The Volatility Smile Dynamics Implied by Smile-Consistent Option Pricing Models and Empirical Data

MSc Thesis

Author: Besiana Rexhepi

Supervisers: Dr. Drona Kandhai Drs. Qiu Guangzhong

Commitee members: Prof. Dr. Peter van Emde Boas Prof. Peter M.A. Sloot Dr. Drona Kandhai Drs. Qiu Guangzhong

Date of defense: 22nd of August, 2008

Submitted to the Faculty of Science in partial fulﬁllment of the requirements for the degree Master of Science in Grid Computing (Free Track Programme) 2

Abstract

It is well-known that the fair value of options can be determined by using the Black-Scholes model. However, for liquidly traded options, i.e. instruments for which the market price is known, there is clear evidence that the Black-Scholes model is not correct. This is reflected in the existence of the volatility smile phenomenon which is one the most challenging problems in financial economics. Re- cently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). However, the quantification of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. Peo- ple have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. We find that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor fitting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other hand, was shown to be able to capture the smile dynamics very closely to the empirically observed dynamics. Acknowledgements

I would like to express my deepest respect and gratitude for my supervisers, Dr. Drona Kandhai and Drs. Qiu Guangzhong. I am very happy to contribute to the computational finance research activities of the section of computational science. I highly appreciate your lectures, very interesting discussions, advice and mostly your personality. My interest in derivative finance started one year ago, during the course Computational Finance, lectured by Dr. Drona Kandhai. From there on I continued to take courses in finance (at the Faculty of Economics at Universiteit van Amsterdam), stochastic calculus (at Vrije University) and computational aspects of stochastic modeling (at TU Delft). Next, for about seven months I did this research with extensive help from my supervisors. One important aspect of this group is their approach to tackle a certain problem in finance. Qiu, on one hand, through his research on micro-simulation of options markets tries to understand the behavior of options traders. From his discussions I gained a lot of insight in market anomalies and possible reasons accounting for them. Drona, on the other hand, being a quantitative finance professional too, tutored me in option pricing and more importantly on modern market practice. This kept my excitement high at all stages. In addition, I would like to thank Joost Geerdink for his useful suggestions and help, and mostly for his friendship. Many thanks go to my classmates, Anton Bossenbroek and Andrea Sottoriva for their discussions and support during the course of my studies at Universiteit van Amsterdam. Finally, I am grateful to my boyfriend, Astrit, and my family.

3 4 Contents

1 Introduction 7

2 The Black-Scholes Model 11 2.1 Binomial Trees ...... 11 2.2 Continuous-time Processes ...... 14 2.3 Derivation of the Black-Scholes-Merton Differential Equation ...... 17 2.4 Properties of Option Prices ...... 19 2.5 Implied Volatility ...... 20

3 Stylized Facts Observed in Options on the S&P500 Index 23 3.1 Liquidity Imbalance ...... 23 3.2 The Implied Volatility Smile ...... 25 3.3 Implied Volatility Dynamics ...... 34

4 Smile-Consistent Option Pricing Models 39 4.1 The Displaced Diffusion Model ...... 40 4.2 The Constant Elasticity of Variance Model ...... 42 4.2.1 Similarity between displaced diffusion and CEV model ...... 43 4.3 SABR Stochastic Volatility Model ...... 45

5 Comparison of Implied Volatility Dynamics of Smile-Consistent Models and Empirical Data 49 5.1 Fitting to Market Data ...... 51 5.1.1 Displaced diffusion model ...... 51 5.1.2 CEV model ...... 52 5.1.3 SABR model ...... 52 5.2 Smile Dynamics ...... 59 5.2.1 Displaced diffusion model ...... 59 5.2.2 CEV model ...... 65 5.2.3 SABR model ...... 67

6 Discussion and Future Work 75

A Appendix 77 A.1 Additional Results on Liquidity Imbalance ...... 77 A.2 Data Smoothing ...... 81 References ...... 82

5 6 CONTENTS Chapter 1

Introduction

In financial markets a wide range of derivatives are traded such as bonds, futures and options. Options markets are similar to futures markets, in that they give the holder the ability to enter into a future transaction, where typically one can buy or sell an underlying commodity for a specific price. The main difference between options and futures is that options contracts only give the right to buy or sell the underlying commodity, whereas the holder of a futures contract has the obligation to buy or sell the underlying commodity. Options contracts are typically traded to make a profit by taking a speculative view on the market or for hedging purposes. A hedger uses options to eliminate (hedge) potential risk, whereas a speculator uses them to bet on future movements in the price of the underlying asset. A third type of trader, arbitrageurs, try to take advantage of discrepancies between prices. An option contract specifies the underlying security, the expiration date and the strike (exercise) price. It can be a call option, in which case the holder has the right to buy the underlying asset, or a put option where the buyer has the right to sell the underlying asset. An option that can only be exercised at expiration date is called European, while an option that can be exercised before or at expiration date is called an American style option. When a trader buys an options contract, he pays an up front fee which is known as the option’s premium. Three main criteria are used to determine an option’s premium:

• In, at, or out-of-the-money, i.e. moneyness. An option is in-the-money, if it is already in profit; it is out-of-the-money when it is not yet in profit; and it is at-the-money otherwise. If an option is in-the-money, its premium will have additional value since the profit will be immediately available to the buyer of the option. If an option is at-the-money or out-of-the-money, its premium will not have any additional value, therefore it always costs less than an in-the-money option.

• Time value. All options contracts have an expiration date, after which they become worthless. The more time that an option has before its expiration date, the more time there is available for it to come into proﬁt, so its premium will have additional time value. The less time that an option has until its expiration date, the less time there is available for the option to come into proﬁt, so its premium will be lower.

• Volatility. If the price of the underlying is highly volatile, the premium will be higher, because the option has the potential to be more proﬁtable. Conversely, if the price is less volatile, the premium will be lower.

7 8 CHAPTER 1. INTRODUCTION

Option pricing theory is a model-based approach for calculating the fair value of an option. One of the most important inventions in ﬁnancial theory is the Black-Scholes model (BS73), developed in 1973 by Fisher Black, Robert Merton and Myron Scholes. It is still widely used today and can be regarded as a basic building block of more sophisticated approaches that have been proposed in the past few years. Their approach is as follows. A trader who sold an option is exposed to risk. The fundamental idea of the Black-Scholes model is to hedge this risk by taking positions in the underlying. In technical terms this is referred to as setting up a replicating portfolio. Details are discussed in chapter 2. The fair value of this option equals the cost of dynamically hedging the position. Given certain assumptions, they proposed that this cost could be known in advance. The price of a European option given by Black-Scholes formula takes as input the stock price, the strike price, the expiration time, the risk-free interest rate and the volatility of the stock price. All input variables in their formula are observable, with the exception of the volatility, which is assumed to be constant. The assumptions underlying the Black-Scholes formula (see chapter 2), give rise to so-called stylized facts observed in options markets. The most important one is what is known as the implied volatility smile. Implied volatility (IV) is the volatility that needs to be plugged in the Black-Scholes formula such that the market price of the option is recovered. When plotted against different strikes, implied volatility typically forms a smile-shaped curve, hence the name implied volatility smile. The European calls and puts considered thus far are also called vanilla or plain vanilla options. Their payoffs depend only on the ﬁnal value of the underlying asset. Plain vanilla options are highly liquid1, and as a result their prices are determined by supply and demand principles. As such, implied volatility obtained from these options is an indicator of the current sentiment of the market. This means that pricing other vanilla options is straightforward: the volatility that is needed as input in the Black-Scholes formula is the implied volatility taken from other vanilla options in the market. However, the problems arise with illiquid options. Examples include way (deep) out-of-the-money or way in-the-money European options. Even more complicated is the pricing of exotic options, which typically have complex payoffs and/or early exercise features. Examples include American options, Barrier options or callable structures where the payoff depends on embedded options. The value of these options does not only depend on the volatility smile observed today, but also on its time evolution. Besides valuation, it is important to keep in mind that in order to hedge positions taken in these exotic options, accurate and reliable estimation of hedge sensitivities and consistent pricing of plain vanilla hedging instruments in time is required. The existence of the smile requires the development of new pricing models that capture the dynamic distortions of the implied volatility smile. One way is to add another degree of freedom to the process of the underlying asset or to the volatility process. For example, the constant volatility case can be relaxed by allowing the volatility to be a deterministic function of the asset price and time. Such models are often referred to as local volatility models. Examples include the Displaced Diffusion and Constant Elasticity of Variance. Other models allow for a stochastic volatility setting. In stochastic volatility models, in addition to the uncertainty which drives the asset price, a second stochastic process is introduced that governs the volatility dynamics. An example is the SABR Stochastic Volatility model. The presence of highly liquid option markets has brought up a new practice in option pricing: a trader takes the prices of plain vanilla options as given and tries to extract information about the volatility directly from the observed option prices, which is then employed to price and hedge other derivative products. Models used in this case are called smile-consistent volatility models since Eu-

1An options market is liquid, if there is a large number of buyers and sellers of options. 9 ropean options priced with these models approximately (within calibration errors) reproduce the IV smile observed empirically (Fen05). Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02; Fen05). However, the quantiﬁcation of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. Peo- ple have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. The material in this work is organized as follows. Chapter 2 introduces the reader to the main principles used in option pricing theory, e.g. risk neutrality and arbitrage. The most important model in ﬁnancial theory is discussed and derived, namely the Black-Scholes (BS) model. The chapter ends with an outline of the shortcomings of the model and stylized facts observed in options markets resulting from these. Market stylized facts, such as implied volatility smile, are presented in more detail in chapter 3, through an empirical study of a major equity options market, namely options on the S&P 500 index. Furthermore, the dynamics of implied volatility surface is assessed by applying a principal component analysis on the time-series of the volatility smile. In chapter 4, we introduce extensions of BS model that are supposed to incorporate the smile. The main part of this project constitutes chapter 5, where we assess the ability of smile-consistent models to capture the market volatility dynamics. Discussion of the results and ideas for future work are given in chapter 6. 10 CHAPTER 1. INTRODUCTION Chapter 2

The Black-Scholes Model

Instruments traded in financial markets can be divided into two groups: underlying assets such as shares, bonds, commodities, foreign currencies; and their derivatives, claims that promise some pay- ment in the future contingent on the behavior of one or more underlying assets. Derivatives are traded in exchanges and over the counter between financial institutions or fund managers. They are used in different ways, depending on the type of trader. For example, a hedger uses derivatives to hedge or eliminate risk, whereas a speculator uses them to bet on future movements in the price of the underlying. An arbitrageur on the other hand, tries to take advantage of discrepancies between prices. In general, the relation between the underlying and the derivative may be rather complex with un- certain payoffs since future values of the underlying are random. Based on methods developed in stochastic calculus and fundamental principals from finance, it is possible to formulate methodologies to calculate the fair value of a derivative. This chapter discusses a well-known model that is used to price options. Based on distributional assumptions of the behavior of the underlying, it will be shown that a claim can be valued as an expectation in a probability space that ignores risk preferences of traders. The intuition behind this model will be highlighted by starting with the relatively simple discrete-time model, which will be later generalized to a continuous-time setting. Next, the derivation of the BS model will be presented with a clear description of its main assumptions . These issues are reflected in the existence of a complex implied volatility surface observed in empirical data. As mentioned in chapter 1, the characteristics of these surfaces as observed in empirical data and implied by sophisticated models will be the main subject of this work.

