Pricing Lookback Options under Multiscale

Stochastic

CHAN Chun Man

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Philosophy

•“ in

Statistics

‘ ⑥The Chinese University of Hong Kong

July 2005

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate

School. ifTTDTir)! iSi^^一扇 system/-^ Abstract of thesis entitled:

Pricing Lookback Options under Multiscale

Submitted by CHAN Chun Man for the degree of Master of Philosophy in Statistics at The Chinese University of Hong Kong in July 2005.

ABSTRACT

This thesis investigates the valuation of lookback options and dynamic fund protection under the multiscale stochastic volatility model. The underlying as- set price is assumed to follow a Geometric Brownian Motion with a stochastic volatility driven by two stochastic processes with one persistent factor and one fast rrieari-reverting factor. Semi-analytical pricing formulas for lookback options are derived by means of multiscale asymptotic techniques. Effects of stochastic volatility to options with lookback payoffs are examined. By calibrating effective parameters from the volatility surface of vanilla options, our model improves the valuation of lookback options. We also develop model-independent parity rela- tion between the price functions of dynamic fund protection and quanto lookback options. This enables us to investigate the impact of iriiiltiscale volatility to the price of dynamic fund protection.

i 摘要

本文研究回顧選擇權和動態基金保障的定價。傳統以來，回顧選擇權的定

價都是根據Black Scholes模型中的資産價格行為而定°本文使用更符合現實的

模型-多尺度隨機波幅模型。我們假設資産價格行為依循有隨機波幅的幾何布

朗運動。模型内的隨機波幅由兩組隨機過程控制，一組為持續性的因素，另一

組為快速平均數復歸的因素。利用多尺度漸近的技術，我們求出回顧選擇權的

半解分析定價公式，並且分析了多尺度隨機波幅對回顧選擇權的影響。透過歐

式選擇權的微笑波幅，我們可以校正有效的參數，從而改良回顧選擇權的定

價。我們也建立了回顧選擇權和動態基金保障之間的獨立模型關係。透過這個

關係，我們可以探討多尺度隨機波幅對動態基金保障的影響。

ii ACKNOWLEDGEMENT

I thank my God who gives me an opportunity to study a master degree at this place and at this moment, so that I can meet my supervisor, Professor Wong

Hoi-Yirig. I would like to express gratitude to my supervisor, Professor Wong

Hoi-Yirig, for his invaluable advice, generosity of encouragement and supervision on private side during the research program. I also wish to acknowledge my fellow classmates and all the staff of the Department of Statistics for their kind assistance.

iii Contents

1 Introduction 1

2 Volatility Smile and Stochastic Volatility Models 6

2.1 Volatility Smile 6

2.2 Stochastic Volatility Model 9

2.3 Multiscale Stochastic Volatility Model 12

3 Lookback Options 14

3.1 Lookback Options 14

3.2 Lookback Spread 15

3.3 Dynamic Fund Protection 16

3.4 Floating Strike Lookback Options under Black-Scholes Model . . 17

4 Floating Strike Lookback Options under Multiscale Stochastic

Volatility Model 21

4.1 Multiscale Stochastic Volatility Model 22

4.1.1 Model Settings 22

4.1.2 Partial Differential Equation for Lookbacks 24

4.2 Pricing Lookbacks in Multiscale Asymtoeics 26

..4.2.1 Fast Tirriescale Asyiiitotics 28

4.2.2 Slow Tiriiescale Asymtotics 31

iv 4.2.3 Price Approximation 33

4.2.4 Estimation of Approximation Errors 36

4.3 Floating Strike Lookback Options 37

4.3.1 Accuracy for the Price Approximation 39

4.4 Calibration 40

5 Other Lookback Products 43

5.1 Fixed Strike Lookback Options 43

5.2 Lookback Spread Option 44

5.3 Dynamic Fund Protection 45

6 Numerical Results 49

7 Conclusion 53

Appendix 55

A Verifications 55

A.l Formula (4.12) 55

A.2 Formula (4.22) 56

B Proof of Proposition 57

B.l Proof of Proposition (4.2.2) 57

C Black-Scholes for Lookback Options 60

Bibliography 63

V List of Figures

2.1 against 8

2.2 Implied Volatiliy against LMMR 10

2.3 Implied Volatilities on the fitted volatility surface 13

6.1 Fixed strike lookback 50

6.2 Fixed strike lookback 52

vi Chapter 1

Introduction

Lookback options provide oppoitiiiiities for the holders to realize attractive gains in the event of substantial price movement of the underlying assets during the life of the options. For instance, the floating strike lookback call allows the holder to purchase the underlying asset with set as the minimum asset price over a given time period. Investors who speculate on the price volatility of an asset may be interested in the lookback spread option which payoff depends on the rlifForenco l)ctweon maximum and minimum of asset prices over a horizon of time. More exotic forms of lookback payoffs are discussed in El Babsiri and Noel

(1998) arid Dai, Wong and Kwok (2004).

In the insurance industry, lookback features appear in many insurance prod- ucts. Lee (2003) proposed that equity-indexed annuities (EIA) can be embedded with lookback feature. The concept of dynamics fmid protection in insurance was first introduced by Gerber and Shiu (1999). Imai and Boyle (2001) related

1 this concept to the payoff of a lookback option and derived the iriid-contract val- uation. Gerber and Shiu (2003) considered perpetual equity-indexed annuities with dynamic protection and withdrawal right, where the guarantee level is an- other stock index. With logiiorrrial asset price movement, Cliu arid Kwok (2004) analyzed the dynamic fund protection in detail and showed that the proposed scheme of Gerber and Shiu (2003) is indeed related to a lookback option payoff.

The valuation of lookback options presents interesting mathematical chal-

lenges. Under the Black-Scholes (1973) assumptions, the pricing of lookback

options becomes more transparent. For instance, analytic formulas for one-asset

lookback options have been systematically derived by Goldman et al. (1979) and

Coiize and Viswanathaii (1991). He et al. (1998) derived joint density functions

for different combinations of the maximum, the minimum and the tenriiiial asset

values. These density functions are useful in pricing lookback options via iiuirier-

ical scheme or Monte Carlo methods. Dai, Wong and Kwok (2004) derived closed

form solution for quaiito lookback options. Wong and Kwok (2003) proposed a

new pricing strategy for various types of lookback options by means of replicat-

ing portfolio approach. This strategy develops model-independent parity relation

between different lookbacks and derives analytic representation for multi-state

lookback options.

However, the Black-Scholes (BS) assumptions are hardly satisfied in the prac-

tical financial market, especially the constant volatility hypothesis. As volatility

"smile" is commonly observed in financial markets, various methods are pro

2 posed to capture this effect. Two successful models are jump-diffusion models and stochastic volatility (SV) models. The formal approach is more suitable for short maturity options whereas the latter one is more suitable for medium and long maturity options. In this thesis, we concentrate on the latter approach.

A typical SV model assumes volatility to be driven by stochastic process. Hull and White (1987) examined pricing vanilla options with instantaneous variance

modelled by Geometric Browniari motion. Hestoii (1993) obtained analytical

formulas for options on bonds and currency in terms of characteristic functions

by using a mean-reverting stochastic volatility process. Fouqiie et al. (2000)

observed a fast tirriescale volatility factor in S&P 500 high frequency data and

derived a perturbation solution for European and American options in a fast

inean-revertirig stochastic volatility world. The assessirierit of accuracy of their

analytic approximation is reported in Fouque et al. (2003a).

Two advantages of using fast irieaii-reverting SV are that the number of pa-

rameters offcctivoly used in the model can bo much reduccd and ofFectivc param-

eter values can be calibrated by a simple linear regression approach. These make

the model implement able in the practical financial market and hence motivate

research along this line. There have been several theoretical works on pricing

exotic options under the fast irieari-reverting volatility assumption, see for exam-

ple: Fouque and Han (2003), Cotton et al. (2004), Wong and Cheung (2004) and

Ilhaii et al. (2004).

Empirical tests however suggest a modification in stochastic volatility models.

3 The empirical study of Alizedeh et al. (2002) documented that there are two fac- tors governing the evolution of the stochastic volatility with one highly persistent factor and one quickly rneari-revertirig factor. Thus, Fouque et al. (2003b) iriod- ified their early work by considering the multiscale stochastic volatility model,

and managed to calibrate all effective parameters from volatility smiles. As the extension of the multiscale volatility model to path-dependent option pricing is

non-trivial and indispensable, the present paper considers the valuation of look-

back options with this model. This work is also inspired by Imai and Boyle

(2001), who suggested that future research should take a look at the dynamic

fund protection under a two-factor stochastic volatility model.

