Local and Stochastic Volatility Models: An Investigation into the Pricing of Exotic Equity Options A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful¯llment of the requirements of the degree of Master of Science. Abstract The assumption of constant volatility as an input parameter into the Black-Scholes option pricing formula is deemed primitive and highly erroneous when one considers the terminal distribution of the log-returns of the underlying process. To account for the `fat tails' of the distribution, we consider both local and stochastic volatility option pricing models. Each class of models, the former being a special case of the latter, gives rise to a parametrization of the skew, which may or may not reflect the correct dynamics of the skew. We investigate a select few from each class and derive the results presented in the corresponding papers. We select one from each class, namely the implied trinomial tree (Derman, Kani & Chriss 1996) and the SABR model (Hagan, Kumar, Lesniewski & Woodward 2002), and calibrate to the implied skew for SAFEX futures. We also obtain prices for both vanilla and exotic equity index options and compare the two approaches. Lisa Majmin September 29, 2005 I declare that this is my own, unaided work. It is being submitted for the Degree of Master of Science to the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination to any other University. (Signature) (Date) I would like to thank my supervisor, mentor and friend Dr Graeme West for his guidance, dedication and his persistent e®ort. I would also like to give thanks to Professors D.P. Mason and P.S. Hagan as well as Grant Lotter for their additional assistance and to the heads of department, Professors D. Sherwell and D. Taylor. I owe my deepest gratitude to my parents for their unconditional support and kindness. i Contents 1 Introduction 1 2 Local Volatility Models: Implied Binomial and Trinomial Trees 4 3 The Derman and Kani Implied Binomial Tree 6 3.1 Arrow-Debreu Prices . 7 3.2 Upper Tree . 9 3.3 Centre of the Tree . 12 3.3.1 Odd Number of Nodes . 12 3.3.2 Even number of nodes . 12 3.3.3 Lower Tree . 12 3.4 Transition Probabilities . 14 3.5 Local Volatility . 14 3.6 Computational Algorithm . 15 3.6.1 Input Data . 15 3.6.2 Algorithm . 16 3.7 Barle and Cakici Algorithm Modi¯cations . 18 3.7.1 Non-Constant Time Intervals and a Dividend Yield . 23 3.8 Discrete Dividends and a Term Structure of Interest Rates . 23 4 Implied Trinomial Tree of Derman, Kani and Chriss 28 4.1 Introduction . 28 4.2 Constructing the State Space . 32 4.2.1 Term Structure Adjustments . 32 4.2.2 Skew Structure Adjustments . 38 ii 4.2.3 Term and Skew Structure . 40 4.3 Solving for the Transition Probabilities . 40 4.4 Negative Transition Probabilities . 42 4.5 Local Volatility . 43 4.6 Computational Algorithm . 43 4.6.1 Input Data . 43 4.6.2 Constructing the required state space . 44 4.6.3 Non-Constant Time Intervals and a Dividend Yield . 49 5 Characterization of Local Volatility and the Dynamics of the Smile 50 5.1 Introduction . 50 5.2 Kolmogorov Equations . 50 5.3 Relationship between Prices and Distributions . 53 5.4 Local Volatility in terms of Implied volatility . 57 5.5 Dynamics of the Volatility Surface . 61 5.5.1 The Forward Measure . 61 5.5.2 Local Volatility Model . 61 5.5.3 Perturbation Techniques . 62 5.5.4 Solving for Option Prices and Implied Volatility . 64 5.5.5 Incorrect Local Volatility Dynamics . 77 6 Stochastic Volatility Models 79 6.1 Introduction . 79 6.2 Derivative Pricing . 81 6.3 Arbitrage Pricing . 85 6.3.1 Equivalent Martingale Measure . 85 6.3.2 Martingale Representation Theorem . 86 6.3.3 Incomplete Markets . 92 7 Hull-White Model 93 7.1 Introduction . 93 7.2 The Two Factor Model . 93 7.3 Pricing Under Zero Correlation . 94 iii 7.4 Pricing Under Non-Zero Correlation . 96 7.4.1 Monte Carlo Simulation: Antithetic Variates Approach . 97 7.4.2 Hybrid Quasi-Monte Carlo Simulation . 98 8 The Heston Model 100 8.1 Introduction . 100 8.2 The Mean Reverting Ornstein-Uhlenbeck Process . 100 8.3 Stochastic Volatility Model . 101 8.4 Solution Technique: Fourier Transform . 104 8.4.1 The Direct Application of the Fourier Technique: Standard Black-Scholes Model . 104 8.4.2 Application of the Characteristic Function . 108 8.4.3 Solution to the Stochastic Volatility Process . 109 8.5 Computational Procedures . 120 8.5.1 Quasi-Monte Carlo Simulation . 120 8.5.2 Gauss-Legendre Integration . 120 9 SABR Model 123 9.1 Introduction . 123 9.2 Black Volatilities of Vanilla Options Priced with the SABR Model . 123 9.2.1 Multiple Scales Technique . 126 9.2.2 Near-identity Transform Method: Option Price Expansion . 127 9.2.3 Equivalent Normal Volatility . 146 9.2.4 Equivalent Black Volatility . 155 9.2.5 Stochastic ¯ Model.................................... 157 9.3 Monte Carlo under SABR . 163 9.3.1 SDE of the Underlying . 163 9.3.2 Quasi-Monte Carlo . 164 10 Calibration to Market Data 165 10.1 Source Data . 165 10.2 Disk Contents . 165 10.3 Local Volatility . 166 iv 10.4 Stochastic Volatility . 167 10.5 Model Calibration and Pricing Options . 168 10.5.1 SABR Parameters . 168 10.5.2 Vanilla European Option Prices . 169 10.5.3 Exotic Equity Options . 171 10.6 Conclusion . 175 v Chapter 1 Introduction Since the derivation of an arbitrage-free and risk-neutral closed-form solution to European option pricing (Black & Scholes 1973), a number of advancements and modi¯cations to the original modelling techniques have been suggested. These attempt to account for certain behavioural patterns displayed by the under- lying (equity index in our case) which are contrary to the assumptions that have been made in the original lognormal one-factor model. The original model is Markovian in nature and consists of a deterministic drift term (which is the continuously compounded risk free rate in the risk-neutral world) and a term that accounts for random or volatile behaviour. In pricing European options that have a terminal payo® dependent on the underlying, the assumptions that are made pertain to continuous trading, transaction costs, borrowing and lending and the returns distribution of the underlying. At maturity of the option, and throughout the option life, it is assumed that the terminal distribution of the underlying is lognormal with a constant standard deviation (volatility). The focus of this thesis is to examine two classes of models that have been proposed to account for the leptokurtotic terminal distribution of the underlying, alternatively the non-constant volatility feature. The ¯rst class, local volatility models, are deterministic in nature and can be calibrated using all available market data (European options, current spot and risk free rate etc.). They are deemed arbitrage-free and self-consistent yet produce volatility surfaces which, because they are static in nature, do not display the correct dynamics of the implied volatility skew from which they are derived (this will be seen in Chapter 5). This can be explained by considering the analogy between local volatility surfaces and forward rate curves. Given that the arbitrage-free short.