Introduction [chapter]

2021 Discrete-Time Approximations and Limit Theorems  
This book is devoted to various approximations of financial markets, both in time and in relation to basic parameters. Regarding approximation in time, everyone understands that markets operate in discrete time; however, from a computational point of view, it is much easier to work with them if viewed in continuous time, for which the Black-Scholes formula is one of the first and famous pieces of evidence. Therefore, the following questions immediately arise. How wrong are we if we replace
more » ... ete time with continuous? And what is the rate of convergence of asset and option prices in this case? Moreover, if we let the time be continuous in the entire sequence of models, including the limiting model, but the parameters change, tending to a certain limit, then what will happen to prices and capitals? The question of convergence itself has been studied for many years, and for initial acquaintance we recommend to the interested reader, for example, the book [139]. Also, for a preliminary acquaintance with the functional limit theorems and theory of financial markets, we recommend the books [17, 58, 62, 65, 110, 120, 136, 152, 153, 171]. However, the question of choosing a suitable model and a suitable approximation is so wide that there is an urgent need to describe new models from the point of view of limit transitions, for example, models with stochastic volatility and new approximations, and to estimate, if possible, the rate of convergence of prices of basic securities. In general, this is the subject of this book. The book consists of four chapters, each of which solves its own problem related to approximations of financial markets. Chapter 1 is devoted to the approximation of financial markets with continuous time by markets with discrete time. To begin with, we introduce the discrete-time multi-period market and define the basic concepts including the structure of asset prices, self-financing strategies, arbitrage, and completeness. We then consider discrete-time sequences of financial markets as an intermediate step in the transition to continuous time. We consider the traditional Cox-Ross-Rubinstein market model, but then we move away from this model and study arbitrage-free markets with the jump distribution concentrated on some interval and incompleteness of the non-Bernoulli market. After that, we introduce the basic concepts related to financial markets with continuous time. Then we describe the simplest passage to the limit, in which the geometric Brownian motion is the limiting process. In this case we have the Black-Scholes formula as the result of limit transition, weak convergence holds in the Skorokhod topology, and we can immediately conclude that the convergence of option prices holds for some options. But option price is not a unique object whose convergence follows from the weak convergence of the pre-limit markets. Other possible objects are the so-called Greeks, of which we have chosen to consider the Delta of European call options written both in discrete and continuous time and state the conditions of its convergence. After that, we go beyond the limit geometric Brownian motion and consider a much more general situation where the limiting process of the stock price is a geometric diffusion process.
doi:10.1515/9783110654240-201 fatcat:ucy6bhqml5awjjxkywxwtwhpgq