## Stochastic Analysis for Finance with SimulationsThis book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black–Scholes–Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena. The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendices. This work attempts to bridge the gap between mathematics and finance by using diagrams, graphs and simulations in addition to rigorous theoretical exposition. Simulations are not only used as the computational method in quantitative finance, but they can also facilitate an intuitive and deeper understanding of theoretical concepts. Stochastic Analysis for Finance with Simulations is designed for readers who want to have a deeper understanding of the delicate theory of quantitative finance by doing computer simulations in addition to theoretical study. It will particularly appeal to advanced undergraduate and graduate students in mathematics and business, but not excluding practitioners in finance industry. |

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### Contents

2 | |

23 | |

Part III Brownian Motion | 108 |

Part IV Itô Calculus | 157 |

Part V Option Pricing Methods | 236 |

Part VI Examples of Option Pricing | 295 |

Part VII Portfolio Management | 351 |

Part VIII Interest Rate Models | 394 |

B Linear Algebra | 555 |

C Ordinary Differential Equations | 566 |

D Diffusion Equations | 575 |

E Entropy | 582 |

F Matlab Programming | 591 |

Solutions for Selected Problems | 603 |

Glossary | 637 |

646 | |

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Ä t Ä Analysis for Finance Asian option asset price binomial tree Black–Scholes–Merton bond price Brownian bridge cash flow conditional expectation consider constant continuous converges coupon defined Definition Delta denote derivative differential equation equal European call option Example exists expiry date Finance with Simulations formula G.H. Choe geometric Brownian motion given Hence interest rate International Publishing Switzerland interval Itô Itô formula Lebesgue integral Lemma linear martingale matrix maturity Monte Carlo method Note obtain option price payoff plot portfolio probability density function probability measure Proof Publishing Switzerland 2016 put option random numbers random variable respect risk-free risky asset sample paths satisfies sequence ſº solution Springer International Publishing Stochastic Analysis stochastic process strike price subsets Theorem underlying asset uniformly distributed Universitext variance Vasicek model vector volatility zeros