Pricing Lookback Options under Multiscale Stochastic Volatility 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 Stochastic Volatility 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 Option 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 Greeks for Lookback Options 60 Bibliography 63 V List of Figures 2.1 Implied Volatility against Moneyness 8 2.2 Implied Volatiliy against LMMR 10 2.3 Implied Volatilities on the fitted volatility surface 13 6.1 Fixed strike lookback call option 50 6.2 Fixed strike lookback put option 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 strike price 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.