 . (2.3) By (2.2) and (2.3), the original model (2.1) is degenerated into the following form:

dSt = St ⋅dWt . (2.4)

Denote the option value function as V (St ,t) .

Hence, it follows from Ito formula and equation (2.4) that,

∂V ∂V 1 ∂ 2V dV = dt + dS + (dS)2 ∂t ∂S 2 ∂S 2

8 ∂V ∂V 1 ∂ 2V = dt + dS + ( SdW )2 ∂t ∂S 2 ∂S 2

∂V 1 ∂ 2V ∂V =( + S)dt + dS (2.5) ∂t 2 ∂S 2 ∂S Then we consider delta-hedged portfolio, ∂V π = −V + S . (2.6) ∂S Gven the martingale measure Q , thus under arbitrage-free condition, we will arrive at the condition that, dπ = r ⋅π ⋅dt . (2.7) We re-write (2.6) in differential form that,

∂V dπ = −dV + dS . (2.8) ∂S Compare (2.8) with (2.7), then insert (2.5), the corresponding partial differential equation (PDE) for model (2.4) takes the form,

∂V 1 ∂ 2V + S = 0 . (2.9) ∂t 2 ∂S 2

If we let V (St ,t) , Vt (St ,t) and VSS (St ,t) represent the option value, the first-order partial derivative with respect to variable t , the second-order partial derivative with respect to variable

St , respectively, (2.9) can be expressed in the following way, 1 V (S ,t) + SV (S ,t) = 0 . (2.10) t t 2 ss t Model (2.4) is known as one of the local volatility models, whose form can be included into the SDEMRD Model category inside the matlab database.

Creating the Local Volatility Model from Mean-Reverting Drift (SDEMRD) Models

The SDEMRD class derives directly from the SDEDDO class. It provides an interface in which the drift-rate function is expressed in mean-reverting drift form:

α (t) dX t = S(t)[L(t) − X t ]⋅ dt + D(t, X t )⋅V (t)dWt , (2.11) where，

Xt is an NVARS-by-1 state vector of process variables; S is an NVARS-by-NVARS matrix of mean reversion speeds; L is an NVARS-by-1 vector of mean reversion levels;

9 D is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector raised to the corresponding power of α; V is an NVARS-by-NBROWNS instantaneous volatility rate matrix; dWt is an NBROWNS-by-1 Brownian motion vector.

SDEMRD objects provide a parametric alternative to the linear drift form by reparameterizing the general linear drift such that:

A(t) = S(t)L(t), B(t) = −S(t) . (2.12)

Hence, we can create in matlab the model in

dSt = St ⋅dWt . (2.4) by inputing the following command in Matlab. SDEMRD objects display the familiar Speed and Level parameters instead of A and B.

Table 2.1: The Local Volatility Model in Matlab >> obj = sdemrd(0, 0, 0.5, 1) % (Speed, Level, Alpha, Sigma) obj =

Class SDEMRD: SDE with Mean-Reverting Drift ------Dimensions: State = 1, Brownian = 1 ------StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Alpha: 0.5 Sigma: 1 Level: 0 Speed: 0

2.2 The Dupire Model(Method)

Frankly, this Dupire model is more of a method for calculating local volatilities than a pricing model itself. First of all, let us discuss some basic developments on the implied volatilities so far.

10 One of the basic assumption in Black-Scholes is that the volatility of the underlying asset stays constant during the entire time of option's lifespan. Hence, we can know from the Black-Scholes formula for option prices, that, option prices has the following form V = V (S,t;σ , K,T ).

If we quote from the market date, the option price V = V0 and underlying asset price S = S0 of an option with strike price K = K0 and maturity T = T0 at time point t = t0 , we can obtain an equation from Black-Scholes formula for σ ,

V0 = V (S0 ,t0 ;σ , K0 ,T0 ) . (2.13) From Black-Scholes formula, we can calculate the Greeks, in particular, vega,

∂V ν = = Se−qτ φ(d ) τ = Ke−rτ φ(d ) τ > 0. ∂σ 1 2

Hence σ = σ 0 can be uniquely determined by equation (2.13). Since the volatility σ of the underlying asset is constant by assumption of the Black-Scholes model. Then, theoretically the implied volatility σ = σ 0 derived from (2.13) should be a constant,

i.e., independent of the strike price K0 and maturity T0 chosen here. However, in reality, this is contradicted by the existences of volatility smile and volatility skew. In fact, the implied volatility σ inferred from option prices with different strike prices and expiration dates is a function of K,T , σ = σ (K,T ) [1]. The dependence on strike prices can be shown by the following figure 2.1 and figure. 2.2, given a fixed maturity time T = T0 and a fixed initial price S = S0 at time point t = t0 . The curve in figure 2.1 is called the volatility smile, the curve in figure 2.2 is called the volatility skew. Similarly, the dependence on the maturity time T can be illustrated by curve in figure 2.3, given the stock prices S and the strike prices K stay unchanged. This shows the term structure of volatility.

11 Figure 2.1: Volatility Smile

Figure 2.2: Volatility Skew

Figure 2.3: The Term Structure of Volatility

12 To explore the characteristics of implied volatility in a more mathematical way, let us discuss the model analytically. Under risk-neutral measure, the underlying asset price process is dS = (r − q)dt +σ (S,t)dW , (2.14) S t where r is the risk-free interest rate, q is the dividend yield, S is the asset prices, {Wt }0≤t≤T is a Brownian motion(Wiener process), σ is the asset's volatility that depends on asset prices S and time t . Thus, by using the same approach as in section 2.1, we obtain the PDE for this option under Black- Scholes model,

∂V 1 ∂ 2V ∂V + σ 2 (S,t)S 2 + (r − q)S − rV = 0 . (2.15) ∂t 2 ∂S 2 ∂S Adding the terminal and boundary conditions to equation (2.15), we can estabilish the following value problem for option price, in particular, an European call option price.

Definition 2.1 G(S,t;ξ ,T ) is called the fundamental solution of the Black-Scholes equation, if it satisfies the following terminal value problem to the Black-Scholes equation: ⎧ ∂v σ 2 ∂ 2V ∂V ⎪Lv = + S 2 + (r − q)S − rV = 0, (2.16) ⎨ ∂t 2 ∂S 2 ∂S ⎪ ⎩V (S,T ) = δ (S −ξ ), (2.17) where 0 < S < ∞,0 < ξ < ∞,0 < t < T,δ (x) is the Dirac function.. □

Problem 2.2 Let V = V (S,t;σ , K,T ) be a call option price, satisfying the following terminal value problem: ⎧∂V 1 ∂ 2V ∂V + σ 2 (S,t)S 2 + (r − q)S − rV = 0, (2.18) ⎪ ∂t 2 ∂S 2 ∂S ⎪ ⎨ (0 ≤ S ≤ ∞,0 ≤ t ≤ T ) ⎪ + ⎪V (S,T ) = (S − K) . (0 ≤ S < ∞) (2.19) ⎩⎪

* * * Suppose S = S at t = t ,(0 ≤ t < T1)

* * V (S ,t ;σ , K,T ) = F(K,T ) (0 < K < ∞,T1 ≤ T ≤ T2 ) is given as the boundary condition,

find σ = σ (S,t),(0 ≤ S < ∞,T1 ≤ t ≤ T2 ) . □

Before we are ready to discuss the Dupire method, let us familiarize ourselves with some theoretical background [1] beforehand.

13 Theorem 2.3 If the fundamental solution G(S,t;ξ ,η) is regarded as a function of ξ ,η , then it is the fundamental solution of the adjoint equation of the Black-Scholes equation. That is, let

v(ξ ,η) = G(S,t;ξ ,η) , then v(ξ ,η) satisfies ⎧ ∂v σ 2 ∂ 2 ∂ ⎪L∗v = − + (ξ 2v) − (r − q) (ξv) − rv = 0, (2.20) ⎨ ∂η 2 ∂ξ 2 ∂ξ ⎪ ⎩v(ξ ,t) = δ (ξ − S), (2.21) where 0 < ξ < ∞,0 < S < ∞,t <η1. □

Corollary 2.4 Theorem 2.1 indicates, if the fundamental solution of equation (2.18) is

G* (ξ ,η;S,t) , then

G(S,t;ξ ,η) = G* (ξ ,η; S,t). □

The proof of above theorem 2.1 and corollary 2.2 can be referred to Lishang Jiang(1994)[1]. Then, let us move on to discuss the Dupire method in detail. We denote an European call option price as V = V (S,t; K,T ) , define the second derivative of the option prices with respect to strike prices

∂ 2V = G(S,t; K,T) . (2.22) ∂K 2

By equation (2.22) and (2.23), G satisfies the system that ⎧∂G 1 ∂ 2G ∂G ⎪ + σ 2 (S,t)S 2 + (r − q)S − rG = 0, (2.23) ⎨ ∂t 2 ∂S 2 ∂S ⎪ ⎩G(S,T ) = δ (S − K), (2.24) where δ (S − K) is the Dirac function. We know that δ (−x) = δ (x) , thus (2.24) can be written as,

G(S,T ) = δ (S − K) = δ (K − S) . (2.25) Then by Definition 2.1, we know that G(S,t; K,T ) is the fundamental solution to equation (2.18). By Theorem 2.3, G(S,t; K,T ) is the fundamental solution, as a function of K,T ( S,t are paramters), similar to (2.23) and(2.24), thus satisfies the following system,

14 ⎧ ∂G 1 ∂ 2 ∂ − + (σ 2 (K,T )K 2G) − (r − q) (KG) − rG = 0, ⎪ ∂T 2 ∂K 2 ∂K ⎪ ⎨ (0 ≤ K < ∞,t < T ) (2.26) ⎪G(S,t; K,T ) = δ (K − S). (0 ≤ K < ∞) (2.27) ⎪ ⎩⎪

We substitute (2.22) into (2.26), (2.27), then integrate both sides twice with respect to K in interval

[K,∞] . Since we know that, i) given a certain S , if K → ∞ , for a call option, the following items will all tend to 0, i.e., ∂V ∂G ∂ V , K ,σ 2 K 2G, K , (σ 2 K 2G) → 0, ∂K ∂K ∂K ∞ ∞ ∞ ii) dξ δ (η − S)dη = (η − K)δ (η − S)dη ∫K ∫ξ ∫K

∞ + = (η − K) δ (η − S)dη ∫0 =(S − K)+ ,

2 ∞ ∞ ∂ V ∂V iii) G(S,t;ξ ,T)dξ = dξ = − , ∫K ∫K ∂ξ 2 ∂K

∞ ∂ iv) V (S,t;ξ ,T )dξ = −V (S,t; K,T), ∫K ∂ξ

2 ∞ ∞ ∂ V ∂V v) ξG(x,t;ξ ,T )dξ = ξ dξ = −K +V , ∫K ∫K ∂ξ 2 ∂K

2 2 ∞ ∞ ∂ ∂ V vi) dξ (σ 2 (η,T )η 2G)dη = σ 2 (K,T )K 2 . ∫K ∫ξ ∂η 2 ∂K 2

Thus, we can transform the system of (2.33), (2.34) based on G(S,t; K,T ) into the following ⎧ ∂V 1 ∂ 2V ∂V − + K 2σ 2 (K,T ) − (r − q)K − qV = 0, ⎪ ∂T 2 ∂K 2 ∂K ⎪ ⎨ (0 ≤ K < ∞,t < T ) (2.28) ⎪ + ⎪V (S,t; K,T = t) = (S − K) . (0 ≤ K < ∞) (2.29) ⎩⎪

From equation (2.28), we obtain the explicit expression for implied volatility

∂V ∂V + (r − q)K + qV σ (K,T) = ∂T ∂K . (2.30) 1 ∂ 2V K 2 2 ∂K 2

15 This idea of Dupire's of calculating volatility seems to be simple and nice in theory.

However, when it becomes to the reality, when traders want to apply this into real market, the first

∂V ∂V ∂ 2V obstacle we must overcome is calculating the derivatives of option price, i.e., , , . ∂T ∂K ∂K 2 And in fact, there is no simple analytical way to do it but to resort to some numerical approach, for example, finite difference method, etc. Nevertheless, as we are about to see in chapter 3 section 2, the numerical approach is not good enough for calculating this Dupire volatility, as a slight amount change in option value would lead to some significant change in the value of derivatives, thus the volatility value. We can almost say that using (2.30) to calculate implied volatility is ill-posed

[1](we will go to details about this in Chapter 3).

16 Chapter 3. Implied Volatility Models

In this chapter, we compare two different types of volatilities, the local volatility and Dupire volatility, with implied volatilities under the structure of local volatility model.

3.1 The Local Volatility Model

As we establish in Section 2.1 that, given an asset's price process under an EMM Q with the risk free interest rate r = 0 that

dS = S ⋅dW , (3.1) we will have the option pricing problem for an European call option as ⎧ 1 V (S ,t) + SV (S,t) = 0 (0 ≤ t ≤ T ) (3.2) ⎪ t 2 ss ⎪ ⎪V (0,t) = 0, (3.3) ⎨ ⎪V (S,T ) = (S − K)+ , (3.4) ⎪ ⎪ ⎩V (S,t) → S − K ,VS (S,t) →1 as S → ∞. (3.5)

3.1.1 Option Pricing

Since there is no simple analytical solution for the system (3.2)-(3.5), we then have to resort to the numerical way to solve the option terminal value problem for this system. We use software Matlab in this paper to solve numerical problems. After a closer examination, we realize that we have a terminal boundary value problem here instead of an initial one, hence in order to use the built-in initial boundary value solver function in Matlab, we have to substitute some variables in the problem to shift the terminal boundary problem to an initial boundary problem in order fit this problem into the solving range of the built-in function. If we denote the time-to-maturity as τ = T -t , then it becomes obvious that if any one of these three variables( τ ,t,T ) is fixed, the other two will either move in the same direction or in the opposite ones. Thus, for every given τ , we have difference between T and t is fixed, written in the differential form, i.e., dT = -dt . (3.6) Also, in our model here, we have the asset price not dependent on time as shown in (3.1), thus we are ready to substitute the time point variable t with the maturity time parameter T , and regard the maturity time T as a variable as well as treating time-to-maturity τ as a parameter from now on .

17 Thus, system (3.2)-(3.5) can be transformed into the following system, ⎧ 1 −V (S ,T ) + SV (S,T ) = 0 (0 = t ≤ T ≤ t ) (3.7) ⎪ T 2 ss 0 T ⎪ ⎨V (0,T ) = 0, (3.8) ⎪V (S ,T = t ) = (S − K)+ , (3.9) ⎪ 0 ⎩⎪V (S,T ) → S − K , VS (S,T) →1 as S → ∞. (3.10)

Then, we can apply the built-in function pdepe in Matlab to solve the above problem. pdepe is a function that solves initial-boundary value problems for parabolic-elliptic Partial Differential Equations (PDEs) in one-dimension. pdepe solves PDEs of the form:

∂u ∂u −m ∂ ⎛ m ∂u ⎞ ⎛ ∂u ⎞ c(x,t,u, ) = x ⎜ x f (x,t,u, )⎟ + s⎜ x,t,u, ⎟ . (3.11) ∂x ∂t ∂x ⎝ ∂x ⎠ ⎝ ∂x ⎠

The PDE holds for t0 ≤ t ≤ t f and a ≤ x ≤ b . The interval [a,b] must be finite. m can be 0, 1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If m > 0 , then a must be non-negative. ∂u ∂u In (3.11), f (x,t,u, ) is a flux term and s(x,t,u, ) is a source term. The coupling of the ∂x ∂x partial derivatives with respect to time is restricted to multiplication by a diagonal matrix ∂u ∂u c(x,t,u, ) . The diagonal elements of this matrix c(x,t,u, ) are either identically zero or ∂x ∂x positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a parabolic equation, and there must be at least one parabolic equation. An element of c that corresponds to a parabolic equation can vanish at isolated values of x if those values of x are mesh points. Discontinuities in c and/or s due to material interfaces are permitted provided that a mesh point is placed at each interface[2].

For t = t0 and all x , the solution components satisfy initial conditions of the form

u(x,t0 ) = u0 (x) . (3.12)

For all t and either x = a or x = b , the solution components satisfy boundary conditions of the form ∂u p(x,t,u) + q(x,t) f (x,t,u, ) = 0 . (3.13) ∂x

Particularly, in our PDE (3.2) here, if we denote in (3.2), S as x , T as t , V (S,T) as u(x,t) , then (3.7)-(3.10) become

18 ⎧ 1 u = x⋅u (0 = t ≤ t ≤ t ) (3.14) ⎪ t 2 xx 0 T ⎪ ⎨u(0,t) = 0, (3.15) ⎪u(x,t = t ) = (x − K)+ , (3.16) ⎪ 0 ⎩⎪u(x,t) → x,ux (x,t) →1 as x → ∞. (3.17)

In fact, from a mere observation in the real market, we know that underlying stock price S =100 is quite high for an option with strike price K = 10 . Then we can replace the infinity requirement of limits in equation (3.17) by setting stock price to S =100 , given a strike price K = 10 . Then, (3.10) and (3.17) become

V (S,T ) = S - K,VS (S,T) =1 , where S =100 , (3.18)

u(x,t) = x - K,ux (x,t) =1 , where x =100 . (3.19)

Then (3.14)-(3.17) take the new forms of (3.20)-(3.23), ⎧ 1 u = x ⋅u (0 = t ≤ t ≤ t ) (3.20) ⎪ t 2 xx 0 T ⎪ ⎨u(0,t) = 0, (3.21) ⎪u(x,t = t ) = (x − K)+ , (3.22) ⎪ 0 ⎩⎪u(x,t) = x - K ,ux (x,t) =1, where x = 100. (3.23) Now, let us rearrange (3.20) in the following form

∂u −0 ∂ ⎛ 0 1 ⎞ = x ⎜ x ⋅ (x ⋅ux − u)⎟ . (3.24) ∂t ∂x ⎝ 2 ⎠ Comparing (3.24) to (3.11), i.e.

∂u ∂u −m ∂ ⎛ m ∂u ⎞ ⎛ ∂u ⎞ c(x,t,u, ) = x ⎜ x f (x,t,u, )⎟ + s⎜ x,t,u, ⎟ , (3.11) ∂x ∂t ∂x ⎝ ∂x ⎠ ⎝ ∂x ⎠ we find out that. m = 0 , (3.25) ∂u c(x,t,u, ) =1, (3.26) ∂x ∂u 1 f (x,t,u, ) = (x ⋅u − u) , (3.27) ∂x 2 x ∂u s(x,t,u, ) = 0 . (3.28) ∂x Next step, let us identify (3.21)-(3.23) to initial and boundary conditions of the form (3.12)-(3.13). (3.22) itself already complies with the form of (3.12), i.e., the initial condition. Hence, identifying (3.21) and (3.23) to the boundary conditions of the form (3.13), i.e.,

19 ∂u p(x,t,u) + q(x,t) f (x,t,u, ) = 0 , (3.13) ∂x is equivalent to finding pairs of values of function p(x,t,u) and function q(x,t) , which satisfies

∂u 1 the form in (3.13), given the flux function f (x,t,u, ) = (x ⋅u − u) by (3.27). ∂x 2 x If we substitute (3.27) into (3.13), we have 1 p(x,t,u) + q(x,t)⋅ (x ⋅u − u) = 0 . (3.29) 2 x And the boundary conditions (3.21) and (3.23) are ⎪⎧u(0,t) = 0, (3.30) ⎨ ⎩⎪u(x,t) = x - K ,ux (x,t) =1 , where x =100. (3.31) We insert (3.30) into (3.29) at x = 0 , then 1 p(0,t,u) + q(0,t)⋅ (x ⋅u − 0) = 0 . (3.32) 2 x For (3.32) to hold, one option is to put p(0,t,u) and q(0,t) to 0, i.e.,

⎧p(0,t,u) =0, (3.33) ⎨ ⎩q(0,t) =0. (3.34) Similarly, we insert (3.31) into (3.29) at x = 100 , then 1 p(100,t,u) + q(100,t)⋅ (x ⋅u − u) | = 0 . (3.35) 2 x x=100 We simplify (3.35), obtain 1 p(100,t,u) + q(100,t)⋅ (x ⋅u − u) | 2 x x=100 1 = p(100,t,u) + q(100,t)⋅ (x ⋅1− (x − K)) | 2 x=100 1 = p(100,t,u) + q(100,t)⋅ (K) 2 = 0.

This is to say, 1 p(100,t,u) + q(100,t)⋅ K = 0 . (3.36) 2 For (3.36) to hold, we can simply choose a pair of values of p(100,t,u) and q(100,t) ,

⎧ 1 ⎪ p(100,t,u) = - K , (3.37) ⎨ 2 ⎩⎪q(100,t) =1 . (3.38) In conclusion, our boundary conditions now take the form of

20 ⎧ p(0,t,u) = 0, (3.39) ⎪q(0,t) = 0, (3.40) ⎪ ⎨ 1 ⎪ p(100,t,u) = - K, (3.41) ⎪ 2 ⎩⎪q(100,t) =1 . (3.42)

After specifying all the conditions and function forms, we are ready to gather together all the thoughts stated above to write them into a program file pdex_u.m (which is included in Appendix A, Table A.1)in Matlab. We set the values for each one of the variables and parameters, the initial value of maturity T = t0 = 0 , strike price K = 10 , risk free interest rate r = 0 , dividend yield q = 0 , using 201 mesh points in the option price range from 0 to 100 and 51 mesh points in the maturity range from 0 to 5, to simulate the numerical option value at each price level. The option value curves, option values plotted against asset prices, under different time-to-maturity periods τ are shown in Figure 3.1(the more complete series of curves of option value at different levels of time-to-maturity is included in Appendix A). The option price surface with respect to the time-to- maturity τ and asset price is shown in Figure 3.2.

Figure 3.1: The Option Value Curves of European Calls and the Payoff Diagram at Maturity

21 Figure 3.2: The Option Value Surface for European Call Options

As shown in figure 3.1 and figure 3.2, without any unexpected outcome , the option value curve and surface under this local volatility model have no substantial difference to those of a vanilla European call option under a generic Black-Scholes model. The longer the period of time-to- maturity, the more valuable the call options; the higher the stock/asset price, the closer to payoff the option values at maturity. In the following subsection, we try to find out the internal connection between the implied volatilities and the local volatilities.

3.1.2. Implied Volatilities and The Local Volatilities

With the preparations in Section 3.1.1, we can now move on to calculate the implied volatilities of this local volatility model for each mesh point in the two-dimensional space consisted of asset prices and time. As we mentioned in Section 2.2, the volatility is determined by the Black-Scholes formula for option prices uniquely, given the other inputs, such as asset price S , interest rate r , dividend q , maturity T , time t , strike K , option price V and so on. This is to say, we can derive a unique implied volatility σ imp from Black-Scholes model, in other words,

σ imp = σ (S,T; K,t,V ) exists and is unique.

22 The existence of the implied volatility can be observed from the corresponding relationship between the option price V and the implied volatility σ imp . This is true by the formulation of the

Black-Scholes formula for option pricing. The problem of the uniqueness of the implied volatility

σ imp can be solved by the monotonicity of the option price V with respect to the maturity time

T . For a call option value VCall = V (S,T;σ , K,t) , where S,T are variables. σ , K,t are parameters. We know that the one of the Greeks in Black-Scholes formula for call options, vega, ∂V ∂V ν = = - < 0 [3]. Hence, given any value set of (V , S,T; K,t) , we will find a unique ∂T ∂t value for σ , which is called the implied volatility, denoted asσ imp . For example,σ imp = σ 0 , for

an input set of (V , S,T; K,t) = (V0 , S0 ,T0 ; K,t0 ) .

Therefore, we can regard σ as a function of S,T , where K,t are parameters, V is also quoted from market price, i.e. σ = σ (S,T; K,t,V ) . While at the same time, the local volatility

denoted as σ loc can be easily observed from the price processes of this local volatility model, that

1 σ = , for each mesh point in the price axis. Thus, the distance between two corresponding loc S volatilities can be easily calculated. The program for implied volatilities' calculation pdex_imp.m is included in Appendix A Table A.2.

The plots of implied volatilities and local volatilities are shown in the following.

23 Figure 3.3: The Implied Volatility Curves(Plotted against the Stock Price S )

Figure 3.3 is the implied volatility curve plotted against the stock prices at three different time points. As we can see in figure 3.3, the implied volatility of the option is quite large (In fact, when the stock price is close to 0, the implied volatility tends to infinity. We will discuss this in detail at this end of Section 3.1.2) at those points where the stock prices S are close to 0, and as the stock price S goes up, the implied volatilities gradually fall back to a relatively low and stable level.

The implied volatility decreases at a decreasing speed as the stock price increase. From an economic point view, if the stock prices drop to a level close to 0, then the options based on the same stock will be extremely risky, thus the indicator of riskiness will be extremely large, i.e.

σ imp → ∞, as S → 0 . On the contrary, the higher the stock price S , the less risky the call option value V . However, the decreasing of the riskiness of the underlying asset that the option is based on, is not enough to reduce all of the risks that the option is facing, some of which are some systematic risks, such as the macroeconomic status and so on. Hence, as the stock price increases, eventually, the implied volatilities will tend to a stable non-negative level in general. However, in our stock price process here, we assume the risk-free interest rate is 0. This means that investors are

24 not rewarded at an interest r = 0 for taking the systematic risk, this means that the systematic risk is 0. Hence, in our special case here(the risk-free interest rate is 0), when the stock price tends to infinity, the implied volatility tends to 0. The term structure of the implied volatility is shown in figure 3.4.

Figure 3.4: The Implied Volatility Curves(Plotted against the Time-to-maturity τ )

Figure 3.4 shows the term structure of the implied volatilities at different stock price levels. As the maturity time comes closer, the option value will become more volatile, hence the implied volatility will become higher. And as the stock price increases, the curve of implied volatilities plotted against the time-tom-maturity will shift downward as a whole.

25 1 Figure 3.5: The Local Volatility Curve σ = loc S

1 Figure 3.5 shows the local volatility curve that given by σ = which only depends on the loc S 1 1 stock price S . And, we know that σ = → ∞ as S → 0 , as well as σ = → 0 loc S loc S as S → ∞ .

From the illustration of above figure 3.3-3.5, we find out that the implied volatility σ imp and the

local volatility σ loc almost have the same tendency of change. Then, we are more curious to find out exactly how far away they are from each other.

The Distance between Implied Volatilities and Local Volatilities

Since we have calculated the value of implied volatilities σ imp and are aware of that the local

26 1 volatility has the form σ = , then we can find out the distance between σ and σ by loc S imp loc

1 distance function d = σ −σ = σ − . imp loc imp S

We put our theory here into practice by program file pdex_dis_imp_loc.m written in matlab(this program is include in Appendix A). All the parameters and indicators that need to be specified are gathered in the following table 3.1.

Table 3.1: The Initial Variable Set-up for Program pdex_dis_imp_loc.m Price(stock/asset price) 201 mesh points, from 0 to 100. Strike(option strike price) 10 Rate(risk-free interest rate) 0 Time(time-to-maturity) 51 mesh points, from 0 to 5 Value(option value) 51×201 values, calculated in Section 3.1.1 Limit(the upper bound for volatility searching 10 times interval) Yield(dividend yield) 0 Tolerance(calculation accuracy) 10-16 Class(option type) call option

The plots of this section is shown in the following figure 3.6 and figure 3.7.

27 Figure 3.6: The Comparison of Implied Volatility and Local Volatility ( K = 10 )

Figure 3.6 is a demonstration of how the distance between the implied volatility and the local volatility changes as the stock price increases. At first, the the local volatility curve is above the implied volatility curve, then as the stock price increases, the local volatility decreases more rapidly,

then at a certain stock price level, they intersect, and after that the implied volatility curve lies above the the local volatility curve. The change of distance between them is shown by the distance curve marked in black in figure 3.6. Before the stock price reaches the strike price, as the stock prices increases, the distance decreases rapidly; then at the point when the stock price is equal to the strike price, the distance reaches 0; after the stock price exceeds the strike price, the distance gradually increases to a certain relatively low level and stays that way as the stock price continues to increase.

28 Figure 3.7: The Absolute Difference between Implied Volatility and the Local Volatility Curve

σ loc −σ imp (Plotted Against S , K = 10 )

When plotting distance curves between the implied volatilities and the local volatilities solely, we obtain the three curves shown in figure 3.7, each of which represents a different time-to-maturity level. We can hardly tell them apart without magnifying them since they are lying very close to each other in figure 3.7. Nevertheless, we can almost say affirmatively that the distance between these two volatilities are essentially 0 at the point where the stock price S is equal the option strike price K ; while at those points where the price doesn't reach the strike price level from the below, the distance between them are relatively far away. On the other hand, as the stock price goes up from above the strike price, the distance between them then will be maintained at a quite stable level. The more accurate and actual computation results can be read from numerical results included in Appendix B.

29 Figure 3.8: The Absolute Difference between Implied Volatility and the Local Volatility Curve

σ loc −σ imp (Plotted Against T )

In order to show how the distance between the implied volatility and the local volatility changes as time goes by, we plot figure 3.8. Notice in the distance function that 1 d = σ −σ = σ − , imp loc imp S

the only time-sensitive factor in it is the implied volatility σ imp . Hence, at the parts of curves where the maturity time T is far away, the distance curves in figure 3.8 reveal some similar nature to implied volatility curves in figure 3.4, as they both tend to stay at a relative stable level, almost parallel to the time-to-maturity axis. Another interesting fact can be observed in figure 3.8 as well is that: when the stock price S is below the strike price K , the distance curve shifts downward as the stock price S goes up; when the stock price S is above the strike price K , the distance curve shifts upward as the stock price S continues to increase. The distance curve hits the bottom when the stock price S is equal to the strike price K . This observation again, is consistent with our conclusions in figure 3.6 and figure 3.7.

30 The Limits of Implied Volatilities and Local Volatilities at S=0.

One vague statement that we have not really explained in this section is that we say the implied volatility is very large for those points at which the stock prices S are close to 0. Although the plots of the implied volatility curves have indicated that the implied volatilities would more than likely to go to infinity when the stock price tends to 0. However, we still need more concrete evidence to prove our speculation here. We know from initial condition of option pricing that when the stock price falls back to 0, the option value is also 0, meaning that the ownership of this asset is worthless. Then it makes no sense to talk about the implied volatility of the option value. Thus, we choose a small neighbourhood of 0 on the stock price axis S with its left side end open. For example, we choose S = (0,10−10 ] . By usage of the option pricing scheme described in Section 3.1.1, we calculate the option values within this small interval. We modify our previous program for option pricing by equally choosing 201 mesh points on interval S = [0,10−10 ] and adding a single point S =100 to the collection of mesh points. In this way, we can both achieve the pricing for option prices at small stock price points and keep our boundary conditions unchanged. This altered program for option pricing is named as pdex_u_small_s.m, which is included in Appendix A. Then, we use the same scheme for implied volatility calculations as before. The program file of implied volatility computation for small S , pdex_imp_small_s.m is included in Appendix A as well. The more detailed initial parameter set-up for program pdex_imp_small_s.m is displayed in the following table 3.2.

Table 3.2: The Initial Variable Set-up for Program pdex_imp__small_s.m Price(stock/asset price) 202 mesh points, 100 points from 0 to 10-10 , and 1. Strike(option strike price) 10 Rate(risk-free interest rate) 0 Time(time-to-maturity) 51 mesh points, from 0 to 5 Value(option value) 51×202 values Limit(the upper bound for volatility searching 109 times interval) Yield(dividend yield) 0 Tolerance(calculation accuracy) 10-18 Class(option type) call option

The plots of this program are shown in figure 3.9-3.12. Figure 3.9 depicts the implied volatility curves plotted against the stock prices at different time-to-maturities. However, according to figure

31 −10 3.9, the implied volatility σ imp at S = 0.1⋅10 is not a very large number, although the slope is quite steep in the neighbourhood of S = 0 . Notice that the scale of the volatility axis and the scale of the stock price axis are obviously different, the latter is enormously larger comparing to the former. So, if we put both axis to the same measure scale, then the implied volatility curve will be extremely steep in this interval [0,10-10]. From the above argument, we realize that the steepness of the implied volatility curve is not necessary an accurate way to determine whether the implied volatility tends to infinity at S = 0 .

As an alternative of graphical analysis, let us consider the derivative of implied volatility with ∂σ respect to the stock price imp . Notice that our partition of the interval [0,10-10] is enough small, ∂S hence we can use the implied volatility value at each mesh point and the step size of the partition to

∂σ approximate the derivatives imp at each point. For example, we select some results of implied ∂S volatility from program's data(the numerical results of the implied volatilities calculated by pdex_imp_small_s.m, which are included in Appendix B).

