Options, Futures, and Other Derivatives, 10th Edition

28.1 THE MARKET PRICE OF RISK

We start by considering the properties of derivatives dependent on the value of a single variable $\mathit{\theta }$. Assume that the process followed by $\mathit{\theta }$ is

$$\frac{d\mathit{\theta }}{\mathit{\theta }}=mdt+sdz$$(28.1)

where dz is a Wiener process. The parameters m and s are the expected growth rate in $\mathit{\theta }$ and the volatility of $\mathit{\theta }$, respectively. We assume that they depend only on $\mathit{\theta }$ and time t. The variable $\mathit{\theta }$ need not be the price of an investment asset. It could be something as far removed from financial markets as the temperature in the center of New Orleans.

Suppose that $f_1$ and $f_2$ are the prices of two derivatives dependent only on $\mathit{\theta }$ and t. These can be options or other instruments that provide a payoff in the future equal to some function of $\mathit{\theta }$. Assume that during ...