This thesis contributes to the literature in the following aspects. We derive

asymptotic approximation and its accuracy to prices of various types of lookback

options under the two-factor SV model. This enables us to understand how

lookback prices can be adjusted to fit volatility smiles. We also show that floating

strike lookback options are important iristrurrients that can be used to replicate

many lookback options. Specifically, we develop a model-independent result to

replicate dynamic fund protection by quanto floating strike lookbacks. Therefore,

dynamic fund protection under the stochastic volatility model can be analyzed

effectively. To illustrate the use of our model, we provide numerical demonstration

for implementing our model. This allows us to assess the impact of multiscale

volatility on lookback option pricing.

The remaining part of this thesis is organized as follows. In Chapter 2, we

4 introduce volatility smile and SV models. Then we give a brief overview on the iiiiiltiscale stochastic volatility model of Fouque et al. (2003b). Using the S&P

500 index option data, we demonstrate how to perforin calibration to the volatiitly surface. In Chapter 3, we introduce derivatives with lookback features, iiicludirig fixed strike lookback options, floating strike lookback options, lookback spread options and dynamic fund protection. In Chapter 4, we detail the iiiultiscale SV model for pricing lookback options. Specifically, wc derive the partial differential equation (PDE) for lookback options with linear hoiiiogeiious payoff. An asymp- totic solution to the PDE is then established by means of singular perturbation technique, of Fouque et al. (2003b). The accuracy of the analytic approxima-

tion is presented also. In Chapter 5, we develop the inodel-iiidepeiiderit parity

relation to connect price functions of floating strike lookback options with those

products with lookback features. Specifically, we show that dynamic

fund protection can be viewed as a quarito lookback option. To examine the

impact of iiiultiscale volatility to lookback option prices, we perforin nuiiierical

analysis in Chapter 6. With market implied (effective) parameters, we check the

price difference between the Black-Scholes lookback price and our solution. This

enables us to visualize the effect of multiscale volatility model in lookback options

pricing with graphs. Chapter 7 concludes the thesis.

5 Chapter 2

Volatility Smile and Stochastic

Volatility Models

111 this chapter, we introduce the volatility smile and stochastic volatility (SV) models. We will see the application of SV models in capturing volatility surface.

However our focus is the multiscale volatility model proposed by Fouque et al.

(2003b) for pricing European options.

2.1 Volatility Smile

Black and Scholes (1973) assume the following asset price dynamics:

学bt =("-+

where St is the asset price at time t, Wt is the Wiener process, ji, q and a are constant parameters representing drift, dividend yield and volatility respectively.

6 For a call option with payoff max(5r-/v, 0), Black and Scholes derive the pricing formula:

VBs(t,St) = Ste-q�T-t� N�ct)— Ke-T�T-t�N{cn,

hi⑶/幻 + (r - q 土 (T - t) where a = . , cr^T^t where t is the current time, K is the strike price of the option, T is the maturity of the option and r is the risk free interest rate.

In (2.1), the only parameter that is not directly observable is the volatility,

a. Market practitioners usually estimate it by calibrating to traded options data.

That means they set market price to be equal to the BS price and then extract the

volatility. The volatility obtained in this way is called the implied volatility. This

method worked quite well in the early 1980s. However, after the stock market

crashed in Black Monday on 19 October 1987, there is an effect called volatility

skew/smile observed iii the .

After the market crashed, it is discovered that the implied volatility decreases

with the iiiorieyness , the strike price over the current asset price (K/S). This

pattern is shown in Figure 2.1 where the S&P 500 options data are downloaded

from Yahoo on 4 Juiie, 2005. The circles are the implied volatility of call op-

tions with maturity 137 clays. The skow ofFoct is not compatible with the model

assumption that the volatility is a constant.

Rubinstein (1994) suggests that the reason for this effect may be of "crashopho-

bia", the awareness of stock crash like the Black Monday. This results in the

7 0.32 r

0.3-〇

0.28 - 〇

0.26 - o

t 0.24 - °

！ 0.22 - 9b t O t。.2 - O

0.18- O o

0.16- O 〇

〇o 0.14 - O 〇 0 12 I I 1 1 I I ① I ‘0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Moneyness (K/S)

Figure 2.1: Implied Volatility against Moneyness market practitioners believe that returns should not follow a nonrial distribu- tion, rather a distribution that has heavier tails. Therefore two classes of models are proposed to capture the skew effect. They are jump-difTusion models and SV models. The formal approach is more suitable for short maturity options whereas the latter one is more suitable for medium and long maturity options. In this thesis, one of the focus is pricing the dynamic fund protection, an option feature embedded in insurance contract with long maturity, so we are focusing on SV models.

8 2.2 Stochastic Volatility Model

Stochastic volatility model is similar to the Black-Scholes model, except that the volatility is driven by stochastic variable(s). In 1987, Hull and White (1987) introduce the asset price dynamics with a stochastic volatility. They model the instantaneous variance as a Geometric Biowniaii Motion that is indepeiicleiit to the asset return and derive the analytical solution for European options. Stein and Stein (1991) view the volatility itself as a irieari-revertiiig process. Mean reverting process is a process that is pulled backed to the long-run average over time. They obtained an analytical solution by assuming the volatility process to be iiiicorrelated with the asset dynamics.

Heston (1993) relaxed the assumption of Steiii and Stein (1991) to allow cor- relation between assets and volatility. Then, closed form solutions are derived for bond and currency options in terms of characteristic functions. However one has to employ iimrierical Fourier inversion to implement the computation.

Fouqiie et al. (2000) examine the S & P 500 option data and discover that one factor governing the volatility follows a fast mean-revertiiig process. It iriearis that the rriean-revertiiig rate is high. They model the volatility as a positive function with a latent factor, which follows a fast-mean reverting process. They perform perturbation techniques to obtain European and American option prices. Under this framework, a large number of parameters can be reduced into two grouped

parameters only. Moreover, the perturbation solution solely depends on these two

9 0.45 —1 1 1 1 1

-

0.35 - •

f 0.3- ° \ O

5 0.25 - 〇〇〇of〇 -

0.05' ‘ ‘ ‘ ‘ ‘ -2.5 -2 -1.5 -1 -0.5 0 0.5 LMMR

Figure 2.2: Implied Volatiliy against LMMR grouped parameters, the Black-Scholes price and Greeks. The most attractive thing of this approach is that the two grouped parameters can be calibrated with a simple linear regression.

Uiifortiinately their approach has its weakness. Since the skewness of the volatility smiles varies across maturities, a simple linear regression may not be able to capture the whole volatility surface. Under their approach, the implied volatility surface implied by European options is given by

where 07 is the implied volatility and the parameters (斤,b^) are used to obtain the two grouped parameters. ？f is estimated from historical daily index

10 value over one month horizon. After regressing aj on log-irioiieyiiess to maturity ratio (LMMR), we can calibrate a' and Also, LMMR is defined as 冗).In order to show the performance of this approach, we downloaded S&P 500 option prices reported on June 3，2005 from yahoo. The data consists of options with maturity greater than a iiioiith and less than 18 months, and inoiieyiiess between

0.7 and 1.05. Then we make a scatter plot and follow Fouque et al. (2000) to obtain a regression line from the data as shown in Figure 2.2. In this figure, the circles are the implied volatilities plotted against the LMMR and the solid line is the regression line obtained from the simple linear regression. It is observed that simple linear regression cannot capture such situation.

Although the approach of Fouque et al. (2000) is inadequate to capture volatil- ity surface, many researches appeared to price exotic products under this fraiiie- work. For instance, pricing formulas on Asian options, barrier options, lookback options and interest rate derivatives are derived in Fouque and Han (2003), Cot- ton et al. (2004), Wong and Cheuiig (2004) and Ilhaii et al. (2004).

However empirical studies suggest that stochastic volatility should consist of two factors, a slow tirriescale factor and a fast tiniescale factor. Alizeth et al. (2002) perform an empirical study on stochastic volatility model to show the mentioned result empirically. This motivates people to explore the effect of iriiiltiscale SV on derivatives pricing.