Table 3. 3: The Slopes of Implied Volatility Curves

∂σ σ −σ τ σ σ ∆S = S - S imp imp0 imp1 imp0 imp1 0 0 1 = ∂S ∆S0

−12 -102104497982.486 - 0.5⋅10 2.8 3.71288296174587 3.66183071275462 ≈ -1.02⋅1011

−12 -87954343044.2116 - 0.5⋅10 3.8 3.23962470398632 3.19564753246422 ≈ -0.87⋅1011

−12 - 78464029934.4002 - 0.5⋅10 4.8 2.92041541613200 2.88118340116480 ≈ -0.78⋅1011

∂σ As we can see in table 3.3, that the derivative imp is a really large negative number when ∂S

32 −10 the stock price S is 10 . And we know that the implied volatility σ imp is a positive

∂σ number at S =10−10 . Hence, if the value of the slope of the implied volatility curve imp stays ∂S at the current amount, the implied volatility will eventually go to plus infinity as the stock price

∂ 2σ continues to decrease from below S =10−10 . This is to say, if imp ≥ 0 holds, then ∂S 2

2 ∂ σ imp limσ imp = +∞ . Now, all we have to do make sure that ≥ 0 is true. We plot the slopes of S →0 ∂S 2 implied volatilities into curves in figure 3.10, with respect to the stock prices S . And indeed, as it

∂ 2σ is shown in figure 3.10, the slopes of the "slope curves" are truly non-positive, i.e., imp ≥ 0 . In ∂S 2 fact, we can also read from the numerical results of slopes of implied volatility curves to arrive at the same conclusion(see table 3.4). The slopes of implied volatility curves are decreasing as the

∂ 2σ stock price decreases, i.e., imp ≥ 0 . Then, we can say surely that the implied volatility σ ∂S 2 imp tends to infinity as the stock price S goes to 0(The more complete numerical results of slopes are included in Appendix B).

In short, this is to say, because i) σ | > 0 for S = ο(10−10 ) ; imp S =S0 0

∂σ ii) imp | = −1⋅ο(1010 ) , for S = ο(10−10 ) ; ∂S S =S0 0

∂ 2σ iii) imp ≥ 0 for 0 ≤ S ≤ S , S = ο(10−10 ) ; ∂S 2 0 0 that we have limσ imp = ∞ for 0 < S ≤ S0 , and thus limσ imp = ∞ . S→0 S→0

33 Table 3. 4: The Slopes of Implied Volatility Curves

τ \S 0.5⋅10−12 1.0⋅10−12 1.5⋅10−12 2.0⋅10−12

2.8 -102104497982.486 -60234023483.1758 -42968495088.3899 -33463371536.9570

3.8 -87954343044.2116 -51889446183.0994 -37017182609.4161 -28829357814.5602

4.8 -78464029934.4002 -46292532890.8126 -33025342546.6038 -25721008566.0331

Figure 3.9: The Implied Volatility Curves(Against Stock/Asset Price S )

34 Figure 3.10: The Slope Curves of Implied Volatility Curves(Against Stock/Asset Price S )

Now that we know for a fact that the implied volatility tends to infinity as stock price goes to 0, and

1 also that the local volatility tends to infinity as the stock price goes to 0, since σ = . Then, loc S we are even more curious about the their speeds of converging to infinity, in order to make better judgements when approximating the implied volatility by local volatility at small stock prices.

First of all, we draw figures for the local volatility curve and the slope curve, which are shown in figure 3.11 and figure 3.12.

35 1 Figure 3.11: The Local Volatility Curve σ = = S -0.5 (Against Stock/Asset Price S ) S

1 1 1 -1.5 Figure 3.12: The Slope Curve of Local Volatility Curveσ S = - = - ⋅ S 2 S 3 2

36 1 We know from σ loc = , that limσ loc = ∞，limσ loc = 0. And the shape of curves in figure S S→0 S→∞

3.11 and 3.122 confirms that. Notice that the order of magnitude of the vertical axis in figure 3.11 and 3.12 are completely different from that of figure 3.9 and 3.10. So, we can say affirmatively that the speed of the local volatility's converge to infinity is much more faster than that of the implied volatility's. But the question is that how much faster the former is. We know in general, that the implied volatility curve is below the local volatility curve on interval [0,10-10], i.e.,

−0.5 0 < σ imp << σ loc = S . We also know that, the corresponding slope curves of the implied volatilities are above those of the local volatilities due to the fact that their signs are negative, i.e.,

∂σ ∂σ − 0.5⋅ S −1.5 = loc = imp < 0 . These motivate us to find out the exact magnitude of implied ∂S ∂S ∂σ volatility σ and its 1st order derivative imp , or at least the range of it. One way to do it is imp ∂S ∂σ by fitting σ and imp into some certain potential functions' forms. More specifically, imp ∂S we find the value ranges of α and β , 0 < −α lower < −α < −α upper and

αlower αupper βlower < β < βupper < 0 that satisfy 0 < S < σ imp < S and

β ∂S βlower ∂σ ∂S upper − 0.5⋅ < imp < −0.5⋅ < 0 . ∂S ∂S ∂S

We establish our algorithm for searching such ranges by the following way. First of all, from a simple observation of previous plots and some tests on the matlab program, we choose a rough lower bound of searching region of α , -0.5(motivated by the degree of S in local volatility), a upper bound -0.01(motivated by some simple testing on matlab). Also, we choose a rough lower bound of searching region of β , -1.5(motivated by the degree of S in slope of local volatility), a upper bound -0.01(motivated by some simple testing matlab). Set the searching step size to 0.01, then we can begin our search for such suitable values of α and β . The search program pdex_imp_slope_small_s_fit.m is included in Appendix A. And the results of our searching is

37 α lower = −0.03,

α upper = −0.12,

βlower = −1.00,

βupper = −0.90.

Thus, we can estimate the implied volatility and its 1st order derivatives as ∂σ 0 < S −0.03 < σ < S −0.12 and − 0.5⋅ S −1.00 < imp < −0.5⋅ S −0.90 < 0 , for S ∈ (0,10−10 ]. imp ∂S

The plots for demonstrating such estimations are shown in figure 3.13 and figure 3.14.

The result of this estimation is to say that if we were to express the implied volatility σ imp in a

α potential form, that σ imp = S , where − 0.12 < α < −0.03 . Since the local volatility

−0.03 −0.12 −0.5 S σ imp S σ imp σ loc = S is given, hence 0 = lim ≤ lim ≤ lim = 0 , i.e., lim = 0 . S→0 −0.5 S→0 S→0 −0.5 S→0 S σ loc S σ loc

Similarly, if we present the 1st order derivative of the implied volatility with respect to stock/asset

∂σ price imp in a potential form, that σ = S α , where −1.00 < β < −0.90 . Also, knowing ∂S imp

st −1.5 the 1 order derivative of a local volatility σ loc = −0.5⋅ S , we then can compute the limit

∂σ imp ∂σ imp - 0.5⋅ S −1.0 - 0.5⋅ S −0.9 0 = lim ≤ lim ∂S ≤ lim = 0 , i.e., lim ∂S = 0 . S→0 −1.5 S →0 S→0 −1.5 S→0 - 0.5⋅ S ∂σ loc - 0.5⋅ S ∂σ loc ∂S ∂S σ In addition, we can write that lim loc = ∞ . Again, let us look back on the distance between the S→0 σ imp ∂σ implied volatility and the local volatility. Facts are σ = S −0.5 , loc = −0.5⋅ S −1.5 , loc ∂S ∂σ 0 < S −0.03 < σ < S −0.12 , and − 0.5⋅ S −1.00 < imp < −0.5⋅ S −0.90 < 0 for S ∈ (0,10−10 ]. imp ∂S

Hence the distance function d = σ imp −σ loc satisfies

⎧d = σ −σ = σ -σ > S −0.5 − S −0.12 > 0, ⎪ imp loc loc imp ⎨∂d ∂σ − ∂σ ⎪ = loc imp > −0.5⋅ S −1.5 − (−0.5⋅ S −0.9 ) = −0.5⋅(S −1.5 − S −0.9 ), ⎩∂S ∂S where, S ∈ (0,10−10 ].

38 And we know that − 0.5⋅(S −1.5 − S −0.9 ) = -0.5⋅ S −1.5 (1− S 0.6 ) → −∞ and

−0.5 −0.12 −0.5 0.38 S − S = S ⋅(1− S ) → ∞ as S → 0 , which means d = σ imp −σ loc → ∞ as

S → 0 .

This is to say that the distance between these two volatilities is infinitely far away. So, when the stock price is really close to 0, it would be not suitable to approximate the implied volatility by the local volatility due to the fact that their distance function tends to infinity at 0.

39 Figure 3.13: The Fitting to Potential Function of The Implied Volatility Curves

Figure 3.14: The Fitting to Potential Function of The Implied Volatility Slope Curves

40 3.1.3. Implied Volatilities and The Dupire Volatilities

In this section, we concentrate on finding the inner connections between the implied volatilities and the Dupire volatilities. The approach and ideas to solve the implied volatilities for the local volatility model are the same as shown in Section 3.1.2 stated above. So, our focus here is mainly concentrated on finding the Dupire volatilities.

The Dupire method is an approach of calculating volatilities by using market-quoted inputs, such

∂V ∂V ∂ 2V as , , V , and so on. This local volatility is called the Dupire volatility. ∂T ∂K ∂K 2 From formula (2.30) in Chapter 2, we know that

∂V ∂V + (r − q)K + qV σ (K,T) = ∂T ∂K . (3.43) 1 ∂ 2V K 2 2 ∂K 2

And also r = q = 0 , thus

∂V σ (K,T) = ∂T . (3.44) 1 ∂ 2V K 2 2 ∂K 2

Up till now, the option value we have discussed is a function of asset price S and maturity time T , i.e., V (S,T ) , where strike price K is regarded as a parameter, relatively fixed, comparing to S and T .

According to formula (3.44), in order to calculate the Dupire volatility, we have to obtain the values

∂ 2V ∂V for the derivative and first. ∂K 2 ∂T To calculate the derivatives, our plan here is to choose a spectrum of strike prices K , and for each K , we use the same scheme of pricing options as before while the strike price K is treated as if it is a constant. Then we collect all the option values calculated in this way and rearrange them not only by the time-to-maturity indexes and the stock price indexes, but also by the strike price indexes. Hence, the option prices calculated in this way will be a 3-dimensional space composed of time, price and strike. As abstract as it is, this would enable us to implement the finite difference method to calculate the two deorivatives mentioned in (3.44). Then, let us discuss the practical way of implementing the finite difference method.

41 Finite difference method is a numerical method that approximates the solutions to differential equations by replacing derivative expressions with approximately equivalent difference quotients. We use the forward difference algorithm here. Suppose function f (x) 's 1st and 2nd order derivatives at x = a exist, then we can use the following method to approximate f ′(a) and f ′′(a) . ( f + h) − f (x) f ′(a) = ,as h →0 ; (3.45) h ( f + 2h) − 2 f (x + h) + f (x) f ′′(a) = ,as h →0 . (3.46) h2 If we apply this algorithm to our call option here, using the same notation, that, for a given asset price S and a maturity time T , u(x,t; K + 2h) − 2u(x,t; K + h) + u(x,t; K) C (K,T ) = , (3.47) KK h2 u(x,t + h; K) − u(x,t; K) C (K,T ) = , (3.48) T h where the variable x represents asset price S , variable t stands for the maturity time T , and K is a parameter in function u . First of all, let us compute the option values. The parameters' set-up in this subsection is organized in table 3.5.

Table 3.5: The Initial Parameter Set-up for Program pdex_dupire.m Price(stock/asset price) 101 grid points, from 0 to 100. Strike(option strike price) 21 mesh points, from 9.18① to 11. Rate(risk-free interest rate) 0 Time(time-to-maturity) 21 mesh points, from 0 to 2 Value(option value) 21×101×21 values Limit(the upper bound for volatility searching 100 times per annum interval) Yield(dividend yield) 0 Tolerance(calculation accuracy) 10-6 Class(option type) call option

The numerical results of option values are calculated by program pdex_dupire_option.m(the m file is included in Appendix A). The plot of option value surface is shown in figure 3.15 and the option value curve plotted against the strike price K is shown in figure 3.16. Figure 3.15 and figure 3.16 are in accordance with our intuitions, that the option value V is

① The starting point of the mesh point setseems like an unconventional choice. However, this enables us to place the middle point of 21 mesh points on a point with value K=10, i.e., K21=11, K11=10 → K1=9.18. The value of middle point of strike price spectrum is in line with the strike price we use in calculating the implied volatility.

42 positively correlated with the stock price S and negatively correlated with the strike price K . The latter is not quite obvious shown in figure 3.15, but can be observed from figure 3.16. Although the strike price's impact on the option value is not as much as the stock price S , still a higher strike price level will lead to a decrease in option value to a certain extent.

Figure 3.15: The Option Value Surface

43 Figure 3.16: The Option Value Curves(Against the Strike Price K )

Then, we can use the numerical results of the option prices and the algorithms for derivatives calculations to compute the Dupire volatilities. The program code is pdex_dupire.m (included in

Appendix A). The plots from this program are shown in figure 3.17 and figure 3.18. The dependence of the Dupire volatility on the time-to-maturity τ is reflected in figure 3.17. As the maturity time approaches, the Dupire volatility increases at an increasing speed. This pattern, however, is in consistence with the implied volatility curves in figure 3.4. They all display a ∂σ ∂ 2σ property of the volatility curves that < 0, < 0 (if the derivatives exist). ∂T ∂T 2 Besides the above, curves in figure 3.17 also show that the Dupire volatility is sensitive to strike price K as well. A small increment of 0.1 in strike price K leads to a huge upward shift of about 1 in the volatility curve.

44 Figure 3.17 The Dupire Volatility Curve(Plotted Against Time-to-maturity τ )

All other things equal, panel a of figure 3.18 is plotted with a strike K = 9.9 , while panel b is plotted with a strike K = 10 . However, this small difference in choosing strike prices leads to a huge difference in the shapes of Dupire volatility curves, as shown in panel a and panel b of figure 3.18. Figure 3.18, panel a, describes the Dupire volatility curve at K = 9.9 , τ = 1.9 . Figure 3.18, panel b, depicts the Dupire volatility curve at K =10 , τ =1.9 . Their shapes are quite different from the the implied volatility's and the local volatility's. The Dupire volatility curves change the convexity at least once throughout the the range within which the stock price changes. As for the implied volatility and local volatility curves, their convexity don't change on the interval for stock prices, this can be observed from curves in figure 3.6. Besides the differences with other volatility curves, the Dupire volatility curves themselves are different from one another. Graphically, we can see that the convexity changes at least twice in panel a while it only changes at least once in panel b. Moreover, it's quite obvious that the curve is more peaked in panel b since the difference between the extreme point and the lowest point in the Dupire volatility curve is much larger. Without ruling out the impact resulted from choosing the finite difference method in calculation, we can still tell that the Dupire volatility is extremely sensitive to strike prices K.

45 Figure 3.18 a : The Dupire Volatility Curve(K=9.9)

Figure 3.18 b : The Dupire Volatility Curve(K=10)

46 Although, it appears that the Dupire volatility will drop to 0 rather rapidly as the stock price increases.But in fact, the Dupire is not as nice as it seems in the diagram. Recall the algorithms of derivatives and the formula for calculating the Dupire volatility, u(x,t; K + 2h) − 2u(x,t; K + h) + u(x,t; K) C (K,T ) = , (3.47) KK h2 u(x,t + h; K) − u(x,t; K) CT (K,T ) = h . (3.48)

∂C σ (K,T) = ∂T . (3.44) 1 ∂ 2C K 2 2 ∂K 2

CT (K,T ) computed by (3.48) is a positive real number. This is guaranteed by the monotonicity of option value V with respect to the time-to-maturity T . However, we can not say the same for

nd the 2 order derivative CKK (K,T) because (3.47) is not necessarily a positive number since function u(x,t, K) is not necessarily a convex function with respect to K in the interval we choose. Hence, the numerical solution of (3.44) could be a complex number with a non-zero imaginary part. In such scenarios, the Dupire volatility can not be computed by the formulation (3.44) proposed by Dupire. However, the numerical existence problem of the Dupire volatility while using the finite difference method is only part of the difficulties we might encounter in Dupire volatility calculation here. We notice that, in figure 3.18, after the stock price reaches a certain high level, the Dupire volatility begins to display some irregularity and inconsistency in the movement of the volatility curve as the stock price S continues to increase. And eventually, the Dupire volatility vanishes when the stock price reaches 80(take figure 3.18 for an example). In retrospect, during our previous discussion in section 3.1.2, the implied volatility curve in figure 3.3 shows a relatively stable property when the stock price S is at a relatively high level, and this property continues to remain so even when the stock price S continues to increase. And naturally, we would wonder why is that these two different volatilities have so extraordinarily different behavior at high stock prices. To solve this mystery, we need to go back to the Dupire formula for volatility calculations, equation (3.44). Dupire formula states that, the volatility of an option can be calculated by the following,

∂C σ (K,T) = ∂T . (3.44) 1 ∂ 2C K 2 2 ∂K 2

47 ∂C ∂ 2C The derivatives (for computing the Dupire volatility) required here, ， are the essential ∂T ∂K 2 inputs for determining the Dupire volatility's value. ∂C Let us first examine the 1st order derivative of option price with respect to the maturity-time, . ∂T ∂C We know from experience and intuition that is a positive number, our interpretation is that ∂T longer time-to-maturity produces a high option value. In fact, from an observation of the numerical lincluded in Appendix B), we find out that, when the price of underlying asset/stock is high enough, the option value curves lie parallel to the payoff diagram at maturity. This is shown by figure 3.19.

And, another fun fact is that, at large stock prices, the increment between two option values whose ∂C maturity-times are adjacent to each other is a constant. This is to say, (S,T, K) = k , where ∂T k > 0 is a constant and S is quite large comparing to the strike price K . Besides our numerical results included in Appendix B, we can conclude the same results by the curve in figure 3.20.

48 Figure 3.19 : The Option Value Curve(τ = 0 : 2.0 : 0.1)

∂C Figure 3.20 : The Derivative C = T ∂T

49 ∂C Figure 3.21 : The Derivative C = K ∂K

∂ 2C ∂C As for the derivative C = , we consider its corresponding 1st order derivative C = . KK ∂K 2 K ∂K

We can read from the numerical results of program pdex_dupire_option.m that, the 1st oder ∂C derivative C = tends to be piecewise constant when the stock price is relatively quite high K ∂K comparing to the strike price K (say from 81 to 100). The graph to illustrate this is shown in figure

∂ 2C 3.20. Hence, the 2nd order derivative C = is 0 in the region that S ∈[81,100] . This KK ∂K 2 indicates that, the Dupire volatility

∂C σ (K,T ) = ∂T can not be evaluated by the finite difference numerical method for 1 ∂ 2C K 2 2 ∂K 2

S ∈[81,100].

This explains why the Dupire volatility can not be computed for large stock prices when using the

50 finite difference numerical method. And notice that in figure 3.18 that, there is a small jump around S ∈[70,80] , after examining the numerical results around those points, we find out that this is caused by the incompatibility between

∂C ∂ 2C C = and C = . To be more specific, incompatibility is referred to that at those T ∂T KK ∂K 2 ∂C jump points, the value of C = still exists and is positive under the evaluation by finite T ∂T ∂ 2C difference method, while the value of C = has already become 0. In conclusion, this KK ∂K 2 explains why the Dupire volatility curve behaves in this way in figure 3.18.

Distance Between Implied Volatilities and Dupire Volatilities

By distance function d = σ imp −σ dupire , we are able to compute the distance between the implied volatility and the Dupire volatility. Then, we can move on to calculate the distance between implied volatilities and Dupire volatilities. We include the program file pdex_dis_dupire.m in Appendix A. The plots for illustrating the distance between the implied volatilities and the Dupire volatilities are shown in the following figure 3.22 and 3.22. Panel a and panel b in figure 3.22, are the distance curves plotted against the stock price S at τ =1.9 , K = 9.9 and at τ =1.9 , K =10 . Figure 3.23 is a plot of the distance value of mesh points within the interval S ∈[40,49], plotted against the strike price K (The plots for the other groups are included in Appendix A). For the curve in figure 3.22, panel a, on one hand, the distance curve has some similar patterns as opposed to the Dupire volatility curve in figure 3.18 a, i.e., the distance curve is truncated for large stock price. The distance between these two volatilities can not be measured by distance function using finite difference method due to the fact that the Dupire volatility can not be calculated from the finite difference method for large stock prices. The reason of this is stated in the illustration of figure 3.18 a. On the other hand, the distance curve in figure 3.22 has also inherited some properties from the implied volatility curve demonstrated in figure 3.3, that as the stock price goes up the distance between these two volatilities decreases as long as the stock price does not exceed the certain upper boundary for which the Dupire volatility is truncated by using the finite difference numerical method. As for the curve in figure 3.22, panel b, resembles some similar features of the curve in figure 3.18 b. The distance between the Dupire volatility and the implied volatility reaches its maximum point

51 around S = 47 , and hits its minimum around S = 70 at which the difference between the Dupire volatility and the implied volatility is almost 0. And the curve in panel b is more peaked than the curve in panel a. Because of the sensitivity of the Dupire volatility curve to strike price K, hence the absolute difference curve between the Dupire volatility and the implied volatility is highly sensitive to the strike price K as well. Because of these properties, we have to impose more constraints on the usage of approximating the implied volatility by Dupire volatility, since we not only have to make sure the distance between these two volatilities is controllable but also in numerical existence under the finite difference method we use here. Besides these, in fact, there are still more obstacles in applying this approximation scheme. This is because the Dupire volatility calculated by the finite difference numerical method does not always show a nice consistency between adjacent mesh points. We can observe this problem from the numerical results included in Appendix B.

Figure 3.22 a: The Absolute Difference between Implied Volatility and The Dupire Volatility

σ imp −σ dupire (Plotted Against Stock Price S )

52 Figure 3.22 b: The Absolute Difference between Implied Volatility and The Dupire

Volatility(Plotted Against Stock Price S )

53 Figure 3.23 : The Absolute Difference between Implied Volatility and The Dupire Volatility at

MeshPoints(Plotted Against K )

Figure 3.23 shows the distance between the volatilities of each mesh point. The circled points are sparce in figure 3.23, due to the same reason that we stated in the illustration of figure 3.18 that the Dupire volatilities' numerical existence is not guaranteed by our algorithms of finite difference method. We can not evaluate numerically for some points with our current algorithm. To sum up our investigation into the inner connections between the implied volatility and the

Dupire volatility, we draw the following conclusions. If we can make the option value to be convex with respect to the strike price K on a certain finite interval, then our scheme of volatility calculation by finite difference method is practical. Hence, we can compare the volatility calculated by the Dupire method to the one computed by using Black-Scholes formula rebersely. Even if we can obtain a convex function of option value with respect to the strike price K by imposing conditions on the local volatility model or choosing a suitable interval, the distance between the implied volatility still will not be of some specific regularity. As we can see in figure 3.22, the lower bound of the distance seems to be unstable, since every time we change the upper bound of stock prices, the lower bound will be very likely to change accordingly. And as for the upper bound

54 of the distance, we can determine it by comparing the value of the left endpoint of the curve and the value of the peak point of the curve. But still, this upper bound of distance is not fixed either. And according our conclusions at the end of Section 3.1.2 that the implied volatility tends to infinity as

S → 0 , unless we can prove the Dupire volatility has the same speed of convergence to infinity, it would be really hard to approximate the implied volatility by the Dupire volatility. More importantly, the distance curve does not converge to anything or have a certain pattern of changing.

It is described in LiShangJiang (1994) [1] that the Dupire volatility is ill-posed. Based on all of the research in section 3.1.3, the Dupire volatility does seem to a method not very appropriate to simulate the implied volatility under our local volatility model here.

3.1.4. Summary of Three Types of Volatilities

After previous subsections of discussions with three types of volatilities, the local volatility, the implied volatility, the Dupire volatility, we now summarize their features comprehensively. 1 Apparently, the local volatility we discuss here, σ = is determined only by the stock price loc S

S . Given a certain time-to-maturity τ and a strike price K , the implied volatility, however, is determined by the stock price process S (provided that all other parameters, such as risk-free interest rate r and dividend yield q is known). Or, in other words, with all other parameters in the option pricing model given, the stock price process S and the time-to-maturity τ co-

determine the implied volatility σ imp = σ imp (S,τ ) . As for the Dupire volatility, it is determined by the maturity time T and the strike price K , while the stock price S and the time point t and all other parameters influence the Dupire volatility merely as parameters. And it's worth mentioning that the Dupire volatility is very sensitive to the strike price K .

We draw these three types of volatilities in the same diagram. Figure 3.24, panel a, is plotted with a strike price K = 9.9 , and panel b is drawn with a strike price K = 10 .

Throughout the entire interval for stock price S , we conclude the following.

For the local volatility, i) when the price of stock/underlying asset S is below the option's strike price K ,(the option is out-of-the-money) the local volatility is larger than the implied volatility; conversely, when the price of stock/underlying asset S is above the option's strike price K ,(the option is in-the-

55 money) the local volatility is smaller than the implied volatility. ii) as the stock price tends to 0, the local volatility tends to infinity; as the stock price tends to infinity, the local volatility eventually tends to 0. iii) as the stock price increases, the local volatility decreases at a decreasing speed.

For the implied volatility, i) as the stock price tends to 0, the implied volatility tends to infinity at a much slower speed than the local volatility; as the stock price tends to infinity, the implied volatility tends to a stable non- negative level, particularly, when the risk-free interest rate r is set to be 0, then the implied volatility tends to 0 as the stock price tends to infinity. ii) as the stock price increases, the implied volatility decreases at a decreasing speed.

For the Dupire volatility, i) the Dupire volatility curve changes its convexity at least once on the stock price interval. ii) the Dupire volatility is highly sensitive to the value of strike price K . iii) judging by the numerical approach, the finite difference method we use, as the stock price goes up to a relatively high level, the Dupire volatility can not be numerically evaluated after it reaches 0 at certain point in the stock price axis. iv) based on our investigation in this paper, the Dupire volatility is bounded within the range of variation of the stock price S .

56 Figure 3.24. a The Volatility Curves(σ loc ,σ imp ,σ dupire )

Figure 3.24. b The Volatility Curves(σ loc ,σ imp ,σ dupire )

57 Chapter 4. Conclusions and Future Studies

−0.5 In this paper, we mainly consider a local volatility model with the local volatility σ loc = S . We evaluate the model numerically after deriving this option pricing model by using its specified underlying asset's price processes. We focus the study on three types of volatilities, the local volatility specified by the model itself

σ loc , the implied volatility inferred by the Black-Scholes model σ imp , and the so-called Dupire volatility given by the Dupire formula for implied volatilities σ dupire . By showing the absolute difference between the implied volatility and the local volatility, and between the implied volatility and the Dupire volatility, we illustrate the inner connections between these volatilities. From our research here, we conclude that the difference between the local volatility and the implied volatility is bounded for large values of stock price S ; however, the difference between them tends to infinity since the implied volatility tends to infinity at a much slower speed than the local volatility −0.5 σ loc = S as the stock price S tends to 0. At the same time, we also investigate the relationships between the implied volatility and the volatility given by the Dupire formula. For large values of stock price S , the Dupire volatility tends to 0 as well as the implied volatility does; for small small values of stock price S , the results shown by this numerical study is inconclusive. Besides, the Dupire volatility is also time-dependent and highly sensitive to the change of the values of strike price K . We know that, our research here is based on an option pricing model(featured by its stock price processes) that has a implied volatility similar to the one inferred by the Black-Scholes model, yet still processes a price process that does not depend on the assumption of constant volatility, which is more plausible and realistic. Hence, intuitively, we would expect these three volatilities possess some similarities. And our results of this paper verify this speculation, all three types of volatilities tend to 0 for large values of stock price S ; the local volatility and the implied volatility tend to infinity for small values of stock price S although they have difference speeds of convergence towards infinity. However, procedures stated and demonstrated above may have some limitations as well. For example, if we choose another local volatility model other than this one we use in this paper, one question may arise naturally is that whether it is possible to solve the option values for such a model, if the mechanics of this model does not allow the option value to be solved by any built-in function of programming software. An obvious alternative is to resort to some other numerical methods to treat the PDE we might encounter discretely, such as the finite difference method and so on. Nevertheless, the accuracy of such approximations might be questionable. To avoid such "vains",

58 one useful shortcut is that we can observe the situation in the real market, make guesses such as the one in Heston model[4] that the stock's volatility might be negatively correlated to the stock price, to find suitable models guided by experiential knowledge and intuitions. For future studies, we can also studies other local volatility models other than this simple one we use here in this paper. Moreover, we can always search for another numerical option pricing approach other than the one we use here. Besides the alternative approaches and models for implied volatility study. Another future study direction can be looked into is the investigation of the tendency of change of the Dupire volatility as the stock price tends to 0, whether it is really bounded on the stock price axis, whether it tends to infinity as the implied volatility and the local volatility do, still needs to be verified. Furtherore, other numerical approaches that can fix the truncation problem of the Dupire volatility at large values of stock price S can be studied as well. This concludes our conclusions of this paper and directions for future study references to this topic.

59 Notations

SDE Stochastic Differential Equation PDE Partial Differential Equation EMM Equivalent Martingale Measure K Strike Price S Stock/Asset Price t Time point T Maturity Time τ Time-to-maturity Brownian Motion, Wiener Process {Wt }0≤t≤T

V Option Value G The 2nd Order Derivative of Option Price with Respect to the Strike C Call Option Value The Implied Volatility σ imp

The Local Volatility σ loc

The Dupire Volatility σ loc r The Risk-free Interest Rate The Stochastic Volatility ν t

ρ The Correlation between Brownian Motions

60 Appendix A

Program codes and plots

1. pdex_u.m

Program code Table A.1 pdex_u.m functionvalue=pdex_u(m,T,x,tau) %Functioncalculatesthedifferencebetweentwotypes ofvolatilities. i=51;%Thenumberofmeshpoints inthe range ofthe maturitytime T. t0=0;%Theinitial value ofmaturitytimeT. tT=5;%Theendpoint value ofmaturitytimeT. j=201;%Thenumberofmeshpointsinthe range ofthe assetprices S. K=10;%ThestrikepriceK. x=linspace(0,100,j);%Themeshpointsintherangeofasset prices S. T=linspace(t0,tT,i);%Themeshpointsintherangeofmaturity timeT. t=linspace(t0,tT,i);%Theactualvariablenotationweusein pdepe tau=linspace(0,tT-t0,i);%Time-to-maturity tau.tau=T-t0 m=0;%Parameterinbuilt-in function pdepe,indicatesthePDEisslab. sol=pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Usingthebuilt-infunctionpdepeto solvethePDEfor option valueV. u=sol(:,:,1); %SolvethePDE,denotethefirstsolution astheoption value. value=u; %------%Thecurveofdistanceplotting againsttheassetprice figure; plot(x,value); colormaphsv; xlabel('StockPrices'); ylabel('OptionValue'); title('EuropeanCallOptions'); figure; plot(x,value(51,:),'Color','red');holdon; plot(x,value(11,:),'Color','blue');holdon; plot(x,value(1,:),'Color','black');holdon;

61 xlabel('Stock Prices'); ylabel('Option Value'); title('European Call Options'); % ------% The surface consists of the distance between two kinds of volatilities figure; surf(x,tau,value); colormap hsv; title('Distance between two kinds of volatilities'); xlabel('Stock Prices'); ylabel('Time-to-maturity \tau'); zlabel('Implied Volatilities'); title('The European Call Option Value Surface'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K),0); end % ------function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) K = 10; pl = ul; ql = 0; pr = -1/2*K; qr = 1; end end

Plots

62 Figure A.1

2. pdex_imp.m

Program code

Table A.2 pdex_imp.m function value = pdex_imp(m, T, x, tau) %Function calculates the difference between two types of volatilities. i = 51; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 5; %The endpoint value of maturity time T. j = 201; %The number of meshpoints in the range of the asset prices S. K = 10; %The strike price K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 imp_v = ones(i,j);%Implied volatilities m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.

63 sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u = sol(:,:,1); %Solve the PDE, denote the first solution as the option value.

Limit = 100; Yield = 0; Tolerance = 1e-18; Class={'Call'}; for i=1:51 for j=1:201 imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield, Tolerance, Class); end end value =imp_v; % ------% The curve of implied volatilities figure; plot(x,value(29,:),'.','Color','black','LineWidth',2);hold on; plot(x,value(39,:),'Color','blue','LineWidth',2);hold on; plot(x,value(49,:),'Color','magenta','LineWidth',2); xlabel('Stock Prices'); ylabel('Implied Volatilities'); title('European Call Options'); % ------% The Term Structure of Implied Volatilities for European Call Options figure; plot(tau,value(:,2),'Color','black','LineWidth',2);hold on; plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on; plot(tau,value(:,22),'Color','red','LineWidth',2);hold on; plot(tau,value(:,32),'Color','blue','LineWidth',2);hold on; plot(tau,value(:,42),'Color','black','LineWidth',2);hold on; plot(tau,value(:,52),'.','Color','magenta','MarkerSize',6); colormap hsv; xlabel('Time-to-maturity \tau'); ylabel('Implied Volatilities'); title('The Term Structure of Implied Volatilities for European Call Options'); % ------% The implied volatility surface figure; surf(x,tau,value);

64 colormap hsv; xlabel('Stock Prices'); ylabel('Time-to-maturity \tau'); zlabel('Implied Volatilities'); title('The Implied Volatility Surface of European Call Options'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K),0); end % ------function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) K = 10; pl = ul; ql = 0; pr = -1/2*K; qr = 1; end end

65 Plots

Figure A.2

3. pdex_dis_imp_loc.m

Program code

Table A.3 pdex_dis_imp_loc.m function value = pdex_dis_imp_loc(m, T, x, tau) %Function calculates the difference between two types of volatilities. i = 51; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 5; %The endpoint value of maturity time T. j = 201; %The number of meshpoints in the range of the asset prices S. K = 10; %The strike price K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0

66 sig = ones(i,j);% The local volatilities imp_v = ones(i,j);%Implied volatilities dis = ones(i,j);% The distance between local & implied volatilities m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab. sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u = sol(:,:,1); %Solve the PDE, denote the first solution as the option value.

