Under the physical measure dS = mu S dt + sigma S dW. Write the Radon-Nikodym derivative taking you to the risk-neutral measure with constant rate r, state the market price of risk, and explain precisely why sigma is unchanged.

Under the physical measure dS = mu S dt + sigma S dW. Write the Radon-Nikodym derivative taking you to the risk-neutral measure with constant rate r, state the market price of risk, and explain precisely why sigma is unchanged.

Approach: Choose the drift shift that turns the discounted stock into a martingale, write the corresponding stochastic exponential, then argue about volatility from the pathwise nature of quadratic variation.

dQ/dP = exp(-lambda W_T - lambda^2 T/2) with lambda = (mu - r)/sigma the market price of risk, under which W~_t = W_t + lambda t is a Q-Brownian motion and dS = r S dt + sigma S dW~. Girsanov's theorem says that under the measure with this Radon-Nikodym derivative the shifted process W~ is standard Brownian motion, and substituting W = W~ - lambda t into the original equation replaces mu by mu - sigma lambda = r, so the discounted stock becomes a martingale. Volatility survives the change because the quadratic variation of ln S over [0,T] is sigma^2 T, and quadratic variation is a limit in probability of sums of squared increments computed path by path. Equivalent measures agree on which events have probability zero, so they agree on that limit, and no reweighting of paths can change it. Only the drift, which is a first order property of the average path, responds to the measure change.

Follow-up: In a stochastic volatility model driven by a second Brownian motion, which parameters are left undetermined by no-arbitrage?

Key concepts: Girsanov theorem, market price of risk, Radon-Nikodym derivative, quadratic variation.