11 2.3 Multiscale Stochastic Volatility Model

Fouque et al. (2004) develop a framework to price European options under the multiscale SV eriviroiirneiit. Similarly, the advantage of their approach is that grouped parameters can be calibrated easily. They model the volatility as a pos- itive function of two latent variables, one follows a fast rneari-reverting process and the other one follows a slow meaii-revertiiig process. They derive analytical formulas for European option prices by using perturbation technique. The so- lution is expressed in terms of four grouped parameters, the Black-Scholes price arid Greeks. The four grouped parameters can be calibrated through multiple linear regression.

The approach of Fouque et al. (2004) outperforms that of Fouque et al.

(2000) on capturing volatility surface. Under their approach, the volatility surface implied by European option data is given by

cTi ~ a(z) + + b\T - t)] + [a' + a\T - t)] T — t where the parameters (a(-2；), l/) are related to the four effective grouped parameters. a(z) is estimated from historical daily index value over one month horizon. After regressing cr； on time to maturity, log-moneyness and the interac- tion term, LMMR, we obtain a', If, a^ and

By using the same dataset described in previous section, we produce Fig- ure 2.3. In this figure, the circles are the market implied volatilities plotted against time to maturity and log-riiorieyness in a three-dimensional space. The

12 0, 4 0.4、 ^ ) ^ 〇 〕 3 )- 震。.3、 、\ \、j

10.2、 、 . %

...... 、:.

0.8

1.4 2 Moneyness (K/S) Time to Maturity (Years)

Figure 2.3: Implied Volatilities on the fitted volatility surface surface plotted is obtained from the multiple regression stated in (2.1) The R- square obtained is very close to one and all the points are fitted to the surface with a high quality. This approach can successfully capture the whole volatil- ity surface. Recently, Wong (2005) also generalizes the result to the riiultistate option pricing problem and obtained a closed form solution.

In this thesis, we investigate the impact of niultiscale SV on the derivatives with lookback feature by using Fouque et al. (2004) approach. In chapter 4, we give full details of this niultiscale SV model and the implementation of calibration procedures is discussed too.

13 Chapter 3

Lookback Options

In this chapter, we introduce lookback features of different financial securities.

These products include some popular lookback options in the market, lookback spread option and dynamic fund protection. This chapter ends with a review on the derivation of floating strike lookback option pricing formulas.

3.1 Lookback Options

The payoffs for lookback options involve maximum value or minimuni value of the underlying asset price over a period of time. In the financial market, lookback options arc of two types, floating strike lookbacks and fixed strike lookbacks.

The floating strike call (put) option gives holder the right, but not obligation, to buy(sell) the underlying asset at its iiiiiiiinum(iiiaxiiriuin) value observed during the life of the option. The fixed strike call (put) is a call(put) option on the realized rnaxiiriurn(iriiiiiinum) price over the life of the option. The payoff of the

14 options can be written out mathematically. We use the following notations to indicate the extreme values:

M厂=

rriT = iriiii S” t

Payoff functions of four popular lookback options arc:

1. Floating strike lookback call: Cfi(T, St, mj) = St - mj;

2. Floating strike lookback put: Pfi(T, St, M(f) 二 M^ — St]

3. Fixed strike lookback call: Cfix{T, St, Mq) = max(A/『—K, 0);

4. Fixed strike lookback put: Pf ix{T, St, rri^) = max(A' - mj, 0),

where all payoff functions depend on one lookback variable only.

3.2 Lookback Spread Option

Lookback spread option has payoff depends on both the realized maximum value and rriiriimum value of the underlying asset over a certain period of time. A typical lookback spread option has a payoff:

LspiT, St, Mo^, mj) = max(M『-mj - K, 0)

As the payoff depends on the difference between maximum and minimum of the asset prices, this product allows investors to speculate on the volatility of the underlying asset.

15 3.3 Dynamic Fund Protection

Dynamic fund protection is a protection feature added on a fund. The dynamic fund protection feature ensures that the fund value is upgraded if it ever falls below a certain threshold level. In some insurance policy, there is involving saving terms. The premium paid by the policy holder not only paid for protection premium, but also invested in a underlying fund for savings purpose. In order to increase the attractiveness of the policy, dynamic fund protection can be added into the policy. This protection coiiccpt was first proposed by Gerber and Shiu

(1999).

The iiiechanisin of dynamic fuiid protection can be demonstrated through an example. Suppose an investor holds one unit of underlying fund and protects it with the dynamic fuiid protection. Let K be the protection floor level which can be considered similar to the strike price of a put option. The return rate of

the protected fund will remain the same when the fund lies above the level K.

However ones the fund value goes down below the protection floor K, additional

cash is added instantaneously to the portfolio to bring its value up to protection

level K.

Let P{t) denote the value of the protection. Denote F{t) is the underlying

fund, the terminal payoff of the protected portfolio should be given by,

F{T) max j 1, max —^ \ . 、’ \ 0

The value of the dynamic protection at maturity should be given by subtract-

16 irig the naked fund from the protected one. Hence, we have, see Iiriai and Boyle

(2001),

nT) = F(T)max|l, max^^} - F{T). (3.1)

3.4 Floating Strike Lookback Options under Black-

Scholes Model

In this section, we discuss the pricing of floating strike lookback option under the Black-Scholes assumption. The pricing of lookback options is challenging, siiicc the payoff functions involve the realized cxticiiic value of the underlying

asset over a certain period of time. Recall that Black and Scholes (1973) describe

the asset price dyrianiics under the risk-iieutral measure by using a Geometric

Browiiiaii Motion,

(ISt , 、

where St is the asset price at time t, Wt is the Wiener process, r, q and a

are constant parameters representing risk free interest rate, dividend yield and

volatility respectively. Denote U(t, S, S*) as the price function for a floating

strike lookback option where S^ represents m[) or Mq. The corresponding payoff

function is denoted as H{S, S*).

By Goldman et al. (1979), U{t, S, S*) should satisfy the governing partial

17 differential equation (PDE),

芸+ 广' + ,厂"=0’ 0 < ^ < T, S

The terminal payoff condition is given by U{T, St, Mq) = H{St, S^). Since lookback derivatives involve the path dependent lookback variable too, the price function is needed to satisfy one more boundary condition,

£L,0. (3.3)

The last boundary condition is intuitive. It is because if the current asset price of the lookback put (call) option is the same as the realized maximum(miniiriurii) ill the cuiieiit moment,，one should expect, the probability that the realized iiiax- imum(minimmn) at the maturity remains the same as current asset price is zero.

The value of the floating strike lookback put (call) option should be insensitive to infinitesimal changes in MQ(mf)). Goldiriaii et al. (1979) showed that the change ill option value with respect to iriarginal changes in MQ^rriQ) is proportional to the probability that A/(5(mf)) will still be realized iriaxiiriuiri(iiiiiiimurii) at the maturity. For a mathematical proof on this detail, one may consult Dai, Wong and Kwok (2004).

For lookbacks with linear homogeneous property, i.e.

H(t,SuMl,) = SH(tM(S*/S)) (3.4)

for some function H(-). We can reduce the iiuiriber of independent variables by one with using trarisforriiatiori of variables. One should notice that floating

18 strike lookback put option payoff has this linear homogeneous property. Let

X = \n{Ml^/St) and V = U/S, then the function V{t,x) satisfies the PDE with

Neumann Boundary condition,

CbsV = 0，0 < i < T, x>0, (3.5)

V{T,x) = e^-l,

彻 0

where Cbs is an operator defined as ”

Now, we can reach the solution of floating strike lookback put option. By solving the PDE (3.5) and trarisfonriing back the solution to U{t, St, Mo), the

Black-Scholcs price for floating strike lookback put is given by

p'fi = Mje-r(了-。iV(-Se-収-。AV4,) (3.6)

+ &+…)一^力-广鳴)’

In 杀+ 土引(:r —力） 2(r - a) ,

where N(-) is the cumulative distribution function of a standard normal random variable.

Similarly, we can derive the solution for the floating strike call option. The lookback floating call terminal payoff satisfies the linear homogenous condition and by trarisfonriing x = lri(mf)/5t) arid V = U/S, the function V{t,x) satisfies

19 another PDE:

CbsV = 0, 0

V{T,x) = l-e,

彻 x=o 0

One should notice that the differences between PDE (3.5) and PDE (3.7) are that

the PDEs are defined in different domains of x and with different terminal condi-

tion. After solving the PDE (3.7) and transforming V(t, x) back to U(t, St, 112^), the Black-Scholes formula, cj；, happens to be

= Se普t�聰- mtoe-r�T-t�N((rj (3.8)

+ “卜[i广明-e刷N�’

"m 二 , (Im = dm VT - t. G\J L — t a

We concentrate on pricing floating strike lookback options under the Black-

Scholes Model in this chapter, because these options are fmidameiital iiistruirieiits to replicate other lookback products. The pricing of fixed strike lookback options, lookback spread option and dynamic fund protection are discussed in Chapter 5.

