## Option Theory with Stochastic Analysis: An Introduction to Mathematical FinanceSince 1972 and the appearance of the famous Black & Scholes option pric ing formula, derivatives have become an integrated part of everyday life in the financial industry. Options and derivatives are tools to control risk ex posure, and used in the strategies of investors speculating in markets like fixed-income, stocks, currencies, commodities and energy. A combination of mathematical and economical reasoning is used to find the price of a derivatives contract. This book gives an introduction to the theory of mathematical finance, which is the modern approach to analyse options and derivatives. Roughly speaking, we can divide mathematical fi nance into three main directions. In stochastic finance the purpose is to use economic theory with stochastic analysis to derive fair prices for options and derivatives. The results are based on stochastic modelling of financial as sets, which is the field of empirical finance. Numerical approaches for finding prices of options are studied in computational finance. All three directions are presented in this book. Algorithms and code for Visual Basic functions are included in the numerical chapter to inspire the reader to test out the theory in practice. The objective of the book is not to give a complete account of option theory, but rather relax the mathematical rigour to focus on the ideas and techniques. |

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adapted Algebra approximation arbitrage arbitrage opportunities arbitrage-free price aS(t Asian option assume Black & Scholes bond calculate call option chooser option conditional expectation contingent claim dB(s dB(t defined delta derivatives discounted distributed random variable dS(t e-rTEQ empirical equivalent martingale measure exercise FTSE index geometric Brownian motion Girsanov's theorem hedging portfolio Hence increments interval investment Ito integral Ito's formula Levy process logreturns martingale with respect measure Q Monte Carlo algorithm multi-dimensional NASDAQ NIG distribution normal inverse Gaussian normal random variable normally distributed random option contract option pricing parameters partial differential equation path of Brownian payoff function price P(0 pricing and hedging probability Q put option quantiles reader replicating portfolio respect to Q risk-free risk-neutral probability Scholes formula Scholes model Sect self-financing semimartingale simulated solution standard normal stochastic analysis stochastic process strike price theorem theory underlying stock variance Visual Basic zero

### Popular passages

Page 158 - Follmer, H. and Schweizer, M.: Hedging of contingent claims under incomplete information. In: MHA Davies and RJ Elliot (eds) Applied Stochastic Analysis, 389-414, Gordon and Breach, New York (1991) 27.