Martingale Methods in Financial Modelling
The book provides a comprehensive, self-contained and up-to-date treatment of the main topics in the theory of option pricing. The first part of the text deals with simple discrete models of financial markets, including the Cox-Ross-Rubinstein binomial model. No knowledge of probability and stochastic processes is assumed at this stage, while most of the concepts from modern mathematical finance are explained at a very elementary mathematical level. The passage from the discrete to the continuous market models, done in the Black-Scholes model setting, assumes familiarity with basic ideas and results from stochastic calculus such as Wiener process and Ito formula; however, an appendix containing all the necessary results is included. The Black-Scholes setting is later generalized to cover standard and exotic options involving several assets and/or currencies. Numerous examples of exotic options are analysed. An outline of a general theory of arbitrage pricing is presented. A very substantial part of the text is devoted to term structure modelling and to the pricing of interest rate options. The HJM framework is discussed in detail. Models based on the forward LIBOR and forward swap rates are introduced. The main emphasis is on models that can be made consistent with the market pricing practice.
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An Introduction to Financial Derivatives
The CoxRossRubinstein Model
Finite Security Markets
17 other sections not shown
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absence of arbitrage American call option American put American put option arbitrage opportunities arbitrage price arbitrage-free arbitrary assume assumption Black-Scholes cash contingent claim continuous-time cross-currency defined Definition derivative discounted dividend domestic currency dynamics equivalent European call option exchange rate exercise expiry date finite foreign market forward contract forward LIBOR rates forward measure forward price forward rate forward swap futures contracts futures options futures price Gaussian given interest rate investor Lemma market model martingale measure maturity no-arbitrage optimal option price payoff portfolio price process probability measure probability space Proof Proposition put option random variable real numbers replicating strategy respectively risk-free satisfies savings account self-financing trading strategy semimartingale settlement date short-term rate standard Brownian motion stochastic process stock price strictly positive strike price swap rate swaption term structure terminal theorem underlying asset unique valuation formula wealth process yield zero-coupon bond