20 Chapter 4

Floating Strike Lookback Options under Multiscale Stochastic

Volatility Model

This chapter is divided into two parts. In the first part, we derive the seriii- aiialytical pricing formula for lookback options with linear homogenous payoffs by using asymptotic technique. Then we present specific results for floating strike lookback options. After that, we demonstrate effective grouped parameters cali- bration by multiple regression with European option data.

21 4.1 Multiscale Stochastic Volatility Model

4.1.1 Model Settings

Denote St as the underlying asset price at time t. We assume that St follows a

Geometric Browriiari Motion where the volatility is a stochastic variable depend- ing on a fast mean-reverting process Yt and a persistent process Zi. Under the physical probability measure, the benchmark stochastic processes for St, Yt and

Zt are: ‘

“ clYt = -(m - Yt)dt + "^dWl^^ (4.1) e Ve

dZt = Sc(Zt)dt + VScj(Zt)d\\f\

where m, e and <) are constant parameters,

,W^'/i) and H/j⑵ are Wiener processes, f[Y, Z) is a positive function representing the volatility of the stock.

When t and 8 are small, the stochastic variable Yt is the fast irieaii-revertiiig factor and the stochastic variable Zt is the persistent factor. We allow a general correlation structure arriong three Wiener processes H^/o), H/•广）and Vl^/?) so that ( \ ( \

1^(0) I I 1 0 0

⑴=Pi yr^ 0 W, (4.2)

� y y P2 p\2 \/l -P2 -Pl2 乂 where Wt is a standard three-dimensional Browriian motion, and the constant correlation coefficients pi, p2, pu satisfy \pi\ < 1 and pg + 离2 < 1. As we

22 can choose any positive function f(Y, Z) to model the volatility, this foriiiulatiori provides sufficient flexibility for describing various types of SV models.

To obtain a pricing formula with reasonable growth at its limiting points, we should impose some regularity conditions for functions, c(Z), g(Z) and /(F, Z), in

(4.1). Specifically, the f(Y, Z) is a smooth function that is bounded and bounded away from zero. The two functions c(Z) arid g{Z) are assumed to be smooth and at most lineally growing at infinity.

No arbitrage pricing theory states that options valuation should be done un- der a risk-neutral probability measure or equivalent martingale measure. With constant volatility, the market is complete and there is a unique risk-neutral mea- sure. However, the present thesis takes into account the iioii-tradable stochastic volatility so that a family of pricing measures is obtained through parameteriz- ing the market price of volatility risk. The market uses one of them in pricing securities. To access the choice of the market, one has to calibrate (effective) risk-neutral parameters from market price and characteristics. This explains why market people requires option prices to fit volatility smiles.

The choice of the market can be described by stochastic differential equations which characterize the process of (4.1) under the risk-neutral measure. Specifi- cally, we define

I ("- /•)肌幻、

W卜 Wt+/ 如, Jo

23 where 7(7/, z) and ^(y, z) are smooth bounded functions of y and z. Hence, the market priccs of volatility risk arc defined as

A('"’'）二 加)

F("，:）= "〉((; + Pl2l{y, z) + yjl- p\-远彻’ z).

The risk-neutral measure used by the market is completely reflected by the func- tional forms of A and F. Calibration to effective parameters implied by A and F will be detailed in the later part of this paper. With A and F fixed, the processes of (4.1) under the risk-neutral measure become

‘dSt 二+

dYt = dt + "^dW^'^* (4.3) e ve 」

dZt = [SciZt) - VSg{Zt.)riYt, Z,)] (It + y/dg(Zt)dWl:^^\

where r is the interest rate, q is the dividend yield and the correlation structure of Wj remains the same as (4.2).

4.1.2 Partial Differential Equation for Lookbacks

Denote U{t, 5, 5*, Y, Z) be the price function for a lookback option where S; represents mf, or ML The corresponding payoff function is denoted as H[S, S*).

Then, risk-neutral valuation asserts that

"‘ U(t, S, S\ y, Z) = E^ s*)^，

24 where Q is the risk-neutral measure under which the processes of (4.3) are de- fined. By the Fcyniiiaii-Kac formula, the partial differential equation (PDE) for lookback options can be obtained as an initial boundary value problem.

In this paper, the PDE formulation focuses on lookback options whose payoffs have the linear homogeneous property stated in equation (3.4), i.e. H{S, S*)=

SH{\n(S*/S)) for some function H(-). One immediately recognizes that floating strike lookback options follow this property but fixed strike lookback options do not. However, I shall demonstrate in the next chapter that fixed strike lookback options can be valued through price functions of floating strike lookback options.

Thus, our consideration has rich enough application.

For lookback options with linear homogeneous payoffs, the niiiiiber of inde- pendent variable involved in the PDE can be reduced by one. Specifically, we present the PDE as follows.

V = 0, 0 < i < T,

V(T,x,Y,Z) = H(x) (4.4)

^ — 0 彻 x=o — ’ where

x = In (575), V = U/S

+ \/ a' = -£e o + ^CVe , + £2 + V^A^i + V e

, ( � a 2 炉 •• A) 二 + Q2 Q £1 = — V%yi"/('"，20^^ + \/^(Pl"/('"，2；) -"A('"，2:))@，

25 Ml = (P2g(z)f(y, z) - g{z)r(y, z)) — - f)2(Az)f{y,

Q2

The governing equation (4.4) is defined for 工〉0 if 二 MJ and for x < 0 if

57 = mf). It may be worth mentioning that the governing equation of (4.4) is the consequence of the Feyniriari-Kac formula followed by transformation of variables.

4.2 Pricing Lookbacks in Multiscale Asymtoeics

111 this section, we value lookback options by solving (4.4) in asymptotic ex- pansions under the assumption that 0 < 《1. This can be achieved by considering the pricing function of the form

oo V = E 科"％ (4.6) »j=o

=+ v^Vo,! + y/SVi^o + v^Vi’i + eVo’2 + SV2,o + …•’

where Vq (or V。’。）and Vij are functions of {t, x, Y, Z) that will be solved one by one until certain accuracy attained.

Substituting (4.6) into (4.4) and collecting 0(l/e) terms, we end up with

= 0’

which confirms V^) be a function independent to y. The next step collects the

26 terms of to yield

CoVo,i + C.Vo = 0 => CoVo,i = 0,

since Ci involves '"-differentials. It concludes that Vq.i is a function iridepeiideiit of y also. Contiiiiiirig the process to obtain (9(1) terms, we have

CoVo,2 + CiVo,i + C2V0 = 0 CoVo,2 + C2V0 = 0. (4.7)

Given the function Vq, equation (4.7) is a first order linear ordinary differential equation (ODE) that unique solution exists within at most polynomially growing to infinity if

Ey{C2Vo) = 0,"〜AA(m,"2)’

where the distribution of y is derived from the operator Cq. This fact is known as the Fredholiri solvability for Poissoii equations. Since V^ is a function independent to y, the cxpcctatioii only takes cffcct on the operator £2 through the function

/(-y, z). Specifically,

Ey{C2Vo) = Ey(C2)Vo = 0, 0

Vo{T,x)=则，尝=0. X—0

A more explicit expression for (4.8) can be obtained through denoting

Ey{f{y,z)') = a{z)\ y �hfijn.i?). (4.9)

27 It follows that the Ey(C2) in (4.8) is the standard BS operator^ with volatility

？f(z). Hence, wc define

:=五“乙2) = ~ + 去刚2‘ - (r - (I + •咖2)基 1.. (4.10)

In fact, the parameter a{z) is the short term volatility of the underlying asset

price since the distribution of y is the invariant distribution of Yt with Zt being fixed. This leads to the following consequence.

Proposition 4.2.1. The zeroth order approximation for any lookback option with linear homogeneous payoff is the Black-Scholes pricing formulas of that option with a short-term volatility, defined in (4.9). In fact, Uq = SVq.

4.2.1 Fast Timescale Asymtotics

After the zeroth order appioxiinatioii is settled, I then derive higher order cor- rection terms. Consider 0{y/e) terms,

that leads to the result

Ey(CoVo,3) = 0 2) + Ey(C2)Vo,, = 0’

after applying the Fredholin solvability and recognizing Vq,! be independent to y.

