Soloveiko O. Limit theorems for the prices of derivatives

Thesis is devoted to investigate the properties of convergence of derivative securities. The models, considered in this thesis, generalize the classical Black-Scholes-Merton models. The discussed problems can be divided into two parts. The first one is investigation of stability, i.e., robustness, of the fair prices of options with respect to changes of market parameters, such as interest rate, drift, volatility, or errors in their measurements. The second one is determination of convergence conditions for options fair prices when the market model with discrete time converges to the model with continuous time. The boundary value problem containing Black-Scholes equation for the prices of European call and put options is solved for complete arbitrage free market model that involves parameters depending on time. After some technical transformations, the equation in partial derivatives for any of such prices is reduced to the heat equation. Explicit form of fair prices of European call and put options with time dependent parameters is established. The conditions of stability of share prices, fair prices of call and put European options and barrier European "up-and-out'' call options with respect to changes or errors of measurements are found in two ways: using the explicit form of option prices and without using explicit form solution by probabilistic method. The model of financial market with discrete time is considered under assumption that jump of share price is uniformly distributed on some time interval. The rate of convergence of fair prices of European call and put options in such a model to the corresponding Black-Scholes price of the model with continuous time is established. This is done by using the theorem about the asymptotic decomposition of distribution functions of sums of independent identically distributed random variables. This model is natural because in many cases we can only predict the interval for the share price jump size, but not the value of this jump. Discrete market is incomplete, although the limit market is complete. We choose one of the possible martingale measures in incomplete market and estimate that the rate of convergence of fair prices of European call and put options to the Black-Scholes price equals one divide square root of n, where n is the number of periods in the discrete model. The general discrete market and the discrete binomial market with time dependent interest rate and drift are considered. The rates of convergence of the fair prices of barrier European "up-and-out'' call options on these markets to appropriate prices for continuous market are established. It is proved that the rate of convergence is O of logarithm n divide square root of n. Numerical examples (modeling) are given. Parameter convergence of fair price of contingent claim is proved for the market with jumps. Some a priori estimates for solution of backwards stochastic differential equation with Poisson component are found. With their help the convergence of solutions of backwards stochastic differential equation with Poisson component is proved. The result is applied in the financial market with jumps to prove the convergence of prices of contingent claims.