2.1 Binomial Trees

In ﬁnance, the binomial pricing model provides a numerical method for the valuation of options. Es- sentially, the method uses a binomial tree to model the stochastic behavior of the underlying ﬁnancial instrument. Before going into further details, we recall an important fact from economics, namely the time value of money. If we are in January, then one dollar in coming December is worth less than one dollar now. Interest rates are the formal acknowledgment of this fact. From now on, we assume that for any time T , the value today of a dollar to be payed at time T is given by exp(−rT ) for some constant r > 0. The rate r is the so-called continuously compounded interest rate corresponding to this period. We start with a simple two-state economy where the stock price can move in two possible directions. The time span considered here is denoted with ∆t. The binary branch starts with a stock with initial

11 12 CHAPTER 2. THE BLACK-SCHOLES MODEL price S0 and an option on this stock with current price f. This is illustrated in ﬁgure 2.1. Suppose that the option has maturity time equal to ∆t and during this time tick, the stock price can either move up from S0 to uS0 with probability q, or down to dS0 with probability 1 − q. If the stock price moves up, we denote the payoff from the option with fu; if the stock price goes down, we denote the payoff of the option with fd. Our objective is to determine a fair premium f to be charged for this option today

Figure 2.1: Stock and option prices in a one-step binomial tree.

(t = 0). Note here that values of fu and fd are known at time t = ∆t since they are dependent on the corresponding future value of the underlying, uS0 and dS0, respectively. However, today the value of the option is inherent to risk since the underlying is random.

The fundamental idea behind option pricing is to consider a risk-free portfolio where the risk inherent to the claim is killed by taking positions into the same underlying. We consider a portfolio consisting of: • a long1 position in ∆ shares (stocks) • a short position in one option It is possible to ﬁnd the value of ∆ such that this portfolio bears no risk. In case the stock price moves up, the value of the portfolio at maturity is:

uS0 ∆ − fu (2.1)

while, if the stock price moves down to dS0, the value of the portfolio is:

dS0 ∆ − fd (2.2) A risk-free portfolio is obtained when these expressions are equal: f − f ∆ = u d (2.3) uS0 − dS0 Since this portfolio is riskless, it must earn the risk-free2 interest rate, otherwise there would be an arbitrage opportunity3. It follows that the present value of the portfolio is: −r∆t e (uS0 ∆ − fu) (2.4)

1Taking a long position in an asset means buying the asset. Similarly being short in an asset means selling it. 2The risk-free interest rate is the interest rate that can be obtained by investing in financial instruments with no default risk. The rate used in option pricing methods is the continuously compounded rate. 3In finance theory, an arbitrage is a ”free lunch” - a transaction or portfolio that makes a profit without risk. A fundamental law in finance is that arbitrage opportunities cannot exist. We emphasize that this is an assumption and in practice arbitrage opportunities do exist. However, due to fast electronic trading these opportunities do not exist very long. 2.1. BINOMIAL TREES 13

On the other hand, the cost of setting up our portfolio at time t = 0 is:

S0 ∆ − f (2.5)

Again, by using the same arbitrage argument, the expressions 2.4 and 2.5 must be equal:

−r∆t e (uS0 ∆ − fu) = S0 ∆ − f (2.6) or −r∆t fu − fd −r∆t fu − fd f = S0∆ − e (S0u∆ − fu) = S0 − e S0 u − fu = S0u − S0d S0u − S0d f − f (f − f )u − f (u − d) f − f uf − df = u d − e−r∆t u d u = u d + d u e−r∆t = u − d u − d u − d u − d

er∆t(f − f ) uf − df er∆tf − er∆tf + uf − df = e−r∆t u d + d u = e−r∆t u d d u = u − d u − d u − d

er∆t − d u − er∆t = e−r∆t ( )f + ( )f = e−r∆t [pf + (1 − p)f ] (2.7) u − d u u − d d u d

where er∆t − d p = . (2.8) u − d The price of the option in a one-step binomial tree is given by equations 2.7 and 2.8. The most important thing to note is that the real world probabilities of up and down movements of the stock price, q and 1 − q, respectively, are not relevant when valuing the option. Although this seems coun- terintuitive, it can be explained by the fact that otherwise the option premium would depend on the probabilities that trades would have in mind and thus there would not be a single fair premium. In equation 2.7 we can interpret p as the probability of an up movement of the stock price. If we do so, than 1 − p is the probability of a down movement of the stock price and the expression:

pfu + (1 − p)fd (2.9) is the expected payoff from the option. As a result, equation 2.7 means that the value of the option today is the expected future payoff in a risk-neutral probability measure, discounted at the risk- free rate. This is the most important result in option pricing theory which stipulates that the value of the option is independent of risk preferences amongst different investors. The expected value of the stock price at maturity E(S∆t), under the risk-neutral measure is given by:

E(S∆t) = pS0u + (1 − p)S0d, (2.10) which can be simpliﬁed by substituting equation 2.8 to the following form:

r∆t E(S∆t) = S0 e . (2.11)

Thus, the stock price grows at the risk-free rate under the risk-neutral measure. We can extend this simple model to a two-step binomial tree, which is illustrated in ﬁgure 2.2. Again the initial stock price is S0. During the ﬁrst time step it can go up to uS0 or it can move down to dS0. Similar to the one-step case, the payoff of the option when the stock moves up is denoted 14 CHAPTER 2. THE BLACK-SCHOLES MODEL

Figure 2.2: Stock and option prices in a two-step binomial tree.

with fu and when it moves down it is fd. In the second time step the stock price can again move up or down. If during the first time step the stock price went down to dS0, then at expiry of the option, the stock price can move up to udS0 or down to ddS0. In the first case the option’s payoff is denoted by fud and in the second case it is fdd. Assuming that the length of the time step is ∆t and using equation 2.7 we can compute the option price after the first time step:

−r∆t fu = e [pfuu + (1 − p)fud] −r∆t fd = e [pfud + (1 − p)fdd]

and at the beginning of the time interval

−r∆t −2r∆t 2 2 f = e [pfu + (1 − p)fd] = e [p fuu + 2p(1 − p)fud + (1 − p) fdd] (2.12)

2 2 If we interpret p as the probability of the upper movement of S0 to u S0, 2p(1 − p) as the probability of the middle movement and (1 − p)2 as the probability of the lower movement, than the last formula is consistent with the risk-neutral valuation: the option price is equal to the expected value of its future payoff discounted at the risk-free interest rate. In a similar way, we can continue to build multi-step binomial trees as a model for the stock price process.

2.2 Continuous-time Processes

Despite the fact that the stock price can take only discrete values and the price changes in reality can be seen only when the exchange is open (i.e. discrete times), the pricing of options and other more complicated derivatives is usually modeled with continuous-time processes. Before defining a suitable continuous-time stochastic process as a model for the stock price, we are first going to list some desirable properties of the stock price. In the following, we list some important definitions adapted from the book by (KPS94).

A sequence of random variables X1,X2,...,Xn,... that describes the evolution of a probabilistic system over discrete instants of time t1 < t2 < . . . < tn < . . . is called a discrete time stochastic process. Stochastic processes can also be defined for all time instants in an interval, in which case 2.2. CONTINUOUS-TIME PROCESSES 15 we call them continuous-time stochastic processes. If the time under consideration is denoted by T and there is a common underlying probability space (Ω, A, P ), then the stochastic process X = {X(t), t ∈ T } is defined as the function X : T × Ω → R of two variables, where X(t) = X(t, ·) is a random variable for each t ∈ T and X(·, ω): T → R is a sample path or trajectory of the stochastic process for each ω ∈ Ω (i.e. set of all possible paths), A is the sigma-algebra (KPS94) and P is the probability measure. An example of a stochastic process with specific temporal relationships is a Markov process:

P (Xn+1 = x|Xn = xn,...,X1 = x1) = P (Xn+1 = x|Xn = xn) (2.13) Here we have written the Markov property for a discrete time stochastic process: given the present state of the process, the future states are independent from the past. Stock prices are usually assumed to follow a Markov process: the probability distribution of the price at any future time is independent of the particular path followed by the price in the past. Another important process is the Wiener process, also known as the Brownian motion. The standard Wiener process W = {W (t), t ≥ 0} is deﬁned as: W (0) = 0

E(Wt+∆t − Wt) = 0

V AR(Wt+∆t − Wt) = ∆t (2.14) where E(·) is the expectation operator and V AR(·) is the variance. If we denote the Wiener increment with ∆Wt = Wt+∆t − Wt, then it follows from the law of large numbers that ∆Wt is a gaussian variable with mean zero and variance ∆t:

∆Wt ∼ N(0, ∆t) (2.15)

This is illustrated in ﬁgure 2.3. In addition, the increments W (t2) − W (t1) and W (t4) − W (t3) are

Figure 2.3: Mean and variance of the standard Wiener process. This is generated by simulating a number of paths and then computing the mean and variance. As the number of paths increases the mean converges to zero and the variance to the straight line t. independent for all 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4. As a result we can write: √ Wt+∆t = Wt + ∆t ∼ N(0, 1) (2.16) 16 CHAPTER 2. THE BLACK-SCHOLES MODEL

In order to generalize to continuous processes, we go from small changes to the limit as the small changes approache zero. By taking the limit as ∆t goes to zero, we can write ∆Wt as dWt. We can say that the standard Wiener increment dWt is a stochastic process with a drift rate of zero and a variance rate of 1. This is the starting point in building our model for the stock price. Next, we deﬁne a generalized form of this process by making the drift and variance rates equal to some constants:

dXt = a dt + b dWt (2.17)

A simulated path of this process is given in ﬁgure 2.4 with a = 0.3 and b = 2.

Figure 2.4: Generalized Wiener process: a sample path of the process with a = 0.3 and b = 2 (left) and the mean and variance of the process (right).