In order to solve Ko,i from the above equation, we express K)’2 in terms of Vo，i

iRemark: If we let x = In 5, then the coefficient of 嘉 in the BS operator is r — g — fT^/2.

However, we now have x = hi(S*/S) so that the coefficient of 悬 in the BS operator becomes

28 with the equation (4.7) to reach

CLS^O,! = 0

VoAT,x,z) 二 0, ’ =0.

似 x=0

The right hand side of (4.11) is a known quantity as Vq lias been obtained by the early steps. Therefore, the PDE of (4.11) becomes the standard BS equation for lookback options with a source, the function in the right hand side of (4.11).

Appendix A shows that the source term of (4.11) takes the following explicit form:

一 Ey(C,Co'C2Vo) (4.12)

/ d 炉、 d ( d d^ \ =(P3(.) - ⑷）+ ⑩;K。- P3⑷石+ K

='p^jz) - P2{z)] dVo Ps(z) d avo —_ -cf(z){T -t) \ da{z) ~ 7f(z){T -t)'^d7f{z)' where P2(z) and P^iz) are effective parameters that can be calibrated from the volatility smile of vanilla options, see Fouque et al. (2003b). The expression

(4.12) shows that the source term of PDE (4.11) is a linear combination of higher order differentials of the zeroth order term. Vq. It is easy to show that functions of the form: are homogeneous solutions to the PDE (4.11). Therefore, the source term is a linear combination of homogeneous solutions of the PDE (4.11).

It is important to check whether (4.11) can be solved in closed form for x > 0 and X' < 0 respectively. However, PDEs like (4.11) are relevant to slowtiirie scale analysis also. Thus, we develop a proposition for solving a more general form of

PDE during x' > 0 arid a: < 0.

29 Proposition 4.2.2. Consider the PDE:

n

JC^bsV = 办,之)G 办’ 0

If Gi are homogeneous solutions of the above PDE, i.e. C^s^i 二 0 for all i = 1, 2, • • • , n, tiicii the solution for different doiiiairi of x can be expressed ill one equation by using a variable P. When domain of the PDE is a: > 0, is defined as 1. Otherwise, ,3 is defined as — 1.

The solution of the PDE consists of three terms, W{t, x, z), \]/(/3’ t, x, 2, g(., z)} and J(J3, t, z,g{-,x)) , i.e.

Vit,x,z) = VV(t, X-, z) + t, X, 2, z)) + t, z, z)). (4.13)

where

"/ /-T \

W{t, x,z) = h、s, z)ds Gi(t, X, z), (4.14)

^(j3,Lx,z.g(;z)) = Jt g{e, z)de, (4.15)

j(/u�"(、劝=义 V") [蟲+小

�職;；；，"(•’劝- f州，—劝.(4.16)

with r](9, X, 2), h(j3,9. x, 2,g(-,z)) and a are given by:

八 � L J

[N(a(0 -t,x,z))- N{a{9 - t, 0, z))], (4.17)

30 where a{-f,x,z)=

fT 細小了-s) 1[ . f h(p,e.x,z.g(-,z)) = / e •⑷v/rrgj z)ds , ( and a = I r - g H j .

The remaining function, g(.，z), is independent of x and it can be determined

from z):

"(.’…^ . (4.18) U 丄 x=0

The proof is given in appendix B.

Applying Proposition (4.2.2) to PDE (4.11), I obtain

= Wo,i(t, X, z) + L X, z. (•, z)) + J{(5, t, z, (4.19)

where t, x, z, (/o,i(-, z)) and J(j3, t, z)) are defined in Proposition (4.2.2)

and

IV (十.，、 刚-刚 dVo Psjz) d'Vo 輩工’力=—W)^顾 + 雨^^^， （4.2。）

抓 1(.,⑷ 二 I

It may be worth riieritionirig that the function is related to the Black-

Scholes Vega of the corresporidirig lookback option.

4.2.2 Slow Timescale Asymtotics

From the 0(\/^/e) terms of equation (4.4) and (4.6):

jCoVifi = 0, V^olbT = 0，

31 we know that Vi^ is a function independent to y. This makes us confirm that

Vi,i is a function independent to y also. To see this, collect terms:

A)K’1+AVi’O + M3K) = 0’

which implies

A)Vi’i = 0’

because Vq and Vi’o are functions iiidependerit to y and operators Ci and M3 involve y differential. The message of Vi，i independent to y is clearly followed.

The goal of this subsection is to determine the slowscale correction term

Collect 0{y/S) terms from equation (4.4):

CoVi,2 + A Vi，i + £2^^1,0 + MiVo + M.Vo,! = 0.

Since and Vo,i are functions independent to y, the above governing equation is reduced to

A)V^i，2 + /:2VVo + A^iVb = 0，

which is a Poissori equation on V"i，2. Applying Fredholiri solvability condition and recognizing Vi，o be a function independent to "，we obtain a PDE for the slowscale correction term:

Ey(C2)Vi,o = C^sVifi = -Ey(Mi)Vo, (4.21)

1/1 n 3 巧，0 n .. u丄 工=0

32 Appendix A shows that the source term of the PDE (4.21) is: = 2 p^]尝+ 2譜基(声（,22) =�-P i�)(-£+£)v�

1 dx \ dx dx"^) • where PQ{Z) arid P\{z) are effective parameters to be calibrated.

One immediately recognizes that this source term is a horriogerieous solution of (4.21). We can use Proposition (4.2.2) to solve the problem. The result is

= X, z) + t, X, z. z)) + t, 2, 2)), (4.23)

where vI/(/j, /,，:,;’ z、"i’o(-，z)) and /,’ 2’ .奶’ o(-, 2)) arc do fined in Proposition (4.2.2) and

仍，o(-,之）= ^ L 」口 0

4.2.3 Price Approximation

According to (4.6) and the transfoririatiori that U = SV, the first order price approximation, [/, for a lookback option with linear homogeneous payoff is

U = s(yo^ V~tVo,i + 哉 0) ：二 (4.25)

where Vq, Vo,i, are given by Proposition (4.2.1), (4.19) and (4.23)，respectively.

In order to interpret the price formula, we inverse the transformation to make the price function in terms of S and S* instead of x alone. To simplify matter and

33 connect effective parameters to calibration, we introduce the following notations

Po" = V^n(^), P^ = V~dPi(z), (4.26)

Pi = ⑷，尸二 WP3⑷，

where Po{z), f\(z),戶2⑷ and Pslz) are defined in appendix A. Then, we have the following proposition.

Proposition 4.2.3. For any lookback options with linear homogeneous payoff, the first order price approximation is given by

〜 f dURq\ \d(j(z)J ‘ fT「队 /^/VroM -sp / 宰= rae, in(sys)^z) de (4.27)

似t, z,如’i(.’ 2O) + StV~dJi3, /，z,仍’。(.’ z)),

where ri{0, - .z), J{P, L, z, f/o,i(-, z)) and J{/3, L, z, 2)) arc dofinod in Proposi- tion (4.2.2), z) and gi’o(-’2) are defined in equation (4.20) and (4.24). At,

Be are operators that

乂F 由+(T - ” (-Pi喊):.

召。=由[(巧-p

The proof is omitted since it is just a straight forward computation traris- forrriirig back the price function of (4.25) into a function of S and S*.

Remarks:

34 1. Since UBS is linearly homogeneous in 5, the term

fdUssX -S I游⑷力

is a function of the time variable 0 only.

2. On page 254 of Kwok (1998), it is seen that •^ii{OJn�S*/S), z) is propor-

tional to the probability density function of the first passage time that the

asset price S hits the value S* under constant volatility. Therefore, the in-

tegration in Proposition (4.2.3) is indeed rofiocting the impact of multiscale

stochastic volatility on the first passage time density.

In words, Proposition (4.2.3) says that the first order approximation for look- back options comprises with four parts. Specifically,

Looback option price under SV

= Black-Sclioles fonriula for lookback option

+ Correction of Vega

+ Correction of Delta-Vega

+ Correction of first passage time distribution.

In Fouque et al. (2003b), the first order approximation for European options does not have the last part, the first passage time adjustment. This is because lookback options involve an additional path dependent variable S* • We see that pricing exotic options under the multiscale SV model can be rriiich more sophisticated

than pricing path independent options.

