Pricing Financial Derivatives Under the Regime-Switching Models Mengzhe Zhang A thesis in fulfilment of the requirements the degree of Doctor of Philosophy School of Mathematics and Statistics Faculty of Science, University of New South of Wales, Australia 19th April 2017 Acknowledgement First of all, I would like to thank my supervisor Dr Leung Lung Chan for his valuable guidance, helpful comments and enlightening advice throughout the preparation of this thesis. I also want to thank my parents for their love, support and encouragement throughout my graduate study time at the University of NSW. Last but not the least, I would like to thank my friends- Dr Xin Gao, Dr Xin Zhang, Dr Wanchuang Zhu, Dr Philip Chen and Dr Jinghao Huang from the Math school of UNSW and Dr Zhuo Chen, Dr Xueting Zhang, Dr Tyler Kwong, Dr Henry Rui, Mr Tianyu Cai, Mr Zhongyuan Liu, Mr Yue Peng and Mr Huaizhou Li from the Business school of UNSW. i Contents 1 Introduction1 1.1 BS-Type Model with Regime-Switching.................5 1.2 Heston's Model with Regime-Switching.................6 2 Asymptotics for Option Prices with Regime-Switching Model: Short- Time and Large-Time9 2.1 Introduction................................9 2.2 BS Model with Regime-Switching.................... 13 2.3 Small Time Asymptotics for Option Prices............... 13 2.3.1 Out-of-the-Money and In-the-Money.............. 13 2.3.2 At-the-Money........................... 25 2.3.3 Numerical Results........................ 35 2.4 Large-Maturity Asymptotics for Option Prices............. 39 ii 2.5 Calibration................................ 53 2.6 Conclusion................................. 57 3 Saddlepoint Approximations to Option Price in A Regime-Switching Model 59 3.1 Introduction................................ 59 3.2 Saddlepoint Approximation....................... 62 3.3 Apply Saddlepoint Approximation to a Regime-Switching Model... 66 3.4 Numerical Results............................. 72 3.4.1 Design of the Test........................ 72 3.4.2 Results of the Test........................ 73 3.4.3 Sensitivity of Approximation................... 79 3.5 Extension................................. 79 3.5.1 European-Style Barrier Options................. 79 3.6 Conclusion................................. 81 4 Pricing European Options in Heston's Model with Regime Switch- ing: A Saddlepoint Method 83 4.1 Lugannani-Rice (LR) Formula...................... 85 4.2 Pricing European Options under Heston's Model with Regime-Switching 86 iii 4.3 Numerical Results............................. 90 4.4 Conclusion................................. 92 5 Pricing Volatility Swaps in Heston's Stochastic Volatility Model with Regime Switching: An Alternative Saddlepoint Approxima- tion Method 95 5.1 Introduction................................ 95 5.2 Volatility Swaps.............................. 100 5.3 Saddlepoint Approximation Methods.................. 102 5.3.1 Approximation to the Fourier Inversion Integrals....... 103 5.4 The Cumulant Generating Functions of the Realised Variance.... 106 5.5 Numerical Results............................. 114 5.6 Conclusion................................. 120 6 Discussion and Conclusion 121 iv List of Figures 2.1 In-the-money Call Options (S0=100, K=80, r1 = 0.2, r2=0.5, σ1=σ2=0.2, λ1=λ2=1)................................. 36 2.2 At-the-money Call Options (S0=100, K=100, r1=0.1, r2=0.1, σ1=0.3, σ2=0.5, λ1=λ2=1)............................ 38 2.3 PDF of T1 (T=100)............................ 51 4.1 Option Values with Regime-switching and without Regime-switching 93 4.2 Volatility Smile (T=1).......................... 93 v List of Tables 2.1 In-the-money Call Options (S0=100, K=100, r1=0.2, r2=0.5, σ1=σ2=0.2, λ1=λ2=1)................................. 37 2.2 At-the-money Call Options (S0=100, K=100, r1=0.1, r2=0.1, σ1=0.3, σ2=0.5, λ1=λ2=1)............................ 38 2.3 At-the-money Call Options (S0=100, K=100, r1=0.1, r2=0.1, σ1=0.3, σ2=0.5, λ1=λ2=1)............................ 39 3.1 Lugannani-Rice method, the parameters are: S0 = 100, K = 90, λ1 = λ2= 1, r = 0.1, σ1 = 0.2, σ2 = 0.3.................... 74 3.2 Lugannani-Rice method, the parameters are: S0 = 100, K = 90, λ1 = λ2= 1, r = 0.1, σ1 = 0.9, σ2 = 1..................... 74 3.3 Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = λ2= 1, r = 0.1, σ1 = 0.2, σ2 = 0.3....................... 75 3.4 Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = λ2= 1, r = 0.1, σ1 = 0.9, σ2 = 1........................ 75 vi 3.5 Lugannani-Rice method and Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = λ2= λ3= 1, r = 0.1, σ1 = 0.1, σ2 = 0.2, σ3 = 0.3...................................... 77 3.6 Lugannani-Rice method and Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = λ2= λ3= 1, r = 0.1, σ1 = 0.8, σ2 = 0.9, σ3 = 1. 77 3.7 Lugannani-Rice method and Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = 1, λ2= 5, λ3= 10, r = 0.1, σ1 = 0.1, σ2 = 0.2, σ3 = 0.3................................... 78 3.8 Lugannani-Rice method and Lieberman method, the parameters are: S0 = 100, K = 90, λ1 = 1, λ2= 5, λ3= 10, r = 0.1, σ1 = 0.8, σ2 = 0.9, σ3 = 1.................................... 