A stock has mu = 12%, r = 3% and sigma = 25% under the physical measure. Compute the value of the Radon-Nikodym derivative dQ/dP on a one year path that ends with W_1 = 1.

A stock has mu = 12%, r = 3% and sigma = 25% under the physical measure. Compute the value of the Radon-Nikodym derivative dQ/dP on a one year path that ends with W_1 = 1.

Approach: Compute the market price of risk from the excess return over volatility, then substitute into the stochastic exponential evaluated at the given terminal Brownian value.

0.6539. The market price of risk is lambda = (mu - r)/sigma = (0.12 - 0.03)/0.25 = 0.36. Girsanov's theorem gives the Radon-Nikodym derivative of the risk-neutral measure as dQ/dP = exp(-lambda W_T - lambda^2 T/2), so with W_1 = 1 and T = 1 the exponent is -0.36 - 0.36^2/2 = -0.36 - 0.0648 = -0.4248 and the value is exp(-0.4248) = 0.6539. A value below 1 means this path, which had a favourable Brownian shock, is downweighted under Q, consistent with the risk-neutral measure removing the compensation for bearing that risk. The average of dQ/dP over all paths is 1 by construction.

Follow-up: What is E_P[(dQ/dP)^2] here, and what does its size say about the stability of importance sampling between the two measures?

Key concepts: market price of risk, Radon-Nikodym derivative, Girsanov theorem, risk-neutral measure.