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Cover Art
PRINTED BOOKS
Author Higham, Desmond J., 1964-

Title An introduction to financial option valuation : mathematics, stochastics, and computation / Desmond J. Higham.

Published New York : Cambridge University Press, 2004.

Copies

Location Call No. Status
 UniM Giblin Eunson  332.6453 HIGH    AVAILABLE
Physical description xxi, 273 pages ; 25 cm
Bibliography Includes bibliographical references and index.
Contents 1 Options 1 -- 1.2 Why do we study options? 2 -- 1.3 How are options traded? 4 -- 1.4 Typical option prices 6 -- 1.5 Other financial derivatives 7 -- 2 Option valuation preliminaries 11 -- 2.1 Motivation 11 -- 2.2 Interest rates 11 -- 2.3 Short selling 12 -- 2.4 Arbitrage 13 -- 2.5 Put-call parity 13 -- 2.6 Upper and lower bounds on option values 14 -- 3 Random variables 21 -- 3.1 Motivation 21 -- 3.2 Random variables, probability and mean 21 -- 3.3 Independence 23 -- 3.4 Variance 24 -- 3.5 Normal distribution 25 -- 3.6 Central Limit Theorem 27 -- 4 Computer simulation 33 -- 4.1 Motivation 33 -- 4.2 Pseudo-random numbers 33 -- 4.3 Statistical tests 34 -- 5 Asset price movement 45 -- 5.1 Motivation 45 -- 5.2 Efficient market hypothesis 45 -- 5.3 Asset price data 46 -- 5.4 Assumptions 48 -- 6 Asset price model: Part I 53 -- 6.1 Motivation 53 -- 6.2 Discrete asset model 53 -- 6.3 Continuous asset model 55 -- 6.4 Lognormal distribution 56 -- 6.5 Features of the asset model 57 -- 7 Asset price model: Part II 63 -- 7.1 Computing asset paths 63 -- 7.2 Timescale invariance 66 -- 7.3 Sum-of-square returns 68 -- 8 Black-Scholes PDE and formulas 73 -- 8.1 Motivation 73 -- 8.2 Sum-of-square increments for asset price 74 -- 8.3 Hedging 76 -- 8.4 Black-Scholes PDE 78 -- 8.5 Black-Scholes formulas 80 -- 9 More on hedging 87 -- 9.1 Motivation 87 -- 9.2 Discrete hedging 87 -- 9.3 Delta at expiry 89 -- 9.4 Large-scale test 92 -- 9.5 Long-Term Capital Management 93 -- 10 Greeks 99 -- 10.1 Motivation 99 -- 10.2 Greeks 99 -- 10.3 Interpreting the Greeks 101 -- 10.4 Black-Scholes PDE solution 101 -- 11 More on the Black-Scholes formulas 105 -- 11.1 Motivation 105 -- 11.2 Where is [mu]? 105 -- 11.3 Time dependency 106 -- 11.4 Big picture 106 -- 11.5 Change of variables 108 -- 12 Risk neutrality 115 -- 12.1 Motivation 115 -- 12.2 Expected payoff 115 -- 12.3 Risk neutrality 116 -- 13 Solving a nonlinear equation 123 -- 13.1 Motivation 123 -- 13.2 General problem 123 -- 13.3 Bisection 123 -- 13.4 Newton 124 -- 13.5 Further practical issues 127 -- 14 Implied volatility 131 -- 14.1 Motivation 131 -- 14.2 Implied volatility 131 -- 14.3 Option value as a function of volatility 131 -- 14.4 Bisection and Newton 133 -- 14.5 Implied volatility with real data 135 -- 15 Monte Carlo method 141 -- 15.1 Motivation 141 -- 15.2 Monte Carlo 141 -- 15.3 Monte Carlo for option valuation 144 -- 15.4 Monte Carlo for Greeks 145 -- 16 Binomial method 151 -- 16.1 Motivation 151 -- 16.3 Deriving the parameters 153 -- 16.4 Binomial method in practice 154 -- 17 Cash-or-nothing options 163 -- 17.1 Motivation 163 -- 17.2 Cash-or-nothing options 163 -- 17.3 Black-Scholes for cash-or-nothing options 164 -- 17.4 Delta behaviour 166 -- 17.5 Risk neutrality for cash-or-nothing options 167 -- 18 American options 173 -- 18.1 Motivation 173 -- 18.2 American call and put 173 -- 18.3 Black-Scholes for American options 174 -- 18.4 Binomial method for an American put 176 -- 18.5 Optimal exercise boundary 177 -- 18.6 Monte Carlo for an American put 180 -- 19 Exotic options 187 -- 19.1 Motivation 187 -- 19.2 Barrier options 187 -- 19.3 Lookback options 191 -- 19.4 Asian options 192 -- 19.5 Bermudan and shout options 193 -- 19.6 Monte Carlo and binomial for exotics 194 -- 20 Historical volatility 203 -- 20.1 Motivation 203 -- 20.2 Monte Carlo-type estimates 203 -- 20.3 Accuracy of the sample variance estimate 204 -- 20.4 Maximum likelihood estimate 206 -- 20.5 Other volatility estimates 207 -- 20.6 Example with real data 208 -- 21 Monte Carlo Part II: variance reduction by antithetic variates 215 -- 21.1 Motivation 215 -- 21.2 Big picture 215 -- 21.3 Dependence 216 -- 21.4 Antithetic variates: uniform example 217 -- 21.5 Analysis of the uniform case 219 -- 21.6 Normal case 221 -- 21.7 Multivariate case 222 -- 21.8 Antithetic variates in option valuation 222 -- 22 Monte Carlo Part III: variance reduction by control variates 229 -- 22.1 Motivation 229 -- 22.2 Control variates 229 -- 22.3 Control variates in option valuation 231 -- 23 Finite difference methods 237 -- 23.1 Motivation 237 -- 23.2 Finite difference operators 237 -- 23.3 Heat equation 238 -- 23.4 Discretization 239 -- 23.5 FTCS and BTCS 240 -- 23.6 Local accuracy 246 -- 23.7 Von Neumann stability and convergence 247 -- 23.8 Crank-Nicolson 249 -- 24 Finite difference methods for the Black-Scholes PDE 257 -- 24.1 Motivation 257 -- 24.2 FTCS, BTCS and Crank-Nicolson for Black-Scholes 257 -- 24.3 Down-and-out call example 260 -- 24.4 Binomial method as finite differences 261.
Summary This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of first-year calculus. Written in a series of short chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black-Scholes equation are provided, along with a presentation of appropriate computational techniques including binomial, finite differences and, in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock-market data. Solutions to exercises are available from solutions@cambridge.org.
Subject Options (Finance) -- Valuation -- Mathematical models.
Options (Finance) -- Prices -- Mathematical models.
Derivative securities.
ISBN 0521838843
0521547571 (pb.)