In our model for the stock price we need to make the drift rate and the variance rate proportional to the stock price. In addition, a stock price must be positive (note that this is not the case for a generalized Wiener process, ﬁgure 2.4). A geometric Brownian motion serves this purpose:

dSt = µ St dt + σ St dWt (2.18)

Writing 2.18 in discrete form as ∆St = µSt∆t + σSt∆Wt and substituting equation 2.16 yields: √ ∆St = µ St ∆t + σSt ∆t (2.19)

or ∆S t ∼ N(µ∆t, σ2∆t) (2.20) St The parameter µ is the continuously compounded return from the stock per year. The parameter σ is the volatility of the stock price, interpreted as the standard deviation of the return during one year (Hul00). Next, we need to price an option on the stock that follows a geometric Brownian motion. There is a lemma that makes it possible to write a differential equation for a function of the process that follows equation 2.18, namely Ito’s lemma. According to Ito’s lemma, if the stochastic process is given by: dX = a(X, t) dt + b(X, t) dWt (2.21) 2.3. DERIVATION OF THE BLACK-SCHOLES-MERTON DIFFERENTIAL EQUATION 17 and G is a function of X and t, then

∂G ∂G 1 ∂2G dG = dt + dX + dt (2.22) ∂t ∂X 2 ∂X2 which can be further simpliﬁed to

∂G ∂G 1 ∂2G ∂G dG = a + + b2 dt + b dW . (2.23) ∂X ∂t 2 ∂x2 ∂X t

Note that in equation 2.22, we have included second-order terms which can be intuitively understood since dXdX ≈ dt. For a more rigorous derivation of the Ito’s lemma, we refer the reader to standard text books in stochastic calculus. To summarize, this lemma implies that a stochastic differential equation for G can be derived and more importantly it is subject to the same source of randomness as X, namely dWt. Furthermore, Ito’s lemma is used to get insight into the distribution of the stock price at some future time (e.g. expiry time of the option). Applying it to the logarithm of the stock price that follows 2.18, it can be shown that ln St follows a normal distribution. As a result, St follows a log-normal distribution (see ﬁgure 2.5).

Figure 2.5: Probability density function of a log-normal distribution.

2.3 Derivation of the Black-Scholes-Merton Differential Equation

In 1973, Fischer Black and Myron Scholes published their groundbreaking paper entitled ”The pricing of options and corporate liabilities”. In deriving their formula for the option value, they assumed that the price of the underlying instrument St follows a geometric Brownian motion with constant drift µ and volatility σ, and the price changes are log-normally distributed. Following the notation in (Hul00), in this section we show how (BS73) derived their option valuation formula. Let’s denote the value of the option with f. For simplicity, we leave out the time subscript. Writing the discrete form of the stock price process as

∆S = µ S ∆t + σ S ∆W (2.24) 18 CHAPTER 2. THE BLACK-SCHOLES MODEL and applying Ito’s formula to f(S, t) yields4: ∂f ∂f 1 ∂2f ∂f ∆f = µS + + σ2S2 ∆t + σ S ∆W . (2.25) ∂S ∂t 2 ∂S2 ∂S t ∂f ∂f Here we note that S = S(ti), ∂S = ∂S (ti) and similar for the other higher-order terms. Under such assumptions one can set up a risk-less portfolio of: ∂f • a long position in ∂S shares which is ﬁxed in the interval (ti, ti+1) • a short position in one option Denoting the value of the portfolio with V it follows that: ∂f V = −f + S (2.26) ∂S Given that the portfolio is self-ﬁnancing5, the change in the value of the portfolio after ∆t time has passed is given by: ∂f ∆V = −∆f + ∆S (2.27) ∂S Substituting equations 2.24 and 2.25 into equation 2.28 yields: ∂f 1 ∂2f ∆V = − − σ2S2 ∆t (2.28) ∂t 2 ∂S2 This means that the change of portfolio value during time ∆t is independent of uncertainty caused by ∆W . Since this portfolio bears no risk it must earn the risk-free rate during ∆t: ∆V = rV ∆t (2.29) where r is the risk-free rate. Substituting from equations 2.26 and 2.28, we have: ∂f 1 ∂2f ∂f − − σ2S2 ∆t = r −f + S ∆t (2.30) ∂t 2 ∂S2 ∂S Simplifying the last equation yields the Black-Scholes-Merton partial differential equation: ∂f ∂f 1 ∂2f + rS + σ2S2 = rf (2.31) ∂t ∂S 2 ∂S2 This equation has many solutions depending on the boundary conditions that are used. A European call option imposes a boundary condition of the form:

f = max(ST − K, 0) (2.32) and a put option has the boundary:

f = max(K − ST , 0) (2.33) where K is the strike price. The important note in equation 2.31 is that the portfolio has no risk during the period ∆t only: as S ∂f and t change, ∂S changes as well. Therefore, it is necessary to frequently change the proportions of the options and shares in the portfolio.

4Ito’s formula is applied in continuous time only. Here we write the derivations in discrete form for simplicity. 5A portfolio is self-ﬁnancing if and only if the change in its value only depends on the change in the asset prices. The existence of such a portfolio, when the stock price is a geometric Brownian motion, is proved in most books in ﬁnancial calculus (See for example, (BR96)). 2.4. PROPERTIES OF OPTION PRICES 19

In case of a European call option that pays no dividends, BS equation can be solved analytically and the price is given explicitly by the following formula:

−rT c = S0N(d1) − Ke N(d2) (2.34) where

σ2 ln(S0/K) + (r + 2 )T d1 = √ σ T √ d2 = d1 − σ T and spot price is denoted with S0, K is the strike price, T is the expiry time, r is the risk-free interest rate, σ is the volatility of the stock price and N(·) is the standard normal distribution function. The price for a European put is given by:

−rT p = Ke N(−d2) − S0N(−d1) (2.35)

In deriving their formula, Black and Scholes assumed ”ideal conditions” in the market for the stock and for the option. We list some of them here: • The interest rate is constant through time.

• The volatility of the stock is constant.

• The stock pays no dividends.

• There are no transaction costs in buying or selling the stock or the option.

2.4 Properties of Option Prices

The right to buy a stock can never be worth more than the stock itself. Therefore, the upper bound of a European call option is the stock price, S0. A lower bound for the price of a European call option −rT on a non-dividend-paying stock is max(S0 − Ke , 0). This can be explained by considering two portfolios: Portfolio I: one European call option plus an amount of cash equal to Ke−rT , Portfolio II: one stock. In portfolio I, it is possible to invest the cash at the risk-free interest rate, and it will grow to K at time T . If ST > K, the call option is exercised at time T and this portfolio is worth ST . If ST < K, the call option expires worthless and the porfolio is worth K. Hence, at time T portfolio I is worth max(ST ,K). In the other hand, portfolio II is worth ST at time T . It follows that portfolio I is always worth at least as much as portfolio II at time T . It follows that it must be worth at least as much as portfolio −rT −rT II today. Hence c + Ke ≥ S0 or c ≥ S0 − Ke . Since in the worst case an option can expire worthless, its value is always non-negative. Therefore

−rT c ≥ max(S0 − Ke , 0).

Similarly, a put option at time T cannot be worth more than the strike price, K. It means that today, the option cannot be worth more than the present value of the strike: p ≤ e−rT K. To illustrate the lower bound for a put option on a non-dividend-paying stock we consider again two portfolios: 20 CHAPTER 2. THE BLACK-SCHOLES MODEL

Portfolio I: one European put option and one share, Portfolio II: an amount of cash equal to Ke−rT . If ST < K, portfolio I is worth K. If ST > K, the put option expires worthless, and the portfolio is worth ST at time T . Hence portfolio I is worth

max(ST ,K)

at time T . The portfolio II, assuming that the cash is invested at the risk-free interest rate, is worth K at time T . Therefore, portfolio I is always worth as much as portfolio II at time T . In absence of arbitrage, it −rT must hold that portfolio I is worth at least as much as portfolio II today: p + S0 ≥ Ke or

−rT p ≥ Ke − S0.

An important relationship between put and call prices is the so-called put-call parity. For an option on a non-dividend-paying stock, put-call parity can be illustrated by considering two portfolios: Portfolio I: one European call option plus an amount of cash equal to Ke−rT , Portfolio II: one European put option plus one share. At expiration time T , both portfolios are worth the same, namely max(ST ,K). The portfolios must therefore have identical values today. This means that

−rT c + Ke = p + S0.

These properties will be used in the remaining part of the text.

2.5 Implied Volatility

All parameters in the BS formula are observable with the exception of the volatility, σ, which is assumed to be independent of the strike price and time to maturity. The options described above are traded liquidly and therefore a market price is available in practice. Moreover, usually traders do not quote the price, but instead the implied volatility, σ˜, the input volatility for which the BS price and the market price are equal. More precisely, in case of a call option:

σ˜ : CBS(S0,K, σ˜) = C(S0,K)

where CBS(S0,K, σ˜) is the BS price and C(S0,K) is the market price of the call option. If we plot implied volatilities of options on the same underlying against strikes we end up with a smile shaped curve, which is called a volatility smile. In ﬁgure 2.6 a sample smile is illustrated. Note that for the sake of convenience we deﬁne new variables, time to maturity τ = T − t, and moneyness, m = K/St. It has been documented in a number of studies that in addition to moneyness m, IV smile varies over time to maturity, τ. The implied volatilities for different m and τ is referred to as an implied volatility surface (IVS):

IVS :(t, m, τ) → σ˜t(m, τ) which is illustrated in ﬁgure 2.7. 2.5. IMPLIED VOLATILITY 21

Figure 2.6: Implied volatility as a function of moneyness (τ is ﬁxed). The ﬂat line is implied by the BS model, while the smile shaped curve is what is typically observed in options on foreign exchange rate.

Figure 2.7: Implied volatility as a function of moneyness and maturity. Volatility surface as implied by market prices is shown in the left picture while the one implied by the BS model is shown in the right picture. 22 CHAPTER 2. THE BLACK-SCHOLES MODEL

In summary, this chapter introduced the reader to the main principles used in option pricing theory. We started with a very simple model for the underlying behavior and priced a claim in a discrete-time setting. A remarkable result was that although both the underlying and the option price are random, when valuing the option, we can cancel out the randomness by carefully choosing a risk-free portfolio. Even though a discrete-time world serves good for intuition, the continuous-time processes are usually employed in financial modeling. In section 2.3, it was shown that by assuming a geometric Brownian motion as a model for the underlying, a claim can be valued as an expectation under the risk-neutral measure. Furthermore, for any claim one can write a partial differential equation (PDE) for its price with appropriate boundary conditions depending on the payoff of the claim. The price of a European call was given in a closed form and assumptions underlying the BS formula were highlighted. Among these assumptions, the one that is empirically rejected the most is the constant volatility hypothesis. It is common practice to use market prices and find the corresponding value for the volatility such that the BS price is equal to the market price. Empirical studies for many different markets show that volatility varies across strike and time to maturity. This is well-known as the volatility smile phenomenon. In the next chapter, we will describe a detailed empirical study of European equity options of the S&P 500 index to confirm some of the stylized facts observed in options markets. Chapter 3

Stylized Facts Observed in Options on the S&P500 Index

In this chapter we will present an empirical study of equity options on the S&P5001 index traded during a period of three years, namely from Dec. 1999 to Oct. 2002. We are going to look at the following features of these European style options:

• Liquidity imbalance which is reﬂected in the traded volume.

