Option Theory with Stochastic Analysis

1 Introduction.- 1.1 An Introduction to Options in Finance.- 1.1.1 Empirical Finance.- 1.1.2 Stochastic Finance.- 1.1.3 Computational Finance.- 1.2 Some Useful Material from Probability Theory.- 2 Statistical Analysis of Data from the Stock Market.- 2.1 The Black & Scholes Model.- 2.2 Logarithmic Returns from Stocks.- 2.3 Scaling Towards Normality.- 2.4 Heavy-Tailed and Skewed Logreturns.- 2.5 Logreturns and the Normal Inverse Gaussian Distribution.- 2.6 An Alternative to the Black & Scholes Model.- 2.7 Logreturns and Autocorrelation.- 2.8 Conclusions Regarding the Choice of Stock Price Model.- 3 An Introduction to Stochastic Analysis.- 3.1 The Itô Integral.- 3.2 The Itô Formula.- 3.3 Geometric Brownian Motion as the Solution of a Stochastic Differential Equation.- 3.4 Conditional Expectation and Martingales.- 4 Pricing and Hedging of Contingent Claims.- 4.1 Motivation from One-Period Markets.- 4.2 The Black & Scholes Market and Arbitrage.- 4.3 Pricing and Hedging of Contingent Claims X= f(S(T)).- 4.3.1 Derivation of the Black & Scholes Partial Differential Equation.- 4.3.2 Solution of the Black & Scholes Partial Differential Equation.- 4.3.3 The Black & Scholes Formula for Call Options.- 4.3.4 Hedging of Call Options.- 4.3.5 Hedging of General Options.- 4.3.6 Implied Volatility.- 4.4 The Girsanov Theorem and Equivalent Martingale Measures.- 4.5 Pricing and Hedging of General Contingent Claims.- 4.5.1 An Example: a Chooser Option.- 4.6 The Markov Property and Pricing of General Contingent Claims.- 4.7 Contingent Claims on Many Underlying Stocks.- 4.8 Completeness, Arbitrage and Equivalent Martingale Measures.- 4.9 Extensions to Incomplete Markets.- 4.9.1 Energy Markets and Incompleteness.- 5 Numerical Pricing and Hedging of Contingent Claims.- 5.1 Pricing and Hedging with Monte Carlo Methods.- 5.1.1 Pricing and Hedging of Contingent Claims with Payoff of the Form f(ST).- 5.1.2 The Accuracy’ of Monte Carlo Methods.- 5.1.3 Pricing of Contingent Claims on Many Underlying Stocks.- 5.1.4 Pricing of Path-Dependent Claims.- 5.2 Pricing and Hedging with the Finite Difference Method.- A Solutions to Selected Exercises.- References.