Limit = 10; Yield = 0; Tolerance = 1e-16; Class={'Call'}; for i=1:51 for j=1:201 sig(i,j)=sqrt(1/x(j)); imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield, Tolerance, Class); dis(i,j)=abs(imp_v(i,j)-sig(i,j)); end end value =dis; % ------% The curve of distance plotting against the asset price figure; plot(x,sig); xlabel('Stock Prices'); ylabel('Local Volatility'); title('European Call Options') % ------% The curve of distance plotting against the asset price figure; plot(x,value(31,:),'.','Color','red','LineWidth',2);hold on; plot(x,value(41,:),'Color','blue','LineWidth',2);hold on; plot(x,value(51,:),'Color','black','LineWidth',2);hold on; xlabel('Stock Prices'); ylabel('Distance between Two Types of Volatilities'); title('European Call Options'); % ------% The curve of distance plotting against the time-to-maturity figure; plot(tau,value(:,2),'Color','red','LineWidth',2);hold on;

67 plot(tau,value(:,3),'Color','blue','LineWidth',2);hold on; plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on; plot(tau,value(:,21),'Color','green','LineWidth',2);hold on; plot(tau,value(:,42),'Color','black','LineWidth',2);hold on; xlabel('Time-to-maturity'); ylabel('Distance between Two Types of Volatilities'); title('European Call Options'); % ------% The surface consists of the distance between two kinds of volatilities figure; surf(x,tau,value); title('Distance between Two Types of Volatilities'); xlabel('Stock Prices'); ylabel('Time-to-maturity'); zlabel('Distance between Two Types of Volatilities'); title('The Distance Surface of European Call Options'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K),0); end % ------function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) K = 10; pl = ul; ql = 0; pr = -1/2*K; qr = 1; end end

68 Plots

Figure A.3: The Distance Surface of European Call Options

4. pdex_u_small_s.m

Program code

Table A.4 pdex_u_small_s.m function value = pdex_u_small_s(m, T, x, tau) %Function calculates the difference between two types of volatilities. i = 51; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 5; %The endpoint value of maturity time T. j = 202; %The number of meshpoints in the range of the asset prices S. K = 10; %The strike price K. x = ones(1,j); xx = linspace(0,10^-100,j-1); for j=1:201 x(1,j)=xx(j); end x(1,202)=100;

69 %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); %The actual variable notation we use in pdepe tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab. sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u = sol(:,:,1); %Solve the PDE, denote the first solution as the option value. u_small_s = ones(i,j-1); for i=1:51 for j=1:201 u_small_s(i,j)= u(i,j); end end value =u_small_s; % ------% The curve of distance plotting against the asset price figure; plot(xx,value); colormap hsv; xlabel('Stock Prices'); ylabel('Option Value'); title('European Call Options'); figure; plot(xx,value(51,:),'Color','red'); hold on; plot(xx,value(11,:),'Color','blue'); hold on; plot(xx,value(1,:),'Color','black'); hold on; xlabel('Stock Prices'); ylabel('Option Value'); title('European Call Options'); % ------% The surface consists of the distance between two kinds of volatilities figure; surf(x,tau,u); colormap hsv; title('Distance between two kinds of volatilities'); xlabel('Stock Prices'); ylabel('Time-to-maturity \tau'); zlabel('Implied Volatilities');

70 title('The European Call Option Value Surface'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K),0); end % ------function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) K = 10; pl = ul; ql = 0; pr = -1/2*K; qr = 1; end end

5. pdex_imp_small_s.m

Program code

Table A.5 pdex_imp_small_s.m function value = pdex_imp_small_s(m, T, x, tau) %Function calculates the difference between two types of volatilities. i = 51; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 5; %The endpoint value of maturity time T. j = 202; %The number of meshpoints in the range of the asset prices S. K = 10; %The strike price K. K=1,S=20 imp_1 K=10,S=100,imp_2 x = ones(1,j); xx = linspace(0,10^-10,j-1); for j=1:201 x(1,j)=xx(j); end

71 x(1,202)=100; %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 imp_v = ones(i,j);%Implied volatilities m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab. sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u = sol(:,:,1); %Solve the PDE, denote the first solution as the option value.

Limit = 1000000000; Yield = 0; Tolerance = 1e-18; Class={'Call'}; for i=1:51 for j=1:202 imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield, Tolerance, Class); end end imp_v_small_s = ones(i,j-1); for i=1:51 for j=1:201 imp_v_small_s(i,j)= imp_v(i,j); end end value =imp_v_small_s; % ------% The curve of implied volatilities figure; plot(xx,value(29,:),'Color','black','LineWidth',2);hold on; plot(xx,value(39,:),'Color','blue','LineWidth',2);hold on; plot(xx,value(49,:),'Color','magenta','LineWidth',2); xlabel('Stock Prices'); ylabel('Implied Volatilities'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------slope = ones(i-1,j-2);

72 xxx = ones(1,j-2); for i=1:50 for j=1:199 slope(i,j)=(imp_v_small_s(i+1,j+1)-imp_v_small_s(i+1,j+2))/(xx(j)- xx(j+1)); xxx(1,j)=xx(j+1); end slope(i,200)=slope(i,199); end xxx(1,200)=xx(201); % ------% The curve of implied volatility curves' slope figure; plot(xxx,slope(29,:),'Color','black','LineWidth',2);hold on; plot(xxx,slope(39,:),'Color','blue','LineWidth',2);hold on; plot(xxx,slope(49,:),'Color','magenta','LineWidth',2); xlabel('Stock Prices'); ylabel('The Slope of Implied Volatility Curves'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------loc = ones(i,j-1); for i=1:51 for j=1:201 loc(i,j)=sqrt(1/(xx(j))); end end % ------% The curve of local volatilities figure; plot(xx,loc,'Color','blue','LineWidth',2); xlabel('Stock Prices'); ylabel('Local Volatilities'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------slope_loc = ones(i-1,j-2); for i=1:50 for j=1:200 slope_loc(i,j) =-(1/2)*(xxx(1,j)^(-3/2)); end end % ------% The curve of local volatility curves' slope figure; plot(xxx,slope_loc,'.','Color','black','LineWidth',2);

73 xlabel('Stock Prices'); ylabel('The Slope of Local Volatility Curves'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K),0); end % ------function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) K = 10; pl = ul; ql = 0; pr = -1/2*K; qr = 1; end end

6. pdex_imp_slope_small_s_fit.m

Program code

Table A.6 pdex_imp_slope_small_s_fit.m load local_volatility_imp_small_s.mat imp= ans; xx = linspace(0,10^-10,201); load local_volatility_imp_small_s_slope.mat xxx = linspace(xx(2),10^-10,200); x=ones(1,8); % ------%lower bound for implied volatility curve,tau=0.1 x(1)=-0.5; for j=1:201 while imp(2,j)

74 end % ------%upper bound for implied volatility curve,tau=0.1 x(2)=-0.01; for j=1:201 while imp(2,j)>xx(j)^x(2) x(2)=x(2)-0.01; end end % ------%lower bound for implied volatility curve,tau=5 x(3)=-0.5; for j=1:201 while imp(51,j)xx(j)^x(4) x(4)=x(4)-0.01; end end % ------%lower bound for implied volatility curve,tau=0.1 x(5)=-1.5; for j=1:200 while slope(1,j)>-0.5*(xxx(j))^(x(5)); x(5)=x(5)+0.01; end end % ------%upper bound for implied volatility curve,tau=0.1 x(6)=-0.01; for j=1:200 while slope(1,j)<-0.5*(xxx(j))^(x(6)); x(6)=x(6)-0.01; end end % ------%lower bound for implied volatility curve,tau=5 x(7)=-1.5;

75 for j=1:200 while slope(50,j)>-0.5*(xxx(j))^(x(7)); x(7)=x(7)+0.01; end end % ------%upper bound for implied volatility curve,tau=5 x(8)=-0.01; for j=1:200 while slope(50,j)<-0.5*(xxx(j))^(x(8)); x(8)=x(8)-0.01; end end % ------value=x; % ------% The curves of implied volatilities figure; plot(xx,imp(2,:),'.','Color','cyan','LineWidth',2);hold on; plot(xx,imp(51,:),'.','Color','black','LineWidth',2);hold on; xlabel('Stock Prices'); ylabel('Implied Volatilities'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------loc=ones(1,201); % ------for j=1:201 loc(j)=(xx(j))^(x(1)); end plot(xx,loc,'Color','green','LineWidth',4);hold on; for j=1:201 loc(j)=(xx(j))^((x(1)+x(2))/2); end plot(xx,loc,'Color','magenta','LineWidth',2);hold on; for j=1:201 loc(j)=(xx(j))^(x(2)); end plot(xx,loc,'Color','blue','LineWidth',4);hold on; for j=1:201 loc(j)=(xx(j))^(x(3)); end plot(xx,loc,'Color','blue','LineWidth',2);hold on; for j=1:201 loc(j)=(xx(j))^((x(3)+x(4))/2);

76 end plot(xx,loc,'Color','red','LineWidth',2);hold on; for j=1:201 loc(j)=(xx(j))^(x(4)); end plot(xx,loc,'Color','green','LineWidth',2);hold on; % ------% The curve of slopes figure; plot(xxx,slope(1,:),'Color','black','LineWidth',4);hold on; plot(xxx,slope(50,:),'Color','blue','LineWidth',4);hold on; xlabel('Stock Prices'); ylabel('The Slope of Implied Volatility Curves'); title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2'); % ------locc=ones(1,200); for j=1:200 locc(j)=-0.5*(xxx(j))^(x(5)); end plot(xxx,locc,'Color','red','LineWidth',2);hold on; for j=1:200 locc(j)=-0.5*(xxx(j))^(x(6)); end plot(xxx,locc,'Color','magenta','LineWidth',2);hold on; % ------for j=1:200 locc(j)=-0.5*(xxx(j))^(x(7)); end plot(xxx,locc,'Color','green','LineWidth',2);hold on; for j=1:200 locc(j)=-0.5*(xxx(j))^(x(8)); end plot(xxx,locc,'Color','cyan','LineWidth',2);hold on; % ------

7. pdex_dupire_option.m

Program code

Table A.7 pdex_dupire_option.m function value = pdex_dupire_option(m, K, x, tau) %Function calculates the difference between two types of volatilities.

77 i = 21; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 2; %The endpoint value of maturity time T. j = 101; %The number of meshpoints in the range of the asset prices S. h = 21; %The number of meshpoints in the range of the strike prices K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K. sig_2 = ones(i,j,h);%The square of implied volatility imp_v = ones(i,j,h);%The implied volatilities. sig_dupire = ones(i,j,h);%The Dupire volatility dis = ones(i,j,h);%The distance between Dupire&implied volatilities. u = ones(i,j,h); m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab. for h=1:21 sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u(:,:,h) = sol(:,:,1); %Solve the PDE, denote the first solution as the option value. end value = u; % ------%The Option Value Surface figure; uu = ones(21,101); for h=1:21 for j=1:101 uu(h,j)=u(21,j,h); end end surf(x,K,uu); xlabel('Stock/Asset Prices'); ylabel('Strike Prices'); zlabel('Option Value'); title('The Option Value Surface,\tau=2'); % ------%The Option Value Curve(Agaist the Strike K)

78 figure; uu = ones(21,101); for h=1:21 for j=1:101 uu(h,j)=u(21,j,h); end end plot(K,uu(:,1),'Color','red','LineWidth',2);hold on; plot(K,uu(:,51),'Color','blue','LineWidth',2);hold on; plot(K,uu(:,101),'Color','black','LineWidth',2);hold on; xlabel('Strike Prices K'); ylabel('Option Value V'); title('The Option Value Curve,\tau=2'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K(h)),0); end function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = -1/2*K(h); qr = 1; end end

8.pdex_dupire.m

Program code Table A.8 pdex_dupire.m function value = pdex_dupire(m, K, x, tau) %Function calculates the difference between two types of volatilities. i = 21; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 2; %The endpoint value of maturity time T. j = 101; %The number of meshpoints in the range of the asset prices S.

79 h = 21; %The number of meshpoints in the range of the strike prices K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K. u = ones(i,j,h); %The option values sig_2 = ones(i,j,h);%The square of implied volatility imp_v = ones(i,j,h);%The implied volatilities. sig_dupire = ones(i,j,h);%The Dupire volatility dis = ones(i,j,h);%The distance between Dupire&implied volatilities. m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.

Limit = 100; Yield = 0; Tolerance = 1e-18; Class={'Call'};

C_T = ones(i,j,h); C_KK = ones(i,j,h); for h=1:21 sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u(:,:,h) = sol(:,:,1); %Solve the PDE, denote the first solution as the option value. end for j=1:101 for i=1:20 for h=1:19 C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i)); C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2); sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h)); if sig_2(i,j,h)>=0 sig_dupire(i,j,h)=sqrt(sig_2(i,j,h)); else sig_dupire(i,j,h)=NaN; end end end

80 end dupire = ones(i-1,j-1,h-2); for i=1:20 for j=1:100 for h=1:19 dupire(i,j,h)=sig_dupire(i,j+1,h); end end end value = dupire; vol= ones(j-1,h-2); xx = linspace(x(2),100,100); KK = linspace(9.18,K(h-2),h-2); TT= linspace(T(1),T(i-1),i-1); % ------for j=1:100 for h=1:19 vol(j,h)=dupire(20,j,h); end end figure; plot(xx,vol(:,9),'.','Color','red'); hold on; xlabel('Stock Prices S'); ylabel('Dupire Volatility'); title('European Calls,Dupire Volatility Curve, \tau=1.9, K=9.9');%K=9.908 figure; plot(xx,vol(:,10),'.','Color','red'); hold on; xlabel('Stock Prices S'); ylabel('Dupire Volatility'); title('European Calls,Dupire Volatility Curve, \tau=1.9, K=10');%K=9.999 % ------vol_t=ones(20,19); for h=1:19 for i=1:20 vol_t(i,h)=dupire(i,20,h); end end tt=linspace(T(1),T(20),20); figure; plot(tt,vol_t(:,9),'.','Color','red'); hold on; plot(tt,vol_t(:,10),'.','Color','blue'); hold on;

81 xlabel('Time-to-maturity \tau'); ylabel('Dupire Volatility'); title('European Calls,Dupire Volatility Curve, S=20'); % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K(h)),0); end function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = -1/2*K(h); qr = 1; end end

9.pdex_dis_imp_dupire.m Program code

Table A.9 pdex_dis_imp_dupire.m function value = pdex_dis_imp_dupire(m, T, x, tau) %Function calculates the difference between two types of volatilities. i = 21; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 2; %The endpoint value of maturity time T. j = 101; %The number of meshpoints in the range of the asset prices S. h = 21; %The number of meshpoints in the range of the strike prices K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. t = linspace(t0,tT,i); tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0 K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K. u = ones(i,j,h); %The option values V. sig_2 = ones(i,j,h);%The square of implied volatility

82 imp_v = ones(i,j,h);%The implied volatilities. sig_dupire = ones(i,j,h);%The Dupire volatility dis = ones(i-1,j,h-2);%The distance between Dupire&implied volatilities. m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.

Limit = 100; Yield = 0; Tolerance = 1e-18; Class={'Call'};

C_T = ones(i,j,h); C_KK = ones(i,j,h); for h=1:21 sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t); %Using the built-in function pdepe to solve the PDE for option value V. u(:,:,h) = sol(:,:,1); %Solve the PDE, denote the first solution as the option value. for i=1:21 for j=1:101 imp_v(i,j,h) = blsimpv(x(j), K(h), 0, tau(i), u(i,j,h), Limit, Yield, Tolerance, Class); end end end for h=1:19 for j=1:101 for i=1:20 C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i)); C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2); sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h)); if sig_2(i,j,h)>=0 sig_dupire(i,j,h)=sqrt(sig_2(i,j,h)); else sig_dupire(i,j,h)=NaN; end dis(i,j,h)=abs(imp_v(i,j,h)-sig_dupire(i,j,h)); end end end

83 distance= ones(i-1,j-1,h-2); TT= linspace(T(1),T(i-1),i-1); xx = linspace(x(2),100,100); KK = linspace(9.18,K(h-2),h-2); for i=1:20 for j=1:100 for h=1:19 distance(i,j,h)=dis(i,j+1,h); end end end value = distance; % ------function [c,f,s] = pdex1pde(x,t,u,DuDx) c = 1; f = 1/2*(x*DuDx-u); s = 0; end % ------function u0 = pdex1ic(x) u0 = max((x-K(h)),0); end function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t) pl = ul; ql = 0; pr = -1/2*K(h); qr = 1; end end

10.Dupire_plots_dis.m Program code

Table A.10 dupire_plots_dis.m for type=1:2 if type==1 load local_volatility_dis_dupire.mat; distance; %The distance between the implied volatility and the dupire

84 %volatility % ------i = 21; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 2; %The endpoint value of maturity time T. j = 101; %The number of meshpoints in the range of the asset prices S. h = 21; %The number of meshpoints in the range of the strike prices K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. % ------TT= linspace(T(1),T(i-1),i-1); xx = linspace(x(2),100,100); % ------distance_1 = ones(20,19); %For plotting distance surface,S=29. distance_2 = ones(20,19); %For plotting distance surface,S=31. distance_3 = ones(20,19); %For plotting distance surface,S=33. distance_4 = ones(20,19); %For plotting distance surface,S=35. distance_5 = ones(20,19); %For plotting distance surface,S=35. distance_6 = ones(20,19); %For plotting distance surface,S=35. ttau = linspace(0,1.9,20); %Time-to-maturity,time-to-maturity=linspace(0.1,2,20). KK = linspace(9.1,10.9,19); %Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19). % ------for i=1:20 for h=1:19 distance_1(i,h)=distance(i,55,h);%S=55 distance_2(i,h)=distance(i,60,h);%S=60 distance_3(i,h)=distance(i,65,h);%S=65 distance_4(i,h)=distance(i,70,h);%S=70 distance_5(i,h)=distance(i,75,h);%S=75 distance_6(i,h)=distance(i,80,h);%S=80 end end % In order to demonstrate the relationships between %strike prices K and distancetance between the volatilities, we select some %effective data from the computing results by picking column 1-20 in time %meshpoints, column 11-70 in asset price meshpoints(eliminating those

85 %out-of-money points), column 2-20 in strike prices meshpoints, i.e., %time-to-maturity=linspace(0.1,2,20), %price=linspace(10,69,60),strike=linspace(9.1,10.9,19). % ------figure; plot(KK,distance_1,'o','Color','red'); hold on; plot(KK,distance_2,'o','Color','red'); hold on; plot(KK,distance_3,'o','Color','red'); hold on; xlabel('Strike Prices K');ylabel('Distance');title('European Calls, S=55:65:1, \tau=0:1.9:0.1'); figure; plot(KK,distance_4,'o','Color','blue'); hold on; plot(KK,distance_5,'o','Color','blue'); hold on; plot(KK,distance_6,'o','Color','blue'); hold on; xlabel('Strike Prices K');ylabel('Distance');title('European Calls, S=70:80:1, \tau=0:1.9:0.1'); % ------figure; plot(xx,distance(20,:,10),'.','Color','blue'); hold on; xlabel('The Stock Prices S'); ylabel('The Distance'); title('The Distance Curve \tau=1.9, K=9.9'); % ------end if type==2 load local_volatility_dis_dupire1.mat; distance=ans; %The distance between the implied volatility and the dupire %volatility % ------i = 21; %The number of meshpoints in the range of the maturity time T. t0 = 0; %The initial value of maturity time T. tT = 2; %The endpoint value of maturity time T. j = 101; %The number of meshpoints in the range of the asset prices S. h = 21; %The number of meshpoints in the range of the strike prices K. x = linspace(0,100,j); %The meshpoints in the range of asset prices S. T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T. % ------TT= linspace(T(1),T(i-1),i-1); xx = linspace(x(2),100,100); % ------distance_1 = ones(20,19); %For plotting distance surface,S=29.

86 distance_2 = ones(20,19); %For plotting distance surface,S=31. distance_3 = ones(20,19); %For plotting distance surface,S=33. distance_4 = ones(20,19); %For plotting distance surface,S=35. distance_5 = ones(20,19); %For plotting distance surface,S=35. distance_6 = ones(20,19); %For plotting distance surface,S=35. ttau = linspace(0,1.9,20); %Time-to-maturity,time-to-maturity=linspace(0.1,2,20). KK = linspace(9.1,10.9,19); %Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19). % ------for i=1:20 for h=1:19 distance_1(i,h)=distance(i,55,h);%S=55 distance_2(i,h)=distance(i,60,h);%S=60 distance_3(i,h)=distance(i,65,h);%S=65 distance_4(i,h)=distance(i,70,h);%S=70 distance_5(i,h)=distance(i,75,h);%S=75 distance_6(i,h)=distance(i,80,h);%S=80 end end % In order to demonstrate the relationships between %strike prices K and distancetance between the volatilities, we select some %effective data from the computing results by picking column 1-20 in time %meshpoints, column 11-70 in asset price meshpoints(eliminating those %out-of-money points), column 2-20 in strike prices meshpoints, i.e., %time-to-maturity=linspace(0.1,2,20), %price=linspace(10,69,60),strike=linspace(9.1,10.9,19).

% ------figure; plot(KK,distance_1,'o','Color','red'); hold on; plot(KK,distance_2,'o','Color','red'); hold on; plot(KK,distance_3,'o','Color','red'); hold on; xlabel('Strike Prices K');ylabel('Distance');title('European Calls, S=55:65:1, \tau=0:1.9:0.1'); figure; plot(KK,distance_4,'o','Color','blue'); hold on;

87 plot(KK,distance_5,'o','Color','blue'); hold on; plot(KK,distance_6,'o','Color','blue'); hold on; xlabel('Strike Prices K');ylabel('Distance');title('European Calls, S=70:80:1, \tau=0:1.9:0.1'); % ------figure; plot(xx,distance(20,:,10),'.','Color','blue'); hold on; xlabel('The Stock Prices S'); ylabel('The Distance'); title('The Distance Curve \tau=1.9, K=10'); % ------end end

Plots

Figure A.4

88 Appendix B

Numerical results

1. pdex_dis_european.m

Table B.1 Distance between Implied Volatilities and Local Volatilities

tau=0.1:1:10; K=10; S=0.5:100:200; r=0，q=0.

S\tau 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.381217 0.562026 0.637643 0.681383 0.709304 0.729117 0.742915 0.753640 0.761925 0.768356 0.5 10463 28141 49171 62441 00844 19886 56128 93769 00543 03124 0.156847 0.297276 0.354790 0.387215 0.421240 0.430712 0.437811 0.443132 0.447167 1 4.07E-01 2335 30276 60181 67973 72249 34359 49542 12615 81166 0.079851 0.196688 0.243554 0.269281 0.295276 0.302272 0.307340 0.311044 0.313802 1.5 2.85E-01 112732 22929 7783 66874 19812 81675 66609 2074 31805 0.043809 0.143776 0.182987 0.203900 0.224242 0.229539 0.233257 0.235915 0.237863 2 2.16E-01 557089 21278 53735 49192 33557 45373 42891 91075 93361 0.024745 0.111435 0.144600 0.161739 0.177780 0.181818 0.184575 0.186508 0.187903 2.5 1.72E-01 94886 58702 0573 64357 36096 87993 21156 62371 45719 0.014203 0.089826 0.117945 0.131986 0.144623 0.147694 0.149740 0.151150 0.152150 3 1.40E-01 857135 0792 22421 01408 0234 58832 48042 22207 57439 0.008449 0.074479 0.098240 0.109669 0.119555 0.121870 0.123379 0.124399 0.125110 3.5 1.16E-01 1081817 836628 347451 33365 94756 45214 02905 7755 08085 0.005585 0.063066 0.082975 0.092172 0.099811 0.101527 0.102622 0.103347 0.103839 4 9.70E-02 8472887 055416 577613 906451 442784 20649 16219 6191 74336 0.004553 0.054246 0.070705 0.077984 0.083773 0.085013 0.085785 0.086283 0.086609 4.5 8.17E-02 3637491 253429 784348 865619 880683 609228 923341 607658 017504 0.004712 0.047194 0.060540 0.066169 0.070431 0.071291 0.071810 0.072130 0.072327 5 6.89E-02 6469886 877022 937843 042355 2401 874146 535036 824443 792519 0.005653 0.041372 0.051903 0.056112 0.059114 0.059673 0.059992 0.060174 0.060272 5.5 5.81E-02 8076325 734601 26799 61 391901 824716 805149 579524 535422 0.007096 0.036409 0.044400 0.047395 0.049361 0.049683 0.049846 0.049920 0.049942 6 4.87E-02 7849401 200015 245524 403057 988459 789077 465072 628511 812564 0.008834 0.032035 0.037753 0.039718 0.040845 0.040982 0.041023 0.041015 0.040980 6.5 4.05E-02 2403076 861076 63071 598468 36912 073006 930092 623378 723531 0.010693 0.028046 0.031758 0.032864 0.033225 0.033324 0.033319 0.033269 0.033199 0.033123 7 03823 981074 792998 376975 244523 246804 580777 87766 90549 053761 0.012501 0.024276 0.026261 0.026672 0.026701 0.026619 0.026509 0.026392 0.026278 0.026171 7.5 325247 244109 945981 000066 507477 142471 45314 418869 157234 507635 0.014052 0.020585 0.021148 0.021022 0.020805 0.020593 0.020408 0.020244 0.020100 0.019973 8 756095 76494 046921 32082 855795 371306 605068 541551 37908 545563 0.015063 0.016865 0.016334 0.015826 0.015439 0.015140 0.014905 0.014711 0.014549 0.014409 8.5 536815 705996 160232 438994 353051 736457 557566 667948 148614 806611 0.015129 0.013041 0.011764 0.011016 0.010525 0.010177 0.009912 0.009702 0.009531 0.009385 9 590141 900052 958072 796266 756632 103033 4032809 8458776 0413388 5427985 0.013730 0.009083 0.007407 0.006541 0.006004 0.005634 0.005358 0.005144 0.004970 0.004824 9.5 267484 1345279 91223 7578271 5460935 9789657 7646869 5833034 7869967 3885744 0.010406 0.004994 0.003242 0.002362 0.001825 0.001459 0.001186 0.000976 0.000806 0.000663 10 040359 5820073 5736833 7687265 7235286 545036 8861975 85756307 94483241 98776611 0.005160 0.000810 0.000747 0.001552 0.002051 0.002394 0.002651 0.002849 0.003011 0.003147 10.5 389942 50255125 86909528 6631874 516054 6005087 1013317 8912811 1923524 1124541 0.001305 0.003422 0.004572 0.005232 0.005661 0.005965 0.006195 0.006377 0.006525 0.006651 11 9134515 241281 3422008 1319763 7628258 3676034 482706 1527103 9062484 8516811

89 0.008313 0.007652 0.008239 0.008700 0.009033 0.009284 0.009480 0.009639 0.009772 0.009886 11.5 3524762 1224059 8686139 0494039 7945192 8025767 2628419 8184484 7311917 3242765 0.015408 0.011833 0.011758 0.011975 0.012192 0.012380 0.012534 0.012667 0.012781 0.012881 12 50842 934932 341859 683396 629057 042065 511146 475864 791537 106418 0.022340 0.015928 0.015132 0.015074 0.015159 0.015274 0.015382 0.015485 0.015578 0.015662 12.5 759103 928666 151877 86044 820894 164483 940499 458329 761692 291003 0.028990 0.019906 0.018363 0.018011 0.017953 0.017987 0.018046 0.018115 0.018185 0.018252 13 507838 798096 816975 322582 45945 04686 501523 557493 686826 226043 0.035313 0.023746 0.021455 0.020796 0.020588 0.020535 0.020543 0.020576 0.020621 0.020670 13.5 118715 903873 587684 979258 740211 916997 130669 512578 633129 147192 0.041303 0.027437 0.024410 0.023442 0.023078 0.022935 0.022888 0.022884 0.022903 0.022932 14 919531 723908 298008 098458 63793 706712 355026 468891 157154 735512 0.046977 0.030975 0.027231 0.025955 0.025434 0.025199 0.025095 0.025053 0.025044 0.025054 14.5 771882 367047 831216 655362 416284 369031 698235 41487 670193 551154 0.052357 0.034361 0.029925 0.028345 0.027666 0.027338 0.027176 0.027095 0.027058 0.027048 15 625122 798768 332871 716354 001814 190131 979634 552705 752796 355536 0.057468 0.037603 0.032497 0.030619 0.029782 0.029362 0.029142 0.029021 0.028956 0.028925 15.5 438526 089227 202861 778034 27942 075098 570478 593306 431687 365949 0.062334 0.040707 0.034954 0.032785 0.031791 0.031279 0.031001 0.030840 0.030747 0.030695 16 21219 846011 911837 032742 333255 794296 624261 991838 417323 471679 0.066976 0.043685 0.037306 0.034848 0.033700 0.033099 0.032762 0.032562 0.032440 0.032367 16.5 740184 927164 699812 550969 640722 189668 280519 141424 303343 421914 0.071415 0.046547 0.039561 0.036817 0.035517 0.034827 0.034431 0.034192 0.034042 0.033948 17 265637 475603 2159 380303 223359 345203 840259 536166 733123 988042 0.075666 0.049302 0.041727 0.038698 0.037247 0.036470 0.036016 0.035738 0.035561 0.035447 17.5 552972 265676 154201 568419 757471 72641 913411 909582 539522 102274 0.079745 0.051959 0.043812 0.040499 0.038898 0.038035 0.037523 0.037207 0.037002 0.036867 18 143719 316108 931147 124418 647128 293137 540525 351955 862946 975407 0.083663 0.054526 0.045826 0.042225 0.040476 0.039526 0.038957 0.038603 0.038372 0.038217 18.5 649072 706727 433953 937761 062846 589252 291722 409198 251515 196949 0.087434 0.057011 0.047774 0.043885 0.041985 0.040949 0.040323 0.039932 0.039674 0.039499 19 215469 535637 852134 676083 950732 811999 345954 165472 746106 820557 0.091062 0.059419 0.049664 0.045484 0.043434 0.042309 0.041626 0.041198 0.040914 0.040720 19.5 448777 96276 588793 681701 018692 863306 553283 311677 952366 437202 0.094562 0.061757 0.051501 0.047028 0.044825 0.043611 0.042871 0.042406 0.042097 0.041883 20 727534 304776 238828 882183 707651 385185 482336 201744 101424 237931 0.097779 0.064028 0.053289 0.048523 0.046166 0.044858 0.044062 0.043559 0.043225 0.042992 20.5 148496 133239 616548 72445 156076 781467 454724 898396 100799 067653 0.100936 0.066236 0.055033 0.049974 0.047460 0.046056 0.045203 0.044663 0.044302 0.044050 21 42391 422444 816098 136011 165238 228459 568095 209776 576835 471111 0.103052 0.068385 0.056737 0.051384 0.048712 0.047207 0.046298 0.045719 0.045332 0.045061 21.5 75848 506871 289279 512401 170879 677394 709579 718077 909804 732002 0.213200 0.070478 0.058402 0.052758 0.049926 0.048316 0.047351 0.046732 0.046319 0.046028 22 71636 602084 930072 726704 224817 851668 561625 801224 262696 906104 0.210818 0.072518 0.060033 0.054100 0.051105 0.049387 0.048365 0.047705 0.047264 0.046954 22.5 51068 153653 158968 156339 987844 241651 602408 648547 604528 849144 0.208514 0.074506 0.061629 0.055411 0.052254 0.050422 0.049344 0.048641 0.048171 0.047842 23 41406 370792 999657 720577 733619 099457 103011 271487 728923 24004 0.206284 0.076442 0.063195 0.056695 0.053375 0.051424 0.050290 0.049542 0.049043 0.048693 23.5 24925 776523 151912 925885 362192 435469 123418 510406 268617 600079 0.204124 0.078332 0.064730 0.057954 0.054470 0.052397 0.051206 0.050412 0.049881 0.049511 24 14523 350451 065224 911958 420906 017712 50898 038581 706546 308499 0.202030 0.080175 0.066235 0.059190 0.055542 0.053342 0.052095 0.051252 0.050689 0.050297 24.5 50891 842065 881994 497931 130522 374657 888596 364462 38413 614902 0.081665 0.067713 0.060404 0.056592 0.054262 0.052960 0.052065 0.051468 0.051054 25 0.2 618899 849378 233472 414445 801387 675352 833123 507385 648916 0.198029 0.083062 0.069164 0.061597 0.057622 0.055160 0.053803 0.052854 0.052221 0.051784 25.5 5086 450688 544621 419351 929811 368978 06995 627699 151443 427521 0.196116 0.085190 0.070588 0.062771 0.058635 0.056036 0.054625 0.053620 0.052949 0.052488 26 13514 237614 864638 213136 097848 936075 066841 771397 264065 860464