• The static stylized characteristics of the option prices with respect to moneyness and expiry, and their dynamic behavior.

The source database contains the following daily information for each traded option: the date, the type (call or put), the expiration date, the strike price, the number of contracts traded (the so-called transaction volume) and the price. In fact, information is available regarding the opening, closing, lowest and highest price observed during that day. It should be noted that in our studies we will use the closing prices. In the ﬁrst stage, the raw data is screened for data errors where options violating well-known lower and upper bounds for the price are eliminated. Next, the three features mentioned above will be examined in more detail.

3.1 Liquidity Imbalance

Figure 3.1 shows the distribution of the number of observations as a function of the level of moneyness. As illustrated in this ﬁgure, the number of traded options decreases as we move further away from at-the-money. Figure 3.2 shows that the number of traded options decreases as time to maturity increases.

1The Standard & Poor’s 500 index is a capitalization-weighted index of 500 stocks from a broad range of industries. The component stocks are weighted according to the total market value of their outstanding shares. The purpose of the S&P 500 Stock Price Index is to portray the pattern of common stock price movement. The S&P 500 is the most widely accepted barometer of the market. Its stocks represent about 75% of the total U.S. equities market (www.standardandpoors.com).

23 24 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.1: Distribution of call and put observations with respect to moneyness - accumulated for all maturities, S&P500 index, Dec. 1999 - Oct. 2002.

Figure 3.2: Distribution of call and put observations with respect to maturity (in years) - accumulated for all moneyness, S&P500 index, Dec. 1999 - Oct. 2002. 3.2. THE IMPLIED VOLATILITY SMILE 25

In ﬁgure 3.3 the traded volume is shown for different moneyness. Additional results are included in appendix A.1. It is clear that highest traded volumes are observed for at-the-money options. When comparing puts with calls, the traded volume is higher when they are both out-of-the-money. There- fore in this study we will use puts for moneyness smaller than one (at-the-money level) and calls otherwise.

Figure 3.3: Traded volume of call (blue curve) and put (red curve) options on the S&P500 index, Dec. 1999 - Oct. 2002.

3.2 The Implied Volatility Smile

The IV is calculated by inverting the BS formula (See Chapter 2). The IV smile is the IV of options as a function of the strike price (or moneyness). In ﬁgures 3.4, 3.5 and 3.6, the IV smile is shown for three days (17th Dec 1999, 20th Dec 2000 and 24th Jun 2002). In the left, the IVs of both calls and puts are shown in blue and red respectively for three maturities; the right panel shows the same IV smiles except that now only out-of-the-money options are used. It is clear that IVs obtained from calls and puts fall apart, thus violating the put-call-parity. 26 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.4: IV smiles of call and put options on 1999-12-17. Interest rate equals zero and no dividends are assumed. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 3.2. THE IMPLIED VOLATILITY SMILE 27

Figure 3.5: IV smiles of call and put options on 2000-12-20. Interest rate equals zero and no dividends are assumed. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 28 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.6: IV smiles of call and put options on 2002-06-24. Interest rate equals zero and no dividends are assumed. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 3.2. THE IMPLIED VOLATILITY SMILE 29

As will be shown in the following, the difference between the implied volatility obtained from calls and puts is a result of excluding interest rate and dividends in the analysis. In our analysis presented so far, we have adopted an interest rate equal to zero and have assumed that no dividends were paid during the lifetime of the option. As illustrated in ﬁgure 3.7, call prices increase with increasing interest rates while puts decrease in value. Since in reality interest rate is not equal to zero, by assuming a zero interest rate, we overprice calls and underprice puts. This is reﬂected as a gap in the implied volatility curve.

Figure 3.7: Put prices (red curve) and call prices (blue curve) as a function of risk-free interest rate.

To correct for this, we will now include interest rate and dividend in our analysis. A dividend- paying stock can reasonably be expected to follow the same stochastic process except when the stock goes ex-dividend2. The Black-Scholes formula can therefore be used provided that the stock price is reduced by the present value of all the dividends during the life of the option, the discounting being done from the ex-dividend dates at the risk-free rate (Hul00). Instead of correcting the Black-Scholes formula, we make the adjustment in the interest rate. First, we recall that the stock price at future time t is given by: rτ St = S0 e (3.1) where S0 is the current stock price, τ is the time to maturity of the option and r is the risk-free interest rate. If we denote with Sˆt the dividend paying stock price, we can write:

rτ Sˆt = (S0 − δ) e (3.2) where δ is the present value of dividends. As a proxy for Sˆt, we write:

(r−rδ)τ Sˆt = S0e (3.3) where r − rδ is the effective interest rate and 0 ≤ rδ ≤ r. We use this reduced interest rate, while keeping the same spot price, when inverting the BS formula. As an approximation for the risk free interest rates, we use the average of daily treasury yields3.shown in ﬁgure 3.8 . These rates are adjusted to account for dividends by using an rδ = 2%.

2At this point the stock’s price goes down by an amount reﬂecting the dividend paid per share. As a result the dividend can be interpreted as reduction in the stock price on the ex-dividend date caused by the dividend. 3The yield data is taken from the U.S. Department of Treasury website: www.ustreas.gov. 30 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.8: Daily Treasury Yield Curve Rates, in percentage. The green, blue and red lines show the yield curves for three, six and twelve months maturity bonds, respectively.

The implied volatilities obtained when including these corrections are shown in ﬁgures 3.9, 3.10 and 3.11. Indeed, the gap is reduced signiﬁcantly. It is important to notice, that in all plots, we persistently observe a volatility smile which is usually steeper for short time to maturities. This will be the major stylized fact that we will study in this work. 3.2. THE IMPLIED VOLATILITY SMILE 31

Figure 3.9: IV smiles of call and put options on 1999-12-17. Interest rate is adjusted to account for dividends. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 32 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.10: IV smiles of call and put options on 2000-12-20. Interest rate is adjusted to account for dividends. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 3.2. THE IMPLIED VOLATILITY SMILE 33

Figure 3.11: IV smiles of call and put options on 2002-06-24. Interest rate is adjusted to account for dividends. Left panel, upper to lower graph: IVs of calls, blue colored and IVs of puts, red colored are shown for maturities of three, six and twelve months. Right panel, upper to lower graph: IVs of calls and puts are shown for maturities of three, six and twelve months. In this case, calls are used for moneyness larger than one and puts otherwise. 34 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

In all our studies, we restrict to a speciﬁc range of moneyness, namely from 0.75 to 1.25. This choice can be justiﬁed by looking at the vega of the option.

Vega is the change in price of the option ∂c due to a change in volatility, ∂σ . Vega depends on the moneyness and time to maturity. This is illustrated in ﬁgure 3.12: vega reaches its highest value at the at-the-money strike and increases with longer time to maturity. This means that the implied volatility cal- culation is more sensitive to numerical errors as we move away from at-the-money. This effect is even more pronounced for shorter time to maturities. Therefore, we always use out- of-the-money options with a restriction on the moneyness and time to maturity. Put options are used when m is in the range of [0.75, 1.0] and calls are used otherwise (m ∈ [1.0, 1.25]). The time to maturity, measured in years, is re- stricted in the range from 0.083 (one month) to Figure 3.12: Vega of an option as a function of mon- 1.0 (one year). These we believe are the op- eyness m and maturity τ. tions which contain reliable information about implied volatility movements.

3.3 Implied Volatility Dynamics

Implied volatilities are observed for a discrete set of moneyness and time to expiration. The grid of observations is both irregular and changes with time as the level of the underlying fluctuates. To char- acterize its dynamics, we will analyze in this section the time series of the implied volatility surface. We will follow the approach outlined in (CdF02). As a first step, we smooth the data following the method described in appendix A.2 and estimate an implied volatility surface It(m, τ) for each day t. A sample surface is shown in figure 3.13. Using the smoothed surfaces, we first calculate the sample average over the period of interest. This is illustrated in figure 3.14 (left) as a function of time to maturity in years and moneyness. A decreasing profile in moneyness (skew) as well as a downward sloping term structure can be seen. In the right panel, the sample standard deviation of IVs shows that the surface is dynamic and fluctuates around its average profile. Furthermore, the surface is more volatile for shorter maturities and for moneyness smaller than one. This is inline with the empirical findings using the same data set reported in (CdF02). To further analyze the dynamics of the implied volatility surface, we apply a principal component analysis (PCA). PCA is used for dimensionality reduction in a data set by retaining those characteristics of the data set that contribute most to its variance, thus by keeping lower-order principal components and ignoring higher-order ones (HS07). Such low-order components often contain the most important aspects of the data. For the purpose of this study we consider N days each consisting of a certain number of options (observations). After calculating implied volatility of all the options and interpolating the IVs of every point in the grid, for each day we have the associated implied volatility surface as a function of moneyness level and maturity, It(m, τ). Given the high autocorrelation of the 3.3. IMPLIED VOLATILITY DYNAMICS 35

Figure 3.13: Implied volatility surface on 2000-02-23, S&P 500 Index. Red dots are the actual observations and the surface is obtained using Nadaraya-Watson estimator. The bandwidth in the moneyness direction is h1 = 0.04 and in maturity direction is h2 = 0.3. implied volatility, the PCA is applied on the daily variations of the logarithm of implied volatility:

∆Xt(m, τ) = ln It(m, τ) − ln It−1(m, τ) (3.4)

Therefore for each day in our sample, we compute the logarithm of the implied volatility return. The PCA is applied to the time series of the variable ∆Xt in the following way: • The sample has N days in total, each consisting of M values of the implied volatility return.

• First, the mean IV return is calculated from the sample, a vector with dimensions M × 1.

• Next, the mean is subtracted from each day, resulting in a sample with zero mean. These zero centered returns are used as columns in the data matrix X which has dimensions M × N.

0 0 • The eigenvectors of the covariance matrix C = XX , where X is the transpose of X, are the principal components of the IV returns.

