Rates are zero, the one year forward is 100 and the risk neutral distribution is lognormal with 20% annual volatility. What is the fair value of a claim paying S_T^2 in one year, and how is it replicated statically?

Rates are zero, the one year forward is 100 and the risk neutral distribution is lognormal with 20% annual volatility. What is the fair value of a claim paying S_T^2 in one year, and how is it replicated statically?

Approach: Take the risk neutral expectation of the square under a lognormal law, then expand the payoff to second order in the strike to get the option strip that reproduces it.

10408. Under the risk neutral measure S_T = F*exp(sigma*sqrt(T)*Z - sigma^2*T/2), so the risk neutral expectation of the square is F^2*exp(sigma^2*T) = 10000*exp(0.04) = 10408.1, and with zero rates that expectation is the price. Static replication follows the Carr-Madan formula: a twice differentiable payoff equals f(F) plus f'(F)*(S_T - F) plus the integral of f''(K) against out of the money options at every strike K. Here f''(K) = 2 everywhere, so the claim is 10000 in cash, 200 forwards, and two units of every strike in the option strip, and since one unit of every strike is worth 204.05 that leg is worth 408.1. The weighting is flat in strike, which is what makes the claim replicable with no model for the lognormal parameters, and the same construction with 1/K^2 weights gives the variance swap.

Follow-up: What strip weights replicate a claim paying log(S_T), and why does the variance swap follow from it?

Key concepts: static replication, carr-madan formula, risk neutral expectation, lognormal.