90 0.194257 0.194257 0.071986 0.063926 0.059630 0.056894 0.055428 0.054366 0.053654 0.053169 26.5 17247 17247 435886 484615 129607 16345 462707 130433 668565 754181 0.192450 0.192450 0.073353 0.065063 0.060609 0.057733 0.056214 0.055092 0.054339 0.053828 27 08973 08973 039093 994666 062924 52652 867309 418078 066633 814658 0.190692 0.190692 0.074723 0.066184 0.061572 0.058556 0.056985 0.055801 0.055004 0.054467 27.5 51785 51785 470441 813187 782694 335534 71505 200064 041225 649572 0.188982 0.188982 0.076080 0.067289 0.062522 0.059363 0.057742 0.056493 0.055651 0.055087 28 2365 2365 266942 024716 035096 744843 278546 9005 059925 770135 0.187317 0.187317 0.077387 0.068377 0.063457 0.060156 0.058485 0.057171 0.056281 0.055690 28.5 16232 16232 514192 709068 453351 774282 681704 809883 478497 592785 0.185695 0.185695 0.077291 0.069450 0.064379 0.060936 0.059216 0.057836 0.056896 0.056277 29 33818 33818 740107 212077 512251 32214 913379 091954 545357 441081 0.184114 0.184114 0.079305 0.070505 0.065288 0.061703 0.059936 0.058487 0.057497 0.056849 29.5 92358 92358 588343 697352 693754 181863 840246 793162 40606 547751 0.182574 0.182574 0.079771 0.071541 0.066185 0.062458 0.060646 0.059127 0.058085 0.057408 30 18584 18584 896187 869008 665853 016563 220899 850789 108466 057357 0.181071 0.181071 0.087004 0.072565 0.067070 0.063201 0.061345 0.059757 0.058660 0.057954 30.5 49209 49209 839025 77066 697091 508114 718665 102508 608316 028945 0.179605 0.179605 0.179605 0.073576 0.067942 0.063934 0.062035 0.060376 0.059224 0.058488 31 30203 30203 30203 639701 853421 100385 904292 292176 774314 439507 0.178174 0.178174 0.178174 0.074532 0.068805 0.064656 0.062717 0.060986 0.059778 0.059012 31.5 16127 16127 16127 949771 512361 251906 278902 083227 39539 187481 0.176776 0.176776 0.176776 0.075309 0.069654 0.065368 0.063390 0.061587 0.060322 0.059526 32 6953 6953 6953 773773 694696 491046 25895 061534 184268 096663 0.175411 0.175411 0.175411 0.076204 0.070495 0.066070 0.064055 0.062179 0.060856 0.060030 32.5 60386 60386 60386 601211 334886 952805 281853 745989 785224 920488 0.174077 0.174077 0.174077 0.076149 0.071310 0.066764 0.064712 0.062764 0.061382 0.060527 33 65596 65596 65596 242699 695742 938185 643061 594774 779073 345506 0.172773 0.172773 0.172773 0.074931 0.072135 0.067449 0.065362 0.063342 0.061900 0.061015 33.5 68512 68512 68512 194264 775894 19491 5928 028945 687034 997216 0.171498 0.171498 0.171498 0.083644 0.072939 0.068123 0.066005 0.063912 0.062410 0.061497 34 58514 58514 58514 537939 759719 077224 329381 390431 981796 443139 0.170251 0.170251 0.170251 0.170251 0.073725 0.068789 0.066641 0.064475 0.062914 0.061972 34.5 30615 30615 30615 30615 928047 9149 224116 963429 08256 197127 0.169030 0.169030 0.169030 0.169030 0.074367 0.069441 0.067269 0.065033 0.063410 0.062440 35 85095 85095 85095 85095 925746 304727 863433 106861 364177 721511 0.167836 0.167836 0.167836 0.167836 0.075779 0.070106 0.067892 0.065583 0.063900 0.062903 35.5 27166 27166 27166 27166 983861 580939 760186 981173 164199 440271 0.166666 0.166666 0.166666 0.166666 0.075869 0.070734 0.068509 0.066128 0.064383 0.063360 36 66667 66667 66667 66667 508203 183419 905904 817679 791984 731282 0.165521 0.165521 0.165521 0.165521 0.068854 0.071381 0.069117 0.066667 0.064861 0.063812 36.5 17772 17772 17772 17772 173639 934794 767653 793878 540358 927971 0.164398 0.164398 0.164398 0.164398 0.164398 0.072037 0.069721 0.067201 0.065333 0.064260 37 98731 98731 98731 98731 98731 9876 961366 005372 619575 355088 0.163299 0.163299 0.163299 0.163299 0.163299 0.072599 0.070320 0.067728 0.065800 0.064703 37.5 31619 31619 31619 31619 31619 266602 211563 658784 241786 263865 0.162221 0.162221 0.162221 0.162221 0.162221 0.072923 0.070901 0.068251 0.066261 0.065141 38 42113 42113 42113 42113 42113 478096 149696 626786 616024 906891 0.161164 0.161164 0.161164 0.161164 0.161164 0.073503 0.071491 0.068766 0.066717 0.065576 38.5 59281 59281 59281 59281 59281 225535 226542 172864 933416 543917 0.160128 0.160128 0.160128 0.160128 0.160128 0.074246 0.072066 0.069279 0.067169 0.066007 39 15381 15381 15381 15381 15381 994014 964969 044305 293713 259318 0.159111 0.159111 0.159111 0.159111 0.159111 0.075161 0.072650 0.069786 0.067616 0.066434 39.5 45684 45684 45684 45684 45684 968969 702883 597029 018201 414809 0.158113 0.158113 0.158113 0.158113 0.158113 0.074124 0.073200 0.070297 0.068057 0.066858 40 88301 88301 88301 88301 88301 956738 110884 676537 464037 053506 0.157134 0.157134 0.157134 0.157134 0.157134 0.157134 0.073846 0.070786 0.068495 0.067278 40.5 84026 84026 84026 84026 84026 84026 194044 011646 172483 371312 0.156173 0.156173 0.156173 0.156173 0.156173 0.156173 0.074188 0.071288 0.068928 0.067695 41 76189 76189 76189 76189 76189 76189 108361 388058 06489 193941

91 0.155230 0.155230 0.155230 0.155230 0.155230 0.155230 0.074346 0.071758 0.069358 0.068109 41.5 10514 10514 10514 10514 10514 10514 39182 409278 568427 253444 0.154303 0.154303 0.154303 0.154303 0.154303 0.154303 0.075731 0.072233 0.069786 0.068521 42 34996 34996 34996 34996 34996 34996 467452 767643 044746 43279 0.153392 0.153392 0.153392 0.153392 0.153392 0.153392 0.072844 0.072674 0.070202 0.068926 42.5 99777 99777 99777 99777 99777 99777 024387 047892 323234 471925 0.152498 0.152498 0.152498 0.152498 0.152498 0.152498 0.074552 0.073004 0.070606 0.069334 43 57033 57033 57033 57033 57033 57033 266009 255216 217121 058554 0.151619 0.151619 0.151619 0.151619 0.151619 0.151619 0.151619 0.073455 0.071047 0.069732 43.5 60872 60872 60872 60872 60872 60872 60872 491996 490899 076903 0.150755 0.150755 0.150755 0.150755 0.150755 0.150755 0.150755 0.073631 0.071438 0.070140 44 67229 67229 67229 67229 67229 67229 67229 819994 422195 662694 0.149906 0.149906 0.149906 0.149906 0.149906 0.149906 0.149906 0.072619 0.071856 0.070529 44.5 3378 3378 3378 3378 3378 3378 3378 744278 634599 594396 0.149071 0.149071 0.149071 0.149071 0.149071 0.149071 0.149071 0.074670 0.072348 0.070936 45 1985 1985 1985 1985 1985 1985 1985 154139 638598 47169 0.148249 0.148249 0.148249 0.148249 0.148249 0.148249 0.148249 0.075871 0.072861 0.071311 45.5 86333 86333 86333 86333 86333 86333 86333 271662 995302 125104 0.147441 0.147441 0.147441 0.147441 0.147441 0.147441 0.147441 0.075881 0.072990 0.071701 46 95615 95615 95615 95615 95615 95615 95615 34685 044664 559997 0.146647 0.146647 0.146647 0.146647 0.146647 0.146647 0.146647 0.078938 0.073562 0.072122 46.5 11502 11502 11502 11502 11502 11502 11502 853965 197461 997425 0.145864 0.145864 0.145864 0.145864 0.145864 0.145864 0.145864 0.145864 0.073110 0.072533 47 9915 9915 9915 9915 9915 9915 9915 9915 11719 165928 0.145095 0.145095 0.145095 0.145095 0.145095 0.145095 0.145095 0.145095 0.071692 0.072895 47.5 25002 25002 25002 25002 25002 25002 25002 25002 999001 796666 0.144337 0.144337 0.144337 0.144337 0.144337 0.144337 0.144337 0.144337 0.072354 0.073114 48 5673 5673 5673 5673 5673 5673 5673 5673 695642 381776 0.143591 0.143591 0.143591 0.143591 0.143591 0.143591 0.143591 0.143591 0.074692 0.073545 48.5 63172 63172 63172 63172 63172 63172 63172 63172 942297 428027 0.142857 0.142857 0.142857 0.142857 0.142857 0.142857 0.142857 0.142857 0.142857 0.073546 49 14286 14286 14286 14286 14286 14286 14286 14286 14286 797388 0.142133 0.142133 0.142133 0.142133 0.142133 0.142133 0.142133 0.142133 0.142133 0.073768 49.5 8109 8109 8109 8109 8109 8109 8109 8109 8109 154945 0.141421 0.141421 0.141421 0.141421 0.141421 0.141421 0.141421 0.141421 0.141421 0.074698 50 35624 35624 35624 35624 35624 35624 35624 35624 35624 676925 0.140719 0.140719 0.140719 0.140719 0.140719 0.140719 0.140719 0.140719 0.140719 0.075734 50.5 50895 50895 50895 50895 50895 50895 50895 50895 50895 759239 0.140028 0.140028 0.140028 0.140028 0.140028 0.140028 0.140028 0.140028 0.140028 0.075535 51 0084 0084 0084 0084 0084 0084 0084 0084 0084 325825 0.139346 0.139346 0.139346 0.139346 0.139346 0.139346 0.139346 0.139346 0.139346 0.075638 51.5 60286 60286 60286 60286 60286 60286 60286 60286 60286 685004 0.138675 0.138675 0.138675 0.138675 0.138675 0.138675 0.138675 0.138675 0.138675 0.078415 52 04906 04906 04906 04906 04906 04906 04906 04906 04906 042229 0.138013 0.138013 0.138013 0.138013 0.138013 0.138013 0.138013 0.138013 0.138013 0.138013 52.5 11187 11187 11187 11187 11187 11187 11187 11187 11187 11187 0.137360 0.137360 0.137360 0.137360 0.137360 0.137360 0.137360 0.137360 0.137360 0.137360 53 56395 56395 56395 56395 56395 56395 56395 56395 56395 56395 0.136717 0.136717 0.136717 0.136717 0.136717 0.136717 0.136717 0.136717 0.136717 0.136717 53.5 1854 1854 1854 1854 1854 1854 1854 1854 1854 1854 0.136082 0.136082 0.136082 0.136082 0.136082 0.136082 0.136082 0.136082 0.136082 0.136082 54 76349 76349 76349 76349 76349 76349 76349 76349 76349 76349 0.135457 0.135457 0.135457 0.135457 0.135457 0.135457 0.135457 0.135457 0.135457 0.135457 54.5 0923 0923 0923 0923 0923 0923 0923 0923 0923 0923 0.134839 0.134839 0.134839 0.134839 0.134839 0.134839 0.134839 0.134839 0.134839 0.134839 55 97249 97249 97249 97249 97249 97249 97249 97249 97249 97249 0.134231 0.134231 0.134231 0.134231 0.134231 0.134231 0.134231 0.134231 0.134231 0.134231 55.5 21104 21104 21104 21104 21104 21104 21104 21104 21104 21104 0.133630 0.133630 0.133630 0.133630 0.133630 0.133630 0.133630 0.133630 0.133630 0.133630 56 62096 62096 62096 62096 62096 62096 62096 62096 62096 62096

92 0.133038 0.133038 0.133038 0.133038 0.133038 0.133038 0.133038 0.133038 0.133038 0.133038 56.5 02105 02105 02105 02105 02105 02105 02105 02105 02105 02105 0.132453 0.132453 0.132453 0.132453 0.132453 0.132453 0.132453 0.132453 0.132453 0.132453 57 23571 23571 23571 23571 23571 23571 23571 23571 23571 23571 0.131876 0.131876 0.131876 0.131876 0.131876 0.131876 0.131876 0.131876 0.131876 0.131876 57.5 09468 09468 09468 09468 09468 09468 09468 09468 09468 09468 0.131306 0.131306 0.131306 0.131306 0.131306 0.131306 0.131306 0.131306 0.131306 0.131306 58 43286 43286 43286 43286 43286 43286 43286 43286 43286 43286 0.130744 0.130744 0.130744 0.130744 0.130744 0.130744 0.130744 0.130744 0.130744 0.130744 58.5 09009 09009 09009 09009 09009 09009 09009 09009 09009 09009 0.130188 0.130188 0.130188 0.130188 0.130188 0.130188 0.130188 0.130188 0.130188 0.130188 59 91098 91098 91098 91098 91098 91098 91098 91098 91098 91098 0.129640 0.129640 0.129640 0.129640 0.129640 0.129640 0.129640 0.129640 0.129640 0.129640 59.5 74471 74471 74471 74471 74471 74471 74471 74471 74471 74471 0.129099 0.129099 0.129099 0.129099 0.129099 0.129099 0.129099 0.129099 0.129099 0.129099 60 44487 44487 44487 44487 44487 44487 44487 44487 44487 44487 0.128564 0.128564 0.128564 0.128564 0.128564 0.128564 0.128564 0.128564 0.128564 0.128564 60.5 86931 86931 86931 86931 86931 86931 86931 86931 86931 86931 0.128036 0.128036 0.128036 0.128036 0.128036 0.128036 0.128036 0.128036 0.128036 0.128036 61 87993 87993 87993 87993 87993 87993 87993 87993 87993 87993 0.127515 0.127515 0.127515 0.127515 0.127515 0.127515 0.127515 0.127515 0.127515 0.127515 61.5 34261 34261 34261 34261 34261 34261 34261 34261 34261 34261 0.127000 0.127000 0.127000 0.127000 0.127000 0.127000 0.127000 0.127000 0.127000 0.127000 62 127 127 127 127 127 127 127 127 127 127 0.126491 0.126491 0.126491 0.126491 0.126491 0.126491 0.126491 0.126491 0.126491 0.126491 62.5 10641 10641 10641 10641 10641 10641 10641 10641 10641 10641 0.125988 0.125988 0.125988 0.125988 0.125988 0.125988 0.125988 0.125988 0.125988 0.125988 63 15767 15767 15767 15767 15767 15767 15767 15767 15767 15767 0.125491 0.125491 0.125491 0.125491 0.125491 0.125491 0.125491 0.125491 0.125491 0.125491 63.5 16103 16103 16103 16103 16103 16103 16103 16103 16103 16103

64 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125

0.124514 0.124514 0.124514 0.124514 0.124514 0.124514 0.124514 0.124514 0.124514 0.124514 64.5 56127 56127 56127 56127 56127 56127 56127 56127 56127 56127 0.124034 0.124034 0.124034 0.124034 0.124034 0.124034 0.124034 0.124034 0.124034 0.124034 65 73459 73459 73459 73459 73459 73459 73459 73459 73459 73459 0.123560 0.123560 0.123560 0.123560 0.123560 0.123560 0.123560 0.123560 0.123560 0.123560 65.5 41264 41264 41264 41264 41264 41264 41264 41264 41264 41264 0.123091 0.123091 0.123091 0.123091 0.123091 0.123091 0.123091 0.123091 0.123091 0.123091 66 49098 49098 49098 49098 49098 49098 49098 49098 49098 49098 0.122627 0.122627 0.122627 0.122627 0.122627 0.122627 0.122627 0.122627 0.122627 0.122627 66.5 8679 8679 8679 8679 8679 8679 8679 8679 8679 8679 0.122169 0.122169 0.122169 0.122169 0.122169 0.122169 0.122169 0.122169 0.122169 0.122169 67 44436 44436 44436 44436 44436 44436 44436 44436 44436 44436 0.121716 0.121716 0.121716 0.121716 0.121716 0.121716 0.121716 0.121716 0.121716 0.121716 67.5 12389 12389 12389 12389 12389 12389 12389 12389 12389 12389 0.121267 0.121267 0.121267 0.121267 0.121267 0.121267 0.121267 0.121267 0.121267 0.121267 68 81252 81252 81252 81252 81252 81252 81252 81252 81252 81252 0.120824 0.120824 0.120824 0.120824 0.120824 0.120824 0.120824 0.120824 0.120824 0.120824 68.5 41867 41867 41867 41867 41867 41867 41867 41867 41867 41867 0.120385 0.120385 0.120385 0.120385 0.120385 0.120385 0.120385 0.120385 0.120385 0.120385 69 85309 85309 85309 85309 85309 85309 85309 85309 85309 85309 0.119952 0.119952 0.119952 0.119952 0.119952 0.119952 0.119952 0.119952 0.119952 0.119952 69.5 02878 02878 02878 02878 02878 02878 02878 02878 02878 02878 0.119522 0.119522 0.119522 0.119522 0.119522 0.119522 0.119522 0.119522 0.119522 0.119522 70 86093 86093 86093 86093 86093 86093 86093 86093 86093 86093 0.119098 0.119098 0.119098 0.119098 0.119098 0.119098 0.119098 0.119098 0.119098 0.119098 70.5 26684 26684 26684 26684 26684 26684 26684 26684 26684 26684 0.118678 0.118678 0.118678 0.118678 0.118678 0.118678 0.118678 0.118678 0.118678 0.118678 71 16582 16582 16582 16582 16582 16582 16582 16582 16582 16582

93 0.118262 0.118262 0.118262 0.118262 0.118262 0.118262 0.118262 0.118262 0.118262 0.118262 71.5 4792 4792 4792 4792 4792 4792 4792 4792 4792 4792 0.117851 0.117851 0.117851 0.117851 0.117851 0.117851 0.117851 0.117851 0.117851 0.117851 72 1302 1302 1302 1302 1302 1302 1302 1302 1302 1302 0.117444 0.117444 0.117444 0.117444 0.117444 0.117444 0.117444 0.117444 0.117444 0.117444 72.5 0439 0439 0439 0439 0439 0439 0439 0439 0439 0439 0.117041 0.117041 0.117041 0.117041 0.117041 0.117041 0.117041 0.117041 0.117041 0.117041 73 1472 1472 1472 1472 1472 1472 1472 1472 1472 1472 0.116642 0.116642 0.116642 0.116642 0.116642 0.116642 0.116642 0.116642 0.116642 0.116642 73.5 3687 3687 3687 3687 3687 3687 3687 3687 3687 3687 0.116247 0.116247 0.116247 0.116247 0.116247 0.116247 0.116247 0.116247 0.116247 0.116247 74 63874 63874 63874 63874 63874 63874 63874 63874 63874 63874 0.115856 0.115856 0.115856 0.115856 0.115856 0.115856 0.115856 0.115856 0.115856 0.115856 74.5 88927 88927 88927 88927 88927 88927 88927 88927 88927 88927 0.115470 0.115470 0.115470 0.115470 0.115470 0.115470 0.115470 0.115470 0.115470 0.115470 75 05384 05384 05384 05384 05384 05384 05384 05384 05384 05384 0.115087 0.115087 0.115087 0.115087 0.115087 0.115087 0.115087 0.115087 0.115087 0.115087 75.5 06753 06753 06753 06753 06753 06753 06753 06753 06753 06753 0.114707 0.114707 0.114707 0.114707 0.114707 0.114707 0.114707 0.114707 0.114707 0.114707 76 86694 86694 86694 86694 86694 86694 86694 86694 86694 86694 0.114332 0.114332 0.114332 0.114332 0.114332 0.114332 0.114332 0.114332 0.114332 0.114332 76.5 3901 3901 3901 3901 3901 3901 3901 3901 3901 3901 0.113960 0.113960 0.113960 0.113960 0.113960 0.113960 0.113960 0.113960 0.113960 0.113960 77 57646 57646 57646 57646 57646 57646 57646 57646 57646 57646 0.113592 0.113592 0.113592 0.113592 0.113592 0.113592 0.113592 0.113592 0.113592 0.113592 77.5 36685 36685 36685 36685 36685 36685 36685 36685 36685 36685 0.113227 0.113227 0.113227 0.113227 0.113227 0.113227 0.113227 0.113227 0.113227 0.113227 78 70341 70341 70341 70341 70341 70341 70341 70341 70341 70341 0.112866 0.112866 0.112866 0.112866 0.112866 0.112866 0.112866 0.112866 0.112866 0.112866 78.5 5296 5296 5296 5296 5296 5296 5296 5296 5296 5296 0.112508 0.112508 0.112508 0.112508 0.112508 0.112508 0.112508 0.112508 0.112508 0.112508 79 79009 79009 79009 79009 79009 79009 79009 79009 79009 79009 0.112154 0.112154 0.112154 0.112154 0.112154 0.112154 0.112154 0.112154 0.112154 0.112154 79.5 43082 43082 43082 43082 43082 43082 43082 43082 43082 43082 0.111803 0.111803 0.111803 0.111803 0.111803 0.111803 0.111803 0.111803 0.111803 0.111803 80 39887 39887 39887 39887 39887 39887 39887 39887 39887 39887 0.111455 0.111455 0.111455 0.111455 0.111455 0.111455 0.111455 0.111455 0.111455 0.111455 80.5 64252 64252 64252 64252 64252 64252 64252 64252 64252 64252 0.111111 0.111111 0.111111 0.111111 0.111111 0.111111 0.111111 0.111111 0.111111 0.111111 81 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 0.110769 0.110769 0.110769 0.110769 0.110769 0.110769 0.110769 0.110769 0.110769 0.110769 81.5 75512 75512 75512 75512 75512 75512 75512 75512 75512 75512 0.110431 0.110431 0.110431 0.110431 0.110431 0.110431 0.110431 0.110431 0.110431 0.110431 82 52607 52607 52607 52607 52607 52607 52607 52607 52607 52607 0.110096 0.110096 0.110096 0.110096 0.110096 0.110096 0.110096 0.110096 0.110096 0.110096 82.5 37651 37651 37651 37651 37651 37651 37651 37651 37651 37651 0.109764 0.109764 0.109764 0.109764 0.109764 0.109764 0.109764 0.109764 0.109764 0.109764 83 25999 25999 25999 25999 25999 25999 25999 25999 25999 25999 0.109435 0.109435 0.109435 0.109435 0.109435 0.109435 0.109435 0.109435 0.109435 0.109435 83.5 13103 13103 13103 13103 13103 13103 13103 13103 13103 13103 0.109108 0.109108 0.109108 0.109108 0.109108 0.109108 0.109108 0.109108 0.109108 0.109108 84 94512 94512 94512 94512 94512 94512 94512 94512 94512 94512 0.108785 0.108785 0.108785 0.108785 0.108785 0.108785 0.108785 0.108785 0.108785 0.108785 84.5 65864 65864 65864 65864 65864 65864 65864 65864 65864 65864 0.108465 0.108465 0.108465 0.108465 0.108465 0.108465 0.108465 0.108465 0.108465 0.108465 85 22891 22891 22891 22891 22891 22891 22891 22891 22891 22891 0.108147 0.108147 0.108147 0.108147 0.108147 0.108147 0.108147 0.108147 0.108147 0.108147 85.5 61409 61409 61409 61409 61409 61409 61409 61409 61409 61409 0.107832 0.107832 0.107832 0.107832 0.107832 0.107832 0.107832 0.107832 0.107832 0.107832 86 7732 7732 7732 7732 7732 7732 7732 7732 7732 7732

94 0.107520 0.107520 0.107520 0.107520 0.107520 0.107520 0.107520 0.107520 0.107520 0.107520 86.5 66611 66611 66611 66611 66611 66611 66611 66611 66611 66611 0.107211 0.107211 0.107211 0.107211 0.107211 0.107211 0.107211 0.107211 0.107211 0.107211 87 25348 25348 25348 25348 25348 25348 25348 25348 25348 25348 0.106904 0.106904 0.106904 0.106904 0.106904 0.106904 0.106904 0.106904 0.106904 0.106904 87.5 49676 49676 49676 49676 49676 49676 49676 49676 49676 49676 0.106600 0.106600 0.106600 0.106600 0.106600 0.106600 0.106600 0.106600 0.106600 0.106600 88 35818 35818 35818 35818 35818 35818 35818 35818 35818 35818 0.106298 0.106298 0.106298 0.106298 0.106298 0.106298 0.106298 0.106298 0.106298 0.106298 88.5 80069 80069 80069 80069 80069 80069 80069 80069 80069 80069 0.105999 0.105999 0.105999 0.105999 0.105999 0.105999 0.105999 0.105999 0.105999 0.105999 89 788 788 788 788 788 788 788 788 788 788 0.105703 0.105703 0.105703 0.105703 0.105703 0.105703 0.105703 0.105703 0.105703 0.105703 89.5 28452 28452 28452 28452 28452 28452 28452 28452 28452 28452 0.105409 0.105409 0.105409 0.105409 0.105409 0.105409 0.105409 0.105409 0.105409 0.105409 90 25534 25534 25534 25534 25534 25534 25534 25534 25534 25534 0.105117 0.105117 0.105117 0.105117 0.105117 0.105117 0.105117 0.105117 0.105117 0.105117 90.5 66625 66625 66625 66625 66625 66625 66625 66625 66625 66625 0.104828 0.104828 0.104828 0.104828 0.104828 0.104828 0.104828 0.104828 0.104828 0.104828 91 48367 48367 48367 48367 48367 48367 48367 48367 48367 48367 0.104541 0.104541 0.104541 0.104541 0.104541 0.104541 0.104541 0.104541 0.104541 0.104541 91.5 6747 6747 6747 6747 6747 6747 6747 6747 6747 6747 0.104257 0.104257 0.104257 0.104257 0.104257 0.104257 0.104257 0.104257 0.104257 0.104257 92 20703 20703 20703 20703 20703 20703 20703 20703 20703 20703 0.103975 0.103975 0.103975 0.103975 0.103975 0.103975 0.103975 0.103975 0.103975 0.103975 92.5 04898 04898 04898 04898 04898 04898 04898 04898 04898 04898 0.103695 0.103695 0.103695 0.103695 0.103695 0.103695 0.103695 0.103695 0.103695 0.103695 93 16947 16947 16947 16947 16947 16947 16947 16947 16947 16947 0.103417 0.103417 0.103417 0.103417 0.103417 0.103417 0.103417 0.103417 0.103417 0.103417 93.5 538 538 538 538 538 538 538 538 538 538 0.103142 0.103142 0.103142 0.103142 0.103142 0.103142 0.103142 0.103142 0.103142 0.103142 94 12463 12463 12463 12463 12463 12463 12463 12463 12463 12463 0.102868 0.102868 0.102868 0.102868 0.102868 0.102868 0.102868 0.102868 0.102868 0.102868 94.5 89997 89997 89997 89997 89997 89997 89997 89997 89997 89997 0.102597 0.102597 0.102597 0.102597 0.102597 0.102597 0.102597 0.102597 0.102597 0.102597 95 83521 83521 83521 83521 83521 83521 83521 83521 83521 83521 0.102328 0.102328 0.102328 0.102328 0.102328 0.102328 0.102328 0.102328 0.102328 0.102328 95.5 90202 90202 90202 90202 90202 90202 90202 90202 90202 90202 0.102062 0.102062 0.102062 0.102062 0.102062 0.102062 0.102062 0.102062 0.102062 0.102062 96 07262 07262 07262 07262 07262 07262 07262 07262 07262 07262 0.101797 0.101797 0.101797 0.101797 0.101797 0.101797 0.101797 0.101797 0.101797 0.101797 96.5 31971 31971 31971 31971 31971 31971 31971 31971 31971 31971 0.101534 0.101534 0.101534 0.101534 0.101534 0.101534 0.101534 0.101534 0.101534 0.101534 97 61651 61651 61651 61651 61651 61651 61651 61651 61651 61651 0.101273 0.101273 0.101273 0.101273 0.101273 0.101273 0.101273 0.101273 0.101273 0.101273 97.5 93671 93671 93671 93671 93671 93671 93671 93671 93671 93671 0.101015 0.101015 0.101015 0.101015 0.101015 0.101015 0.101015 0.101015 0.101015 0.101015 98 25446 25446 25446 25446 25446 25446 25446 25446 25446 25446 0.100758 0.100758 0.100758 0.100758 0.100758 0.100758 0.100758 0.100758 0.100758 0.100758 98.5 54437 54437 54437 54437 54437 54437 54437 54437 54437 54437 0.100503 0.100503 0.100503 0.100503 0.100503 0.100503 0.100503 0.100503 0.100503 0.100503 99 78153 78153 78153 78153 78153 78153 78153 78153 78153 78153 0.100250 0.100250 0.100250 0.100250 0.100250 0.100250 0.100250 0.100250 0.100250 0.100250 99.5 94142 94142 94142 94142 94142 94142 94142 94142 94142 94142

100 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

0.797032 0.798728 0.800062 0.058421 0.058380 0.058430 0.069773 0.068321 0.068097 0.5 5885 39101 03177 198807 448734 848972 410164 552429 528298

95 0.462825 0.463132 0.463182 0.058753 0.058707 0.058754 0.069880 0.068390 0.068162 1 69744 36767 36462 630368 446687 19211 48598 357442 873428 0.323452 0.323330 0.323038 0.059075 0.059023 0.059066 0.069987 0.068458 0.068227 1.5 31995 41299 25858 216352 522235 604492 022884 419318 549034 0.244054 0.243745 0.243332 0.059386 0.059329 0.059368 0.070093 0.068525 0.068291 2 83574 3126 8894 443722 134921 54327 17364 766881 586583 0.191884 0.191479 0.191022 0.059687 0.059624 0.059660 0.070198 0.068592 0.068355 2.5 49292 78716 40988 776563 720747 442242 722454 428209 016057 0.154629 0.154173 0.153702 0.059979 0.059910 0.059942 0.070303 0.068658 0.068417 3 71908 9141 35096 657554 693765 713431 95884 427841 866157 0.126526 0.126044 0.125572 0.060262 0.060187 0.060215 0.070408 0.068723 0.068480 3.5 95587 04722 24877 509297 44753 748537 761787 793608 164515 0.104485 0.103990 0.103524 0.060536 0.060455 0.060479 0.070513 0.068788 0.068541 4 96707 20347 82309 735531 356443 920269 201694 549773 937398 0.086688 0.086188 0.085732 0.060802 0.060714 0.060735 0.070616 0.068852 0.068603 4.5 056293 421911 642971 722234 776987 583578 965989 718816 210011 0.071987 0.071489 0.071044 0.061060 0.060966 0.060983 0.070720 0.068916 0.068664 5 223162 628078 897361 83862 048861 076785 372395 322703 006419 0.059622 0.059131 0.058697 0.061311 0.061209 0.061222 0.070823 0.068979 0.068724 5.5 553137 000071 887289 438055 49603 722627 534596 381334 349686 0.049067 0.048584 0.048163 0.061554 0.061445 0.061454 0.070926 0.069041 0.068784 6 446384 688748 334936 858881 427695 82922 340254 919614 261573 0.039944 0.039472 0.039063 0.061791 0.061674 0.061679 0.071028 0.069103 0.068843 6.5 996607 92392 27575 425171 139182 690944 913007 951854 763214 0.031977 0.031517 0.031119 0.062021 0.061895 0.061897 0.071130 0.069165 0.068902 7 810076 686311 614033 447422 912771 58927 7718 502281 874942 0.024956 0.024509 0.024122 0.062245 0.062111 0.062108 0.071232 0.069226 0.068961 7.5 866985 493605 846768 223183 018457 793515 548174 583523 615223 0.018721 0.018287 0.017911 0.062463 0.062319 0.062313 0.071334 0.069287 0.069020 8 457908 295929 926472 037628 714659 561547 2782 20334 002788 0.013145 0.012725 0.012360 0.062675 0.062522 0.062512 0.071434 0.069347 0.069078 8.5 840819 104192 872561 164099 24887 140431 993754 399666 055119 0.008130 0.007722 0.007369 0.062881 0.062718 0.062704 0.071534 0.069407 0.069135 9 1196913 847002 6201315 864575 858265 767039 820421 176576 788791 0.003593 0.003199 0.002857 0.063083 0.062909 0.062891 0.071635 0.069466 0.069193 9.5 8566773 9664173 6128752 390149 770261 668599 504547 541119 219623 0.000528 0.000909 0.001240 0.063279 0.063095 0.063073 0.071735 0.069525 0.069250 10 49552442 16394868 7785741 981431 203036 063221 817388 521056 363191 0.004290 0.004658 0.004979 0.063471 0.063275 0.063249 0.071833 0.069584 0.069307 10.5 9201189 5667695 5951881 868948 366016 160372 899015 123948 233824 0.007738 0.008093 0.008403 0.063659 0.063450 0.063420 0.071934 0.069642 0.069363 11 3170743 1712271 7724928 273532 460315 161326 52838 358668 845521 0.010908 0.011250 0.011550 0.063842 0.063620 0.063586 0.072031 0.069700 0.069420 11.5 342901 636425 991424 406621 67916 259577 408726 245088 211811 0.013832 0.014162 0.014453 0.064021 0.063786 0.063747 0.072131 0.069757 0.069476 12 831985 790673 100694 470646 208272 641223 840759 787997 345954 0.016538 0.016856 0.017137 0.064196 0.063947 0.063904 0.072227 0.069815 0.069532 12.5 889462 725864 213164 659315 226225 485332 498006 006398 259917 0.019049 0.019355 0.019626 0.064368 0.064103 0.064056 0.072318 0.069871 0.069587 13 74439 654211 561604 157908 904784 964268 038847 894132 967844 0.021385 0.021679 0.021941 0.064536 0.064256 0.064205 0.072411 0.069928 0.069643 13.5 422911 584354 173743 143609 409217 244007 504823 47107 480616 0.023563 0.023845 0.024098 0.064700 0.064404 0.064349 0.072503 0.069984 0.069698 14 28469 859918 409891 785703 898582 484427 56851 740788 811531 0.025598 0.025869 0.026113 0.064862 0.064549 0.064489 0.072616 0.070040 0.069753 14.5 454097 592259 39497 24599 526002 839582 610327 745594 971711 0.027504 0.027764 0.027999 0.065020 0.064690 0.064626 0.072710 0.070096 0.069808 15 169631 011011 368541 678853 438919 457948 784096 472577 977763 0.029292 0.029540 0.029767 0.065176 0.064827 0.064759 0.072826 0.070151 0.069863 15.5 069539 750196 97058 231734 779321 482668 263573 878099 841942