• The IVS is then projected into each of the ”most important” eigenvectors. The importance of an eigenmode is determined by the percentage of variance it can explain: λ i (3.5) λ1 + ··· + λM

where λi is the eigenvalue associated with mode i, and M is the total number of eigenmodes - the dimension of the covariance matrix. In the work by (SHC99), they use a number of criteria to determine the number of components (modes) to be retained. They find that only two eigenmodes are sufficient to explain the dynamics of implied volatility surface for options on futures. In contrast, (CdF02) find that three modes are needed 36 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX

Figure 3.14: The average IVS (left) and the standard deviation IVS (right).

to account for most of the variance in the IV surface for equity options. Since our data comprises of equity options, we follow the latter result and directly look at the interpretation of these modes. The shape of the dominant eigenmode is illustrated in figure 3.15. This eigenmode contains around 75% of the daily variance of implied volatility. All the components of this mode are positive: a positive shock in the direction of this eigenmode results in a global increase of all the implied volatilities, thus it can be interpreted as a ’level’ factor. The projection of the implied volatility surface into the second eigenmode is shown in figure 3.16. This mode accounts for 17% of the variance and changes sign at-the-money: A positive shock following this mode, increases the volatilities of out-of-the-money calls while decreases those corresponding to out-of-the-money puts. This mode, interpreted as the skew, exhibits a systematic slope with respect to expiry, which accounts for the term structure observed of the standard deviation surface in figure 3.14.

Figure 3.15: First eigenmode of daily implied volatility variations for S&P500 index options. 3.3. IMPLIED VOLATILITY DYNAMICS 37

Figure 3.16: Second eigenmode of daily implied volatility variations for S&P500 index options.

The third eigenmode, illustrated in ﬁgure 3.17, represents changes in the curvature/convexity of the surface in the moneyness direction. Here the term structure is more pronounced. This mode explains 3% of the remaining daily variation of the implied volatility.

Figure 3.17: Third eigenmode of daily implied volatility variations for S&P500 index options.

To summarize, volatility smile is a stylized fact that is observed in markets on indices, futures, foreign currencies and commodities. It appears in different shapes, sometimes as a skew, other times as a smile. By analyzing equity options on S&P 500 index, we showed that implied volatility surface is a randomly ﬂuctuating surface over time driven by a number of factors. Those were interpreted as the ”level”, ”skew” and ”smile”. Out of these, the most important one is the level factor, meaning that the surface changes mostly in its level, followed by changes in skew and ﬁnally a small daily variation in smile. This fact is strongly rejecting the constant volatility suggested by the BS model. In 38 CHAPTER 3. STYLIZED FACTS OBSERVED IN OPTIONS ON THE S&P500 INDEX the following chapter, we will discuss different models that try to cope with this shortcoming of the widely used BS model. Chapter 4

Smile-Consistent Option Pricing Models

The existence of the smile requires the development of new pricing models that capture the dynamic distortions of the implied volatility surface. One way is to add another degree of freedom to the process of the underlying asset or to the volatility process. For example, the constant volatility case can be relaxed by allowing the coefﬁcient of the stochastic differential equation, which describes the stock price evolution, to be a deterministic function in the asset price and time:

dSt = µ(St, t) St dt + σ(St, t) St dWt (4.1)

Such models are often referred to as local volatility models. Examples include the Displaced Diffusion model in which an amount of St + a where a is a positive constant, follows a geometric Brownian motion; another one is the Constant Elasticity of Variance in which volatility is a function of stock price. Other models allow for a stochastic volatility setting. In stochastic volatility models, in addition to the Brownian motion which drives the asset price, a second stochastic process is introduced that governs the volatility dynamics. An example is the SABR stochastic volatility model. The presence of highly liquid option markets has brought up a new practice in option pricing: a trader takes the prices of plain vanilla options as given and tries to extract information about the volatility directly from the observed option prices, which is then employed to price and hedge other derivative products. Models used in this case are called smile-consistent volatility models since Eu- ropean options priced with these models approximately (within calibration errors) reproduce the IV surface observed empirically, see (Fen05). This approach is mostly justiﬁed by a practical point con- cerning exotic options: positions in these options need to be hedged, and this hedge will be sought by employing plain vanilla options. In this chapter we introduce three smile-consistent models: Displaced Diffusion, Constant Elas- ticity of Variance and SABR model along with their capability to produce volatility smiles and skews. We exclude models based on jump processes since jumps are highly relevant for options with very short maturities (e.g. a few days, see (Reb04)) which are excluded in this work (see previous chapter).

39 40 CHAPTER 4. SMILE-CONSISTENT OPTION PRICING MODELS

4.1 The Displaced Diffusion Model

Displaced diffusion is a very simple variation on the geometric diffusion that was introduced ﬁrst by (Rub83). The difference is that in a displaced diffusion process the quantity St + a, rather than St, should follow a geometric Brownian motion:

d(St + a) = µadt + σadZt (4.2) St + a where µa is the percentage drift, σa is the volatility of the quantity St + a and the constant a is called the displacement coefficient. The quantity St can be interpreted to be a spot stock price (in our case the index level). Displaced diffusions are useful because they can include a skew as will be seen in the remainder of this section, while retaining the simplicity of the log-normal setting. The drawback of the model is that only St + a (and not St) is guaranteed to be positive. This is illustrated in figure 4.1. With displaced diffusion the stock price can take values from −a to infinity, therefore there is some probability mass associated with negative values. This corresponds to higher implied volatilities for lower strikes, which is reflected as a skew. Below we analyse the displaced diffusion process (4.2) as

Figure 4.1: The probability density of a stock price that follows Black-Scholes (left) and displaced diffusion (right). These were generated with a Monte Carlo simulation. a mixture of a normal and lognormal responsiveness to the same Brownian motion. Leaving out the drift term, we start from:

dSt = σabsdZt + σlogStdZt (4.3) where σabs is the absolute responsiveness to the Brownian motion and σlog is the responsiveness to the Brownian motion proportional to St. Equation (4.3) can be written as:

σabs dSt = σlog(St + )dZt (4.4) σlog

Since d(St + a) = dSt, equation (4.4) can be written as:

σabs d(St + σ ) log = σ dZ σabs log t (4.5) (St + ) σlog 4.1. THE DISPLACED DIFFUSION MODEL 41

σabs Equation (4.2) is obtained by equating = a and σlog = σa. It can be seen that as a goes to zero, σlog the process will behave similar to a log-normal process and as a goes to inﬁnity the process becomes a normal diffusion. With displaced diffusion model as the underlying process, option prices are easy to obtain (Reb04). They are given by the Black-Scholes formula with the exception that the stock price is St + a and the strike is K + a. The price of a call option is:

rτ CBS(St + a, t) = (St + a)φ(d1) − (K + a)e φ(d2) 2 ln St+a + (r + σ )τ d = K+a √ 2 1 σ τ √ d2 = d1 − σ τ where as usual: St denotes the asset price at time t, K is the price of the option, τ denotes the time to maturity, σ is the volatility, r is the riskless interest rate and φ(x) is the cumulative unit normal density function. The main purpose of introducing this extension of the Black-Scholes model is that it is able to produce the implied volatility skew observed in options markets (See Chapter 3). Below we are going to look at the skew that the displaced diffusion can generate with different values of the displacement coefficient. The value of coefficient a is usually chosen to be a small percentage of the stock price1. Figure 4.2 illustrates the smile for a = 0.17S, 0.33S and 0.5S: for small value of a, the model generates an implied volatility that is almost flat with respect to moneyness (or strike price). As a increases the skew of the implied volatility curve is more pronounced.

Figure 4.2: Implied volatility curve as a function of moneyness, with different values of a: small dashed curve a = 0.17S0, medium dashed a = 0.33S0 and the solid curve a = 0.5S0. Here a is expressed as percentage of the stock price. The other input values are: S0 = 30, σ = 0.2, r = 0 and τ = 1.0.

1This is because once the value of a is ﬁxed, the stock price can take values from −a to ∞. By restricting a to take a small value we make sure that the stock price is rarely negative. 42 CHAPTER 4. SMILE-CONSISTENT OPTION PRICING MODELS

4.2 The Constant Elasticity of Variance Model

A number of empirical studies on different markets show that the level of the underlying and the associated implied volatility display a strong inverse relationship. The constant elasticity of variance (CEV) model assumes that the risk-neutral process for a non-dividend paying stock price, St, is:

α dSt = r St dt + σ St dZt (4.6) where r is the risk-free rate, Zt is the Brownian motion, σ is a volatility parameter and α is a positive α−1 constant. The volatility of the stock price is σSt . When α = 1 the CEV stock model is equivalent to a geometric Brownian motion. When α < 1, the volatility increases as the stock price decreases. This creates a probability distribution similar to that observed for equities with a heavy left tail and a less heavy right tail: as the stock price decreases, the volatility increases making even lower stock prices more likely; when the stock price increases, the volatility decreases making higher stock prices less likely. This corresponds to a volatility smile where the implied volatility is a decreasing function of the strike price (Hul00). When α > 1, the volatility and stock price increase together, giving a probability distribution with a heavy right tail and less heavy left tail. This type of behavior is sometimes observed for options on futures and commodities. For European options, (Sch89) has expressed the CEV option pricing formula in terms of the non-central chi-square distribution. However, in this work we adapt the formulas from the book by (Hul00) which are rewritten in slightly different form but are essentially identical to the original formulas. Under the CEV model the price of a European call c and put p, for 0 < α < 1, is given by

−qT 2 −rT 2 c = S0e [1 − χ (a, b + 2, c)] − Ke χ (c, b, a) −rT 2 −qT 2 p = Ke [1 − χ (c, b, a)] − S0e χ (a, b + 2, c) (4.7)

where K2(1−α) 1 (Se(r−q)T )2(1−α) a = , b = , c = (4.8) (1 − α)2σ2T 1 − α (1 − α)2σ2T

When α > 1 the prices are given by

−qT 2 −rT 2 c = S0e [1 − χ (c, −b, a)] − Ke χ (a, 2 − b, c) −rT 2 −qT 2 p = Ke [1 − χ (a, 2 − b, c)] − S0e χ (c, −b, a) (4.9)

Here χ2(z, ν, k) is the cumulative probability that a variable with a noncentral χ2 distribution with noncentrality parameter ν and k degrees of freedom is less than z. The CEV model is considered to be a smile-consistent model which means that it is supposed to produce the implied volatility smile/skew as observed in options market. Here we examine the smile curves generated for different values of α parameter. Figure 4.3 illustrates the smile for α = 0.01, 0.1 and 0.5: for small values of α, the model generates an implied volatility that is skewed with respect to moneyness (or strike price). As α increases the implied volatility curve ﬂattens out2.

