Quantitative Finance

15Solutions to Complex Models Arising in the Pricing of Financial Options

15.1 Introduction

In this chapter we will first analyze a market model where the assets are driven by stochastic volatility models, and trading the assets involves paying proportional transaction costs. The stochastic volatility model is an enhancement of the Black–Scholes model for the pricing of financial options. In the Black–Scholes model, the volatility of the underlying security is assumed to be constant. However in stochastic volatility models, the price of the underlying security is a random variable. Allowing the price to vary helps improve the accuracy of calculations and forecasts.

In this chapter, we will show how the price of an option written on this type of equity may be obtained as a solution to a partial differential equation (PDE). We will obtain the option pricing PDE for the scenario where the volatility is a traded asset. In this case all option prices may be found as solutions to the resulting nonlinear PDE. Furthermore, hidden within this scenario is the case where the option depends on two separate assets, and the assets are correlated in the same form as and . The treatment of the option in this case is entirely equivalent with the case discussed in this chapter.

15.2 Option Pricing ...