96 0.030972 0.031210 0.031429 0.065329 0.064961 0.064889 0.072904 0.070207 0.069918 16 427271 083358 476433 045144 68398 051771 124626 035343 577857 0.032554 0.032781 0.032992 0.065479 0.065092 0.065015 0.072987 0.070261 0.069973 16.5 346957 118079 991328 253123 284651 298376 033894 956313 204824 0.034045 0.034261 0.034466 0.065626 0.065219 0.065138 0.073105 0.070316 0.070027 17 926488 957841 61244 983419 708267 350894 938967 720054 744743 0.035454 0.035659 0.035857 0.065772 0.065344 0.065258 0.073220 0.070371 0.070082 17.5 394061 837469 564779 357638 077128 333207 864796 140671 219399 0.036786 0.036981 0.037172 0.065915 0.065465 0.065375 0.073290 0.070425 0.070136 18 222859 237033 315753 491524 509068 364839 866093 233643 657834 0.038047 0.038231 0.038416 0.066056 0.065584 0.065489 0.073389 0.070479 0.070191 18.5 227759 978124 672232 495118 117627 561121 373454 216953 097093 0.039242 0.039417 0.039595 0.066195 0.065700 0.065601 0.073444 0.070533 0.070245 19 647262 305615 863127 47323 012199 033346 310266 195945 575068 0.040377 0.040541 0.040714 0.066332 0.065813 0.065709 0.073656 0.070586 0.070300 19.5 213294 957433 609899 525277 298193 888904 475566 474256 150905 0.041455 0.041610 0.041777 0.066467 0.065924 0.065816 0.073680 0.070640 0.070354 20 210996 22438 186983 745864 077146 23143 148242 208746 895192 0.042480 0.042626 0.042787 0.066601 0.066032 0.065920 0.073660 0.070693 0.070409 20.5 530204 001646 473732 224801 446875 160927 220464 449213 892393 0.043456 0.043592 0.043748 0.066733 0.066138 0.066021 0.073642 0.070746 0.070465 21 710001 833362 999225 047232 501622 773891 095562 687704 262829 0.044386 0.044513 0.044664 0.066863 0.066242 0.066121 0.073836 0.070799 0.070521 21.5 977409 951261 981072 293505 332122 163429 654874 731695 144415 0.045274 0.045392 0.045538 0.066992 0.066344 0.066218 0.073994 0.070852 0.070577 22 281133 308372 359121 04058 025782 419364 014592 875144 718799 0.046121 0.046230 0.046371 0.067119 0.066443 0.066313 0.073833 0.070906 0.070635 22.5 321077 608445 824852 360691 666749 628352 779299 273465 244718 0.046930 0.047031 0.047167 0.067245 0.066541 0.066406 0.074222 0.070959 0.070694 23 57424 331758 84711 322895 336012 873973 847592 917134 027358 0.047704 0.047796 0.047928 0.067369 0.066637 0.066498 0.074520 0.071013 0.070754 23.5 317496 757781 694682 991702 111577 236836 462211 381636 452294 0.048444 0.048528 0.048656 0.067493 0.066731 0.066587 0.074332 0.071068 0.070817 24 647691 985162 456202 429115 068441 794669 41794 364456 050865 0.049153 0.049229 0.049353 0.067615 0.066823 0.066675 0.074892 0.071123 0.070882 24.5 499411 949373 057718 692091 278856 622411 99361 396518 464253 0.049832 0.049901 0.050020 0.067736 0.066913 0.066761 0.073325 0.071181 0.070951 25 660728 438345 278269 83877 81221 792291 784302 633726 565154 0.050483 0.050545 0.050659 0.067856 0.067002 0.066846 0.073051 0.071239 0.071025 25.5 787184 106343 763709 916685 735305 373919 447299 071682 406178 0.051108 0.051162 0.051273 0.067975 0.067090 0.066929 0.072176 0.071301 0.071105 26 414239 486315 039018 979579 112396 434374 217389 149908 382548 0.051707 0.051755 0.051861 0.068094 0.067176 0.067011 0.075347 0.071366 0.071193 26.5 968368 000905 51929 068491 005187 038262 758329 117092 188956 0.052283 0.052323 0.052426 0.068211 0.067260 0.067091 0.071478 0.071436 0.071291 27 776974 972294 519547 231302 473037 247804 477472 441766 064253 0.052837 0.052870 0.052969 0.068327 0.067343 0.067170 0.072975 0.071513 0.071401 27.5 077263 631014 263542 505479 572903 122905 846373 843508 683167 0.053369 0.053396 0.053490 0.068442 0.067425 0.067247 0.073254 0.071605 0.071528 28 024192 123857 891655 933921 359571 721241 2702 788105 358541 0.053880 0.053901 0.053992 0.068557 0.067505 0.067324 0.102328 0.071702 0.071675 28.5 697614 520994 467994 555509 885712 098291 90202 504331 192652 0.054373 0.054387 0.054474 0.068671 0.067585 0.067399 0.073808 0.071827 0.071846 29 108696 822376 986785 3922 201718 307441 061064 538701 953945 0.054847 0.054855 0.054939 0.068784 0.067663 0.067473 0.077940 0.071974 0.072049 29.5 205699 963517 378153 488574 356032 400038 810257 327726 298247 0.055303 0.055306 0.055386 0.068896 0.067740 0.067546 0.101534 0.072151 0.072288 30 879197 820723 513326 88007 395316 425457 61651 358172 530015 0.055743 0.055741 0.055817 0.069008 0.067816 0.067618 0.075601 0.072372 0.072571 30.5 966782 215824 209359 583601 363929 431117 441211 47855 458586

97 0.056168 0.056159 0.056232 0.069119 0.067891 0.067689 0.101015 0.072647 0.072905 31 257321 920459 233414 631144 304966 462627 25446 791442 307897 0.056577 0.056563 0.056632 0.069230 0.067965 0.067759 0.100758 0.072977 0.073296 31.5 49481 659982 306639 027803 259395 563788 54437 284618 905316 0.056972 0.056953 0.057018 0.069339 0.068038 0.067828 0.078907 0.073390 0.073752 32 38186 116995 1077 840422 26654 776641 327269 069657 40366 0.057353 0.057328 0.057390 0.069449 0.068110 0.067897 0.075723 0.073888 0.074276 32.5 582869 93458 275996 047301 363663 141567 867936 660394 446336 0.057721 0.057691 0.057749 0.069557 0.068181 0.067964 0.075998 0.074476 0.074871 33 726885 719237 414584 711285 587697 69731 458814 375812 910211 0.058077 0.058042 0.058096 0.069665 0.068251 0.068031 33.5 410219 043569 092854 831699 972963 48099

2. pdex_dupire_option.m

Table B.2 The Option Prices(Dupire)

K=9.9

S\tau 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0 0 0 0 0 0 0 0 0 0

1 0 1.91E-14 1.16E-11 4.51E-10 5.69E-09 3.76E-08 1.66E-07 5.62E-07 1.56E-06 3.72E-06 7.91E-06

0.000127 2 0 3.34E-12 1.05E-09 2.73E-08 2.53E-07 1.31E-06 4.76E-06 1.36E-05 3.23E-05 6.75E-05 1315745 1 0.000155 0.000320 0.000590 0.000995 3 0 2.73E-10 4.44E-08 7.67E-07 5.24E-06 2.15E-05 6.43E-05 0685927 5967105 6817072 8489472 6 6 4 1 0.000218 0.000541 0.001119 0.002027 0.003331 0.005082 4 0 1.37E-08 1.14E-06 1.31E-05 6.68E-05 1228494 9990702 5159533 6522572 9099577 3861226 5 3 0.000584 0.000152 0.001536 0.003206 0.005739 0.009218 0.013675 0.019099 5 0 4.67E-07 1.99E-05 6230228 9940418 2392532 6480711 4902939 6656366 218611 755736 6 0.000246 0.001278 0.003724 0.007983 0.014202 0.022346 0.032273 0.043794 0.056707 6 0 1.13E-05 0219253 43345 382041 3820041 33193 168737 776654 600344 864116 1 0.000195 0.002210 0.007906 0.017892 0.031870 0.049188 0.069161 0.091179 0.114744 0.139463 7 0 1902394 4430681 7555402 985955 633401 119641 427185 873385 12381 97533 9 0.002422 0.014596 0.036921 0.066552 0.100792 0.137758 0.176223 0.215411 0.254838 0.294195 8 0 8679091 28447 319922 088053 80743 0672 6819 59448 49606 21557

0.021201 0.071080 0.132141 0.195994 0.259353 0.321028 0.380617 0.438087 0.493525 0.547045 9 0 244573 129065 81253 56598 7999 19311 37414 612 14727 15082

0.127182 0.255396 0.368037 0.467518 0.557108 0.639099 0.715086 0.786195 0.853226 0.916791 10 0 51664 14581 20922 18768 78826 29461 57869 2856 20406 31498

0.502546 0.680540 0.814852 0.927223 1.025801 1.114702 1.196337 1.272194 1.343334 1.410543 11 0.2 58508 72381 9031 78072 9866 802 1659 0973 9919 9168

1.270918 1.379004 1.482297 1.576938 1.664000 1.744767 1.820326 1.891533 1.959020 2.023288 12 1.2 019 7998 3701 6601 2525 9443 5546 1414 0488 6323

2.213152 2.256446 2.314611 2.377751 2.441712 2.504837 2.566432 2.626285 2.684366 2.740709 13 2.2 3359 9337 7901 8806 8517 372 2316 4401 814 0605

3.202050 3.215542 3.241841 3.276997 3.317467 3.360945 3.406005 3.451770 3.497715 3.543504 14 3.2 4235 7474 3202 1539 0412 3015 0077 5118 0331 7105

4.200280 4.203830 4.213947 4.230940 4.253614 4.280564 4.310620 4.342874 4.376657 4.411488 15 4.2 0583 4393 9811 928 0062 3801 708 1778 7964 5486

98 5.200034 5.200861 5.204301 5.211638 5.223130 5.238420 5.256948 5.278130 5.301440 5.326443 16 5.2 5827 4544 4327 5076 0041 4904 7148 1403 0373 1079

6.200003 6.200179 6.201240 6.204130 6.209487 6.217534 6.228207 6.241289 6.256500 6.273556 17 6.2 9579 6243 7488 4536 5721 8948 8995 6199 4065 0509

7.200000 7.200035 7.200337 7.201392 7.203719 7.207690 7.213487 7.221144 7.230593 7.241713 18 7.2 4282 1994 9194 7363 7328 4905 4775 1007 6167 9098

8.200000 8.200006 8.200087 8.200449 8.201400 8.203253 8.206244 8.210518 8.216138 8.223102 19 8.2 0445 5599 6153 0089 6567 4083 1326 2613 1489 3717

9.200000 9.200001 9.200021 9.200139 9.200508 9.201332 9.202806 9.205094 9.208309 9.212515 20 9.2 0045 1748 7855 2154 7933 1126 6765 4859 1978 9653

10.20000 10.20000 10.20004 10.20017 10.20052 10.20122 10.20240 10.20418 10.20664 21 10.2 10.2 0204 5229 1737 9037 9587 8011 7585 3346 3087

11.20000 11.20000 11.20001 11.20006 11.20020 11.20052 11.20111 11.20206 11.20345 22 11.2 11.2 0035 1219 2161 1268 5032 4264 2385 2876 9329

12.20000 12.20000 12.20000 12.20002 12.20007 12.20021 12.20050 12.20099 12.20176 23 12.2 12.2 0006 0277 346 0465 7521 8896 3439 7913 9738

13.20000 13.20000 13.20000 13.20000 13.20002 13.20008 13.20022 13.20047 13.20089 24 13.2 13.2 0001 0062 0965 6695 87 9584 3589 4283 0572

14.20000 14.20000 14.20000 14.20001 14.20003 14.20009 14.20022 14.20044 25 14.2 14.2 14.2 0014 0265 2152 0431 6013 762 179 1364

15.20000 15.20000 15.20000 15.20000 15.20001 15.20004 15.20010 15.20021 26 15.2 15.2 15.2 0003 0072 0682 3731 425 1971 2193 5674

16.20000 16.20000 16.20000 16.20000 16.20000 16.20001 16.20004 16.20010 27 16.2 16.2 16.2 0001 0019 0213 1316 5561 7799 646 4032

17.20000 17.20000 17.20000 17.20000 17.20000 17.20002 17.20004 28 17.2 17.2 17.2 17.2 0005 0066 0459 2144 7458 0868 9588

18.20000 18.20000 18.20000 18.20000 18.20000 18.20000 18.20002 29 18.2 18.2 18.2 18.2 0001 002 0158 0818 3092 9273 3383

19.20000 19.20000 19.20000 19.20000 19.20000 19.20001 30 19.2 19.2 19.2 19.2 19.2 0006 0054 031 127 4081 0919

20.20000 20.20000 20.20000 20.20000 20.20000 20.20000 31 20.2 20.2 20.2 20.2 20.2 0002 0018 0116 0518 1781 5054

21.20000 21.20000 21.20000 21.20000 21.20000 21.20000 32 21.2 21.2 21.2 21.2 21.2 0001 0006 0043 021 0772 2321

22.20000 22.20000 22.20000 22.20000 22.20000 33 22.2 22.2 22.2 22.2 22.2 22.2 0002 0016 0084 0332 1059

23.20000 23.20000 23.20000 23.20000 23.20000 34 23.2 23.2 23.2 23.2 23.2 23.2 0001 0006 0034 0142 048

24.20000 24.20000 24.20000 24.20000 35 24.2 24.2 24.2 24.2 24.2 24.2 24.2 0002 0014 0061 0216

25.20000 25.20000 25.20000 25.20000 36 25.2 25.2 25.2 25.2 25.2 25.2 25.2 0001 0005 0026 0097

26.20000 26.20000 26.20000 37 26.2 26.2 26.2 26.2 26.2 26.2 26.2 26.2 0002 0011 0043

27.20000 27.20000 27.20000 38 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 0001 0005 0019

28.20000 28.20000 39 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 0002 0009

29.20000 29.20000 40 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 0001 0004

30.20000 41 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 0002

31.20000 42 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 0001

99 43 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2

44 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2

45 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2

46 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2

47 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2

48 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2

49 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2

50 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2

51 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2

52 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2

53 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2

54 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2

55 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2

56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2

57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2

58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2

59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2

60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2

61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2

62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2

63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2

64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2

65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2

66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2

68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2

69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2

100 70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2

71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2

72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2

73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2

74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2

75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2

76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2

77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2

78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2

79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2

80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2

81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2

82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2

83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2

84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2

85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2

86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2

87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2

88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2

89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2

91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2

92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2

93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2

94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2

95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2

96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2

101 97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2

98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2

99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2

100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2

S\tau 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0 0 0 0 0 0 0 0 0 0 0

0.000113 0.000166 0.000237 0.000329 0.000444 0.000585 1 1.53E-05 2.75E-05 4.64E-05 7.41E-05 4117520 9755010 8252206 1065341 0143586 7693150 3 8 3 4 8 4 0.000220 0.000358 0.000551 0.000810 0.001145 0.001568 0.002086 0.002709 0.003444 0.004296 2 6799962 4030721 2013948 0869391 7390057 229717 8021025 7940127 434106 9095229 5 3 8 9 0.001565 0.002324 0.003297 0.004502 0.005953 0.007660 0.009631 0.011867 0.014371 0.017139 3 2088983 88863 7621876 5225042 4442069 7407391 0373156 604641 152736 996515

0.007313 0.010045 0.013286 0.017034 0.021278 0.026005 0.031194 0.036825 0.042874 0.049318 4 3693226 431675 442016 14809 814468 274828 706009 32307 400968 134385

0.025456 0.032693 0.040748 0.049553 0.059043 0.069153 0.079825 0.091000 0.102628 0.114660 5 259835 663426 602748 884084 489635 917451 036475 508178 22289 10092

0.070821 0.085961 0.101966 0.118701 0.136047 0.153904 0.172185 0.190815 0.209731 0.228877 6 922558 01798 71941 3653 50413 38334 07854 64588 50689 83337

0.165034 0.191213 0.217819 0.244711 0.271777 0.298932 0.326105 0.353242 0.380301 0.407248 7 3899 80706 45044 038 98073 61882 38537 61563 1638 38531

0.333279 0.371961 0.410168 0.447860 0.485010 0.521604 0.557636 0.593111 0.628038 0.662429 8 49703 02974 57603 96342 96665 13185 40856 59408 5518 30213

0.598780 0.648864 0.697427 0.744582 0.790428 0.835056 0.878548 0.920980 0.962421 1.002930 9 15632 3985 61247 71417 56811 123 5402 86942 26459 2018

0.977371 1.035345 1.091007 1.144601 1.196335 1.246387 1.294906 1.342019 1.387837 1.432456 10 33948 0975 4945 3934 7375 318 1453 3722 5375 734

1.474397 1.535351 1.593749 1.649875 1.703972 1.756242 1.806854 1.855951 1.903657 1.950077 11 8845 6444 1291 5595 8958 7195 8214 7387 1594 9387

2.084749 2.143733 2.200503 2.255279 2.308250 2.359579 2.409405 2.457847 2.505011 2.550988 12 8321 3738 5423 3736 494 3557 3611 9574 7684 1865

2.795388 2.848498 2.900144 2.950417 2.999401 3.047175 3.093814 3.139387 3.183956 3.227580 13 1903 6869 0864 2772 267 449 4107 0433 7533 5384

3.588922 3.633834 3.678168 3.721885 3.764958 3.807376 3.849138 3.890252 3.930729 3.970585 14 9355 3919 249 0034 6179 8611 8833 1136 2634 8084

4.447007 4.482944 4.519107 4.555356 4.591583 4.627704 4.663657 4.699394 4.734881 4.770095 15 6526 5508 5823 7004 6472 9374 2088 2632 5406 0423

5.352783 5.380166 5.408359 5.437173 5.466460 5.496095 5.525977 5.556026 5.586174 5.616366 16 3394 7899 2574 9808 2629 1561 6824 6707 3215 6786

6.292192 6.312170 6.333279 6.355336 6.378188 6.401704 6.425772 6.450295 6.475190 6.500388 17 3489 5847 2344 5626 8092 8966 1849 2711 9441 9063

7.254359 7.268380 7.283622 7.299944 7.317218 7.335327 7.354168 7.373649 7.393687 7.414210 18 612 0178 7218 9248 1824 155 2318 3369 6814 0109

8.231368 8.240870 8.251523 8.263238 8.275924 8.289496 8.303871 8.318973 8.334731 8.351078 19 2922 3412 9824 5601 7159 2838 4703 4649 0098 1727

9.217740 9.223980 9.231210 9.239388 9.248465 9.258388 9.269102 9.280551 9.292682 9.305442 20 4412 6698 2531 0442 1028 1461 1847 5674 4661 4593

10.20984 10.21382 10.21860 10.22417 10.23051 10.23760 10.24542 10.25392 10.26308 10.27286 21 6408 9142 5226 1708 3553 7175 3382 8492 757 409

102 11.20536 11.20784 11.21092 11.21462 11.21896 11.22393 11.22952 11.23572 11.24251 11.24987 22 9943 6592 5443 7948 314 0318 1484 229 5045 8617

12.20288 12.20438 12.20632 12.20873 12.21164 12.21505 12.21898 12.22342 12.22838 12.23383 23 0984 4885 5569 6578 1147 3697 1403 4839 0201 944

13.20152 13.20241 13.20361 13.20515 13.20706 13.20936 13.21208 13.21521 13.21877 13.22276 24 2162 5682 3966 3583 3892 7458 073 4376 4537 3121

14.20079 14.20131 14.20203 14.20300 14.20423 14.20576 14.20761 14.20979 14.21231 14.21518 25 2817 3162 9117 468 9682 946 4843 2224 3954 8679

15.20040 15.20070 15.20113 15.20173 15.20251 15.20351 15.20475 15.20624 15.20801 15.21005 26 7476 4971 713 263 8419 9013 6054 8202 091 6714

16.20020 16.20037 16.20062 16.20098 16.20148 16.20212 16.20294 16.20395 16.20517 16.20660 27 6849 408 7212 8825 1432 6695 4793 4223 1219 9951

17.20010 17.20019 17.20034 17.20055 17.20086 17.20127 17.20180 17.20248 17.20331 17.20431 28 3807 6363 2433 888 3456 4106 834 2997 3542 4154

18.20005 18.20010 18.20018 18.20031 18.20049 18.20075 18.20110 18.20154 18.20210 18.20279 29 1547 2049 5186 3024 8933 7064 1821 7638 8308 6983

19.20002 19.20005 19.20009 19.20017 19.20028 19.20044 19.20066 19.20095 19.20133 19.20180 30 5349 2548 9271 3848 5972 6367 6393 7864 2495 1837

20.20001 20.20002 20.20005 20.20009 20.20016 20.20026 20.20040 20.20058 20.20083 20.20115 31 2355 6832 2787 5798 2673 1268 0237 8895 6814 3725

21.20000 21.20001 21.20002 21.20005 21.20009 21.20015 21.20023 21.20035 21.20052 21.20073 32 5974 3597 7863 2409 1885 1884 8806 9771 2351 4472

22.20000 22.20000 22.20001 22.20002 22.20005 22.20008 22.20014 22.20021 22.20032 22.20046 33 2867 6843 4609 8482 1563 7733 1609 8484 4191 5003

23.20000 23.20000 23.20000 23.20001 23.20002 23.20005 23.20008 23.20013 23.20020 23.20029 34 1367 3423 7613 5386 8761 0377 3487 1938 0114 2861

24.20000 24.20000 24.20000 24.20000 24.20001 24.20002 24.20004 24.20007 24.20012 24.20018 35 0648 1703 3946 8266 5955 8769 8956 9255 2892 3532

25.20000 25.20000 25.20000 25.20000 25.20000 25.20001 25.20002 25.20004 25.20007 25.20011 36 0306 0843 2036 4419 8806 6345 8563 7373 5105 4477

26.20000 26.20000 26.20000 26.20000 26.20000 26.20000 26.20001 26.20002 26.20004 26.20007 37 0144 0416 1046 2352 4838 9244 6588 8186 5692 1088

27.20000 27.20000 27.20000 27.20000 27.20000 27.20000 27.20000 27.20001 27.20002 27.20004 38 0067 0204 0535 1247 2647 5206 9592 6698 768 396

28.20000 28.20000 28.20000 28.20000 28.20000 28.20000 28.20000 28.20000 28.20001 28.20002 39 0031 01 0273 0659 1443 292 5525 9853 6702 7078

29.20000 29.20000 29.20000 29.20000 29.20000 29.20000 29.20000 29.20000 29.20001 29.20001 40 0015 0049 0139 0347 0784 1632 3171 5793 0041 6618

30.20000 30.20000 30.20000 30.20000 30.20000 30.20000 30.20000 30.20000 30.20000 30.20001 41 0007 0024 0071 0182 0425 091 1814 3394 6016 0163

31.20000 31.20000 31.20000 31.20000 31.20000 31.20000 31.20000 31.20000 31.20000 31.20000 42 0003 0012 0036 0096 023 0506 1035 1983 3593 6196

32.20000 32.20000 32.20000 32.20000 32.20000 32.20000 32.20000 32.20000 32.20000 32.20000 43 0001 0006 0018 005 0124 028 0589 1155 214 3766

33.20000 33.20000 33.20000 33.20000 33.20000 33.20000 33.20000 33.20000 33.20000 33.20000 44 0001 0003 0009 0026 0067 0155 0334 0671 1271 2283

34.20000 34.20000 34.20000 34.20000 34.20000 34.20000 34.20000 34.20000 34.20000 45 34.2 0001 0005 0014 0036 0086 0189 0389 0753 1381

35.20000 35.20000 35.20000 35.20000 35.20000 35.20000 35.20000 35.20000 35.20000 46 35.2 0001 0002 0007 0019 0047 0107 0225 0445 0833

36.20000 36.20000 36.20000 36.20000 36.20000 36.20000 36.20000 36.20000 47 36.2 36.2 0001 0004 001 0026 006 013 0263 0501

37.20000 37.20000 37.20000 37.20000 37.20000 37.20000 37.20000 37.20000 48 37.2 37.2 0001 0002 0006 0014 0034 0075 0155 0301

103 38.20000 38.20000 38.20000 38.20000 38.20000 38.20000 38.20000 49 38.2 38.2 38.2 0001 0003 0008 0019 0043 0091 0181

39.20000 39.20000 39.20000 39.20000 39.20000 39.20000 39.20000 50 39.2 39.2 39.2 0001 0002 0004 0011 0025 0054 0108

40.20000 40.20000 40.20000 40.20000 40.20000 40.20000 51 40.2 40.2 40.2 40.2 0001 0002 0006 0014 0031 0065

41.20000 41.20000 41.20000 41.20000 41.20000 52 41.2 41.2 41.2 41.2 41.2 0001 0003 0008 0018 0039

42.20000 42.20000 42.20000 42.20000 42.20000 53 42.2 42.2 42.2 42.2 42.2 0001 0002 0005 0011 0023

43.20000 43.20000 43.20000 43.20000 54 43.2 43.2 43.2 43.2 43.2 43.2 0001 0003 0006 0014

44.20000 44.20000 44.20000 44.20000 55 44.2 44.2 44.2 44.2 44.2 44.2 0001 0002 0004 0008

45.20000 45.20000 45.20000 56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 0001 0002 0005

46.20000 46.20000 46.20000 57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 0001 0001 0003

47.20000 47.20000 58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 0001 0002

48.20000 59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 0001

49.20000 60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 0001

61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2

62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2

63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2

64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2

65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2

66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2

68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2

69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2

70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2

71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2

72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2

73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2

74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2

75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2

104 76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2

77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2

78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2

79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2

80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2

81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2

82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2

83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2

84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2

85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2

86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2

87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2

88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2

89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2

90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2

91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2

92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2

93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2

94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2

95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2

96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2

97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2

98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2

99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2

100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2

105 3. pdex_dupire.m

Table B.3 The Dupire Volatilities

S=1:100 K = 9.9，τ = 1.9.

S=1:10 S=11:20 S=21:30 S=31:40 S=41:50 S=51:60 S=61:70 S=71:80 S=81:90 S=91:100

0.1103875 0.0997995 0.1008571 0.1054613 0.1063929 0.1005420 0.0927723 0.0851126 σ dupire NaN NaN 1351 08443 9287 1299 1169 3662 37099 23971

0.1076708 0.0997230 0.1011916 0.1058786 0.1060496 0.0997735 0.0920217 0.0731887 σ dupire NaN NaN 6022 90679 7192 9067 9887 64282 31358 71445

0.1056060 0.0997211 0.1015750 0.1062409 0.1056361 0.0989937 0.0913724 0.0814369 σ dupire NaN NaN 4521 77438 5219 3047 0557 26871 05838 46953

0.1041218 0.0997553 0.1020028 0.1065384 0.1051580 0.0982061 0.0904617 0.0677596 σ dupire NaN NaN 578 88519 7325 4185 4199 27539 37785 35682

0.1030239 0.0998121 0.1024683 0.1067633 0.1046217 0.0974167 0.0899351 0.0714249 σ dupire NaN NaN 5962 17284 1984 7199 8906 90179 55032 27393

0.1021527 0.0998936 0.1029625 0.1069097 0.1040338 0.0966223 0.0892097 σ dupire Inf NaN NaN 2677 68878 3212 8776 4167 86341 0572

0.1014207 0.1000051 0.1034749 0.1069737 0.1034008 0.0958330 0.0882457 0.0451730 σ dupire NaN NaN 1648 7119 6498 5558 074 65625 96623 90455

0.1007983 0.1001518 0.1039938 0.1069533 0.1027290 0.0950573 0.0863168 σ dupire Inf NaN NaN 6181 7943 0133 181 1956 49174 13945

0.1003138 0.1003395 0.1045064 0.1068483 0.1020247 0.0942440 0.0868018 σ dupire Inf NaN NaN 6197 3131 1173 8764 7141 41036 02914

0.0999892 0.1005734 0.1049998 0.1066605 0.1012934 0.0934875 0.0834390 σ dupire 0 NaN NaN 99656 5679 4282 6525 5784 21274 47803

4. pdex_dis_dupire.m

Table B.4 Distance between Implied Volatilities and the Dupire Volatilities

tau=1.9;K=10. x Distance x Distance x Distance x Distance x Distance 0.038940 0.068873 0.102688 1 0 8 0 15 22 29 004132 053982 79475 0.009629 0.075617 0.105574 2 0 9 0 16 23 30 5917963 101167 78798 0.005573 0.011647 0.081526 0.107758 3 0 10 17 24 31 1734841 847367 241573 81617 0.033925 0.028040 0.086754 0.108238 4 0 11 18 25 32 944179 893949 277933 5101 0.643299 0.041173 0.091414 0.138504 5 0 12 19 26 33 59398 752426 513072 78093

106 0.169428 0.051991 0.095590 6 0 13 20 27 34 NaN 58353 380509 702894 0.083711 0.061090 0.099342 7 0 14 21 28 87034 850366 677508

5. pdex_imp_small_s.m

Table B.5 Implied Volatilities Volatilities for Small Stock Prices σ imp τ = 0.1:1: 0.1

S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

17.06737 12.37480 10.26173 8.988598 8.113080 7.462801 6.954783 6.543451 6.201265 5.910785 0.5 4456 8848 2387 5531 5179 0286 1538 2861 8136 4141 16.80791 12.18978 10.10990 8.856650 7.994740 7.354531 6.854356 6.449357 6.112427 5.826398 1 5572 099 5584 0881 2955 1423 2506 7404 0276 13 16.65495 12.08068 10.02037 8.778839 7.924951 7.290678 6.795127 6.393862 6.060030 5.776625 1.5 1304 344 7406 3657 4453 8408 6005 8779 152 7294 16.54587 12.00288 9.956529 8.723345 7.875177 7.245137 6.752883 6.354281 6.022657 5.741124 2 7707 3154 3611 7874 5815 9284 5684 2586 6181 6589 16.46095 11.94230 9.906815 8.680135 7.836420 7.209676 6.719988 6.323459 5.993555 5.713479 2.5 8653 8056 5815 8652 4939 281 6499 1426 3914 4798 16.39136 11.89266 9.866072 8.644722 7.804655 7.180611 6.693027 6.298196 5.969702 5.690820 3 8347 4854 3467 1475 6713 9612 7045 8286 5155 6547 16.33238 11.85058 9.831536 8.614703 7.777729 7.155974 6.670173 6.276782 5.949482 5.671612 3.5 4569 6173 6027 3947 559 7029 2129 0665 378 589 16.28118 11.81405 9.801554 8.588642 7.754353 7.134585 6.650331 6.258190 5.931927 5.654936 4 1594 6928 8616 614 3712 3953 4813 1385 4897 2737 16.23593 11.78177 9.775058 8.565610 7.733694 7.115681 6.632795 6.241758 5.916412 5.640197 4.5 2943 4575 2899 9033 0002 8209 5033 6221 4196 6148 16.19538 11.75284 9.751315 8.544972 7.715181 7.098742 6.617081 6.227034 5.902509 5.626990 5 9228 809 7264 8167 5258 5504 6156 3513 3054 1858 16.15865 11.72664 9.729804 8.526274 7.698408 7.083395 6.602844 6.213693 5.889912 5.615023 5.5 7868 0897 7959 3574 7626 0405 2555 5511 4495 6033 16.12507 11.70268 9.710139 8.509179 7.683074 7.069363 6.589827 6.201496 5.878395 5.604083 6 8854 2275 2194 8063 6052 8011 8831 7921 7896 1291 16.09415 11.68061 9.692025 8.493433 7.668950 7.056439 6.577838 6.190261 5.867787 5.594005 6.5 0304 4273 2539 8658 0762 333 1697 9962 4123 4608 16.06548 11.66015 9.675234 8.478837 7.655856 7.044458 6.566723 6.179847 5.857953 5.584663 7 1626 8307 2837 8405 9635 5829 8684 4509 5156 498 16.03876 11.64109 9.659584 8.465234 7.643653 7.033292 6.556364 6.170140 5.848787 5.575956 7.5 2836 3281 8691 0351 8279 1417 9398 6833 905 3647 16.01374 11.62324 9.644930 8.452495 7.632226 7.022835 6.546664 6.161050 5.840204 5.567802 8 378 0771 5857 1865 5243 5728 5087 9242 8775 65 15.99021 11.60645 9.631151 8.440517 7.621481 7.013003 6.537543 6.152503 5.832134 5.560135 8.5 9649 4692 5484 0969 5954 367 2565 8543 2619 6968 15.96802 11.59061 9.618148 8.429213 7.611341 7.003724 6.528935 6.144437 5.824517 5.552900 9 0631 3922 3551 3712 5506 6238 4154 851 8747 2353 15.94700 11.57561 9.605837 8.418511 7.601741 6.994939 6.520785 6.136801 5.817306 5.546049 9.5 4359 691 6668 5747 4208 8993 8447 2483 9336 9256 15.92705 11.56137 9.594148 8.408350 7.592626 6.986598 6.513047 6.129550 5.810460 5.539545 10 0293 7673 9195 3759 1953 8605 855 2995 1329 5306 15.90805 11.54782 9.583021 8.398677 7.583948 6.978658 6.505681 6.122647 5.803942 5.533353 10.5 5465 2766 8366 3845 8832 5124 5634 6348 1884 5325 15.88993 11.53488 9.572404 8.389447 7.575669 6.971081 6.498652 6.116061 5.797722 5.527445 11 1225 8951 5197 4919 0245 8353 6298 078 7194 0719 15.87260 11.52252 9.562251 8.380621 7.567751 6.963836 6.491931 6.109762 5.791775 5.521795 11.5 0695 1389 963 5802 5323 7257 2744 726 3797 1215