2This is expected since for α close to 1, the model becomes the Black-Scholes model and it is known that it assumes a constant volatility across strike prices. 4.2. THE CONSTANT ELASTICITY OF VARIANCE MODEL 43

Figure 4.3: Implied volatility curve as a function of moneyness, with different values of α: small dashed curve α = 0.01, medium dashed α = 0.1 and the solid curve α = 0.5. The solid straight line is produced with α = 1 which is equivalent to Black-Scholes volatility. The other input values are: S = 30, σ = 0.2, r = 0 and τ = 1.0.

4.2.1 Similarity between displaced diffusion and CEV model Writing the differential equation of CEV without the drift term, yields

α dSt = St σ dZt. (4.10)

α Expanding the function f(St) = St to the ﬁrst order around S0 results in

∂f(S0) 2 f(St) = f(S0) + (St − S0) + O(St ), (4.11) ∂St or α α α−1 α α−1 St = S0 + α S0 (St − S0) = (1 − α) S0 + α S0 St. (4.12) Substituting equation 4.12 into equation 4.10 yields

α α−1 α−1 α−1 dSt = [(1 − α)S0 + αS0 St] σ dZt = [(1 − α)S0 S0 + αS0 St] σ dZt, (4.13)

α−1 and writing γ = α S0 , equation 4.13 becomes

dSt = [(1 − γ) S0 + γ St] σ dZt. (4.14)

If we denote with ξ = (1 − γ) S0, equation 4.14 reduces to

ξ dS = [ + S ] γ σ dZ . (4.15) t γ t t 44 CHAPTER 4. SMILE-CONSISTENT OPTION PRICING MODELS

Recall that leaving out the drift term, the displaced diffusion model is given by

dSt = (St + a) σa dZt. (4.16)

Therefore, one can write ξ S + a = S + , (4.17) t t γ and ξ (1 − γ) S = 0 = a (4.18) γ γ

In addition, σa is given by the expression

σa = γ σ, (4.19) hence the similarity between the two models. 4.3. SABR STOCHASTIC VOLATILITY MODEL 45

4.3 SABR Stochastic Volatility Model

The failure of non-parametric local volatility models3 to capture the volatility smile dynamics moti- vated (HKLW02) to develop a new model for the pricing of options. Observing that most markets experience both relatively quiescent and relatively chaotic periods, they choose the volatility to be a random function of time. (HKLW02) have introduced a two factor model with stochastic volatility, given by:

β dSˆ =α ˆ Sˆ dW1

dαˆ = ν αˆ dW2

dW1 dW2 = ρdt (4.20) with

Sˆ(0) = f αˆ(0) = α where Sˆ is the stock price process and αˆ is the volatility process. Here, the two Wiener increments are correlated through the coefﬁcient ρ. The developers of this model claim that the model is able to ﬁt the implied volatility curves to those observed in the market for any exercise date. More importantly, they claim that the model predicts the correct dynamics of the implied volatility curves, making the SABR model an effective means to manage the smile risk (HKLW02). To obtain the prices of European options, they use singular perturbation techniques. From these prices, the options’ implied volatility σ(f, K) is then obtained. Their main result is that, under the SABR model the price of a European call option is given by the Black Scholes formula,

rτ CBS(St, t) = Stφ(d1) − Ke φ(d2) σ2 log St + (r + B )τ d = K √ 2 1 σ τ B√ d2 = d1 − σB τ (4.21) where St is the stock price, K is the strike price, r is the risk-free interest rate, τ is time to maturity of the option and the implied volatility σB(f, K) is given by

α z σB(f, K) = × (1−β)/2 n (1−β)2 2 f (1−β)4 4 f o x (z) (fK) 1 + 24 log K + 1920 log K + ... (1 − β)2 α2 1 ρβνα 2 − 3ρ2 × 1 + + + ν2 τ + ... (4.22) 24 (fK)1−β 4 (fK)(1−β)/2 24 where ν f z = (fK)(1−β)/2 log (4.23) α K

3In non-parametric local volatility models, the volatility is assumed to be a function of spot and time to maturity, which is determined by calibrating to market quotes. The displaced diffusion and CEV model discussed earlier, are parametric forms of local volatility models. 46 CHAPTER 4. SMILE-CONSISTENT OPTION PRICING MODELS and ( ) p1 − 2ρz + z2 + z − ρ x(z) = log (4.24) 1 − ρ To make the model’s qualitative behavior apparent, implied volatility in equation 4.22 is approxi- mated with: α 1 K 1 K σ (f, K) = 1 − (1 − β − ρλ) log + (1 − β)2 + (2 − 3ρ2)λ2 log2 + ... (4.25) B f 1−β 2 f 12 f provided that the strike K is not too far from the current forward. The parameter ν λ = f 1−β (4.26) α is the ratio that measures the volatility of volatility at the current forward. The curve that at-the-money volatility traces is known as the backbone, while the smile and the skew refer to the implied volatility σB(f, K) as a function of strike K for a fixed forward price f. When the strike equals the forward price, the implied volatility is given by the first term in equation 4.25, namely α σ (f, f) = (4.27) B f 1−β This means that the backbone is almost fully determined by the β exponent. For β = 0, the backbone −1 σB(f, f) = αf is downward sloping while for β = 1 the backbone is nearly flat. The second term 1 K − (1 − β − ρλ) log (4.28) 2 f represents the skew, the slope of the implied volatility with respect to the strike K. It comprises of 1 K two parts: the beta skew − 2 (1 − β) log f and the skew caused by the correlation between the volatil- 1 K ity and the forward price, 2 ρλ log f . The beta skew is downward sloping since 0 ≤ β ≤ 1. Since usually the volatility and the asset price are negatively correlated it follows that volatility would decrease (increase) when the forward f increases (decreases). Therefore a negative correlation ρ causes a downward sloping skew. 1 2 2 K The third term in equation 4.25 also contains two parts: 12 (1 − β) log f which appears as a 1 2 2 2 K quadratic term (a smile) and 12 (2 − 3ρ )λ log f is the smile induced by the so called volga ef- fect4. The SABR model (equation 4.20) has four parameters: α, β, ν and ρ. The three parameters α, ρ and ν have different effects on the curve: the parameter α mainly controls the overall height of the curve, changing the correlation ρ controls the curve’s skew and changing the vol of vol ν controls how much smile the curve exhibits. Parameter β changes the implied volatility curve in similar ways as ρ. Below we show these effects graphically.

4This smile arises since unusually large movements of the forward happen more often when the volatility α increases, and less often when α decreases. 4.3. SABR STOCHASTIC VOLATILITY MODEL 47

Figure 4.4: Implied volatility curve with different values of α: small dashed curve α = 0.1, medium dashed α = 0.2 and the solid curve α = 0.3. The other input values are: S = 30, β = 1, ρ = −1 and ν = 0.3.

Figure 4.5: Implied volatility curve with different values of ν: small dashed curve ν = 0.9 , medium dashed ν = 1.5 and the solid curve ν = 1.9. The other input values are: S = 30, β = 1, ρ = 0.1 and α = 0.2. 48 CHAPTER 4. SMILE-CONSISTENT OPTION PRICING MODELS

Figure 4.6: Implied volatility curve with different values of ρ: small dashed curve ρ = −1 , medium dashed ρ = −0.5 and the solid curve ρ = 0.5. The other input values are: S = 30, β = 1, ν = 0.3 and α = 0.2.

In summary, this chapter introduced three smile-consistent option pricing models. Among these, only the SABR model can produce an ”extreme” skew which is seen in our data set (see chapter 3). In the next chapter we examine the same models from a different point of view: how well they can ﬁt observed market prices and what dynamics of IV surface do they predict for the future. Chapter 5

Comparison of Implied Volatility Dynamics of Smile-Consistent Models and Empirical Data

Liquidity of plain vanilla options in markets ensures that all market participants know the value of these options. Therefore, liquid options can be priced by observing market values and generating the corresponding implied volatilities. Black-Scholes can then be used safely to price these trades. However, the problems arise with illiquid options. Examples include deep out-of-the-money or deep in-the-money European options. A common solution is to fit smile-consistent models to observed option prices and then interpolate prices over other strikes and expiries. The calibrated model can then be used to price these options. Even more complicated is the pricing of exotic options, which typically have complex payoffs and/or early exercise features. Examples include American options, Barrier options or callable structures where the payoff depends on embedded options. The value of these options does not only depend on the volatility smile observed today, but also on its time evolution. Besides valuation, it is important to keep in mind that in order to hedge positions taken in these exotic options, accurate and reliable estimation of hedge sensitivities and consistent pricing of plain vanilla hedging instruments in time is required. Typically, when a certain model is employed to price an exotic option, the implied smile dynamics is therefore intrinsically used in the pricing and estimation of hedge sensitivities. Different models will typically yield different prices. It is therefore important, not only to assess whether a model is able to fit the smile today, but in addition to verify whether the smile dynamics implied by these models, is correct. To the best of our knowledge, quantification of the smile dynamics has not been done rigorously so far. People has addressed this by looking at evolution of the smile based on asymptotic analysis and qualitative investigations. A lot of work has been done on the comparison of models for prices of for example barrier options (Reb04; HKLW02). Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). In this work, we use similar statistical techniques to quantify the smile dynamics that is implied by the smile-consistent models described in the previous chapter. To obtain a time-series of the model implied volatility smile, we carry out the following numerical experiment:

• At each day we calibrate the model to the smile that is observed on that day;

• The model parameters are frozen, and a set of options along strike and expiry are priced on

49 50 CHAPTER 5.

the next day with the corresponding spot of that day. These prices are then used to extract the evolution of the implied volatility surface.

• This is repeated for every day in the sample.

Using the time-series that is obtained in this way, we analyze in this chapter the smile dynamics following the approach used in the empirical studies. 5.1. FITTING TO MARKET DATA 51

5.1 Fitting to Market Data

There are different approaches used to fit models to options data. One way, that is considered the least data polluting, is to fit directly to observed option prices. Another approach is to fit to implied volatilities. It should be noted that in the following, we use the latter approach to fit the models under consideration, namely, displaced diffusion, CEV and the SABR model. Similar to the quantification of the empirical data we apply a smoothing procedure here. Using the interpolation technique described in A.2, the small number of observable implied volatilities are converted into a smooth surface of implied volatilities as a function of moneyness and time to maturity. In theory, if the model has n parameters, and it is correctly specified, the prices of a set of n options would uniquely determine the values of parameters. In practice, a model is not expected to be fully correct, and therefore we use more options prices than parameters to find a best fit. The fit is usually achieved by a non-linear minimization procedure in an n-dimensional space that can have several local minima. If the model with which we are trying to reproduce the observed prices has few parameters, the fit is likely to be poor. In case a model has a large number of parameters then the fit will be good but it is likely to have several minima. However, each parameter in the model should have an intuitive meaning, and therefore smile-consistent models proposed in practice typically consist of only a few parameters. More precisely, the problem is defined as follows. Let us assume that we can observe a set of 1 traded implied volatilities , yi, for i = 1, 2,...,M with moneyness mi and expiry τj, where j = 1, 2,...,N. Consider the model implied volatility f as a function R2 → R parameterized by n coefficients, β = (β1, β2, . . . , βn), that enables us to associate an implied volatility to an arbitrary 2 j pair (mi , τj). The task is to find the coefficients β, such that some discrepancy between the model and real implied volatilities is minimized. This distance is chosen to be the sum of squared differences between the model and the market implied volatilities:

M X j 2 d(β) = [f(mi , β) − yi] (5.1) i=1

A numerical solution to the problem of minimizing a function can be obtained by the Levenberg- Marquardt nonlinear least squares minimization algorithm3 (Mar63). In the following we will inves- tigate the ﬁtting that is obtained for each model.

