By equating this to the price of an atmf straddle under the

Question

Normal versus lognormal distributions Market participants often convert between normal volatility ψ and lognormal volatility σ by adopting the approximate equivalence ψ ? σF(t, T).

We will investigate two others. (a) Setting K = F(t, T) = S_te^r(T-t) , and using the Taylor series for Φ(·), show that the Black-Scholes price at t of an at-the-money-forward straddle is approximated by

By equating this to the price of an ATMF straddle under the normal model with standard deviation ψ, find an equivalence between σ and ψ.

(b) Calculate Var(S_T) under the Black-Scholes model (Question 2(a) in the preface may be useful), and hence find an equivalence between ψ and σ by equating ψ²(T - t), which is Var(S_T) under the normal model, to your answer.

Question 2 (a)

A knockout option

(a) Use your results from Question 1 to prove that, for λ > 0, one call with strike λF(t, T) has the same value as λ puts with strike F(t, T)/λ.