107 15.85599 11.51067 9.552524 8.372165 7.560165 6.956895 6.485491 6.103728 5.786077 5.516381 12 6774 2215 8828 5049 7807 1616 5033 2226 1722 8357 15.84006 11.49929 9.543188 8.364049 7.552884 6.950232 6.479310 6.097936 5.780607 5.511186 12.5 0545 9399 7858 285 878 5377 4922 1817 904 0324 15.82473 11.48836 9.534213 8.356246 7.545885 6.943827 6.473368 6.092367 5.775349 5.506190 13 9993 5839 2201 452 0831 1315 0904 7221 7476 7765 15.80998 11.47783 9.525571 8.348733 7.539145 6.937659 6.467646 6.087006 5.770286 5.501381 13.5 8968 8611 1662 5209 3309 6685 4186 0912 8848 0419 15.79576 11.46768 9.517238 8.341489 7.532646 6.931712 6.462129 6.081836 5.765405 5.496743 14 6334 8371 5399 5578 8443 9682 5395 3556 2155 4341 15.78203 11.45788 9.509193 8.334495 7.526372 6.925971 6.456803 6.076845 5.760692 5.492265 14.5 5267 8849 7808 8225 8141 6503 1857 147 1171 9625 15.76876 11.44841 9.501417 8.327735 7.520308 6.920421 6.451654 6.072020 5.756136 5.487937 15 267 6435 5118 4725 1328 8916 5346 4503 2449 8502 15.75591 11.43924 9.493892 8.321193 7.514439 6.915051 6.446672 6.067351 5.751727 5.483749 15.5 8691 9833 2532 3142 1725 2225 0189 4271 3652 3756 15.74347 11.43036 9.486602 8.314855 7.508753 6.909848 6.441845 6.062828 5.747456 5.479691 16 6305 9763 1826 5948 5969 3558 1678 2662 214 7388 15.73141 11.42175 9.479532 8.308709 7.503240 6.904803 6.437164 6.058442 5.743314 5.475756 16.5 0973 8719 9316 8251 2038 041 473 0582 379 9481 15.71970 11.41340 9.472671 8.302744 7.497888 6.899905 6.432621 6.054184 5.739294 5.471937 17 0342 0755 4126 6291 789 9416 2733 6876 1974 7245 15.70832 11.40528 9.466005 8.296949 7.492690 6.895148 6.428207 6.050048 5.735388 5.468227 17.5 3998 1307 6711 6158 032 5293 6574 7415 6696 4187 15.69726 11.39738 9.459524 8.291315 7.487635 6.890522 6.423916 6.046027 5.731591 5.464619 18 3239 7033 7579 2681 396 9928 3794 4302 385 9408 15.68650 11.38970 9.453218 8.285832 7.482717 6.886022 6.419740 6.042114 5.727896 5.461109 18.5 0898 5684 6198 8474 0428 1601 7862 5198 4572 699 15.67602 11.38222 9.447078 8.280494 7.477927 6.881639 6.415674 6.038304 5.724298 5.457691 19 1172 5989 0041 311 758 4303 7543 2725 4684 5469 15.66580 11.37493 9.441094 8.275292 7.473260 6.877368 6.411712 6.034591 5.720792 5.454360 19.5 9486 7546 3757 2401 887 714 6351 3955 421 7368 15.65585 11.36783 9.435259 8.270219 7.468710 6.873204 6.407849 6.030970 5.717373 5.451112 20 2365 0745 8448 7764 2781 3826 207 996 6954 8803 15.64613 11.36089 9.429567 8.265270 7.464270 6.869141 6.404079 6.027438 5.714038 5.447943 20.5 733 6681 103 5676 2332 2222 6333 5421 0122 9121 15.63665 11.35412 9.424009 8.260438 7.459935 6.865174 6.400399 6.023989 5.710781 5.444850 21 2797 709 3681 7181 464 3938 4254 8278 3999 0597 15.62738 11.34751 9.418580 8.255718 7.455701 6.861299 6.396804 6.020620 5.707600 5.441827 21.5 7999 4289 3342 7466 0538 3988 4098 9424 1661 8152 15.61833 11.34105 9.413274 8.251105 7.451562 6.857512 6.393290 6.017328 5.704490 5.438873 22 2905 1124 1281 5481 4233 0469 6996 2438 8718 9118 15.60947 11.33473 9.408085 8.246594 7.447515 6.853808 6.389854 6.014108 5.701450 5.435985 22.5 816 0919 2709 3599 3005 4292 6685 3342 3087 3017 15.60081 11.32854 9.403008 8.242180 7.443555 6.850184 6.386492 6.010958 5.698475 5.433159 23 5022 7436 6432 7319 6937 8932 9283 0387 4789 137 15.59233 11.32249 9.398039 8.237860 7.439679 6.846638 6.383202 6.007874 5.695563 5.430392 23.5 5309 4839 454 4999 8674 0209 3081 3862 577 7528 15.58403 11.31656 9.393173 8.233629 7.435884 6.843164 6.379979 6.004854 5.692711 5.427683 24 1354 7658 2139 7612 3208 609 8366 5923 974 6515 15.57589 11.31076 9.388405 8.229484 7.432165 6.839761 6.376822 6.001896 5.689918 5.425029 24.5 5964 076 7094 8534 7689 6515 7251 0441 2025 4893 15.56792 11.30506 9.383732 8.225422 7.428521 6.836426 6.373728 5.998996 5.687179 5.422428 25 2378 932 9814 335 1246 3237 3533 2862 9442 0638 15.56010 11.29948 9.379151 8.221438 7.424947 6.833155 6.370694 5.996153 5.684495 5.419877 25.5 4234 8802 3048 9679 4832 9679 2555 0084 0176 3026 15.55243 11.29401 9.374657 8.217531 7.421442 6.829948 6.367718 5.993364 5.681861 5.417375 26 5541 4929 1705 7018 1085 0802 1093 0342 3678 2536 15.54491 11.28864 9.370247 8.213697 7.418002 6.826800 6.364797 5.990627 5.679277 5.414920 26.5 0647 3669 2687 6597 42 2995 724 3111 0565 0757

108 15.53752 11.28337 9.365918 8.209934 7.414625 6.823710 6.361931 5.987940 5.676740 5.412510 27 4216 1216 4747 1254 9809 396 0313 9009 2538 0304 15.53027 11.27819 9.361667 8.206238 7.411310 6.820676 6.359116 5.985302 5.674249 5.410143 27.5 1205 397 8345 5311 4881 2622 0762 9719 2298 4744 15.52314 11.27310 9.357492 8.202608 7.408053 6.817695 6.356351 5.982711 5.671802 5.407818 28 6842 8527 5534 4474 7624 9042 0089 7905 3479 8529 15.51614 11.26811 9.353389 8.199041 7.404853 6.814767 6.353634 5.980165 5.669398 5.405534 28.5 6605 1661 9841 5732 7394 4331 077 7149 0578 6932 15.50926 11.26320 9.349357 8.195535 7.401708 6.811889 6.350963 5.977663 5.667034 5.403289 29 6211 0316 6169 7269 4624 0581 6195 1885 8896 5989 15.50250 11.25837 9.345393 8.192088 7.398616 6.809059 6.348338 5.975202 5.664711 5.401082 29.5 1592 1588 07 838 074 08 0599 7342 4485 2451 15.49584 11.25362 9.341494 8.188698 7.395574 6.806275 6.345755 5.972782 5.662426 5.398911 30 8886 2723 0813 9401 8104 8846 9009 9489 4097 3732 15.48930 11.24895 9.337658 8.185364 7.392582 6.803537 6.343215 5.970402 5.660178 5.396775 30.5 442 1099 5001 1639 9946 9375 7189 4986 5138 7868 15.48286 11.24435 9.333884 8.182082 7.389639 6.800843 6.340716 5.968060 5.657966 5.394674 31 4701 4225 2803 7305 0311 7787 1595 1144 5623 3473 15.47652 11.23982 9.330169 8.178852 7.386741 6.798192 6.338255 5.965754 5.655789 5.392605 31.5 6403 9727 4734 9462 4004 0179 9331 5877 4141 9709 15.47028 11.23537 9.326512 8.175673 7.383888 6.795581 6.335833 5.963484 5.653645 5.390569 32 6354 5342 2225 197 6548 3303 8106 7667 9818 6243 15.46414 11.23098 9.322910 8.172541 7.381079 6.793010 6.333448 5.961249 5.651535 5.388564 32.5 1531 8913 7567 9433 4132 4521 6198 5531 2281 3223 15.45808 11.22666 9.319363 8.169457 7.378312 6.790478 6.331099 5.959047 5.649456 5.386589 33 9048 8381 3857 7159 3577 1775 2418 8986 1632 1246 15.45212 11.22241 9.315868 8.166419 7.375586 6.787983 6.328784 5.956878 5.647407 5.384643 33.5 6149 1779 4953 1114 2294 3545 6081 8017 8416 1329 15.44625 11.21821 9.312424 8.163424 7.372899 6.785524 6.326503 5.954741 5.645389 5.382725 34 0199 7227 5424 7887 8253 8822 6973 3056 3596 4887 15.44045 11.21408 9.309030 8.160473 7.370251 6.783101 6.324255 5.952634 5.643399 5.380835 34.5 8679 2927 0513 4648 9944 7076 5326 4947 8529 371 15.43474 11.21000 9.305683 8.157563 7.367641 6.780712 6.322039 5.950557 5.641438 5.378971 35 9178 716 6097 912 6356 8229 1791 4933 4943 9941 15.42911 11.20598 9.302383 8.154694 7.365067 6.778357 6.319853 5.948509 5.639504 5.377134 35.5 9385 8276 865 9547 6941 2631 7416 4622 4917 6052 15.42356 11.20202 9.299129 8.151865 7.362529 6.776034 6.317698 5.946489 5.637597 5.375322 36 709 4698 5212 4661 1593 1032 3621 5975 0859 4834 15.41809 11.19811 9.295919 8.149074 7.360025 6.773742 6.315572 5.944497 5.635715 5.373534 36.5 0168 491 3355 3659 0621 4567 218 1286 5491 9372 15.41268 11.19425 9.292752 8.146320 7.357554 6.771481 6.313474 5.942531 5.633859 5.371771 37 6586 7462 1157 6177 4728 4728 52 3157 1831 3032 15.40735 11.19045 9.289626 8.143603 7.355116 6.769250 6.311404 5.940591 5.632027 5.370030 37.5 439 0958 7173 2267 499 3348 5107 4491 3174 9446 15.40209 11.18669 9.286542 8.140921 7.352710 6.767048 6.309361 5.938676 5.630219 5.368313 38 1701 406 041 2373 2834 2584 4623 8469 3083 2498 15.39689 11.18298 9.283497 8.138273 7.350335 6.764874 6.307344 5.936786 5.628434 5.366617 38.5 6718 5479 0306 7312 0023 4898 6756 8538 537 631 15.39176 11.17932 9.280490 8.135659 7.347989 6.762728 6.305353 5.934920 5.626672 5.364943 39 7706 3981 6703 8255 8638 3042 4782 8399 4088 5231 15.38670 11.17570 9.277521 8.133078 7.345674 6.760609 6.303387 5.933078 5.624932 5.363290 39.5 2997 8373 9832 6709 1059 0046 2235 1991 3512 3825 15.38170 11.17213 9.274590 8.130529 7.343386 6.758515 6.301445 5.931258 5.623213 5.361657 40 0985 751 0288 4499 9954 9199 2889 3481 8136 6859 15.37676 11.16861 9.271693 8.128011 7.341127 6.756448 6.299527 5.929460 5.621516 5.360044 40.5 0125 0289 9017 3753 8261 4039 075 7253 2655 9295 15.37187 11.16512 9.268832 8.125523 7.338895 6.754405 6.297632 5.927684 5.619839 5.358451 41 8928 5649 7295 689 9177 8343 0044 7894 196 6279 15.36705 11.16168 9.266005 8.123065 7.336690 6.752387 6.295759 5.925930 5.618182 5.356877 41.5 5957 2563 6715 6599 6147 611 5204 0191 1127 3133

109 15.36228 11.15828 9.263211 8.120636 7.334511 6.750393 6.293909 5.924195 5.616544 5.355321 42 983 0045 917 5833 2849 1556 0865 9114 5407 5346 15.35757 11.15491 9.260450 8.118235 7.332357 6.748421 6.292080 5.922481 5.614926 5.353783 42.5 921 7142 6838 7794 3183 91 1849 9811 022 8565 15.35292 11.15159 9.257721 8.115862 7.330228 6.746473 6.290272 5.920787 5.613326 5.352263 43 281 2934 2173 592 1265 3356 3159 7602 1143 8592 15.34831 11.14830 9.255022 8.113516 7.328123 6.744546 6.288484 5.919112 5.611744 5.350761 43.5 9385 6534 7886 3877 141 9123 9972 7964 3909 137 15.34376 11.14505 9.252354 8.111196 7.326041 6.742642 6.286717 5.917456 5.610180 5.349275 44 7733 7084 6939 5545 813 138 7627 6533 4395 2984 15.33926 11.14184 9.249716 8.108902 7.323983 6.740758 6.284970 5.915818 5.608633 5.347805 44.5 6694 3756 2531 5012 6122 5273 1623 9089 8616 9649 15.33481 11.13866 9.247106 8.106633 7.321948 6.738895 6.283241 5.914199 5.607104 5.346352 45 5145 5749 8088 6563 0258 611 7608 1554 2723 7706 15.33041 11.13552 9.244525 8.104389 7.319934 6.737052 6.281532 5.912596 5.605591 5.344915 45.5 2001 2288 725 4671 5581 9355 1371 9983 2991 3617 15.32605 11.13241 9.241972 8.102169 7.317942 6.735230 6.279840 5.911012 5.604094 5.343493 46 6212 2624 3869 3989 7294 0618 8843 0562 5817 3961 15.32174 11.12933 9.239446 8.099972 7.315972 6.733426 6.278167 5.909443 5.602613 5.342086 46.5 6761 6032 1991 9344 0756 5653 6082 9599 7715 5424 15.31748 11.12629 9.236946 8.097799 7.314022 6.731642 6.276511 5.907892 5.601148 5.340694 47 2666 1809 5857 5726 1474 0349 9275 3519 5309 4802 15.31326 11.12327 9.234472 8.095648 7.312092 6.729876 6.274873 5.906356 5.599698 5.339316 47.5 2973 9278 9884 8285 5095 0722 4727 8861 5333 8988 15.30908 11.12029 9.232024 8.093520 7.310182 6.728128 6.273251 5.904837 5.598263 5.337953 48 676 7777 8669 2322 7406 2916 886 2271 4618 4976 15.30495 11.11734 9.229601 8.091413 7.308292 6.726398 6.271646 5.903333 5.596843 5.336603 48.5 3133 6671 6972 3283 432 3192 8206 05 0097 9849 15.30086 11.11442 9.227202 8.089327 7.306421 6.724685 6.270057 5.901844 5.595436 5.335268 49 1224 534 9716 6753 1879 7928 9404 0398 8796 0782 15.29681 11.11153 9.224828 8.087262 7.304568 6.722990 6.268484 5.900369 5.594044 5.333945 49.5 0193 3184 1976 8452 6241 3609 9194 8911 7829 5035 15.29279 11.10866 9.222476 8.085218 7.302734 6.721311 6.266927 5.898910 5.592666 5.332635 50 9225 9621 8977 4228 3682 6827 4413 3075 4398 995 15.28882 11.10583 9.220148 8.083194 7.300918 6.719649 6.265385 5.897465 5.591301 5.331339 50.5 7529 4087 6083 0051 0588 4274 1993 0015 579 2948 15.28489 11.10302 9.217842 8.081189 7.299119 6.718003 6.263857 5.896033 5.589949 5.330055 51 4337 6034 8796 2013 3451 274 8957 694 9369 1525 15.28099 11.10024 9.215559 8.079203 7.297337 6.716372 6.262345 5.894616 5.588611 5.328783 51.5 8902 4929 275 6314 8866 911 2414 1142 2577 3252 15.27714 11.09749 9.213297 8.077236 7.295573 6.714758 6.260846 5.893211 5.587285 5.327523 52 0502 0257 3702 927 3523 0355 9554 999 2929 5768 15.27331 11.09476 9.211056 8.075288 7.293825 6.713158 6.259362 5.891821 5.585971 5.326275 52.5 8433 1515 7532 7297 4211 3537 7652 0926 8011 678 15.26953 11.09205 9.208837 8.073358 7.292093 6.711573 6.257892 5.890443 5.584670 5.325039 53 2011 8216 0238 6916 7807 5797 4055 1468 5479 406 15.26578 11.08937 9.206637 8.071446 7.290378 6.710003 6.256435 5.889077 5.583381 5.323814 53.5 0573 9884 7926 4743 1277 4359 6189 9201 3053 5442 15.26206 11.08672 9.204458 8.069551 7.288678 6.708447 6.254992 5.887725 5.582103 5.322600 54 3471 6061 6815 7489 1669 6523 1547 1778 8516 8822 15.25838 11.08409 9.202299 8.067674 7.286993 6.706905 6.253561 5.886384 5.580837 5.321398 54.5 0079 6297 3224 1956 6115 9664 7695 6917 9713 215 15.25472 11.08149 9.200159 8.065813 7.285324 6.705378 6.252144 5.885056 5.579583 5.320206 55 9785 0156 3574 5033 1823 1226 2263 2397 4547 3437 15.25111 11.07890 9.198038 8.063969 7.283669 6.703863 6.250739 5.883739 5.578340 5.319025 55.5 1996 7214 4384 3691 6078 8725 2944 6059 0979 0743 15.24752 11.07634 9.195936 8.062141 7.282029 6.702362 6.249346 5.882434 5.577107 5.317854 56 6131 7059 2263 4986 6236 9743 7497 5798 7023 2183 15.24397 11.07380 9.193852 8.060329 7.280403 6.700875 6.247966 5.881140 5.575886 5.316693 56.5 1628 9288 3913 6047 9724 1924 3735 9569 0747 592

110 15.24044 11.07129 9.191786 8.058533 7.278792 6.699400 6.246597 5.879858 5.574675 5.315543 57 7939 351 6121 4084 4038 2977 9533 5378 0268 0167 15.23695 11.06879 9.189738 8.056752 7.277194 6.697938 6.245241 5.878587 5.573474 5.314402 57.5 453 9343 5759 6375 6737 0668 2819 1285 3753 3182 15.23349 11.06632 9.187707 8.054987 7.275610 6.696488 6.243896 5.877326 5.572283 5.313271 58 0881 6418 9778 0272 5445 2822 1574 5398 9417 3269 15.23005 11.06387 9.185694 8.053236 7.274039 6.695050 6.242562 5.876076 5.571103 5.312149 58.5 6483 4371 5208 3192 7844 7321 3834 5876 5521 8775 15.22665 11.06144 9.183697 8.051500 7.272482 6.693625 6.241239 5.874837 5.569933 5.311037 59 0845 2851 9155 2618 1678 2097 7681 0921 0368 809 15.22327 11.05903 9.181717 8.049778 7.270937 6.692211 6.239928 5.873607 5.568772 5.309934 59.5 3483 1513 8798 6099 4747 5139 1248 8785 2305 9643 15.21992 11.05664 9.179754 8.048071 7.269405 6.690809 6.238627 5.872388 5.567620 5.308841 60 3928 0022 1385 1243 4905 4482 2713 776 9722 1904 15.21660 11.05426 9.177806 8.046377 7.267886 6.689418 6.237337 5.871179 5.566479 5.307756 60.5 1722 8052 4233 5717 0059 8212 03 6181 1046 3378 15.21330 11.05191 9.175874 8.044697 7.266378 6.688039 6.236057 5.869980 5.565346 5.306680 61 6419 5284 4727 7246 8169 4461 2277 2426 4744 261 15.21003 11.04958 9.173958 8.043031 7.264883 6.686671 6.234787 5.868790 5.564222 5.305612 61.5 7583 1407 0311 3611 7243 1407 6952 491 9321 8178 15.20679 11.04726 9.172056 8.041378 7.263400 6.685313 6.233528 5.867610 5.563108 5.304553 62 4788 6117 8495 2644 5339 7272 2677 2087 3316 8697 15.20357 11.04496 9.170170 8.039738 7.261929 6.683967 6.232278 5.866439 5.562002 5.303503 62.5 7619 9118 6847 2234 0559 032 7841 2448 5306 2812 15.20038 11.04269 9.168299 8.038111 7.260469 6.682630 6.231039 5.865277 5.560905 5.302460 63 5671 0121 2992 0315 1053 8858 0872 4521 3899 9202 15.19721 11.04042 9.166442 8.036496 7.259020 6.681305 6.229809 5.864124 5.559816 5.301426 63.5 8549 8843 4614 4872 5013 1231 0236 6868 7737 6577 15.19407 11.03818 9.164599 8.034894 7.257583 6.679989 6.228588 5.862980 5.558736 5.300400 64 5866 5009 9447 3937 0673 5823 4434 8084 5494 3677 15.19095 11.03595 9.162771 8.033304 7.256156 6.678684 6.227377 5.861845 5.557664 5.299381 64.5 7245 8349 5281 5589 6308 1056 2003 6797 5875 9272 15.18786 11.03374 9.160956 8.031726 7.254741 6.677388 6.226175 5.860719 5.556600 5.298371 65 2316 8601 9956 7949 0235 539 1511 1667 7614 2159 15.18479 11.03155 9.159156 8.030160 7.253336 6.676102 6.224982 5.859601 5.555544 5.297368 65.5 072 5507 1361 9181 0806 7316 1562 1385 9475 1164 15.18174 11.02937 9.157368 8.028606 7.251941 6.674826 6.223798 5.858491 5.554497 5.296372 66 2105 8815 7433 7494 6414 5364 0789 467 0249 5139 15.17871 11.02721 9.155594 8.027064 7.250557 6.673559 6.222622 5.857390 5.553456 5.295384 66.5 6126 8281 6155 1133 5484 8094 7857 0271 8755 296 15.17571 11.02507 9.153833 8.025532 7.249183 6.672302 6.221456 5.856296 5.552424 5.294403 67 2447 3664 5556 8385 6481 4099 1462 6966 3837 3531 15.17273 11.02294 9.152085 8.024012 7.247819 6.671054 6.220298 5.855211 5.551399 5.293429 67.5 0738 4729 3709 7574 79 2004 0327 3559 4367 578 15.16977 11.02083 9.150349 8.022503 7.246465 6.669815 6.219148 5.854133 5.550381 5.292462 68 0679 1246 8728 7062 8272 0462 3203 8879 924 8656 15.16683 11.01873 9.148626 8.021005 7.245121 6.668584 6.218006 5.853064 5.549371 5.291503 68.5 1954 2992 8769 5244 6158 8159 8871 1784 7375 1134 15.16391 11.01664 9.146916 8.019518 7.243787 6.667363 6.216873 5.852002 5.548368 5.290550 69 4255 9746 2029 0554 0151 3806 6135 1153 7716 221 15.16101 11.01458 9.145217 8.018041 7.242461 6.666150 6.215748 5.850947 5.547372 5.289604 69.5 7282 1293 6742 1456 8876 6144 3828 5891 923 0901 15.15814 11.01252 9.143531 8.016574 7.241146 6.664946 6.214631 5.849900 5.546384 5.288664 70 0739 7423 1181 6449 0984 3941 0804 4927 0902 6245 15.15528 11.01048 9.141856 8.015118 7.239839 6.663750 6.213521 5.848860 5.545402 5.287731 70.5 4338 7929 3656 4063 5157 5989 5946 7213 1745 7303 15.15244 11.00846 9.140193 8.013672 7.238542 6.662563 6.212419 5.847828 5.544427 5.286805 71 7796 2611 251 2859 0105 1108 8157 172 0787 3152 15.14963 11.00645 9.138541 8.012236 7.237253 6.661383 6.211325 5.846802 5.543458 5.285885 71.5 0836 1271 6126 1429 4564 8141 6364 7444 7079 2891

111 15.14683 11.00445 9.136901 8.010809 7.235973 6.660212 6.210238 5.845784 5.542496 5.284971 72 3189 3716 2914 8393 7296 5956 9517 3401 9693 5637 15.14405 11.00246 9.135272 8.009393 7.234702 6.659049 6.209159 5.844772 5.541541 5.284064 72.5 4588 9756 1323 2401 709 3444 6588 8626 7717 0524 15.14129 11.00049 9.133653 8.007986 7.233440 6.657893 6.208087 5.843768 5.540593 5.283162 73 4775 9206 9829 2129 2759 9519 6568 2175 0259 6707 15.13855 10.99854 9.132046 8.006588 7.232186 6.656746 6.207022 5.842770 5.539650 5.282267 73.5 3494 1884 6942 6281 314 3117 847 3123 6448 3355 15.13583 10.99659 9.130450 8.005200 7.230940 6.655606 6.205965 5.841779 5.538714 5.281377 74 0496 7613 1201 3588 7093 3194 1327 0563 5425 9655 15.13312 10.99466 9.128864 8.003821 7.229703 6.654473 6.204914 5.840794 5.537784 5.280494 74.5 5537 6218 1174 2804 3502 873 4192 3609 6354 481 15.13043 10.99274 9.127288 8.002451 7.228474 6.653348 6.203870 5.839816 5.536860 5.279616 75 8377 7528 5459 2711 1274 8723 6135 1388 8411 8039 15.12776 10.99084 9.125723 8.001090 7.227252 6.652231 6.202833 5.838844 5.535943 5.278744 75.5 8782 1375 2681 2112 9335 2193 6246 3048 0792 8577 15.12511 10.98894 9.124168 7.999737 7.226039 6.651120 6.201803 5.837878 5.535031 5.277878 76 6522 7597 1491 9837 6635 8176 3633 7753 2707 5675 15.12248 10.98706 9.122623 7.998394 7.224834 6.650017 6.200779 5.836919 5.534125 5.277017 76.5 1371 603 0569 4735 2142 573 742 4682 338 8595 15.11986 10.98519 9.121087 7.997059 7.223636 6.648921 6.199762 5.835966 5.533225 5.276162 77 3108 6519 8617 568 4847 3929 6749 3032 2054 6617 15.11726 10.98333 9.119562 7.995733 7.222446 6.647832 6.198752 5.835019 5.532330 5.275312 77.5 1516 8907 4366 1569 3757 1868 078 2014 7983 9034 15.11467 10.98149 9.118046 7.994415 7.221263 6.646749 6.197747 5.834078 5.531442 5.274468 78 6382 3043 6569 1316 7901 8655 8686 0855 0437 5152 15.11210 10.97965 9.116540 7.993105 7.220088 6.645674 6.196749 5.833142 5.530558 5.273629 78.5 7498 8778 4003 3859 6325 342 9658 8796 87 429 15.10955 10.97783 9.115043 7.991803 7.218920 6.644605 6.195758 5.832213 5.529681 5.272795 79 4659 5967 5468 8155 8094 5304 2902 5092 2069 578 15.10701 10.97602 9.113555 7.990510 7.217760 6.643543 6.194772 5.831289 5.528808 5.271966 79.5 7663 4464 9788 3181 2289 3469 7639 9014 9854 8967 15.10449 10.97422 9.112077 7.989224 7.216606 6.642487 6.193793 5.830371 5.527942 5.271143 80 6315 4131 5808 7932 801 7091 3103 9844 1379 321 15.10199 10.97243 9.110608 7.987947 7.215460 6.641438 6.192819 5.829459 5.527080 5.270324 80.5 042 4829 2394 1424 4374 5359 8545 6881 598 7875 15.09949 10.97065 9.109147 7.986677 7.214321 6.640395 6.191852 5.828552 5.526224 5.269511 81 9788 6422 8434 269 0511 7481 3228 9432 3004 2346 15.09702 10.96888 9.107696 7.985415 7.213188 6.639359 6.190890 5.827651 5.525373 5.268702 81.5 4234 8777 2835 0779 5571 2677 6428 6822 1812 6012 15.09456 10.96713 9.106253 7.984160 7.212062 6.638329 6.189934 5.826755 5.524527 5.267898 82 3574 1764 4526 476 8717 0181 7436 8384 1776 8279 15.09211 10.96538 9.104819 7.982913 7.210943 6.637304 6.188984 5.825865 5.523686 5.267099 82.5 7629 5254 2454 3717 9129 9242 5555 3465 2279 8559 15.08968 10.96364 9.103393 7.981673 7.209831 6.636286 6.188040 5.824980 5.522850 5.266305 83 6221 9122 5585 6752 5999 9123 01 1424 2715 6278 15.08726 10.96192 9.101976 7.980441 7.208725 6.635274 6.187101 5.824100 5.522019 5.265516 83.5 9179 3244 2904 2982 8537 9099 0399 1631 249 087 15.08486 10.96020 9.100567 7.979216 7.207626 6.634268 6.186167 5.823225 5.521193 5.264731 84 6331 7499 3413 1541 5965 8458 5791 3467 1019 178 15.08247 10.95850 9.099166 7.977998 7.206533 6.633268 6.185239 5.822355 5.520371 5.263950 84.5 751 1766 6134 1577 7518 6501 5629 6325 7728 8463 15.08010 10.95680 9.097774 7.976787 7.205447 6.632274 6.184316 5.821490 5.519555 5.263175 85 2553 5929 0104 2253 2447 2542 9276 9607 2054 0384 15.07774 10.95511 9.096389 7.975583 7.204367 6.631285 6.183399 5.820631 5.518743 5.262403 85.5 1298 9874 4378 2747 0013 5906 6104 2727 3442 7017 15.07539 10.95344 9.095012 7.974386 7.203292 6.630302 6.182487 5.819776 5.517936 5.261636 86 3587 3486 8026 2252 9493 5929 55 5109 1349 7843 15.07305 10.95177 9.093644 7.973195 7.202225 6.629325 6.181580 5.818926 5.517133 5.260874 86.5 9262 6654 0137 9975 0175 1961 6858 6186 524 2356

