Chapter 5

Estimation of stochastic models for finance

In this chapter we review some of the basics models used to model asset prices, or market indexes, interest rates, etc. These processes may have different structures and we start from the very fundamental model of asset dynamics of the Black and Scholes (1973) and Merton (1973) model.

5.1 Geometric Brownian Motion

Let us denote by {St = S(t), t ≥ 0} a stochastic process which represents the value of an asset (the asset price) at time t ≥ 0. The process S is called geometric Brownian motion process if it is the solution to the following stochastic differential equation:

with some initial value S0, and some constants μ, σ > 0. As seen in Chapter 1, the fundamental idea was to describe the returns of St in the interval [t, t + dt) in terms of two components:

where the deterministic contribution is assumed to be proportional to time, i.e. μdt, and the stochastic part is assumed to be of Gaussian type, i.e. σdWt. A simple rewriting of

gives the stochastic differential Equation (5.1). The constant μ represent the drift of the process and σ2 is called volatility. We will see that, in more advanced models, but μ and σ can be deterministic ...