5.1.1 Displaced diffusion model In chapter 4, we have documented that displaced diffusion model is capable of producing a skewed implied volatility across moneyness. However, for the skew to be apparent, it was shown that the displacement coefficient must take values as large as the stock price itself. We have argued that in this case, the probability mass of the stock price shifts to negative values: the larger the displacement coefficient, the larger the probability that the stock takes negative values. Figures 5.1 and 5.2, illustrate a typical fit that is obtained with this model. The fit is poor in general, showing a small improvement for long maturity options. This is expected since long maturity options display a less pronounced

1We can talk of implied volatilities in the same way we talk about option prices, since there is a one-to-one relationship between the two. 2 Notice the dependence of mi in j. This reflects the fact that the fit is applied separately for each expiry. 3In our work, we use levmar package which is a C/C++ implementation of the Levenberg-Marquardt algorithm. It is distributed under the GNU General Public License. 52 CHAPTER 5. skew. The mean absolute error of the fit is 3.1 %. This error is large since implied volatility typically lies in the range from 20% to 40%. The inability of displaced diffusion to produce heavy skews, is inline with findings documented by (Reb04) as well as those by (Mar99).

5.1.2 CEV model Similarity between displaced diffusion and CEV is seen in their ability to fit to observed implied volatilities. The quality of the fit, for the same day as shown in the case of displaced diffusion, is illustrated in figure 5.3 and 5.4. A poor fit is observed, which can be explained by looking at figures in chapter 4: the maximum skew that CEV can generate, is yet smaller than that observed for options with short time to maturity. However, it shows some improvement compared to displaced diffusion: for every time to maturity, the fit is slightly better for CEV. As time to maturity increases, the fit improves since the empirical skew in these cases is flatter. Fits for other days in our sample look similar. The mean absolute error is 2.4%.

5.1.3 SABR model A relatively large number of parameters enables SABR model to fit very closely to observed implied volatilities: each parameter in the model has a unique role, giving the model the flexibility to have certain implied volatility level, skew and slope. The fits obtained for implied volatility surface on 15th of June, 2001 are given in figures 5.5 and 5.6. Fits obtained for other days in our data set, look similar. The mean absolute error is 0.3%. 5.1. FITTING TO MARKET DATA 53

Figure 5.1: Displaced diffusion quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (a, σa), are: (80% S0, 0.233); (100% S0, 0.231); and (100% S0, 0.228), respectively for τ = 0.083, τ = 0.24 and τ = 0.4. 54 CHAPTER 5.

Figure 5.2: Displaced diffusion quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (a, σa), are: (100% S0, 0.225); (100% S0, 0.221); and (100% S0, 0.219), respectively for τ = 0.55, τ = 0.7 and τ = 0.85. 5.1. FITTING TO MARKET DATA 55

Figure 5.3: CEV quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (α, σ), are: (0.02, 0.222); (0.01, 0.231); and (0.01, 0.227), respectively for τ = 0.083, τ = 0.24 and τ = 0.4. 56 CHAPTER 5.

Figure 5.4: CEV quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (α, σ), are: (0.01, 0.225); (0.01, 0.222); and (0.01, 0.220), respectively for τ = 0.55, τ = 0.7 and τ = 0.85. 5.1. FITTING TO MARKET DATA 57

Figure 5.5: SABR quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (α, ν, ρ), are: (0.21, 1.42, −0.49); (0.21, 1.28, −0.51); and (0.21, 1.11, −0.533), respectively for τ = 0.083, τ = 0.24 and τ = 0.4. 58 CHAPTER 5.

Figure 5.6: SABR quality of fit on 15th of June, 2001. Figures in the left part present implied volatilities as a function of moneyness, for fixed time to maturity τ. Associated option prices are shown in the right figures, in a log-log graph. The green line is the empirical implied volatility (price) and the red dots denote the model implied volatility (price). Estimated values for parameters (α, ν, ρ), are: (0.21, 0.94, −0.55); (0.21, 0.76, −0.57); and (0.21, 0.64, −0.57), respectively for τ = 0.55, τ = 0.7 and τ = 0.85. 5.2. SMILE DYNAMICS 59

5.2 Smile Dynamics

The previous section shows that it is possible to tune parameters of the models such that model IVs match closely those observed in the market. The quality of the fit is usually a result of the number of parameters, the more parameters a model has, the better the fit. For example, displaced diffusion and CEV model cannot produce the ”extreme” skew exhibited by options in the equity market. Among the three models, only SABR can generate a volatility surface that matches the one implied empirically. In this section, we will assess the quality of the models in their ability to predict future volatility surfaces. Let St be the closing stock price on day t. There are M implied volatilities with various moneyness and maturities, traded on this stock during day t. Using a fitting algorithm as described in the previous section, n model parameters are estimated. Implied volatilities of the next day, t + 1, are obtained using the model with input stock price St+1 and parameter values β1, β2, ..., βn estimated from day t. When this procedure is applied for every day in our sample, the result is a time-series of daily IV surfaces. These volatilities are said to be predicted by the model one day in the future. Similar to the empirical case (see chapter 3), an analysis of variance is applied to predicted IVS time-series. Since the resulting eigenmodes describe the dynamics of surfaces that are predicted by a model, we refer to them as ”modes implied by the models”. In the remainder of this section we are going to examine the dynamics of IVs as predicted by the models and compare it to market dynamics. In the following, we show the actual stock price (index level) during the period of our analysis (see figure 5.7). The price is volatile, with highest value reaching 1514.09 and the lowest one equal to 797.7. The at-the-money volatility during the same period is shown in figure 5.8.

Figure 5.7: S&P500 index level, during the period of analysis.

5.2.1 Displaced diffusion model In the previous section, we have seen that displaced diffusion model ﬁts poorly to out-of-the-money IVs. This will directly impact its ability to predict volatilities in the future. Figure 5.9 illustrates the 60 CHAPTER 5.

Figure 5.8: Empirical at-the-money volatility, during the period of analysis.

first eigenmode implied by displaced diffusion. It accounts for 98% of daily IV variation. All its components are positive, thus it is interpreted as a level mode. Other eigenmodes are of little importance compared to the first one, reflecting the fact that displacement diffusion is unable to produce a highly skewed IV. The second mode, illustrated in figure 5.10, explains 1% of the remaining variance. It captures the change in slope of the surface in maturity direction4. In empirical IVs, this mode is the sixth most important, accounting for a very small portion of variance. The change in curvature in maturity direction is captured by the third mode, illustrated in figure 5.11. The mode that captures the IV skew (in the moneyness direction), comes fourth in the order of importance, explaining a tiny portion of variance. It is given in figure 5.12. The fact that all the variation of IV is captured by a level mode only, indicates that displaced diffusion is unable to predict future volatilities in markets where skew is a persistent stylized fact.

4This is not a merit of displaced diffusion, but is merely due to the fact that the ﬁtting is done separately for each maturity. 5.2. SMILE DYNAMICS 61

Figure 5.9: Displaced diffusion ﬁrst eigenmode, interpreted as the level. This mode accounts for most of the variance of implied volatility surface, namely 98%.

Figure 5.10: Displaced diffusion second eigenmode, interpreted as the slope in maturity direction. This mode accounts for 1% variation of daily implied volatility. 62 CHAPTER 5.

Figure 5.11: Displaced diffusion third eigenmode, interpreted as the curvature in maturity direction. This mode accounts for 0.05% variation of daily implied volatility.

Figure 5.12: Displaced diffusion fourth eigenmode, capturing the skew. This mode accounts for 0.01% variation of daily implied volatility. 5.2. SMILE DYNAMICS 63

In the following, we will compare the time-series of at-the-money IV predicted by displaced diffusion and that observed in the market. Since at-the-money IV changes in level only, this test will show if the model is at least capable of predicting the level correctly. Figure 5.13 illustrates the deviation of predicted at-the-money IV from that observed empirically. Although the prediction is done for only one day in the future, the absolute error is quite large5. In addition, the actual slope and curvature time-series are compared to their empirical counterparts. For fixed maturity, denoting with f(matm) the IV of at-the-money strike, we compute the slope/skew as f(m + h) − f(m − h) f 0 = atm atm , (5.2) 2h and curvature as f(m + h) + f(m − h) − 2f(m ) f 00 = atm atm atm . (5.3) (2h)2 Figure 5.14 illustrates the time-series of the slope predicted by displaced diffusion, along with the empirical one. A very small variation in the predicted slope, confirms the tiny portion of inertia given to the eigenmode that captures the change in slope. This is remarkable, since displaced diffusion is known as an easy extension of the Black-Schols model to incorporate skew. However, as demonstrated here and in other studies, it can only fit to low skew environments. A similar conclusion can be drawn about the change in curvature, given in figure 5.15: as expected, displaced diffusion predicts no change in curvature of IVs.

Figure 5.13: At-the-money implied volatility predicted one day in the future by displaced diffusion, in red, and that observed empirically (green).

5One possible reason for this is the unconstrained estimation of the volatility parameter. This can be overcome by restricting the ﬁtting procedure such that at-the-money volatilities match. 64 CHAPTER 5.

Figure 5.14: Implied volatility skew predicted one day in the future by displaced diffusion, in red, and that observed empirically (green).

