Exercise: The Market Price of Risk in Black-Scholes

Prerequisites: Girsanov's Theorem, The Derivation of the Black-Scholes Formula

Problem

Under the real-world measure $\mathbb{P}$:

where $S$ is the stock, $B$ is the money-market account, $\mu$ is the real-world expected return, $\sigma$ is volatility, and $r$ is the risk-free rate.

1. Compute $d\tilde S_t$ where $\tilde S_t := S_t/B_t = e^{-rt}S_t$ is the discounted stock price. Show that $\tilde S$ has drift $(\mu - r)\tilde S_t\,dt$ and diffusion $\sigma \tilde S_t\,dW_t$.

2. We want to find $\mathbb{Q} \sim \mathbb{P}$ under which $\tilde S_t$ is a martingale. This requires the drift of $\tilde S_t$ under $\mathbb{Q}$ to vanish. By Girsanov, if $\tilde W_t = W_t + \theta t$ is the $\mathbb{Q}$-BM for some constant $\theta$, then $dW_t = d\tilde W_t - \theta\,dt$. Substitute into the $\tilde S$ SDE and determine the value of $\theta$ that kills the drift.

3. The constant $\theta = (\mu - r)/\sigma$ is called the market price of risk. Interpret its name: for a stock with expected excess return $\mu - r$ and volatility $\sigma$, what does $(\mu - r)/\sigma$ represent in units of "return per unit risk"? Connect to the Sharpe ratio.

4. Verify that under $\mathbb{Q}$, the stock SDE becomes $dS_t = r S_t\,dt + \sigma S_t\,d\tilde W_t$ — the stock's drift is the risk-free rate. What does this mean operationally for pricing derivatives?

Hint

The market price of risk $\theta = (\mu - r)/\sigma$ is a constant here because $\mu, r, \sigma$ are constants. In stochastic-rate / stochastic-vol models, $\theta_t$ becomes a stochastic process — same Girsanov argument, but with time-varying $\theta$.