35 4.2.4 Estimation of Approximation Errors

Since the formula presented in Proposition (4.2.3) is an approximation, it is cru- cial for users to understand its accuracy. This requires us to derive an upper bound for the quantity \U - U\ or \V - V\. To establish the bound we follow the steps of Fouque et al. (2003b) to introduce the higher order approximation for V

V = V + eV2,o + …V3’o + (v^Ki.i + €^2,1) ’ (4.28)

and the corresponding error term R = V — V. Then, it is clear that

< + = + Oie + d + V^). (4.29)

It remains to detenriiiie the order of R. Simple calculation shows that the error

term, R, satisfies

C’6R = eRi + y/76R2 + dR3,

R(T,x,y,z) = 0， (4.30)

f = 0,

where Ri, R2 and R3 take the form ^ ai(d'Vo/dx').

Define X^ as a modified process of Xt such that it has a reflecting boundary

at X = 0. Then, by (4.30), the error term R can be represented as follows:

R = tE 八-jt Ri(s,Xs.ys.Zs)ds Xt.Yt.Zt^ (4.31)

36 It implies that = 0{e + + S) if all expectations are bounded. The equivalent condition is that R:, R〗and are smooth functions independent to

€ and 6, uniformly bounded in t, x, z and at most linearly growing in y. In other words, we require ^^ to be uiiiforrrily bounded in £, x, z for all n. This regularity condition holds if the option payoff, H{x), has the property, H'{0) = 0，such that no singularity happens at (t, x) = (T, 0) in PDE (4.8). Otherwise, the accuracy of the price approximation may be distorted.

For most financial and insurance products, the condition, H'{0) 二 0, does not hold so that differential terms, may blow up as t — T and x —> 0 for some n. The same problem also happens in the case of European options since their payoff functions have an "hiiik"" at the at-the-money point. To obtain the order of accuracy for European options, Fouque et al. (2003a,b) applied singular perturbation techniques to show that the accuracy is of 0(e In e + <5 4- Their approach requires the analytical foririula of VQ to derive the bounds. By modifying the approach of Fouque et al. (2003a,b), we derive the order of accuracy for specific products in next section.

4.3 Floating Strike Lookback Options

We consider the floating strike lookback put option, whose payoff is

- = -ST.

37 This payoff function is linearly homogeneous in the sense that M『—ST =

— 1) where XT 二 > 0. Tlicrcforc, floating strike lookback put has a payoff satisfying all conditions stipulated in equation (3.4).

By Proposition (4.2.1), the zero order term, jj》。is the BS formula for the option. This formula is avaiable in the literature, Goldman et al. (1979) and it is the same as equation (4.32) but with a replaced by a(z).

V]i = _ Se—収-。iV(-4/) (4.32) — 「 2(r-Q) - a(z) N(d+ ) - e-r(T-t) M V TV (rF ) + 2{r-qf ‘ N、dM、e (似《J N {d,,),

+ h 士明(T —”

� nz、vr=i ,

Proposition (4.2.3) can be applied directly to obtain the first order price ap- proximation as

+StVeJ{l, t, 2,如’ 1(•’ z)) + StVSJ{l, t, z, z)), (4.33)

where 7](0, -.z) , A, Be, J{l,t, z, z)) and z, z)) are defined in

Proposition (4.2.2) and Proposition (4.2.3) respectively. To facilitate iiripleirieii-

tatiori, appendix C gives explicit expressions for Vega, Delta-Vega, Gamma-Vega,

go,i{-,z) and ^/i’o(-’2) of floating strike lookback options.

Similarly, floating strike lookback call option can be valued in the same manner

by applying Propositioii(4.2.2) and Propositiori(4.2.3). Without irieiitioiiing the

38 detail procedure, we present the price approximation as follows:

/ \ 厂T「/? / dc^ \ ‘

where

4 = e:-#-。iV(0 (4.34)

+ 药S [e-…）(⑤-錄（）-e-…)(/;),

‘ = ，‘

4.3.1 Accuracy for the Price Approximation

To derive the accuracy for the price approxiniatioii, we concentrate on the put option as the call counterpart is similar. The transformed payoff function is

H(x) = e^ - 1 which implies //'(O) = 1^0. Therefore, it is necessary to derive bounds for the error term in (4.31) to access the accuracy of the price approxiniatioii of (4.33).

By inspection, the BS pricing formula for the floating strike lookback put in

(4.32) can be viewed as

where

Vb = e-八 T-t) Ni^cfM、+ e—q(T-t) Ni^d+M、

39 It is easy (but tedious) to show that both ^^ and ^^ have the same order of blowing up to infinity as that of ^^ for t —> T and x 0, where Vj = CBS/S and CBS is the Black-Scholes price of vanilla call. As Vji {Vji = p^jJS)is the linear combination of VA and Vs, its derivatives have the same order of blowing up to infinity as that of European options. By arguments of Fouque et al. (2003a,b), the error term of R in (4.31) is of 0(e In e + v^ + d) for the floating strike lookback put. Similarly, the floating strike lookback call attains the same order of accuracy.

4.4 Calibration

Fouque et al. (2003b) showed that the market price of European option, denoted by V^, can be expanded in the power of y/t and \/S,

= + + + + + . (4.35)

On the other hand, the market implied volatility function, /, is assumed with another asymptotic expansion, namely,

/ = /o + Velo,! + V^/i，o + + e/o,2 + Sl2,o + ••• . (4.36)

The market implied volatility is obtained by matching the Black-Scholes pric- ing formula to the market price of options. Specifically, the value of I solves the equation"

40 Compare the Taylor expansion of V^f with respect to a at the point IQ and the asymptotic expansion of (4.35) to obtain

V 它=Kf + y^Kfi + 五0 …

=K)，。) + ^^V^/cu + + ….（4.37)

This enables us to relate implied volatility function to sensitivities of options.

Matching zeroth order terms in (4.37) shows that

= = = /o) = a{z). (4.38)

Since the Black-Scholes Vegas of European calls and puts are of positive values, i.e. ^^ > 0, the first order term confirms that

了0,1 = V^, X 閉，7,0 =《X 閱. (4.39)

Therefore, the implied volatilities of European calls and puts can be approximated by the expansion of (4.35) together with identities of (4.38) and (4.39) once the functional forms of Vq^, Vq^^ and V^Q are available.

In the paper of Fouque et al. (2003b), the implied volatility for European call options of (4.35), (4.38) and (4.39) is simplified to a very explicit expression.

Specifically, the implied volatility of European call is approximated by a linear form of the time to maturity, logaritliinic-irioiieyiiess and the interaction effect,

LMMR as follows:

ai ~ aiz) + + b\T - t)] + + a\T — ^仰, (4.40) T _ t

41 where cr； is the implied volatility arid the parameters are related to the group of parameters {P^, Pf, F^, P3) by

o" = -PlKzf, = P^/7f{z) — (r — q — a(z)'/2)，

In our nuirierical results in chapter 6, the parameter a{z) is estimated from historical daily index value over one mouth horizon. Then, we apply simple linear regression to estimate from (4.40) and then extract the group of effective parameters {PQ, Pf, P^、Pf).

42 Chapter 5

Other Lookback Products

In this chapter, we introduce riiodel-iiideperideiit relationships between floating strike lookbacks and other lookback products. By using the relationships, we can price lookback products other than floating strike lookbacks in multiscale asyiiiptotics.

5.1 Fixed Strike Lookback Options

With iriodel-independeiit relationships developed by Wong and Kwok (2003), fixed strike lookback options can be priced in terms of the formula of floating strike lookbcaks. .‘

Basically, fixed strike lookback options cannot be valued with results devel- oped in Chapter 4. The reason is that their payoffs do not observe the linear homogeneity [see equation (3.4)]. To see this, let us take a look at the payoff of fixed strike lookback call, max(M^ — /(，0). This payoff function does not have

43 the linear hoiriogerieous property. Moreover, there is an "liiiik" at the at-tlie- rrioiiey point which requires more calculation in order to obtain accuracy if we go through the whole asymptotic analysis once again.

Fortunately, Wong and Kwok (2003) developed put-call parity relations for lookback options which enable us to price fixed strike lookback options by the pricing formula of floating strike lookbacks. These parity relations are rriodel- independent and hence applicable to stochastic volatility models as well.

Let us consider the fixed strike lookback call. The corresponding parity rela- tion [see Wong and Kwok (2003)] is

Cf.i;[t, S, M^; K) = pfiit, S, iiiax(A/(5, A')) + SV収- 了-…

The right-hand side can be approximated by asymptotic expansion so that the

fixed strike lookback call shares the same order of accuracy as their floating strike

counterpart. It is soon that tho floating strike lookack options are fundeirieiital

iiistriinients to replicate many lookback products.

