Quantitative Finance

3Some Models Used in Quantitative Finance

3.1 Introduction

The Black‐Scholes (B–S) model (see e.g. [24, 57, 101, 104, 152, 211]) is one of the most important models in the evolution of the quantitative finance. The model is used extensively for pricing derivatives in financial markets. As we shall see, using a backward parabolic differential equation or a probabilistic argument, it produces an analytical formula for European type options.

In this chapter, we start by deriving the B‐S differential equation for the price of an option. The boundary conditions for different types of options and their analytical solutions will also be discussed. We present the assumptions for the derivation of the B‐S equation. We then present a series of other models used in finance. Finally, we introduce the concept of volatility modeling.

3.2 Assumptions for the Black–Scholes–Merton Derivation

1. We assume that the risk‐free interest rate is constant and the same for all maturities. In practice, is actually stochastic and this will lead to complex formulas for fixed rate instruments. It is also assumed that the lending and borrowing rates the same.
2. The next assumption is that trading and consequently delta hedging is done continuously. Obviously, we cannot hedge continuously, portfolio rebalancing ...