Devoted to financial markets both with discrete and continuous time, this book also describes how to make the transition from discrete to continuous time in option pricing. Chapter 1 presents the dynamic model of a financial market with discrete time. The general notions of an investor’s portfolio, self-financing strategy, arbitrage opportunity, and completeness are presented. An efficient market hypothesis is discussed. Martingale measures are described and the Martingale criterion of the absence of arbitrage is proved. European contingent claims are described and the problem of hedging is discussed. Binomial trees are introduced and computational formulae for pricing and hedging in the binomial model are deduced as well as the discrete analog of the Black–Scholes formula. The binomial model is considered in the scheme of series and then the scheme of series is generalized to the multiplicative scheme with arbitrary distribution of the multipliers. Incomplete markets are discussed, and lower and upper prices are calculated. American contingent claims are studied from opposing points of view of the buyer and seller. Snell envelopes are constructed and fair prices are produced
In Chapter 2, financial markets with continuous time are described in the general setting. The notions of self-financing admissible strategies and arbitrage opportunities are discussed as well as complete and incomplete markets. The Black-Scholes model as the result of applying the simple functional limit theorem to the discrete multiplicative scheme of series is studied. Partial differential equations are discussed in connection with option pricing, and barrier and other exotic options are also studied.