Two problems on option pricing

The theory of option pricing is focused on the determination of the "fair" (or "arbitrage-free") price for a derivative security whose value depends on some underlying asset. Usually such a price is determined by formulating a model for the financial market and by using a no-arbitrage argument like: "If the price of the option were different from the fair price, then there would be an arbitrage opportunity." We examine two problems arising from the theory of option pricing.; The first problem is about financial models with stochastic volatility. After almost a quarter of a century, the Black-Scholes model is still one of the most widely used for pricing equity options. Most of its strength relies on the simplicity of its assumptions and results. Sometimes, however, some assumptions seem to be too restrictive. Here we focus on the volatility of the underlying asset. We propose a new model for option pricing where the Black-Scholes hypothesis of a constant volatility is removed and instead a stochastic volatility process is considered. The main objective of our construction is to preserve the features of simplicity of the Black-Scholes model. We then analyze a general class of models with stochastic volatility, and give a unifying view on several models proposed in the literature. We conclude by solving a problem of fitting the observed prices with a stochastic volatility model.; The second problem is about the convergence of some approximation algorithms. Sometimes the fair price of an option is the solution of some equation which is hard (or impossible) to solve. Therefore only an approximation of the fair price can be determined. This is the case for an American-style option whose arbitrage-free price is the solution to an optimal stopping-time problem that is usually unsolvable for continuous-time market models. A standard way to get an approximate solution is to solve an analogous problem in a discrete-time setting. Here we determine sufficient conditions for the convergence of the approximation algorithms for the case of American option pricing. Moreover, we propose a new discretization method for continuous-time models which we prove to be more efficient, in some cases, than the standard algorithms. Such an approximation can be used for pricing any derivative security, not only American options.