112 15.07073 10.95011 9.092282 7.972012 7.201163 6.628353 6.180678 5.818081 5.516335 5.260116 87 8173 9269 9812 5135 1358 3361 9585 5402 4588 0055 15.06843 10.94847 9.090929 7.970835 7.200107 6.627386 6.179782 5.817241 5.515541 5.259362 87.5 0167 1225 6171 6966 2356 9499 3097 2209 8878 045 15.06613 10.94683 9.089583 7.969665 7.199057 6.626425 6.178890 5.816405 5.514752 5.258612 88 5098 2414 8345 4714 2491 9757 6819 6069 7601 3059 15.06385 10.94520 9.088245 7.968501 7.198013 6.625470 6.178004 5.815574 5.513968 5.257866 88.5 2819 2734 5484 7638 11 3527 0188 6454 0258 7407 15.06158 10.94358 9.086914 7.967344 7.196974 6.624520 6.177122 5.814748 5.513187 5.257125 89 3188 2082 6749 501 7529 0211 2649 2844 6359 3027 15.05932 10.94197 9.085591 7.966193 7.195942 6.623574 6.176245 5.813926 5.512411 5.256387 89.5 6064 0358 1317 6114 1135 9221 3657 4728 542 9462 15.05708 10.94036 9.084274 7.965049 7.194915 6.622634 6.175373 5.813109 5.511639 5.255654 90 1309 7464 8377 0245 1287 9978 2674 1603 6966 6259 15.05484 10.93877 9.082965 7.963910 7.193893 6.621700 6.174505 5.812296 5.510872 5.254925 90.5 8787 3301 7132 6711 7363 1914 9174 2974 0532 2977 15.05262 10.93718 9.081663 7.962778 7.192877 6.620770 6.173643 5.811487 5.510108 5.254199 91 8364 7776 68 483 8753 447 2637 8356 5657 9178 15.05041 10.93561 9.080368 7.961652 7.191867 6.619845 6.172785 5.810683 5.509349 5.253478 91.5 9908 0793 6609 3933 4854 7095 2553 727 1891 4433 15.04822 10.93404 9.079080 7.960532 7.190862 6.618925 6.171931 5.809883 5.508593 5.252760 92 3291 2261 58 336 5075 9249 842 9246 8788 832 15.04603 10.93248 9.077799 7.959418 7.189862 6.618011 6.171082 5.809088 5.507842 5.252047 92.5 8384 2089 3629 2462 8835 0398 9744 382 5912 0425 15.04386 10.93093 9.076524 7.958310 7.188868 6.617101 6.170238 5.808297 5.507095 5.251337 93 5063 0187 936 0602 5561 0019 6039 0537 2833 0338 15.04170 10.92938 9.075257 7.957207 7.187879 6.616195 6.169398 5.807509 5.506351 5.250630 93.5 3204 6467 2272 7151 469 7597 6827 895 9127 7657 15.03955 10.92785 9.073996 7.956111 7.186895 6.615295 6.168563 5.806726 5.505612 5.249928 94 2687 0843 1654 1491 5668 2624 1636 8617 4378 1987 15.03741 10.92632 9.072741 7.955020 7.185916 6.614399 6.167732 5.805947 5.504876 5.249229 94.5 3391 3229 6806 3015 7948 46 0005 9103 8176 2939 15.03528 10.92480 9.071493 7.953935 7.184943 6.613508 6.166905 5.805172 5.504145 5.248534 95 5199 3542 704 1124 0994 3036 1476 9982 0118 0128 15.03316 10.92329 9.070252 7.952855 7.183974 6.612621 6.166082 5.804402 5.503416 5.247842 95.5 7995 1699 1677 5228 4277 7445 5602 0834 9806 3178 15.03106 10.92178 9.069017 7.951781 7.183010 6.611739 6.165264 5.803635 5.502692 5.247154 96 1666 7619 0052 4748 7277 7354 194 1243 685 1718 15.02896 10.92029 9.067788 7.950712 7.182051 6.610862 6.164450 5.802872 5.501972 5.246469 96.5 61 1222 1507 9113 9481 2292 0056 0804 0865 5381 15.02688 10.91880 9.066565 7.949649 7.181098 6.609989 6.163639 5.802112 5.501255 5.245788 97 1186 2429 5395 7759 0386 1798 952 9113 1472 3808 15.02480 10.91732 9.065349 7.948592 7.180148 6.609120 6.162833 5.801357 5.500541 5.245110 97.5 6815 1162 108 0135 9495 5418 9913 5777 8297 6645 15.02274 10.91584 9.064138 7.947539 7.179204 6.608256 6.162032 5.800606 5.499832 5.244436 98 2881 7346 7934 5694 6317 2704 0817 0406 0973 3542 15.02068 10.91438 9.062934 7.946492 7.178265 6.607396 6.161234 5.799858 5.499125 5.243765 98.5 9278 0904 5341 3899 0373 3215 1825 2617 9139 4155 15.01864 10.91292 9.061736 7.945450 7.177330 6.606540 6.160440 5.799114 5.498423 5.243097 99 5903 1763 2691 4223 1187 6518 2534 2033 2438 8146 15.01661 10.91146 9.060543 7.944413 7.176399 6.605689 6.159650 5.798373 5.497724 5.242433 99.5 2652 9851 9386 6143 8293 2184 2547 8281 0519 5182 15.01458 10.91002 9.059357 7.943381 7.175474 6.604841 6.158864 5.797637 5.497028 5.241772 100 9426 5094 4835 9148 1229 9793 1473 0996 3037 4933

S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

3.712882 3.239624 2.920415 3.393201 2.964191 2.674665 3.339087 2.917554 2.633044 0.5 9617 704 4161 3313 6224 125 6908 361 6465 3.661830 3.195647 2.881183 3.392056 2.963204 2.673784 3.338501 2.917049 2.632593 1 7128 5325 4012 0889 6514 3522 7658 3644 9567

113 3.631713 3.169702 2.858037 3.390927 2.962231 2.672916 3.337920 2.916548 2.632146 1.5 701 8094 1347 0415 6355 0318 0574 0014 5095 3.610229 3.151194 2.841524 3.389813 2.961272 2.672059 3.337342 2.916050 2.631702 2 4535 2181 4634 7336 1822 8133 5048 2197 2581 3.593497 3.136779 2.828663 3.388715 2.960325 2.671215 3.336769 2.915555 2.631261 2.5 7677 5392 9592 7285 9151 3612 0483 9679 1567 3.579782 3.124963 2.818121 3.387632 2.959392 2.670382 3.336199 2.915065 2.630823 3 9147 6574 8757 6076 4735 3534 6299 196 1606 3.568155 3.114946 2.809184 3.386563 2.958471 2.669560 3.335634 2.914577 2.630388 3.5 9727 4047 396 969 5114 4813 1925 8547 226 3.558060 3.106248 2.801424 3.385509 2.957562 2.668749 3.335072 2.914093 2.629956 4 975 8657 2819 4273 6964 4482 6803 896 3098 3.549138 3.098561 2.794565 3.384468 2.956665 2.667948 3.334515 2.913613 2.629527 4.5 4974 4243 3231 6121 7095 9695 0387 2729 3703 3.541142 3.091672 2.788418 3.383441 2.955780 2.667158 3.333961 2.913135 2.629101 5 6427 2561 5479 1674 2439 7716 2142 9393 3661 3.533897 3.085430 2.782848 3.382426 2.954906 2.666378 3.333411 2.912661 2.628678 5.5 7152 009 9317 7511 0049 5914 1544 8499 2571 3.527273 3.079722 2.777756 3.381425 2.954042 2.665608 3.332864 2.912190 2.628258 6 7832 7528 6183 034 7088 1757 8081 9607 0038 3.521172 3.074465 2.773065 3.380435 2.953190 2.664847 3.332322 2.911723 2.627840 6.5 034 3643 6661 6993 0826 281 1249 2283 5675 3.515515 3.069591 2.768716 3.379458 2.952347 2.664095 3.331783 2.911258 2.627425 7 5582 5881 9661 4418 8633 6727 0556 6102 9104 3.510243 3.065048 2.764663 3.378492 2.951515 2.663353 3.331247 2.910797 2.627013 7.5 3131 8376 603 9676 7978 1249 552 0649 9953 3.505306 3.060794 2.760867 3.377538 2.950693 2.662619 3.330715 2.910338 2.626604 8 0336 6715 7128 9934 6419 4196 5667 5515 786 3.500663 3.056794 2.757298 3.376596 2.949881 2.661894 3.330187 2.909883 2.626198 8.5 3816 3391 2924 2461 1603 347 0535 0303 2466 3.496281 3.053019 2.753929 3.375664 2.949078 2.661177 3.329661 2.909430 2.625794 9 9094 0253 6341 4621 1259 7043 9667 462 3424 3.492133 3.049444 2.750740 3.374743 2.948284 2.660469 3.329140 2.908980 2.625393 9.5 573 5701 1831 3871 3195 2961 262 8081 0391 3.488194 3.046050 2.747711 3.373832 2.947499 2.659768 3.328621 2.908534 2.624994 10 626 516 6878 7756 5296 9336 8956 0312 303 3.484444 3.042819 2.744828 3.372932 2.946723 2.659076 3.328106 2.908090 2.624598 10.5 7832 387 5567 3903 5518 4343 8247 0942 1012 3.480866 3.039736 2.742077 3.372042 2.945956 2.658391 3.327595 2.907648 2.624204 11 5784 1357 3646 0021 1886 6222 0072 9609 4013 3.477444 3.036787 2.739446 3.371161 2.945197 2.657714 3.327086 2.907210 2.623813 11.5 8643 7122 4688 3895 2491 3268 4019 5958 1718 3.474166 3.033962 2.736925 3.370290 2.944446 2.657044 3.326580 2.906774 2.623424 12 4188 7257 7066 3382 5488 3834 9684 9641 3815 3.471019 3.031251 2.734506 3.369428 2.943703 2.656381 3.326078 2.906342 2.623037 12.5 6322 1737 1539 6409 909 6328 667 0315 9998 3.467994 3.028644 2.732179 3.368576 2.942969 2.655725 3.325579 2.905911 2.622653 13 2542 2256 9319 0971 1569 9207 4587 7646 9969 3.465081 3.026134 2.729940 3.367732 2.942242 2.655077 3.325083 2.905484 2.622272 13.5 1899 0456 0498 5127 1252 0977 3054 1303 3434 3.462272 3.023713 2.727780 3.366897 2.941522 2.654435 3.324590 2.905059 2.621893 14 332 6489 2758 6995 652 0194 1695 0964 0104 3.459560 3.021376 2.725695 3.366071 2.940810 2.653799 3.324100 2.904636 2.621515 14.5 4226 7827 0309 4755 5801 5457 0142 6312 9695 3.456938 3.019117 2.723679 3.365253 2.940105 2.653170 3.323612 2.904216 2.621141 15 9386 8268 301 664 7576 5408 8034 7034 1931 3.454401 3.016931 2.721728 3.364444 2.939408 2.652547 3.323128 2.903799 2.620768 15.5 9954 7116 5628 0941 037 8732 5016 2825 6538 3.451944 3.014813 2.719838 3.363642 2.938717 2.651931 3.322647 2.903384 2.620398 16 2663 8476 7216 5996 2752 4152 0738 3384 3248

114 3.449560 3.012760 2.718006 3.362849 2.938033 2.651321 3.322168 2.902971 2.620030 16.5 9138 0672 059 0197 3336 0433 486 8418 1798 3.447247 3.010766 2.716227 3.362063 2.937356 2.650716 3.321692 2.902561 2.619664 17 5317 5741 1879 1982 0777 6371 7045 7635 1929 3.445000 3.008829 2.714499 3.361284 2.936685 2.650118 3.321219 2.902154 2.619300 17.5 0951 9006 0143 9835 3766 0802 6961 0752 3388 3.442814 3.006946 2.712818 3.360514 2.936021 2.649525 3.320749 2.901748 2.618938 18 9173 8704 7044 2284 1038 2595 4286 7488 5925 3.440688 3.005114 2.711183 3.359750 2.935363 2.648938 3.320281 2.901345 2.618578 18.5 6135 5672 6562 7899 1359 065 8701 757 9295 3.438618 3.003330 2.709591 3.358994 2.934711 2.648356 3.319816 2.900945 2.618221 19 0682 3068 4747 5293 3534 3899 9891 0726 3256 3.436600 3.001591 2.708039 3.358245 2.934065 2.647780 3.319354 2.900546 2.617865 19.5 4073 6131 9507 3114 6399 1305 7549 6693 7573 3.434632 2.999896 2.706527 3.357503 2.933425 2.647209 3.318895 2.900150 2.617512 20 9743 1971 0419 0054 8824 186 1374 5209 2013 3.432713 2.998241 2.705050 3.356767 2.932791 2.646643 3.318438 2.899756 2.617160 20.5 3082 9386 8566 4835 9712 4583 1067 6019 6347 3.430839 2.996626 2.703609 3.356038 2.932163 2.646082 3.317983 2.899364 2.616811 21 1251 8697 639 6218 7992 8521 6336 8869 0351 3.429008 2.995049 2.702201 3.355316 2.931541 2.645527 3.317531 2.898975 2.616463 21.5 3014 1609 7568 2997 2627 2746 6894 3513 3803 3.427218 2.993507 2.700825 3.354600 2.930924 2.644976 3.317082 2.898587 2.616117 22 8595 1082 6895 3997 2604 6356 2458 9708 6486 3.425468 2.991999 2.699480 3.353890 2.930312 2.644430 3.316635 2.898202 2.615773 22.5 9541 1218 0188 8078 6939 8472 2751 7213 8187 3.423756 2.990523 2.698163 3.353187 2.929706 2.643889 3.316190 2.897819 2.615431 23 8613 7163 4192 4126 4674 824 75 5794 8695 3.422080 2.989079 2.696874 3.352490 2.929105 2.643353 3.315748 2.897438 2.615091 23.5 9675 5016 6506 1059 4876 4828 6435 5219 7804 3.420439 2.987665 2.695612 3.351798 2.928509 2.642821 3.315308 2.897059 2.614753 24 7608 1749 5504 7823 6637 7423 9293 526 5311 3.418831 2.986279 2.694376 3.351113 2.927918 2.642294 3.314871 2.896682 2.614417 24.5 8222 5137 0279 339 907 5237 5813 5693 1016 3.417255 2.984921 2.693164 3.350433 2.927333 2.641771 3.314436 2.896307 2.614082 25 8183 369 058 6759 1314 7499 5741 6298 4723 3.415710 2.983589 2.691975 3.349759 2.926752 2.641253 3.314003 2.895934 2.613749 25.5 4944 6599 6762 6954 2528 346 8823 6859 6237 3.414194 2.982283 2.690809 3.349091 2.926176 2.640739 3.313573 2.895563 2.613418 26 6687 3681 9738 3025 1893 2388 4814 7161 5369 3.412707 2.981001 2.689666 3.348428 2.925604 2.640229 3.313145 2.895194 2.613089 26.5 2264 5331 094 4045 8609 3571 3469 6995 1931 3.411247 2.979743 2.688543 3.347770 2.925038 2.639723 3.312719 2.894827 2.612761 27 115 248 2276 9108 1897 6313 4549 6155 5739 3.409813 2.978507 2.687440 3.347118 2.924476 2.639221 3.312295 2.894462 2.612435 27.5 3397 6557 6097 7333 0997 9935 7817 4437 6611 3.408404 2.977293 2.686357 3.346471 2.923918 2.638724 3.311874 2.894099 2.612111 28 9589 945 5166 7859 5167 3776 3043 1641 437 3.407021 2.976101 2.685293 3.345829 2.923365 2.638230 3.311454 2.893737 2.611788 28.5 0812 3476 2629 9847 3685 7191 9996 7571 8839 3.405660 2.974929 2.684247 3.345193 2.922816 2.637740 3.311037 2.893378 2.611467 29 861 1352 1986 2476 5842 9549 8452 2031 9845 3.404323 2.973776 2.683218 3.344561 2.922272 2.637255 3.310622 2.893020 2.611148 29.5 4961 6165 7069 4947 0951 0234 819 4832 7219 3.403008 2.972643 2.682207 3.343934 2.921731 2.636772 3.310209 2.892664 2.610831 30 2245 135 2021 6478 8337 8646 8992 5786 0791 3.401714 2.971528 2.681212 3.343312 2.921195 2.636294 3.309799 2.892310 2.610515 30.5 3218 0665 1272 6306 7344 4199 0642 4707 0398 3.400441 2.970430 2.680232 3.342695 2.920663 2.635819 3.309390 2.891958 2.610200 31 0986 817 9522 3685 7328 6319 293 1414 5876

115 3.399187 2.969350 2.679269 3.342082 2.920135 2.635348 3.308983 2.891607 2.609887 31.5 8987 8211 1726 7888 7663 4445 5647 5726 7066 3.397954 2.968287 2.678320 3.341474 2.919611 2.634880 3.308578 2.891258 2.609576 32 0965 5397 3075 8203 7735 8032 8588 7468 381 3.396739 2.967240 2.677385 3.340871 2.919091 2.634416 3.308176 2.890911 2.609266 32.5 0955 4589 8983 3935 6944 6544 1551 6464 5952 3.395542 2.966209 2.676465 3.340272 2.918575 2.633955 3.307775 2.890566 2.608958 33 3261 088 5071 4404 4707 9459 4337 2545 3339 3.394363 2.965192 2.675558 3.339677 2.918063 2.633498 33.5 2447 9584 7159 8944 0448 6266

Table B.6 Slopes of Implied Volatility Curves for Small Stock Prices

∂σ imp τ = 0.1:1: 0.1 ∂S

S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

------5.189177 3.700557 3.036536 2.638969 2.366804 2.165397 1.881870 1.776775 1.687745 0.5 2.008538 6734E+1 1602E+1 0519E+1 2989E+1 4473E+1 7265E+1 9149E+1 7202E+1 6831E+1 064E+11 1 1 1 1 1 1 1 1 1 ------3.059285 2.181951 1.790563 1.556214 1.395777 1.277046 1.184573 1.109897 1.047937 1 99544801 3667E+1 0081E+1 5691E+1 4493E+1 0057E+1 0292E+1 0022E+1 2492E+1 5114E+1 152 1 1 1 1 1 1 1 1 1 ------2.181471 1.556005 1.276960 1.109871 1.5 99547727 91081824 84488064 79163238 74745067 71002140 9272E+1 7232E+1 8959E+1 5654E+1 451 878 142 721 703 968 1 1 1 1 ------1.698381 1.211501 2 99427559 86419844 77514175 70923294 65789836 61644231 58204453 55290358 0811E+1 9628E+1 128 277 180 871 985 863 405 241 1 1 ------1.391806 2.5 99286402 81486469 70827435 63529645 58128639 53921890 50524628 47705751 45317650 1196E+1 507 644 477 284 510 733 104 931 112 1 ------1.179675 3 84157362 69071487 60037505 53852224 49274516 45708983 42829524 40440274 38416131 5617E+1 651 939 613 503 618 254 107 981 555 1 ------1.024059 3.5 73058489 59963482 52121561 46752375 42778615 39683463 37183856 35109776 33352630 4972E+1 273 306 486 764 259 169 060 531 546 1 ------4 90497302 64564707 52993143 46063421 41318741 37807148 35071956 32863032 31030140 29477317 476 543 305 283 958 729 076 862 182 880

------4.5 81087430 57852970 47485127 41276173 37024948 33878541 31427775 29448541 27806228 26414857 244 040 035 147 785 010 409 587 353 962

------5 73462720 52414386 43021860 37396918 33545526 30695019 28474720 26681600 25193711 23933164 576 066 936 673 285 725 151 448 878 947

------5.5 67158028 47917243 39331152 34189102 30668314 28062478 26032744 24393517 23033319 21880948 491 416 969 294 876 799 710 982 784 321

------6 61857098 44136003 36227931 31491880 28249058 25848936 23979426 22469591 21216754 20155336 769 537 163 845 067 291 888 719 672 688

------6.5 57337355 40911932 33581940 29192050 26186225 23961500 22228602 20829090 19667793 18683925 840 535 359 597 371 249 694 686 366 628

116 ------7 53437580 38130051 31298829 27207610 24406271 22332882 20717857 19413535 18331221 17414266 211 516 157 829 275 299 024 100 250 602

------7.5 50038112 35705020 29308566 25477697 22854607 20913137 19400862 18179518 17166054 16307429 671 121 723 342 051 794 220 144 960 275

------8 47048262 33572157 27558074 23956179 21489857 19664411 18242504 17094139 16141231 15333906 625 184 762 161 884 671 543 803 238 399

------8.5 44398035 31681541 26006386 22607451 20280089 18557486 17215682 16132006 15232774 14470923 379 218 606 432 660 324 126 748 359 179

------9 42032542 29994024 24621376 21403592 19200259 17569449 16299141 15273205 14421882 13700619 776 416 423 859 550 127 442 252 247 281

------9.5 39908133 28478472 23377494 20322397 18230450 16682077 15475979 14501897 13693601 13008790 398 904 636 586 859 479 336 665 329 085

------10 37989654 27109814 22254165 19345982 17354624 15880696 14732583 13805329 13035889 12383996 989 406 877 882 284 310 274 504 062 177

------10.5 36248480 25867630 21234633 18459785 16559717 15153354 14057867 13173113 12438937 11816921 446 440 749 135 439 225 158 443 910 194

------11 34661059 24735123 20305113 17651823 15834984 14490219 13442710 12596704 11894679 11299900 834 654 433 382 321 165 763 095 446 773

------11.5 33207841 23698348 19454160 16912150 15171503 13883128 12879542 12069006 11396414 10826571 803 583 450 696 269 230 152 761 948 649

------12 31872458 22745630 18672193 16232439 14561805 13325247 12362022 11584081 10938536 10391606 884 617 851 796 301 627 262 897 364 601

------12.5 30641104 21867121 17951131 15605666 13999589 12810812 11884803 11136919 10516312 99905116 278 519 485 048 789 548 610 033 858 87.2

------13 29502049 21054455 17284107 15025862 13479504 12334925 11443343 10723261 10125725 96194693 589 471 731 144 446 855 561 819 616 29.1

------13.5 28445267 20300480 16665252 14487926 12996973 11893400 11033758 10339471 97633386 92752155 645 059 748 247 235 645 294 188 54 87.6

------14 27462134 19599043 16089518 13987470 12548060 11482635 10652707 99824172 94261967 89549432 615 823 129 459 470 900 452 84.3 78 37

------14.5 26545192 18944826 15552537 13520700 12129362 11099517 10297302 96493933 91117442 86562245 640 951 962 071 509 396 232 46.9 72 23.2

------15 25687959 18333205 15050517 13084316 11737920 10741338 99650315 93380464 88177595 83769490 079 230 190 527 725 079 21.6 97.6 53 92.7

------15.5 24884772 17760139 14580141 12675438 11371151 10405733 96537020 90463216 85423022 81152737 043 874 192 844 022 572 79.1 98 54.6 26.4

------16 24130664 17222087 14138502 12291539 11026786 10090629 93613897 87724160 82836700 78695813 395 654 036 515 348 583 45.3 32.3 61.8 36.2

------16.5 23421260 16715927 13723037 11930391 10702829 97941987 90863993 85147411 80403632 76384472 286 102 919 869 485 02.9 74.4 65.2 65.1 94

117 ------17 22752689 16238897 13331483 11590026 10397514 95148247 88272317 82718923 78110554 74206116 653 528 115 578 030 13.7 21.8 17.2 66.5 05.3

------17.5 22121517 15788548 12961826 11268695 10109271 92510729 85825559 80426224 75945692 72149557 143 318 379 495 968 12.5 19.5 73.9 29.3 77.9

------18 21524682 15362696 12612276 10964841 98367065 90016653 83511864 78258208 73898557 70204834 697 563 160 425 58.6 07 66.2 27.5 21.9 85.5

------18.5 20959451 14959391 12281231 10677072 95785695 87654597 81320638 76204946 71959776 68363043 636 448 387 732 54.3 72.5 93.3 60.2 13.2 19.6

------19 20423372 14576884 11967256 10404141 93337419 85414324 79242384 74257540 70120946 66616200 495 182 789 886 66 49.9 37.7 46.8 28.5 68.3

------19.5 19914241 14213602 11669061 10144927 91012177 83286628 77268561 72407988 68374512 64957130 263 489 956 269 40 13.8 90.1 75.8 95 76.1

------20 19430070 13868128 11385483 98984176 88800898 81263209 75391472 70649077 66713664 63379363 909 868 505 70.1 51.3 46.7 91.2 89.4 99 23.6

------20.5 18969065 13539181 11115469 96636990 86695384 79336566 73604158 68974287 65132245 61877049 297 994 810 15.1 02.8 50.8 31.1 19.5 50.4 39.6

------21 18529596 13225600 10858067 94399429 84688204 77499900 71900311 67377707 63624675 60444889 789 733 888 68.8 00.7 93.4 72.3 57.5 06 08.4

------21.5 18110186 12926330 10612412 92263971 82772609 75747037 70274204 65853971 62185885 59078067 922 367 093 05.4 38.7 39.7 66.2 42.7 48.8 81.3

------22 17709489 12640410 10377714 90223764 80942455 74072353 68720621 64398191 60811262 57772202 691 666 322 01.1 67.1 71.5 77.4 94.2 59.4 34.3

------22.5 17326277 12366965 10153255 88272558 79192136 72470720 67234804 63005910 59496596 56523293 027 528 520 46.8 66.6 24.6 59.6 41.5 39.9 41.3

------23 16959426 12105193 99383782 86404640 77516526 70937447 65812402 61673050 58238037 55327684 149 960 76.3 11.7 75.7 06.7 56 33.8 79.6 57

------23.5 16607908 11854362 97324803 84614774 75910930 69468237 64449430 60395877 57032060 54182026 510 191 47.3 18.7 45.4 80.8 17.2 27.2 66.5 17.6

------24 16270780 11613796 95350089 82898156 74371038 68059149 63142229 59170963 55875428 53083243 107 761 88.8 13.2 16.5 19.2 47.6 68.2 66.2 86.7

------24.5 15947172 11382878 93454559 81250368 72892887 66706555 61887437 57995158 54765166 52028510 962 442 67.3 29.7 32.1 46.5 05.8 01.8 03.6 82.3

------25 15636287 11161036 91633531 79667341 71472828 65407117 60681954 56865557 53698531 51015223 615 887 65.2 71.1 11.6 04.1 96.5 46.6 89 32.5

------25.5 15337386 10947745 89882686 78145322 70107493 64157752 59522925 55779483 52672997 50040979 488 891 96.2 32.4 21.5 79.7 00.2 86.6 45 15.5

------26 15049788 10742519 88198034 76680841 68793770 62955615 58407705 54734462 51686225 49103558 008 206 64.5 10.2 93.1 52.4 96 37.2 91.2 46

------26.5 14772861 10544906 86575881 75270687 67528781 61798070 57333853 53728202 50736054 48200906 389 809 07.8 45.5 39.9 12.4 39.4 50.6 55.6 75.4

118 ------27 14506021 10354491 85012802 73911885 66309855 60682674 56299101 52758581 49820478 47331119 978 593 77.8 59.6 36.2 20.9 62 29.1 81.7 79.1

------27.5 14248727 10170886 83505622 72601673 65134515 59607160 55301347 51823628 48937638 46492430 118 405 12 44.3 25.4 78.1 65.6 20.2 07.7 06.6

------28 14000472 99937314 82051385 71337483 64000458 58569422 54338636 50921511 48085802 45683194 436 07.2 65.1 76.4 27.8 76 86.3 71 96.2 75

------28.5 13760788 98226917 80647344 70116927 62905541 57567499 53409150 50050527 47263363 44901884 523 04.7 63.9 27.2 25.6 11.3 08.4 23.5 96.2 87.7

------29 13529237 96574552 79290937 68937777 61847767 56599562 52511192 49209086 46468821 44147075 958 20.7 63.4 46.7 02.4 41.1 09.3 32.1 21.7 64.3

------29.5 13305412 94977307 77979774 67797956 60825272 55663907 51643181 48395706 45700775 43417437 622 84.1 76.4 98.8 19.1 62.5 21.5 90.9 32.5 67.5

------30 13088931 93432464 76711623 66695525 59836316 54758942 50803639 47609004 44957919 42711729 286 06.6 60.3 32.4 12.2 04.9 96.4 56.6 06.6 16.6

------30.5 12879437 91937477 75484396 65628667 58879270 53883176 49991186 46847684 44239029 42028788 429 27.1 39.8 72.2 97.8 18.6 59.8 57.2 92.9 78.2

------31 12676597 90489966 74296138 64595685 57952612 53035215 49204527 46110534 43542963 41367529 269 05.8 53.3 15 72.5 51.8 48.1 76.3 35.2 23.4

------31.5 12480097 89087698 73145018 63594985 57054912 52213753 48442450 45396419 42868646 40726931 974 50.2 07.3 21.2 97.8 04.8 16.6 05.2 60.8 47.1

------32 12289646 87728580 72029316 62625073 56184831 51417562 47703817 44704271 42215073 40106039 046 59.7 28.3 90.8 61.2 53.6 12 54.1 24.4 40.4

------32.5 12104965 86410645 70947419 61684548 55341110 50645492 46987560 44033091 41581298 39503955 846 75.7 00.4 14 04 35 01.8 18.8 04.4 11.3

------33 11925798 85132045 69897808 60772088 54522565 49896459 46292674 43381936 40966432 38919834 257 26.2 82.1 90 10.1 87.4 55.8 93.8 43.4 48.4

------33.5 11751899 83891039 68879057 59886455 53728083 49169446 45618215 42749923 40369640 38352883 463 55.2 91.2 05.5 48.9 40.8 73.3 29.5 29.9 24

------34 11583039 82685990 67889821 59026477 52956616 48463492 44963293 42136216 39790134 37802353 835 28.1 52.8 68.4 67 52 51.2 27.6 16.6 31

------34.5 11419002 81515353 66928832 58191054 52207176 47777693 44327068 41540029 39227171 37267539 912 05.6 03.3 89.3 23.5 79.1 89.7 72.1 71.4 51.5

------35 11259584 80377670 65994893 57379147 51478829 47111196 43708750 40960621 38680052 36747776 472 79.2 45.1 07.6 64.4 92.6 31.8 90.6 58.2 53.6

------35.5 11104591 79271567 65086876 56589772 50770696 46463196 43107590 40397292 38148115 36242436 679 62.5 46.9 57.5 33.7 18.9 32 44.5 44.3 13.5

------36 10953842 78195743 64203713 55822003 50081944 45832930 42522882 39849379 37630735 35750924 294 32.5 85.4 69.2 15.5 13.2 53.4 45.1 31.7 60.4

------36.5 10807163 77148968 63344396 55074963 49411786 45219678 41953958 39316256 37127321 35272680 965 16.6 23.8 02.7 05.9 59.4 88.9 92.2 11 42.2

119 ------37 10664393 76130078 62507968 54347820 48759477 44622759 41400187 38797332 36637313 34807172 557 20.7 25 11.6 10.9 93.2 04.9 34 34.7 10.4

------37.5 10525376 75137970 61693524 53639788 48124311 44041528 40860968 38292044 36160182 34353896 549 94.9 94.9 33.2 68.5 48.6 05.3 46.1 07.7 21.7

------38 10389966 74171601 60900208 52950122 47505620 43475372 40335734 37799861 35695424 33912375 472 33 53.5 02.3 92.5 22.4 14 27.5 95.8 55.1

------38.5 10258024 73229978 60127205 52278113 46902770 42923710 39823946 37320278 35242565 33482157 385 02.7 31.9 88 36 58.9 74.2 12.7 47.5 43.1

------39 10129418 72312160 59373742 51623092 46315157 42385992 39325094 36852815 34801151 33062812 407 05.1 92.2 50.3 72.9 49.3 63.2 98.1 30.3 16.1

------39.5 10004023 71417253 58639087 50984420 45742210 41861694 38838692 36397019 34370752 32653931 264 57.9 68.1 14.6 96 47.2 20.6 81 78.8 58.5

------40 98817198 70544409 57922542 50361490 45183386 41350318 38364277 35952457 33950961 32255127 86.4 04.4 25.5 63.3 30.1 97.4 90.1 11.4 54.4 74.6

------40.5 97623950 69692818 57223443 49753727 44638166 40851392 37901412 35518716 33541389 31866031 27.8 41.8 40.1 42.9 58.7 77.4 71.8 53.5 15.2 66

------41 96459409 68861712 56541159 49160581 44106059 40364465 37449678 35095406 33141665 31486292 12 71.1 91.3 84.9 63.2 50.7 86.4 58.9 95.5 16.2

------41.5 95322549 68050359 55875090 48581531 43586596 39889108 37008678 34682154 32751439 31115574 07.2 63.1 71.2 39.1 74.1 29.8 48.4 47 93.6 84.4

------42 94212392 67258061 55224663 48016078 43079331 39424912 36578032 34278604 32370375 30753561 22.6 42.6 06.4 19.6 32.1 49.9 48.6 95.5 67.5 07

------42.5 93128006 66484152 54589330 47463747 42583837 38971488 36157379 33884419 31998153 30399947 25.5 86.6 93.2 61.1 59.3 50.8 44.3 37.4 38.9 05.1

------43 92068501 65727999 53968573 46924086 42099709 38528464 35746374 33499274 31634468 30054442 79.1 37 43.5 84.5 39.3 66.8 57.9 66.1 02.1 99.5

------43.5 91033029 64988995 53361893 46396663 41626559 38095486 35344688 33122862 31279028 29716772 97.6 25.9 41.3 73.4 05.7 25 82.1 45.9 41.1 30.3

------44 90020780 64266562 52768816 45881065 41164016 37672214 34952007 32754888 30931556 29386670 17.8 12.7 09.5 56.9 37.3 49.7 91.1 29.8 50.3 83.3

------44.5 89030977 63560147 52188887 45376898 40711727 37258325 34568031 32395070 30591786 29063886 86.2 32.6 84.9 01.8 61.2 73.3 58.1 81.6 62.5 20.1

------45 88062882 62869222 51621675 44883784 40269354 36853510 34192472 32043141 30259464 28748177 60.4 54 01.5 11.3 62.3 53.4 78.6 03.9 79.8 13.6

------45.5 87115786 62193282 51066762 44401363 39836573 36457472 33825056 31698841 29934348 28439312 22.9 46.8 82.5 30.4 98 95.3 98.1 70.8 10.8 87.1

------46 86189011 61531843 50523754 43929290 39413076 36069929 33465521 31361926 29616204 28137072 07.4 57.4 37.6 57 18.9 79 44.6 64.4 10 58.4

------46.5 85281908 60884442 49992269 43467235 38998564 35690609 33113614 31032160 29304810 27841244 36.3 93.4 68.1 59.9 94.7 91.4 66.1 16.2 22.9 86

120 ------47 84393856 60250637 49471944 43014882 38592756 35319253 32769095 30709316 28999953 27551627 67.9 14.5 77.9 00.5 44.5 62.9 70.7 51.2 33.2 19.9