Figure 5.15: Implied volatility curvature predicted one day in the future by displaced diffusion, in red, and that observed empirically (green). 5.2. SMILE DYNAMICS 65

5.2.2 CEV model The small improvement in the fit of CEV to out-of-the-money IVs, compared to displaced diffusion, will be reflected in the accuracy of prediction. Short maturity IVs are excluded in the remaining part of analysis, due to their sensitivity to numerical error. In numerical test it was found that the approximation of chi square distribution used by us, is sensitive to numerical errors for very short maturities. Figure 5.16 illustrates the first eigenmode, which accounts for 98% of the daily IV variation. All its components are positive, thus it is interpreted as a level mode. Second and third eigenmodes, explaining 1.4% of the remaining variance, look the same as in the case of displaced diffusion. Again, these modes capture the term structure variation of IV, and thus cannot be attributed to the model. The mean absolute error of IV surfaces, when predicting one day in the future, is equal to 2.6%. The time

Figure 5.16: CEV first eigenmode is interpreted as the level mode. This mode accounts for 98% of daily implied volatility variation. series of at-the-money IV predicted by CEV model deviate from empirical ones (see figure 5.17). The variation in slope predicted by CEV is larger compared to displaced diffusion (see figure 5.18). In terms of IV curvature, CEV does not show any improvement. Hence the time-series of the curvature looks similar to that of displaced diffusion. 66 CHAPTER 5.

Figure 5.17: At-the-money implied volatility predicted one day in the future by CEV, in red, and that observed empirically (green).

Figure 5.18: Implied volatility skew predicted one day in the future by CEV, in red, and that observed empirically (green). 5.2. SMILE DYNAMICS 67

5.2.3 SABR model The ability of SABR model to fit to market IVs very closely, enables SABR model to imply the same dynamics for the IVS as that observed in the market. First we present the results when the prediction is done one day ahead. Figures 5.19, 5.20 and 5.21 illustrate the level, skew and curvature modes implied by SABR, respectively. It is clear that they are very close to eigenmodes obtained from empirical IV surfaces (see chapter 3). This suggests that SABR model is able to capture the dynamics of IV surfaces, therefore it should be able to predict accurately prices in the future. Next, we examine the at-the-money volatility variation as predicted one day ahead by SABR model. Since at-the-money volatility cannot exhibit a skew/smile, this is equivalent with looking at the change in level of the surface. Figure 5.22 illustrates that the prediction is satisfactory. The mean absolute error6 is 1.3%. When volatilities are predicted one day in the future, slope and curvature time-series implied by SABR almost match the empirical ones. This is illustrated in figures 5.23 and 5.24, confirming the resulting egienmodes. In addition, we look at the predicting capabilities when the time lag is varied from one day to one week. When volatilities are predicted one week in the future, level, slope and curvature captured by SABR clearly deviate from empirical ones. This is illustrated in figures 5.25, 5.26 and 5.27. Average absolute error of prediction increases to 2.6%. This is further emphasised when the time lag is set to two weeks, where the level, slope and curvature imposed by SABR deviate from empirical ones even more. This is illustrated in figures 5.28, 5.29 and 5.31. The error in this case, increases to an average of 3.3%. Again, the eigenmodes produced by predicted IV surfaces are the same as before: level mode, accounting for 79% of variance; skew mode which accounts for 15.6%; and the smile, explaining 2.6% of the variance. An example IV surface predicted one day, one week and two weeks in the future, is illustrated in figure 5.30.

Figure 5.19: First eigenmode implied by SABR model, interpreted as a level.

6The error is computed as the absolute error between the model predicted and empirically observed surface, averaged along all days in our sample and strike and expiry axes. 68 CHAPTER 5.

Figure 5.20: Second eigenmode implied by SABR model, interpreted as a skew.

Figure 5.21: Third eigenmode implied by SABR model, interpreted as a curvature. 5.2. SMILE DYNAMICS 69

Figure 5.22: At-the-money implied volatility predicted one day in the future by SABR, in red, and that observed empirically (green).

Figure 5.23: Implied volatility skew predicted one day in the future by SABR, in red, and that observed empirically (green). 70 CHAPTER 5.

Figure 5.24: Implied volatility curvature predicted one day in the future by SABR, in red, and that observed empirically (green).

Figure 5.25: At-the-money implied volatility predicted one week in the future by SABR, in red, and that observed empirically (green). 5.2. SMILE DYNAMICS 71

Figure 5.26: Implied volatility skew predicted one week in the future by SABR, in red, and that observed empirically (green).

Figure 5.27: Implied volatility curvature predicted one week in the future by SABR, in red, and that observed empirically (green). 72 CHAPTER 5.

Figure 5.28: At-the-money implied volatility predicted two weeks in the future by SABR, in red, and that observed empirically (green).

Figure 5.29: Implied volatility skew predicted two weeks in the future by SABR, in red, and that observed empirically (green). 5.2. SMILE DYNAMICS 73

Figure 5.30: Implied volatility surface predicted with SABR, in blue, and empirical implied volatility surface, in green color. From upper to lower graph: the surface is predicted one day, one week and two weeks in the future. 74 CHAPTER 5.

Figure 5.31: Implied volatility curvature predicted two weeks in the future by SABR, in red, and that observed empirically (green).

In summary, this chapter documented the ability of considered models to fit market IVs and predict them in the future. Among the three models, only SABR is able to fit closely to observed IVs. This is then reflected in the models’ ability to predict future IV surfaces. The latter feature was quantified by applying an analysis of variance on the predicted IV surfaces, a technique that is commonly applied to empirical IVS. The resulting dynamics is compared between empirical and the model predicted surfaces. It was found that in markets where options exhibit extreme skew, like the one considered here, displaced diffusion and CEV model cannot be used reliably since they predict the wrong dynamics of IVS. SABR model in the other hand, was shown to be able to capture the right dynamics of IVS. Thus the predicted surfaces by this model remained close enough to empirical ones, for a reasonable time lag used for prediction. This good performance of SABR is attributed to its richness in the number and role of its parameters. Chapter 6

Discussion and Future Work

The previous chapter showed that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor ﬁtting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other hand, was shown to be able to capture the smile dynamics very closely to the empirically observed dynamics. An important point to note is that, although the PCA characterizes the dynamics, it is not the whole story. It was found that when prediction is done further ahead in the future, the difference between the surface predicted by the model and the one obtained empirically increases substantially, while the characteristics in terms of principal components remained similar. This was studied in more detail for the SABR model where the predicting capabilities were found to decrease substantially when the time-lag1 increases. This project can be extended by including other smile-consistent models in the analysis. Examples include Heston model, the mixture of log-normals etc. Another possible extension would be the examination of other options market, e.g. options on futures. In these markets an actual smile, rather than a skew, is the prevailing stylized fact.

1See section 5.2.3.

75 76 CHAPTER 6. DISCUSSION AND FUTURE WORK Appendix A

A.1 Additional Results on Liquidity Imbalance

In this section we show additional results on the traded volume of options. In ﬁgures A.1, A.2 and A.3 the traded volume is shown for different dates, namely 17th Dec. 1999, 20th Dec 2000 and 24th Jun 2002. In the upper plots an interest rate of zero is assumed, while in the lower plots the effect of interest rates is shown when assuming an interest rate of 4 percent1. We clearly see that out-of-the- money options are more liquid. Interest rate introduces a small shift in the moneyness axes, which can be understood from its deﬁnition: K m = rτ (A.1) S0 e where m is the moneyness, K is the strike, S0 is the spot price, r is the interest rate and τ is time to maturity.

1We choose r = 4% as an average interest rate over the years we analyse. This value is not used anywhere else in our analysis.

77 78 APPENDIX A. on 17th Dec. 1999. From left to right, maturities are approximately three months, six 500 P & S Figure A.1: Volume of callmonths and and one put year. options The in taking volumes interest in rate the equal upper to panels 4%.Thethat are blue liquidity created bars is assuming correspond higher an to for interest call at-the-money options rate and whereas equal the out-of-the-money to red options. zero bars and represent their the lower traded counterparts volume are of created put options. It is clear A.1. ADDITIONAL RESULTS ON LIQUIDITY IMBALANCE 79 on 20th Dec. 2000. From left to right, maturities are approximately three months, six 500 P & S Figure A.2: Volume of callmonths and and one put year. options The in taking volumes in interest the rate upper equal panels toclear are 4%. that created liquidity assuming The is an blue higher interest for bars rate at-the-money equal correspond and to to out-of-the zero call money and options. options their whereas lower counterparts the are red created bars represent the traded volume of put options. It is 80 APPENDIX A. on 24th of June 2002. From left to right, maturities are approximately three months, six 500 P & S months and one year. Thetaking volumes in interest the rate upper equal panels toclear are 4%. that created liquidity assuming The is an blue higher interest for bars rate at-the-money equal correspond and to to out-of-the-money zero call options. and options their whereas lower counterparts the are red created bars represent the traded volume of put options. It is Figure A.3: Volume of call and put options in A.2. DATA SMOOTHING 81

A.2 Data Smoothing

The estimation of implied volatility surface (IVS) is a challenge for two reasons:

1. Its rich patterns accross strikes ﬂuctuate through time,

2. The daily IV observations are only available for a limited number of maturities.

As a result it is difﬁcult to justify the use of a parametric form to estimate the IVS. We use a non-parametric estimator that does not enforce any a priori restrictions on the estimate, namely the Nadaraya-Watson estimator ((Fen05)). In non-parametric smoothing one estimates the regression relationship:

yi = r (xi) + i, i = 1 . . . , n. (A.2)

n where we are given a data set {(xi, yi)}i=1 and r is the regression function to be found. The variables xi are called explanatory variables, yi are the response variables and n is the number of observations. In the Nadaraya-Watson method one ﬁts the data to a local constant within a small neighborhood. In the case of IVS xi is a vector of two components, moneyness m and maturity τ, and yi is the respective IV value which is estimated by calculating a weighted average of the data locally:

n 1 X rˆ(x) = ω (x) y (A.3) n i,n i i=1

The weights ωi,n are chosen such that they favor the input corresponding to points closer to x compared to those further away. The different non-parametric techniques differ among each other in the way these weights are computed. This is done through so-called kernel functions K(·). These functions are similar to probability density functions, they integrate to one and furthermore they are symmetric. There are a few kernel functions described in the literature, namely the quartic, Epanechnikov, Gaussian kernel function etc (Fen05). In practice, choosing a speciﬁc kernel function is not of major importance. In this work we use the Gaussian kernel which is given by:

1 −u2 K(u) = √ exp 2 (A.4) 2π The degree of smoothing depends on the bandwidth, which is denoted by h. This is included in n the kernel function for a given data set {(xi, yi)i=1}, as follows: 1 x − x K (x − x ) = K i (A.5) h i h h Since estimating IVS involves two explanatory variables, moneyness and maturity, one has to use two dimensional kernel functions. The kernel used here has the form:

u2 u2 − 1 − 2 1 1 2h2 2h2 K(u1, u2) = exp 1 exp 2 (A.6) h1h2 2π

Variable u1 represents moneyness values and u2 the maturity values. 82 APPENDIX A. Bibliography

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