Solution: The Market Price of Risk in Black-Scholes

Exercise: The Market Price of Risk in Black-Scholes

Part 1

$\tilde S_t = e^{-rt}S_t$ . By Itô's product rule (or by direct Itô on $f(S, t) = e^{-rt}S$ ):

d\tilde S_t = -re^{-rt}S_t\,dt + e^{-rt}\,dS_t = -r\tilde S_t\,dt + e^{-rt}(\mu S_t\,dt + \sigma S_t\,dW_t) = (\mu - r)\tilde S_t\,dt + \sigma \tilde S_t\,dW_t.

Drift $(\mu - r)\tilde S_t$ , diffusion $\sigma \tilde S_t$ . ✓

Part 2

Under $\mathbb{Q}$ with $\tilde W_t = W_t + \theta t$ , we have $dW_t = d\tilde W_t - \theta\,dt$ . Substitute:

d\tilde S_t = (\mu - r)\tilde S_t\,dt + \sigma \tilde S_t(d\tilde W_t - \theta\,dt) = [(\mu - r) - \sigma\theta]\tilde S_t\,dt + \sigma \tilde S_t\,d\tilde W_t.

Setting the drift to zero: $(\mu - r) - \sigma\theta = 0$ , i.e. $\boxed{\theta = (\mu - r)/\sigma}$ .

Part 3

\theta = (\mu - r)/\sigma

is the excess return per unit of volatility of the stock — exactly the Sharpe ratio of the stock under

\mathbb{P}

. In plain English: "how many standard deviations of return you earn above the risk-free rate, per unit time."

The term "market price of risk" reflects the fact that this is the price the market puts on taking one unit of volatility risk: under any arbitrage-free model, the excess expected return must compensate for the volatility exposure at exactly this rate. All assets driven by the same Brownian motion must have the same market price of risk — otherwise an arbitrage exists (go long the higher-MPR asset, short the lower-MPR asset, hedge the Brownian, collect the spread).

Part 4

Under $\mathbb{Q}$ , $dW_t = d\tilde W_t - \theta\,dt$ . Substitute into the stock SDE:

dS_t = \mu S_t\,dt + \sigma S_t(d\tilde W_t - \theta\,dt) = (\mu - \sigma\theta)S_t\,dt + \sigma S_t\,d\tilde W_t = r S_t\,dt + \sigma S_t\,d\tilde W_t.

Operationally: to price a derivative, simulate (or integrate against) the

\mathbb{Q}

-SDE

dS_t = rS_t\,dt + \sigma S_t\,d\tilde W_t

— i.e. replace

\mu

r

and proceed as if the stock earned the risk-free rate. The discounted expected payoff

e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{payoff}(S_T)]

is the fair price.

Crucially, the real-world drift $\mu$ does not appear in the pricing formula — only

r

and

\sigma

. This is what makes Black-Scholes universal: you don't need to estimate

\mu

, which is notoriously hard.

Takeaways

Market price of risk $= (\mu - r)/\sigma =$ Sharpe ratio. In complete markets, this quantity is unique and determined by no-arbitrage.
Girsanov with $\theta = (\mu - r)/\sigma$ is the explicit measure change in Black-Scholes. It moves the stock's drift from real-world $\mu$ to risk-neutral $r$ , leaving volatility unchanged.
$\mu$ drops out of derivative prices. A profound simplification — you don't need to know the "true" drift of the stock to price an option, which is why Black-Scholes has survived. (Your hedge, not the price, is what actually implements the no-arbitrage assumption.)
Incomplete markets have multiple $\theta$ 's. In models with unhedgeable risk factors (stochastic volatility, jumps), there are multiple martingale measures $\mathbb{Q}$ , each with a different $\theta$ . Model calibration to market option prices is effectively choosing a specific $\theta$ .