A desk quotes a contract paying S_T^2 at T = 1 with S_0 = 100, sigma = 20% and r = 5%, no dividends. Give the PDE the price satisfies, the closed form, and the value.

A desk quotes a contract paying S_T^2 at T = 1 with S_0 = 100, sigma = 20% and r = 5%, no dividends. Give the PDE the price satisfies, the closed form, and the value.

Approach: Price as a discounted risk-neutral expectation of the square of a lognormal, then verify the answer satisfies the Black-Scholes equation with the quadratic terminal condition.

10941.74. Under the risk-neutral measure S_T = S_0 exp((r - sigma^2/2)T + sigma W~_T), so the lognormal moments give E[S_T^2] = S_0^2 exp((2r + sigma^2) T), and risk-neutral valuation of the power contract gives V_0 = e^{-rT} E[S_T^2] = S_0^2 exp((r + sigma^2) T). Numerically that is 10000 * exp(0.05 + 0.04) = 10000 * 1.09417 = 10941.74. Feynman-Kac confirms it: the candidate V(t,S) = S^2 exp((r + sigma^2)(T-t)) has V_t = -(r + sigma^2)V, S V_S = 2V and S^2 V_SS = 2V, so V_t + (1/2) sigma^2 S^2 V_SS + r S V_S - r V = (-(r+sigma^2) + sigma^2 + 2r - r) V = 0, and V(T,S) = S^2 matches the payoff. The delta is 2 S exp((r + sigma^2)(T-t)) = 218.83 at inception, so the hedge is more than two shares per contract.

Follow-up: What is the gamma of this contract and what does its constancy in S imply for the variance of a discretely rebalanced hedge?

Key concepts: Feynman-Kac, risk-neutral valuation, lognormal moments, power contract.