------47.5 83524260 59630001 48962430 42571926 38195378 34955612 32431733 30393179 28701429 27268025 54 30.7 89.5 60.3 71.6 07.1 72.1 36.5 13.8 55

------48 82672549 59022128 48463393 42138078 37806171 34599446 32101307 30083541 28409041 26990253 05.4 06.7 65.6 72.9 02.5 65.5 37 32.2 81.5 87.5

------48.5 81838174 58426626 47974512 41713059 37424883 34250528 31777604 29780203 28122603 26718133 66.7 72.2 35.6 59.9 28.9 54.7 36.8 46.2 52.9 73.7

------49 81020611 57843122 47495479 41296601 37051275 33908638 31460421 29482974 27841934 26451493 97.8 37.5 26.4 70.6 54.6 16.7 01.5 91 04.8 91.9

------49.5 80219356 57271255 47025998 40888448 36685117 33573564 31149561 29191672 27566860 26190170 63.1 14.4 97.2 25.7 44.2 73.2 77.1 43.5 35.5 05.9

------50 79433924 56710679 46565787 40488352 36326187 33245105 30844838 28906120 27297216 25934004 27 41.9 78.6 63.9 75.6 81.4 84.6 07.1 29.5 31.2

------50.5 78663849 56161063 46114573 40096077 35974273 32923066 30546071 28626148 27032842 25682845 55.7 15.6 15.1 92.3 95.3 93.2 82.3 75.8 23.2 02.9

------51 77908685 55622087 45672093 39711396 35629171 32607261 30253087 28351596 26773584 25436546 24.5 22.4 10.5 39.1 75.9 16.6 29.8 01.1 74 45.6

------51.5 77168001 55093444 45238095 39334089 35290684 32297508 29965718 28082305 26519296 25194968 31.2 77.6 77.6 08.9 76.5 79.2 54.5 59.6 29.5 45

------52 76441384 54574840 44812338 38963945 34958624 31993636 29683805 27818127 26269835 24957976 13.1 67 89.2 41.6 05.1 93.9 19.3 24.1 00.1 21.1

------52.5 75728435 54065990 44394589 38600762 34632807 31695479 29407192 27558916 26025064 24725440 70.5 90.9 33.7 72.3 83.3 26.4 92.8 34.5 32 03.2

------53 75028772 53566622 43984622 38244345 34313061 31402875 29135733 27304533 25784852 24497235 92.6 12.4 71.2 94.4 12.8 64.9 21 72 82.2 05.6

------53.5 74342026 53076471 43582222 37894507 33999215 31115671 28869283 27054845 25549073 24273241 89 07.9 94 24.4 43.8 91.1 00.4 33.5 94.8 05.4

------54 73667842 52595284 43187181 37551065 33691108 30833719 28607704 26809722 25317605 24053342 24.4 20.8 87.9 68.4 45.6 52.7 52.9 08.2 78.7 21.5

------54.5 73005876 52122817 42799298 37213846 33388583 30556875 28350865 26569039 25090330 23837426 56.6 17.1 96 91.2 77.7 38.1 01.7 55.1 86.5 93.7

------55 72355799 51658834 42418380 36882682 33091490 30285001 28098636 26332677 24867135 23625387 78.5 44 84.7 86.3 63.9 51.6 48.6 81.7 94.3 64.7

------55.5 71717293 51203108 42044241 36557411 32799683 30017964 27850895 26100521 24647911 23417120 62.8 90.5 11.7 47.6 66.6 90.7 52.9 24.2 82.7 61.3

------56 71090051 50755421 41676699 36237876 32513022 29755637 27607523 25872458 24432553 23212525 09.4 50 94.4 43.3 63.3 23.8 11.1 28.2 19.3 77.7

------56.5 70473775 50315560 41315583 35923926 32231372 29497894 27368404 25648381 24220958 23011506 96.4 85.1 82.2 90.2 23.7 69.9 37.2 31 41.3 59.5

121 ------57 69868182 49883322 40960725 35615417 31954601 29244617 27133428 25428186 24013029 22813969 32.8 94 27.7 30 88.1 78.8 45.2 44.3 39.4 88.2

------57.5 69272994 49458510 40611962 35312207 31682585 28995691 26902488 25211773 23808671 22619825 14 78.9 61.8 06.7 47.7 12.6 31.5 38.3 43 66.5

------58 68687944 49040934 40269139 35014160 31415201 28751003 26675480 24999045 23607793 22428987 79.8 15.6 68.2 45.2 24.8 28.1 58.4 26.2 05.2 05.3

------58.5 68112776 48630409 39932105 34721146 31152331 28510446 26452305 24789908 23410305 22241370 74.2 25.2 60.2 30.4 54.6 59.7 39 49.5 89.2 09.6

------59 67547241 48226758 39600714 34433037 30893862 28273917 26232866 24584272 23216124 22056893 07.4 46.8 58.4 88.5 67.9 04.1 22.1 64.5 55.5 67

------59.5 66991097 47829810 39274825 34149712 30639684 28041314 26017069 24382050 23025166 21875479 19.5 11.7 69.7 68.2 74.8 04.6 78.1 29.1 49.2 35.5

------60 66444112 47439398 38954302 33871052 30389691 27812540 25804825 24183156 22837351 21697051 46.4 19.2 66.8 23.3 48.5 37.2 86.2 90.2 88.3 32.4

------60.5 65906061 47055362 38639013 33596941 30143780 27587501 25596047 23987510 22652603 21521536 87.1 12.7 69.5 96.2 10.8 96.7 21.2 71.8 52.7 23.6

------61 65376727 46677546 38328831 33327271 29901851 27366107 25390649 23795032 22470846 21348863 72.6 58.1 26.2 01.8 17.9 84 41.7 64 73.3 13.5

------61.5 64855899 46305801 38023631 33061932 29663808 27148269 25188550 23605646 22292009 21178963 36.2 22.7 96.9 12.8 46.7 93.2 78.9 11.9 21.9 35.5

------62 64343372 45939980 37723296 32800821 29429558 26933903 24989672 23419277 22116021 21011770 85.5 54.6 36.4 45.2 82.2 00.7 25 05.7 01.9 42.8

------62.5 63838950 45579943 37427708 32543838 29199012 26722924 24793937 23235853 21942814 20847219 76 64.1 78.9 44.6 05.1 53 23.6 70.8 38.3 99.4

------63 63342441 45225554 37136757 32290885 28972080 26515254 24601271 23055306 21772323 20685249 84.7 05.5 22.6 73.4 80.1 56.9 59.3 58.6 70.1 72.3

------63.5 62853660 44876679 36850333 32041868 28748680 26310815 24411603 22877568 21604485 20525799 86.7 59.4 16.2 98.3 44.9 69 48.5 37.6 40.9 23

------64 62372428 44533192 36568331 31796696 28528728 26109532 24224863 22702573 21439237 20368810 31.5 16.2 44.6 78.4 99.8 85.7 30.3 85.1 91.4 00.3

------64.5 61898570 44194967 36290650 31555280 28312146 25911333 24040983 22530259 21276521 20214225 20.7 60.8 16.4 54.2 97.1 34.6 57.8 78.9 52.1 33.1

------65 61431917 43861885 36017190 31317534 28098857 25716146 23859898 22360564 21116278 20061990 87.6 56.9 51.4 36.6 31.9 65.1 90.5 90.1 35.4 23.4

------65.5 60972307 43533829 35747856 31083374 27888785 25523904 23681545 22193429 20958452 19912051 76.5 33.1 68.4 96.8 32.8 40.4 85.6 75.2 29.3 39.7

------66 60519581 43210685 35482555 30852721 27681858 25334540 23505862 22028796 20802988 19764357 23.5 68.9 75 56.4 53 29.3 91.5 69.6 90.4 11

------66.5 60073584 42892344 35221197 30625495 27478006 25147989 23332790 21866609 20649835 19618857 38.3 81.9 55.5 78.3 62 98.2 39.6 80.7 37.9 20.4

122 ------67 59634167 42578700 34963694 30401621 27277161 24964191 23162270 21706814 20498940 19475502 86.7 14.7 61.8 57.3 37.6 04.4 38.8 81.6 47.2 99.8

------67.5 59201186 42269648 34709962 30181025 27079256 24783082 22994246 21549359 20350254 19334247 73.5 23.7 02.7 12.1 58.1 88.8 67.7 04.9 44.6 24

------68 58774500 41965088 34459917 29963634 26884227 24604606 22828664 21394191 20203729 19195044 26.9 67 35 76.7 95.3 68.8 69.3 36.9 01.6 06

------68.5 58353971 41664923 34213480 29749380 26692013 24428705 22665471 21241262 20059317 19057848 82.9 93.9 54.4 92.7 07 32.7 44.7 12.3 29.3 91.5

------69 57939468 41369059 33970573 29538196 26502551 24255323 22504615 21090523 19916973 18922618 70.4 34 86.9 02.1 30.8 32.7 47.2 08.4 74 54.5

------69.5 57530861 41077402 33731121 29330014 26315783 24084406 22346046 20941927 19776654 18789310 97.8 87.6 80.5 39.7 77.4 79.9 77.4 40.3 11.9 92.3

------70 57128026 40789865 33495050 29124772 26131653 23915903 22189716 20795429 19638315 18657885 38.8 15.9 97.4 26.6 24.5 37.8 77.6 56.1 44.7 21.7

------70.5 56730840 40506359 33262290 28922407 25950104 23749762 22035578 20650985 19501915 18528301 20 31.8 06.4 63.6 10.8 17.5 26.6 31.9 95.1 74.3

------71 56339185 40226800 33032769 28722860 25771082 23585933 21883585 20508551 19367415 18400521 08.5 91 75.7 24.7 30.7 72.6 35.6 67.5 02.6 92.5

------71.5 55952945 39951107 32806422 28526071 25594535 23424369 21733693 20368086 19234773 18274508 99.9 83.9 66 51.2 28.6 93.6 43 82.1 19.5 26

------72 55572011 39679200 32583183 28331984 25420411 23265024 21585859 20229550 19103952 18150224 07.1 27.1 23.8 46.1 93.8 04 10.2 10.3 06.8 27.9

------72.5 55196271 39411000 32362987 28140543 25248662 23107850 21440040 20092901 18974914 18027634 49.4 56 75.1 68.3 55.5 55.2 17.5 97.8 30.7 51.1

------73 54825621 39146433 32145774 27951695 25079238 22952805 21296195 19958103 18847623 17906704 42 17 19.3 27.4 78.3 22.2 59.8 98.1 59 45

------73.5 54459957 38885424 31931482 27765386 24912093 22799844 21154285 19825118 18722044 17787400 86.2 60.7 23.2 78.6 57.1 99.8 43.1 68.4 57.6 52.2

------74 54099180 38627903 31720053 27581567 24747181 22648927 21014270 19693909 18598142 17669690 59.5 34.9 15.6 17.9 13.3 98.1 80.4 66.3 87 05.2

------74.5 53743192 38373799 31511429 27400186 24584456 22500013 20876113 19564441 18475884 17553541 06.8 77.9 81.7 77.3 90.2 38.9 88.4 46.6 99.6 23.8

------75 53391897 38123046 31305556 27221197 24423877 22353061 20739777 19436679 18355238 17438923 31 12.4 57.9 20.3 49.2 52 83.7 57.9 36.4 12

------75.5 53045203 37875576 31102379 27044551 24265400 22208033 20605226 19310590 18236171 17325805 84.9 39.5 27.3 37.6 65.6 71.6 80.2 39.6 24 55.1

------76 52703021 37631326 30901845 26870203 24108985 22064892 20472425 19186141 18118652 17214159 63 32.6 14 42.9 25.3 32.8 85 18.7 71.9 17.1

------76.5 52365262 37390233 30703902 26698108 23954591 21923600 20341340 19063300 18002652 17103955 93.7 31.7 79.5 69.1 20.8 68.6 96.4 07.5 70 38.3

123 ------77 52031842 37152236 30508502 26528223 23802179 21784123 20211938 18942036 17888141 16995166 31.3 38.7 17.6 64.1 48.1 06.4 99.8 00.2 85.7 32.8

------77.5 51702676 36917276 30315594 26360505 23651712 21646424 20084187 18822318 17775091 16887764 49.5 11.3 50.5 87.5 03.1 65.3 66.1 70.7 61.3 86

------78 51377684 36685294 30125132 26194914 23503151 21510471 19958055 18704118 17663474 16781724 33.6 58.9 24.3 06.6 78.7 52.8 48.1 70 12.3 52.4

------78.5 51056786 36456235 29937069 26031407 23356462 21376230 19833511 18587407 17553262 16677019 74.5 37.2 05.8 93.7 61.3 62.5 78.2 23.4 24.3 53.4

------79 50739906 36230043 29751359 25869948 23211609 21243669 19710526 18472156 17444429 16573624 61.9 43.8 77.9 21.8 28.5 70.9 66 28.9 51.3 75.3

------79.5 50426968 36006665 29567960 25710496 23068557 21112757 19589070 18358338 17336950 16471515 78.1 13.7 36.6 62.3 45.7 35 95.8 54.1 13.2 67

------80 50117899 35786048 29386827 25553015 22927273 20983462 19469116 18245927 17230798 16370668 92 15.2 87 81.8 63.5 89.9 24.3 34.7 94.2 38.6

------80.5 49812628 35568141 29207920 25397469 22787725 20855756 19350634 18134896 17125951 16271059 53.3 45.7 40.4 38.6 15.2 46.5 78.1 72 40.4 58.7

------81 49511084 35352895 29031197 25243821 22649880 20729608 19233599 18025221 17022383 16172666 87.4 27.7 10.7 80.5 14.2 88.5 52.2 31 58.2 53.5

------81.5 49213200 35140261 28856618 25092038 22513707 20604991 19117984 17916876 16920072 16075467 89.4 04.8 11.2 42 51.6 71.2 07.3 38.5 12 04.2

------82 48918910 34930191 28684144 24942085 22379176 20481877 19003762 17809837 16818994 15979439 19.2 38.6 52.1 41.4 93.7 18.5 67.9 81.1 23.1 46

------82.5 48628147 34722640 28513738 24793929 22246258 20360238 18890910 17704082 16719127 15884562 96.8 04.5 37 78.2 79.9 21.2 20.7 03.8 67.6 66.2

------83 48340850 34517561 28345362 24647539 22114924 20240048 18779402 17599586 16620450 15790816 97.1 88.6 60.4 31.3 20.4 34.8 12.4 07.6 74.7 02.5

------83.5 48056957 34314912 28178981 24502882 21985144 20121281 18669214 17496327 16522942 15698179 45.5 84.6 05 56 94.2 77.9 48.2 48.4 25.3 41.8

------84 47776407 34114649 28014558 24359928 21856893 20003913 18560323 17394284 16426581 15606633 13.5 90.1 39.1 81.9 46.9 30.1 89.9 35.1 50.3 18.5

------84.5 47499141 33916731 27852060 24218648 21730142 19887918 18452707 17293435 16331348 15516158 14.4 04 13.7 10.9 89 30.1 54 28 29.5 13.3

------85 47225101 33721115 27691452 24079011 21604866 19773272 18346343 17193759 16237222 15426735 98.9 23.4 60.9 14.8 93.7 74.3 10.8 37.5 89.5 51.7

------85.5 46954233 33527762 27532702 23940989 21481039 19659953 18241208 17095236 16144186 15338347 51.3 40.7 90.6 33.7 95.3 14.7 81.9 22.6 03.2 02.6

------86 46686480 33336633 27375778 23804554 21358636 19547936 18137283 16997845 16052218 15250974 85.7 40.8 88.9 73.4 87.3 57.9 39.7 89.3 87.7 77.4

------86.5 46421790 33147689 27220649 23669680 21237633 19437200 18034546 16901568 15961303 15164601 42.2 98.6 15.6 04.2 20.8 62.8 05.1 89.5 03.3 28.6

124 ------87 46160109 32960894 27067283 23536338 21118005 19327723 17932976 16806386 15871420 15079209 83.3 76.3 02.4 58.5 02.8 39.7 46.7 19.5 52.6 48.3

------87.5 45901387 32776211 26915650 23404504 20999728 19219483 17832554 16712279 15782553 14994782 90.4 21 50.5 29.5 94.8 48.4 79 18.9 78.7 67.6

------88 45645574 32593603 26765722 23274151 20882782 19112459 17733261 16619229 15694685 14911304 60.4 62.2 28.8 68.8 10.7 97.2 61.4 69.2 64.4 55.3

------88.5 45392621 32413037 26617469 23145255 20767142 19006632 17635077 16527219 15607799 14828759 02.6 09.6 72.1 85.8 16.1 41.2 97 92.7 31.3 16.7

------89 45142479 32234477 26470864 23017792 20652787 18901980 17537985 16436232 15521878 14747130 35.5 50.8 78.9 45 26.2 81.4 31.1 51.5 38.2 92.7

------89.5 44895102 32057891 26325880 22891737 20539696 18798485 17441965 16346250 15436906 14666404 83.7 49.1 10.2 65.2 04.9 62.8 50.1 46.1 80.3 58.9

------90 44650445 31883246 26182488 22767068 20427847 18696127 17347000 16257257 15352868 14586565 75 41.6 87.4 17.6 63 73.9 80.7 14.8 88.4 24.4

------90.5 44408463 31710510 26040664 22643761 20317221 18594888 17253073 16169236 15269749 14507598 37.6 36.8 90.4 24.8 57.5 44.9 88.1 32.1 27.7 31.1

------91 44169111 31539652 25900382 22521794 20207797 18494749 17160167 16082172 15187532 14429489 97.3 13.2 56.7 58.8 89.8 46.9 75.8 08.1 96.9 52.9

------91.5 43932348 31370641 25761616 22401146 20099557 18395692 17068265 15996048 15106205 14352224 74.9 16.8 79.3 40.2 04.5 90.9 83.9 87.3 27.1 94.2

------92 43698131 31203447 25624343 22281795 19992479 18297701 16977351 15910851 15025751 14275790 83.8 59.7 05.1 36.6 88.9 26.1 88.3 47.9 81.4 90.1

------92.5 43466420 31038042 25488537 22163720 19886547 18200757 16887410 15826565 14946158 14200174 27.1 18 34.1 61.3 71 39.8 00 00.9 53.7 04.5

------93 43237173 30874396 25354176 22046901 19781742 18104844 16798424 15743174 14867411 14125361 95.5 30.5 17.3 72.3 18.8 55.5 63.8 88.9 67.7 30.3

------93.5 43010353 30712481 25221236 21931318 19678045 18009946 16710380 15660666 14789497 14051339 64.9 96.8 55.7 70.9 39.5 32.5 57.5 85.4 76.7 87.9

------94 42785920 30552271 25089695 21816952 19575439 17916046 16623262 15579026 14712403 13978097 94.2 75.4 98.8 00.8 78 64.8 91.1 94.4 62.2 24.7

------94.5 42563838 30393738 24959532 21703782 19473908 17823129 16537057 15498241 14636116 13905621 23.1 82.7 43.6 46.7 16 80 06 48.9 33.4 14.5

------95 42344068 30236856 24830724 21591791 19373433 17731180 16451748 15418297 14560623 13833899 69.8 91.3 33.1 33.4 71.4 38.9 74.1 10.7 26.8 57

------95.5 42126576 30081600 24703250 21480960 19273999 17640183 16367323 15339180 14485912 13762920 29.3 28.2 55 24.6 96.9 33.9 96.9 69.3 04.8 76.4

------96 41911325 29927943 24577090 21371271 19175590 17550123 16283769 15260879 14411970 13692673 71.1 73.8 41 22.1 79.4 89 05 41.4 55.7 21.7

------96.5 41698282 29775862 24452223 21262706 19078190 17460987 16201070 15183380 14338786 13623145 37.4 60.4 65.3 64.9 38.9 58.3 57.1 70.1 92.6 65.1

125 ------97 41487412 29625332 24328630 21155249 18981783 17372760 16119215 15106672 14266349 13554327 41.4 70.9 43.4 27.8 27.8 25.7 39.3 24.3 52.9 02.4

------97.5 41278682 29476330 24206291 21048882 18886354 17285428 16038190 15030741 14194646 13486206 65.1 37.3 31.5 20.8 30.2 03.8 64.6 97.9 97.9 51.6

------98 41072060 29328832 24085187 20943588 18791888 17198977 15957983 14955578 14123668 13418773 57.9 39.7 25 88.4 60.7 33.5 72 09.4 11.7 52.7

------98.5 40867514 29182816 23965299 20839353 18698371 17113394 15878582 14881169 14053402 13352017 34.8 05.2 58.1 08.1 63.9 83 26.1 00.8 01 67.1

------99 40665012 29038259 23846610 20736158 18605789 17028667 15799974 14807503 13983837 13285928 74.7 06.1 01.9 90.4 13.9 47.2 16.2 37.7 94.6 77.1

------99.5 40464525 28895139 23729100 20633990 18514127 16944782 15722147 14734570 13914965 13220496 18.9 59.5 64.5 77.2 12.9 47.3 55.6 08.3 42.5 85.1

S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8

------1.021044 0.5 87954343 78464029 22904848 19739419 17615455 11718498 10099932 90137952 9798E+1 044 934 89.1 52.6 04.3 98.4 57.9 6.98 1 ------1 60234023 51889446 46292532 22580946 19460317 17366408 11634168 10027259 89489442 483 183 891 57.4 02.2 70 07.5 58.1 1.68

------1.5 42968495 37017182 33025342 22266157 19189066 17124368 11551053 99556346 88850283 088 609 547 66.9 95.5 85.7 39.1 0.98 4.47

------2 33463371 28829357 25721008 21960101 18925341 16889043 11469128 98850350 88220275 537 815 566 82.4 61.1 11.5 73.3 9.14 1.06

------2.5 27429706 23631763 21084167 21662419 18668831 16660155 11388368 98154390 87599221 011 463 010 58.8 28.8 14.1 65.1 9.56 4.62

------3 23253883 20034505 17874959 21372770 18419243 16437443 11308748 97468253 86986932 969 429 359 99.6 07 58.3 41.6 1.44 3.85

------3.5 20189995 17395077 15520228 21090834 18176299 16220661 11230243 96791730 86383223 448 944 121 26.7 70.8 06.7 99.5 3.96 0.95

------4 17844955 15374882 13717917 20816304 17939738 16009573 11152832 96124621 85787913 144 824 639 60.1 58.5 27 02.8 4.18 9.78

------4.5 15991709 13778336 12293550 20548893 17709310 15803958 11076489 95466730 85200830 338 397 358 07.3 75.9 07.7 80.2 5.06 4.07

------5 14489855 12484494 11139232 20288325 17484780 15603604 11001195 94817867 84621802 094 317 447 61.2 09 78.8 23.1 3.48 5.66

------5.5 13247864 11414512 10184626 20034342 17265922 15408313 10926926 94177846 84050665 006 317 756 05.8 41.9 40 83.5 8.45 2.91

------6 12203498 10514777 93819043 19786695 17052524 15217893 10853663 93546488 83487257 320 104 78.6 28.9 82.4 93.2 71.5 9.3 9.08

------6.5 11312951 97475522 86974000 19545150 16844384 15032165 10781385 92923618 82931424 579 56.5 82.5 41.6 92.1 80.4 53.8 3.99 0.82

126 ------7 10544490 90855011 81067262 19309484 16641310 14850957 10710072 92309064 82383011 198 20.8 67.4 03.2 21.8 26.6 51.8 7.49 6.72

------7.5 98745590 85083321 75917803 19079483 16443117 14674104 10639705 91702662 81841872 40.3 42.6 07.7 51.8 52.2 86.2 39.4 0.18 5.9

------8 92853040 80006648 71388407 18854946 16249632 14501452 10570265 91104248 81307862 37.1 37.5 96.1 40.4 37.7 93.3 41.6 6.4 6.69

------8.5 87629444 75506276 67373166 18635679 16060688 14332853 10501734 90513667 80780841 26.5 11.2 60.3 76.2 55.6 16.1 32.6 2.93 5.31

------9 82966728 71489103 63789019 18421499 15876127 14168164 10434094 89930764 80260672 88.1 14.4 44 65.9 57.6 13.8 34.5 7.68 4.66

------9.5 78778939 67881082 60569905 18212230 15695798 14007250 10367328 89355391 79747222 99.2 29.3 72.2 63 25.6 97.2 15.2 8.3 3.11

------10 74996854 64622579 57662623 18007705 15519556 13849984 10301418 88787403 79240361 65.2 56.9 10.9 20.3 29.5 90.8 87.5 0.93 3.36

------10.5 71564095 61665027 55023841 17807763 15347263 13696242 10236350 88226656 78739963 99.7 32.2 89.3 44.3 89.1 99.1 07.5 8.94 1.35

------11 68434283 58968469 52617914 17612252 15178789 13545907 10172105 87673015 78245904 19.4 09.3 71.2 53.6 37.6 73.8 73 1.76 5.19

------11.5 65568908 56499730 50415244 17421026 15014006 13398866 10108670 87126343 77758065 34 67.7 92 39.5 87.6 83.7 22.3 3.75 4.15

------12 62935732 54231038 48391054 17233945 14852796 13255012 10046028 86586510 77276328 71.1 57.8 06.5 29.5 00.1 86.8 33 3.09 7.67

------12.5 60507559 52138963 46524439 17050875 14695041 13114243 99841652 86053388 76800580 90 12 01.5 52.7 54.4 03.5 0.68 0.69 4.43

------13 58261286 50203600 44797642 16871689 14540633 12976458 99230663 85526851 76330709 36.1 53.8 27 08.1 20.8 92.5 7.88 9.25 1.43

------13.5 56177158 48407933 43195481 16696263 14389465 12841566 98627177 85006780 75866606 82 25.7 25.3 34.5 34.5 27.2 2.74 2.2 3.12

------14 54238187 46737324 41704897 16524480 14241436 12709474 98031054 84493054 75408166 77.7 65.7 55.8 82.1 71.6 74.2 8.05 2.81 0.57

------14.5 52429680 45179116 40314597 16356228 14096450 12580097 97442162 83985558 74955285 11.7 42.2 74 86 26.1 73.2 0.12 3.26 0.64

------15 50738863 43722304 39014764 16191399 13954412 12453352 96860367 83484179 74507862 51.1 97.8 64.5 42 88.5 17.7 7.84 3.77 5.21

------15.5 49154582 42357279 37796823 16029888 13815235 12329158 96285544 82988807 74065800 24.2 45.2 63.3 82.7 25.9 37 1.63 1.74 0.42

------16 47667049 41075607 36653251 15871597 13678831 12207439 95717566 82499334 73629001 08.5 92.7 88.8 55.7 63 79.8 2.53 0.96 5.96

------16.5 46267641 39869861 35577421 15716430 13545119 12088122 95156312 82015655 73197373 55 90.4 80.6 03.4 64.7 97.9 1.3 0.82 4.35

127 ------17 44948733 38733471 34563471 15564294 13414020 11971137 94601662 81537667 72770824 49.8 24.1 58.6 43.1 18.8 31.6 7.49 5.52 0.26

------17.5 43703555 37660603 33606197 15415102 13285457 11856414 94053501 81065271 72349263 15.4 56.5 64 49.3 20.7 95.6 8.67 3.4 9.88

------18 42526076 36646063 32700964 15268769 13159357 11743890 93511715 80598368 71932606 09.9 46.1 65.9 35.9 58.7 65.8 9.53 6.18 0.29

------18.5 41410906 35685207 31843630 15125213 13035650 11633501 92976194 80136863 71520764 93.9 76.6 03.6 40.5 99.7 66.7 1.16 8.31 8.84

------19 40353216 34773874 31030480 14984356 12914269 11525187 92446828 79680663 71113657 27 03.4 05.2 09 76.3 59.7 0.24 6.31 2.56

------19.5 39348660 33908319 30258175 14846121 12795148 11418890 91923511 79229676 70711201 23.4 84.5 72.6 80.9 73.9 32.4 8.31 8.12 7.61

------20 38393322 33085171 29523706 14710437 12678225 11314553 91406142 78783814 70313318 48.4 09.2 67.6 76.2 19.8 87.6 1.12 2.51 8.73

------20.5 37483662 32301377 28824351 14577233 12563438 11212124 90894617 78342988 69919930 81.3 72 63.6 81.9 71.3 33.5 7.84 8.47 8.71

------21 36616473 31554175 28157644 14446442 12450731 11111549 90388840 77907115 69530961 02.6 70.8 53.6 40.3 05.7 74.7 0.5 4.66 7.87

------21.5 35788838 30841054 27521345 14317998 12340046 11012780 89888712 77476110 69146337 87.6 28.2 24.6 37.3 10.2 02.9 3.26 8.83 3.58

------22 34998107 30159727 26913414 14191838 12231329 10915766 89394140 77049893 68765984 12.8 53.7 26.5 91.7 72.4 88.8 1.87 7.32 9.8

------22.5 34241856 29508109 26331990 14067903 12124529 10820463 88905031 76628384 68389833 95.5 79.2 72.8 44.4 71.7 74.2 3.01 4.51 6.58

------23 33517875 28884294 25775373 13946133 12019595 10726825 88421295 76211505 68017813 01.8 12.7 26 49.4 70.8 64.3 3.76 2.33 9.65

------23.5 32824133 28286533 25242003 13826472 11916479 10634809 87942844 75799179 67649857 68.4 61.3 22.7 63.7 07.4 20.7 0.98 9.82 9.96

------24 32158771 27713224 24730450 13708866 11815132 10544372 87469591 75391334 67285899 93.5 86.5 06.7 39.3 87.3 54.7 0.86 2.61 3.31

------24.5 31520078 27162893 24239398 13593262 11715511 10455475 87001451 74987895 66925872 58.6 58.6 38.8 14.2 76.4 20.7 8.28 2.47 9.89

------25 30906477 26634181 23767636 13479609 11617571 10368078 86538343 74588791 66569715 49.1 82.3 60 05.1 94.7 10.4 6.42 6.92 3.94

------25.5 30316514 26125836 23314046 13367857 11521271 10282143 86080185 74193953 66217364 47.1 69.8 86.2 99.2 09.1 46.8 6.18 8.77 3.32

------26 29748845 25636700 22877596 13257961 11426568 10197634 85626898 73803313 65868758 73 43.8 18.3 48 28 78.8 5.77 5.72 9.22

------26.5 29202227 25165701 22457328 13149873 11333423 10114516 85178405 73416803 65523839 56.8 50.4 51.4 59.4 94.6 76.1 0.19 9.96 5.74

128 ------27 28675507 24711846 22052357 13043549 11241799 10032755 84734629 73034359 65182547 22.7 70.8 70.9 92.5 82.3 24 0.84 7.8 9.56

------27.5 28167614 24274214 21661861 12938947 11151658 99523171 84295496 72655916 64844826 71.6 17.3 24.5 50.3 88.6 9.04 5.06 9.29 9.65

------28 27677555 23851947 21285074 12836024 11062965 98731706 83860934 72281412 64510620 51.4 04.1 61.2 74.6 30.7 4.11 5.72 7.86 6.89

------28.5 27204404 23444247 20921286 12734741 10975684 97952846 83430872 71910785 64179874 02.3 83.4 27.8 40.4 40.2 4.52 0.81 9.96 3.82

------29 26747297 23050373 20569833 12635058 10889782 97186292 83005239 71543976 63852534 69.5 39.8 17.9 50.2 59 3.78 3.07 4.76 4.28

------29.5 26305431 22669630 20230096 12536938 10805227 96431753 82583967 71180925 63528548 74.9 34.7 65.3 29.5 34.5 9.76 9.56 3.77 3.18

------30 25878054 22301370 19901498 12440344 10721987 95688950 82166991 70821575 63207864 41.1 96.3 78.7 21.5 16 1.02 1.36 0.59 6.13

------30.5 25464462 21944989 19583499 12345240 10640031 94957608 81754243 70465869 62890432 61.6 49.5 11.1 83 50.3 3.26 3.17 0.52 9.28

------31 25063998 21599918 19275591 12251593 10559330 94237464 81345660 70113752 62576203 12.6 81.2 61 79.6 78.2 5.98 2.97 0.34 8.96

------31.5 24676044 21265627 18977302 12159369 10479856 93528262 80941179 69765169 62265129 00.7 37.2 02.2 81.8 30.9 9.28 1.69 7.98 1.46

------32 24300021 20941616 18688185 12068536 10401580 92829755 80540738 69420069 61957161 43.9 46.7 38.3 61 26.2 0.83 2.89 2.23 2.78

------32.5 23935386 20627417 18407823 11979062 10324475 92141700 80144277 69078398 61652253 81.5 72.8 79.7 85.7 65.9 2.92 2.42 2.51 8.38

------3.307775 2.890566 2.608958 33 23581629 20322590 18135824 11890918 10248516 91463864 4337 2545 3339 09.8 85.1 40.6 17.8 32.2 9.65

------33.5 23238267 20026721 17871817 11804073 10173676 90796022 42.2 52.2 54.3 09.2 84.7 4.36

129 Bibliography

[1]Lishang Jiang(1994) Mathematical Modeling and Methods of Option Pricing, 312, Tongji University. [2] Matlab Indexes function pdepe. [3]Wikipedia, Greeks(Finance), http://en.wikipedia.org/wiki/Greeks_(finance) . [4] Steven L. Heston (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, Yale University.

130