Stochastic Volatility - Basic Model

Basic Model

Starting from a constant volatility approach, assume that the derivative's underlying price follows a standard model for geometric brownian motion:

where is the constant drift (i.e. expected return) of the security price, is the constant volatility, and is a standard Wiener process with zero mean and unit rate of variance. The explicit solution of this stochastic differential equation is

.

The Maximum likelihood estimator to estimate the constant volatility for given stock prices at different times is

\begin{align}\hat{\sigma}^2 &= \left(\frac{1}{n} \sum_{i=1}^n \frac{(\ln S_{t_i}- \ln S_{t_{i-1}})^2}{t_i-t_{i-1}} \right) - \frac 1 n \frac{(\ln S_{t_n}- \ln S_{t_0})^2}{t_n-t_0}\\ & = \frac 1 n \sum_{i=1}^n (t_i-t_{i-1})\left(\frac{\ln \frac{S_{t_i}}{S_{t_{i-1}}}}{t_i-t_{i-1}} - \frac{\ln \frac{S_{t_n}}{S_{t_{0}}}}{t_n-t_0}\right)^2;\end{align}

its expectation value is .

This basic model with constant volatility is the starting point for non-stochastic volatility models such as Black–Scholes and Cox–Ross–Rubinstein.

For a stochastic volatility model, replace the constant volatility with a function, that models the variance of . This variance function is also modeled as brownian motion, and the form of depends on the particular SV model under study.

where and are some functions of and is another standard gaussian that is correlated with with constant correlation factor .

Read more about this topic:  Stochastic Volatility

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