Show that the fair strike of a variance swap can be replicated by a static portfolio of European options plus a dynamic stock position, and give the weight assigned to the strike K option.

Show that the fair strike of a variance swap can be replicated by a static portfolio of European options plus a dynamic stock position, and give the weight assigned to the strike K option.

Approach: Apply Ito to the log of the price, subtract it from the return on the stock, and identify what remains as one half the accumulated variance. Then decompose the payoff.

The fair variance strike is twice the price of a log contract, replicated by a strip of out-of-the-money options weighted by 1/K^2 in strike, plus a self-financing position of 1/S_t shares. By Ito's lemma d(log S_t) = (dS_t/S_t) - 0.5*sigma_t^2*dt, so integrating gives the realised variance as 2*integral of (dS_t/S_t) minus 2*log(S_T/S_0). The first term is the dynamic stock position and the second is a static European payoff, the log contract, so no volatility model is needed. Any twice-differentiable payoff f decomposes as a strip of calls and puts weighted by f''(K), and for f = -log(S/S_0) the second derivative is 1/K^2, which puts far more weight on low strikes than high ones. That 1/K^2 weighting is why the fair strike sits above at-the-money implied volatility whenever there is downside skew, and why a missing tail of listed strikes biases the replication low.

Follow-up: How does a single downward jump in the stock break this replication, and in which direction does the hedger lose?

Key concepts: log contract, static replication, variance swap, Ito's lemma, 1/K^2 weighting.