Quantitative Finance

14Approximating General Derivative Prices: Gradient Method

14.1 Introduction

In this chapter, we will study integro‐differential parabolic problems arising in financial mathematics. The solution of these problems models processes with jumps and stochastic volatility. Under suitable conditions, we will prove the existence of solutions in a general domain using the method of upper and lower solutions and a diagonal argument. This type of methodology can be extended to other cases which can be applied to similar problems. We begin this section with some background.

In recent years there has been an increasing interest in solving PDE problems arising in financial mathematics and in particular on option pricing. The standard approach to this problem leads to the study of equations of parabolic type.

Usually the Black–Scholes model (see e.g. [24, 57, 101],[104, 152]) is used for pricing derivatives, by means of a backward parabolic differential equation. In this model, an important quantity is the volatility which is a measure of the fluctuation (risk) in the asset prices and corresponds to the diffusion coefficient in the Black–Scholes equation.

In the standard Black–Scholes model, a basic assumption is that the volatility is constant. However, several models proposed in recent years, such as the model found in [95], have allowed the volatility to be nonconstant or a stochastic variable. In this model, the underlying security follows, as in the Black–Scholes model, a stochastic ...