5.2 Lookback Spread Option

For a more complicated product like the lookback spread option, one can also use

the replication technique to obtain the pricing formula. Tho payoff of tho lookback

spread option is iriax(M^ — rnj^ — K, 0). Wong and Kwok (2003) also derived

a inodel-irideperident relation connecting the currently in-the-money (ITM) or

ATM lookback spread option with floating strike lookback options. Currently

44 ITM or ATM lookback spread means that

ML - ML — K > 0.

Under this situation, lookback spread can be valued as

LsAt, S, A4 mf); K) = CFI(t, S, + PF,(t, 5, M^) - (5.1)

Thcroforo, the asymptotic floating strike lookback option prices can be used to approximate the lookback spread option up to 0(e In e + + (5) accuracy. The price for the currently out-of-the-irioiiey (OTM) lookback spread option can be deteniiiiied in a similar manner with more iiiatlieiiiatics involved, see Wong and

Kwok (2003). The valuation is omitted since this paper concentrates on lookback options with single extreme variable.

5.3 Dynamic Fund Protection

As explained in chaper 3, the terminal payoff for the value of the dynamics fund

protection should be

F(T) = F{T) max |l, max^^} — F(T). (5.2)

Our problem is to estimate the present value of P.

We recognize that the payoff in (5.2) resembles a qiianto lookback option

payoff. To see this, we introduce variables

SPIT) = K/F(t) and M^ = max SF{T). (5.3)

45 Then, the payoff becomes

P(T) = F{T) niax(l, MJ) — F{T) = F{T) iiiax(Mj - 1,0). (5.4)

If we view F as the exchange rate and SP as the underlying asset price in the foreign currency world, then the payoff (5.4) is nothing but the fixed strike look- back call on Sf with unity strike translated back to the domestic currency by the exchange rate F{T). It is well known that the European-style of this option can be simply valued as tlic fixed lookback call in the foreign curreiicy world followed by multiplying the current exchange rate F(t), see Dai, Wong and Kwok (2004).

Ill other words, we establish a iriodel-iiidepeiideiit result that

P(t) = F(t)xcfUt.SF.Ml^). (5.5)

However, one should be aware of the process of SF as it is defined in the foreign currency world. If the process of F{t) is clofiiicd as (4.3), then the interest rate r is the domestic interest rate while the dividend yield q corresponds to the foreign interest rate. In the foreign currency world, Sp is the exchange rate translating from the domestic currency to the foreign currency. Hence, under the foreign currency measure Qf, the process of Sp is given by

CISF = (q — r)SFdt + /(y, ，乂

where W^'^ is a standard Brownian motion under Q^. Alternatively, one can

interpret" Sf as a foreign asset paying a dividend yield of r. Under the Black-

Scholes asset dynamic，i.e. f(x, y) = constant, Dai, Wong and Kwok (2004) gave

46 a complete analysis to American-style quaiito lookback options which should be useful in characterizing the early policy of dynamics fund protection.

However, this paper concentrates on European-style lookback option pricing under niultiscale volatility. Since the pricing formula of cjix is connected to the price of pji, our asymptotic solution (??) also provides an approximation to the price of dynamic fund protection up to hi accuracy. More precisely, our approximated price,尸⑴，is given by

P(f) = F(t)cf^(t, SF, Mp\ interchange the positions of r and q), (5.6)

where c/^ is defined in (??).

To generalize our result, we consider the dynamic fund protection with a gauraiitee level set as an index value. This problem has been studied by Gerber and Shin (2003) arid Chu and Kwok (2004). However, we use an alternative view to analyze this product. Let I denote the index value. The index-linked dynamic fund protection value has the payoff

F(T) rriax|l, max 梨]> -F(T).

�， \ 'o

By viewing F as an exchange rate, the index value and its riiaxiiriuin under in the foreign c.urroncy world arc rcspoctivoly defined as

IF⑴=I{t)/F(t) and M\ = max/^(T).

Under the martingale measure Qf, the process of Ip is given by

dip 二（(/F - qi)lFdt + fj(y, (5.7)

47 where qp is the dividend yield of the naked fund, qi is the dividend yield of the index arid fi{y, z) is the volatility of IF. For the relationship among the volatility of IF, the volatility of I and the volatility of F, we refer to the paper of Wong

(2004). With these notations, the index-linked dynamic fund protection payoff is presented as

Pi{T) = F{T) max(M;T-l，0).

This establishes the model-independent relation between dynamic fund protection and quanto lookback options. Specifically, we write

Pi{t) = F{t)xcfUtJr,Mj).

By recognizing the process of //?, the coriespoiidiiig price approximation becomes

= F(t) X 硕 t, IF, M\- T = (IF, Q = QH f(x, y) = fi(x, y)), (5.8)

where cj^ is found in (??).

48 Chapter 6

Numerical Results

In this chapter, we examine the effect of multiscale SV on fixed strike lookback options. Specifically, wo coinparo the Black-Scholos(BS) priccs of fixed strike lookback put and call with their iriiiltiscale volatility counterparts. In the Black-

Scholes model, we use a(z) as the constant volatility.

Figure 6.1 shows the effect of multiscale stochastic volatility on fixed strike lookback call option price for different strike prices and maturities.

Wo study the effort of miiltiscalo SV on fixed strike lookback call option price through Figure 6.1. The parameters of the pricing model are t = 70/252, r = 0.0363, q = 0.0251, W(z) = 0.099914, S = 1204.04, A/,； = 1229.11. In

the left hand side, the solid line and dashed line show fixed strike lookback call

option prices vary across different strikes under multiscale stochastic volatility

model arid the Black-Scholes model respectively. The graph in the top left hand

corner represents option prices with one year maturity whereas the bottom one

49 T=1 (Year) T=1 (Year) a> 400 p -| 80 I • • 基 m „ 。,…,I Co Full corr. 4= 8 B-S Model ^ 60 八， f 300 Q,, .. [ ou Greeks T3 CL S.V. Model o — « [ •云 40 First pass.

12 ^^^^^^ 5 20 100 0 O ^ CL � qL^ ^ -20 1100 1150 1200 1250 1100 1150 1200 1250 Strike Price Strike Price T=2 (Year) 1=2 (Year) 4001- -n 80 [- n ^ Co -- 访.呂 300 \ r 60 z -D Q. O z z ^^ ^ c 、、^^^^ •g 40 一 一

13 �� -“I �0 O CL J qL -I -20 ‘ ‘ J 1100 1150 1200 1250 1100 1150 1200 1250 Strike Price Strike Price

Figure 6.1: Fixed strike lookback call option represents prices with two years maturity. It is observed that the BS price is always below the multiscale asymptotics. That is, the option contract should be charged for additional premium on the BS price under the multiscale SV assump- tion. Furtherrriore, the additional preiiiiuiri increases with maturity of the option.

In chapter 4，we have derived the multiscale asymptotics which consists of the

Black-Scholos term, the Greeks corroction and the first passage time distribution correction (or boundary correction). So, the Greeks and the boundary corrections should contribute to the price correction. In the right hand side of Figure 6.1, we study how the Greeks correction and the boundary correction contribute to the

whole price correction. In the right hand side, the graphs show the percentage

50 of correction to the Black-Scholes price for the fixed strike lookback call option.

The solid line shows the whole correction, the dashed line shows the contribution of the Greeks terms and the dotted line shows the remainder, i.e. the boundary correction. It is observed that the Greeks correction and the boundary correction move in opposite direction and with opposite sign. The Greeks pull the option price to a higher level whereas the boundary correction pulls it down. Fouque et al. (2003b) derive the multiscale asyrnptotics for European option, the price correction consists of Greeks only. Since lookback call options are path dependent options, so an additional boundary correction term is needed. One should no- tice that the boundary correction is important for lookback options and it offsets some Greeks correction.

Figure 6.2 shows the effect of multiscale SV on fixed strike lookback put op- tion with an extra parameter m^ = 1136.15. In the left hand side, the solid line and dashed line show fixed strike lookback put option prices vary across differ- ent strikes under multiscale SV model and the Black-Scholes model respectively.

Again, the grapli in the top left hand corner represents option prices with one year maturity whereas the bottom one represents prices with two years matu- rity. It is observed that the BS price always below the iriiiltiscale asyiiiptotics.

Furthermore, the additional preiiiiuiii increases with maturity of the option.

