Log-contract The payoff of the log-contract on an underlying asset S is DT = – ln(ST / F0), where F0 is the forward price of S for maturity T.
(a) Draw the payoff function of the log-contract.
(b) Find the fair value D0 of the log-contract in the lognormal model as a function of r, σ and T.
(c) Imagine that, for a given maturity T, we know the prices of an infinite number of European calls and puts struck along a continuum between 0 and infinity. Consider a portfolio which is:
(i) long a quantity dK / K2 of puts struck at K <>0;
(ii) long a quantity dK / K2 of calls struck at K > F0.