78 4.1 Call Option Prices (N=2)........................ 92 4.2 Call Option Prices when T=1, N=2................... 94 5.1 Volatility Swaps × 100 (N=2)...................... 117 5.2 Relative Errors.............................. 118 vii Abstract While several empirical studies find evidence for the existence of regime-switching (RS) effect on stock/future prices, see Vo (2009), Chan (2009), and Ang and Tim- mermann (2012), these studies are focused on finding explanations for the outcome. Their findings do give us useful insights about the statistical relationship between eco- nomic events and price movement behaviour. Others, including Chan and Zhu (2015), Elliott and Chan (2013), and Elliott and Lian (2013) studied financial derivatives, such as options, convertible bounds and swaps, under a regime-switching framework . Overall, the regime-switching modelling approach becomes more popular in the last decade. A well-known advantage of this model is that it can capture major events that affect the market. However, it also has an obvious drawback. Due to the complicated model structure, it is generally difficult to find explicit solutions for the derivative prices. Furthermore, it is often time consuming to solve the problem numerically. Not to mention some numerical instability that one may face. As an attempt to take up these issues, this thesis proposes a modified saddle-point approximation method to study the pricing problem under regime-switching models. One major difficulty in pricing financial derivatives under a RS model is that we need to calculate the total occupation time at each state but this can be normally done by numerical meth- ods, such as Mote Carlo simulation. In Chapter 2, we derive asymptotic expansions to the option prices under the RS Black-Scholes model with special attention paid to studying the changing behaviour of occupation time when the maturity time is short/large. Chapters 3-5 apply the modified saddle-point approximation approach to price various derivative contracts including European options and volatility swaps under various dynamics with regime switching. In addition to the theoretical deriva- tion, a number of numerical examples have been implemented to demonstrate the accuracy and efficiency of the proposed method. ii Chapter 1 Introduction Vo's (2009) study gives the evidence of the existence of regime-switching effect in oil price market, which makes the regime-switching model more popular in recent years. The study has also shown that the regime-switching stochastic volatility model does a good job in capturing major events affecting price dynamics. However, due to the nature of the regime-switching model, it is difficult to find an analytic solution to calculate the price of options or other financial derivatives. Nevertheless, Guo (2001) finds an explicit probability distribution function (PDF) of the occupations time at a given state, which makes various methods other than the Monte Carlo simulation available in the option pricing problem. Unfortunately, Guo's method requires using the Laplace's method to solve a system of ordinary differential equations (ODE) that describes the switching behaviour of the Markov process and this technique is no longer working if the unobservable Markov chain contained in the underlying asset prices has more than two states. Furthermore, the tractability of the pricing methods 1 which include the use of Guo's PDF is usually poor, as most pricing formulas could only be solved numerically. On the other hand, recent papers (Elliott and Chan, 2013; Chan and Zhu, 2015; Elliott and Chan, 2014) try to use the cumulant generating function (CGF) of the underlying asset to pricing options. One aim of this thesis is to find some effective and tractable ways to pricing options under different types of regime-switching models. Although Elliott and Lian (2013) successfully priced the variance swaps under a Heston-type regime-switching model, their method could not be applied to pricing volatility swaps when a realized volatility formula is adapted. Pricing a volatility swap is a highly non-linear problem. Explicit solutions of the prices of volatility swaps are notoriously difficult to find. Therefore, our second aim is to find an approximation method which is less time consuming and with a reasonable accuracy level to pricing volatility swaps under a Heston-type regime- switching model. Beyond that, as we known, the asymptotic formulas for options could be found useful in many places such as model parameter calibration. Although some recent papers give the evidence of the existence of regime-switching effect in volatility indices (Maghrebi, et: al: 2014) and future market (Vo, 2009), there is a gap about the asymptotics under a regime-switching framework. The third aim of this thesis is to find the asymptotic pricing formulas for European options and total occupation time of the Markov chain at a given state, where the price dynamics of the underlying asset are assumed to follow a continuous-time Markov-modulated version of the Black-scholes model.