Chapter 17: Fundamentals of Stochastic Finance
An important technique to price a derivative hinges on transforming the mean of price process by changing its probability measure. Girsanov's theorem provides a general framework for transforming between the original and new probability measures. The approach eliminates the risk premium, so the valuation can take place as if the price process is risk free. A simple yet less useful approach is to simply shift each value by a constant risk premium to alter the mean. Usually, this risk premium is not known before the fair market value is calculated. The chapter concludes with Kolmogorov's backward equation, which describes the evolution backwards in time of a forecasted value as well as the related Feynman–Kac formula. These techniques are applied to pricing an option on a stock as well as pricing a bond.
17.2 Risk-Neutral Pricing
One approach to value a derivative instrument is based on forming risk-free portfolio, for example, an option and the underlying stock. Despite unexpected price movements, the portfolio could remain risk free by continuous fine-tuning of the portfolio weights. A martingale is a stochastic process with zero trend (drift) and no periodicity. A process, such as a stock index, with a positive mean drift is a submartingale and a negative drift is a supermartingale.
An alternate approach to price a derivative instrument (discussed below) is based on finding a modified probability measure of an asset ...