In the right hand side of Figure 6.2, the graphs show the percentage of cor- rection to the Black-Scholes price for the fixed strike lookback put option. The solid line shows the whole correction, the dashed line shows the contribution of

51 T=1 (Years) T=1 (Years) 2501 ; • • 400 p B-S Model Ci" Full corr. g o 200 ——s.V. Model ^ \ ——Greeks 150 ^ I 200 First pass.

so^^：^^ I O 一一 Q- � 0 ^ -200 ^ 1100 1150 1200 1250 1100 1150 1200 1250 Strike Price Strike Price T=2 (Years) T=2 (Years) 250 n n 400 n (D -~. ^ I 200 .~^ C

50 一一一 .g O Q- 」 0 -200 ^ 1100 1150 1200 1250 1100 1150 1200 1250 Strike Price Strike Price

Figure 6.2: Fixed strike lookback put option

the Greeks terms and the dotted line shows the boundary correction.

It is observed that the Greeks correction and the boundary correction move in opposite direction, but they may not have different sign. Since lookback put options are path-dependent options, a boundary correction tenri is needed too.

For both put and call fixed strike options, we observe that the BS price always below the multiscale asyiriptotics. Furtherrnore, the additional preiriiuiii increases with maturity of the option too. To conclude, it is important to adjust the prices for the multiscale SV effect for lookback options. For longer the maturities, the larger the adjustment is needed.

52 Chapter 7

Conclusion

This thesis considers the iiiultiscale volatility model of Fouque et al. . (2003b) for pricing of derivatives with lookback feature. In chapter 2, we show that iiiul-

tiscale volatility model can fit volatility surface very well and attains an efficient

calibration scheme. By irieans of singular-regular perturbation techniques, we

establish price approximation and its accuracy for various types of lookback op-

tions. Specifically, we analytically approximate floating strike lookback options,

fixod strike lookback options, the lookback spread option and dynamic fund pro-

tection under multiscale volatility. By comiecting dynamic fund protection to

quanto lookback options, we conclude that floating strike lookback option is a

fuiidariieiital lookback iristruriieiits that can be used to replicate many lookback

products in the model-iridpeiiderit manner. With calibrated parameters, we see

that lookback option prices are highly sensitive to the realized volatility while

the Black-Scholes may under-value lookback options and hence dyriaiiiic fund

53 protection substantially. In order to correct the Black-Scholes price, an addi- tional premium is needed. The longer the contract maturity, the higher is this premium. Since lookbacks are path dependent options, the premium for the look- backs not only consist of Greeks correction in Fouque et al. (2003b), but also the boundary correction term.

54 Appendix

A Verifications

A.l Formula (4.12)

We realize that £2^0 = (£2 一 Ey (£2)) V'o- Hence, we have

乙二 2 (-五 + � Vo-

Denote (pijj, z) be the solution of the ODE

Co(f> = f{y.zf-a{z)\

Since the term £2^0 only depends on y through f{y, 2), we have

咖々(-l+S)队 which implies

乙 1 乙。-1 诚 二 ；(一("’”-樣劝(-£+£>0’

After defining effective parameters:

_ = and 眷，微f—,

55 we result in the second line of formula (4.12).

To show the last line of formula (4.12), we recognize that VQ satisfies the governing equation C^s^o = 0. Then, we take partial differentiation to both side of the equation with respect to a{z) and yield

It is easy to verify that 當 = 0. This together with the above equation ensure that 當 is the solution of the non-homogeneous convection-diffusion PDE. Since

徵 and ^^ are homogeneous solution of the governing equation, we know that

Sr碰-”(-£+£) 4 � which gives the last line of formula (4.12).

A.2 Formula (4.22)

It is easy to see from the definition of M i that

-M^Vo = (g(z)r(y, z) 一 g(z)p2f(y, z)) ^ + g(z)p2f(i/. z)^,

By formula (1)，we have

After defining effective parameters

Po{z) = \g{z)a{z)^Ey[r{y^z)] (2)

and P,{z) = ly(z)p2cf{z)Ey[f(y,z)]?^,

we establish formula (4.22).

56 B Proof of Proposition

B.l Proof of Proposition (4.2.2)

Consider the PDE:

n

i=l dv V(T.x,z) = 0, — -0.

咖x=0

If Gi are hornogerieous solutions of the above PDE, i.e. C^gGi — 0 for all i = l,2’.-.’n

Let

“/ fT \ W{t,x, z) = - Vt r V-/^/ k,{s,z)ds / (3) which satisfies the PDE:

n

= k办,X-, z), 0 < i < T,

TT"T 、 r腳勝

Define V{t, x', z) = W{t, x, z) + x, z), and it can be shown that x, z) satisfies:

二 0, 00, (4)

((T,x,z) = 0，=g(t,z).

where = ‘ x=0

57 Again, we let

Q(“和 (5)

the equation goveriiiig Q{t, x, z) is :

C%sQ = 0, 0 < i < T, (6)

Q(7>,2：) = 0’ =冲，4

By simple trarisforiiiatioii, h(t, x, z) 二 e—^^ hw^z)^ wj、_ x, z) where q = (r — + and h{t, x, z) satisfies:

dh 1 ,、2 炉/i _ 瓦+ 0 斤际=0’OSkT’

h{T,x,z、= 0，"L=。= e[4++T-〜(t，4

By the solution from Kevorkian (2000)，

( �� /•�":i.e[‘"H(了—s ) ,、， "("’(，工，"(.’。）=L •制广 批 r 2 1

1, if X > 0 where (3 define the solution for different domain of x: = \ -1’ if x<0

Recover C through (5),肌 t, x, z, g{-,z)) = <[>{f3, t, x, z�z) )+ ’J(J3, t, z, z)) where x, 2, z)) = J^ z, z))d^. Using integration by part

58 and doing some simple calculation:

= (3 j: g[e,z)rie,x,z)de (8)

where 秦’ z) = [e-H“’M、e-“一)。] 八 乂 y/27T{9 — t) L J

[N{a{0 - t, x, z)) — N{a{0 - t, 0，z))]

X - ay and a(7’ 工，2)=

One should notice that both (4) and(6) are needed to be satisfied, these force

J{J3, t, z, z)) satisfies an ODE:

—Zt = qJ(pA,z.gi-,z)) + ag{t,z)

—一） ^ ^ U丄 x=0 J(0,T\z』[、z)) = 0 (9)

Solving (9):

聊’ z，於’劝=

X 劝-脊"(M]"。. (10)

Collecting (3), (8) and (10)，V(t,x,z) = W(t,x,z) + +

J(P, t, z, g(-, z)) is solved.

59 C Black-Scholes Greeks for Lookback Options

Black-Scholes Formula for Floating Strike Lookback Put Option

p'fi =-如-収—。iV(-4,) + M“-r(r-。iV(-(ZJ

+ 卜,(4,) - (為)-V N (-‘），

iri 為土 4)(T —力)—+ y H[ — /?r7 ‘ "m — "A/ ^ Vi I, (jvi -1 (y

Greeks for Floating Strike Lookback Put Option (Vega, Delta-Vega, Gaiimia-

Vega)

f =拟f)—利丨字A

士 St六e普t、鴨、

dS OCT J (/• - q) G

— 十(ri) 、M)

+ )丄姊収-0

蔽‘ 二豕 UJ 、M) ^

|81n(蠻I 1 1 紀、2 r21n(f)-2

J StV^ [ a^Vr^t

41n(f)(r-^) — 21n(ftol

a^s/T^t - t)

60 z), gi,o{-, z) for pricing Floating Strike Lookback Put Option

(十、 FsW「 2e刷會-0-

“�P八z) r 2VT^te普t、•臂 九 oM) - 对e

+ 4(,.-」力(-(r - ci …)3 \ H^) J

Black-Scholes Formula for Floating Strike Lookback Call Option

[-e-+e(盖)-V " K),

Greeks for Floating Strike Lookback Call Option (Vega, Delta-Vega, Gariiiria-

Vega)

(r - q)

61 dS da \St) 、、) (r - q) a

41n(碧)(卜叫—

十 (r-q) (爪)

_^經—[4(ri)(ln(f)-l)

\ St y/^ [ a^y/T^t

a^y/T^t c7'(T - t)

yi,o{-j z) for pricing Floating Strike Lookback Call Option

而昨） )

仍，o(力，力二 ^ -徐� Y e I

—4(r - - Q (�(( r- (1 _ 昨)3 V ^ )_

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