An Introduction to Computational Finance Without Agonizing Pain c Peter Forsyth 2021 P.A. Forsyth∗ January 18, 2021 Contents 1 The First Option Trade 4 2 The Black-Scholes Equation 4 2.1 Background . 4 2.2 Definitions . 5 2.3 A Simple Example: The Two State Tree . 5 2.4 A hedging strategy . 6 2.5 Brownian Motion . 6 2.6 Geometric Brownian motion with drift . 12 2.6.1 Ito's Lemma . 14 2.6.2 Some uses of Ito's Lemma . 15 2.6.3 Some more uses of Ito's Lemma . 15 2.6.4 Integration by Parts . 17 2.7 The Black-Scholes Analysis . 17 2.8 Hedging in Continuous Time . 18 2.9 The option price . 19 2.10 American early exercise . 19 3 The Risk Neutral World 20 4 Monte Carlo Methods 22 4.1 Monte Carlo Error Estimators . 24 4.2 Random Numbers and Monte Carlo . 24 4.3 The Box-Muller Algorithm . 26 4.3.1 An improved Box Muller . 27 4.4 Speeding up Monte Carlo . 29 4.5 Estimating the mean and variance . 30 4.6 Low Discrepancy Sequences . 31 ∗Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1, [email protected], www.scicom.uwaterloo.ca/ paforsyt, tel: (519) 888-4567x34415, fax: (519) 885-1208 1 4.7 Correlated Random Numbers . 32 4.8 Integration of Stochastic Differential Equations . 34 4.8.1 The Brownian Bridge . 34 4.8.2 Strong and Weak Convergence . 36 4.9 Matlab and Monte Carlo Simulation . 38 5 The Binomial Model: Overview 41 5.1 A Binomial Model Based on the Risk Neutral Walk . 41 5.2 A No-arbitrage Lattice . 43 5.3 A Drifting Lattice . 45 5.3.1 Numerical Comparison: No-arbitrage Lattice and Drifting Lattice . 47 5.4 Smoothing the Payoff . 48 5.4.1 Richardson extrapolation . 52 5.5 Matlab Implementation . 52 5.5.1 American Case . 54 5.5.2 Discrete Fixed Amount Dividends . 55 5.5.3 Discrete Dividend Example . 56 5.6 Dynamic Programming . 57 6 More on Ito's Lemma 58 7 Derivative Contracts on non-traded Assets and Real Options 61 7.1 Derivative Contracts . 62 7.2 A Forward Contract . 65 7.2.1 Convenience Yield . 66 7.2.2 Volatility of Forward Prices . 66 8 Discrete Hedging 67 8.1 Delta Hedging . 67 8.2 Gamma Hedging . 68 8.3 Vega Hedging . 71 8.4 A Stop-Loss Strategy . 72 8.4.1 Profit and Loss: probability density, VAR and CVAR . 73 8.4.2 Another way of computing CVAR . 74 8.5 Collateralized deals . 76 8.5.1 Hedger . 76 8.5.2 Buyer B . 77 9 Jump Diffusion 78 9.1 The Poisson Process . 80 9.2 The Jump Diffusion Pricing Equation . 82 9.3 An Alternate Derivation of the Pricing Equation for Jump Diffusion . 83 9.4 Simulating Jump Diffusion . 86 9.4.1 Compensated Drift . 87 9.4.2 Contingent Claims Pricing . 88 9.5 Matlab Code: Jump Diffusion . 88 9.6 Poisson Distribution . 90 10 Regime Switching 91 2 11 Mean Variance Portfolio Optimization 92 11.1 Special Cases . 94 11.2 The Portfolio Allocation Problem . 94 11.3 Adding a Risk-free asset . 97 11.4 Criticism . 99 11.5 Individual Securities . 99 12 Some Investing Facts 103 12.1 Stocks for the Long Run? . 103 12.1.1 GBM is Risky . 105 12.2 Volatility Pumping . 107 12.2.1 Constant Proportions Strategy . 108 12.2.2 Leveraged Two Times Bull/Bear ETFs . 110 12.3 More on Volatility Pumping . 111 12.3.1 Constant Proportion Portfolio Insurance . 112 12.3.2 Covered Call Writing . 113 12.3.3 Stop Loss, Start Gain . 114 12.4 Target Date: Ineffectiveness of glide path strategies . 116 12.4.1 Extension to jump diffusion case . 120 12.4.2 Dollar cost averaging . 122 12.5 Bootstrap Resampling . 122 12.5.1 Data and Investment Portfolio . 123 12.5.2 Investment scenario . 124 12.5.3 Deterministic Strategies . 124 12.5.4 Criteria for Success . 124 12.5.5 Bootstrap results . 125 12.6 Maximizing Sharpe ratios . 125 12.6.1 Numerical Examples . 128 12.6.2 Deficiencies of mean-variance (Sharpe ratio) criteria . 130 13 Further Reading 132 13.1 General Interest . ..