One trader states that the replicating portfolio for this

Question

The squared payout A derivative contract has payout at maturity defined by D(T, T) = , and priceD(t, T) at current time t.

(a) One trader states that the replicating portfolio for this derivative is a holding of St stocks, and thus D(t, T) =  . What is wrong with this argument?

(b) Another trader states that the replicating portfolio is a holding of ST stocks, and thus D(t, T) = StS_T. What is wrong with this argument?

(c) Suppose the risk-neutral distribution of S_T conditional on S_t, with respect to the money market numeraire, is lognormal (log S_t + ν(T - t), σ²(T - t)), that is, log S_T | S_t ∼ N(log S_t + ν(T - t), σ²(T - t)). Use a martingale condition for S_T/M_T to find ν in terms of σ and r, the constant, continuously compounded interest rate.

(d) Use the fundamental theorem to write down, but do not solve, an integral expression for D(t, T) involving a normal probability density function.

(e) Suppose you are also told that the risk-neutral distribution of S_T conditional on S_t, with respect to the stock numeraire, is lognormal (log S_t + γ (T - t), σ²(T - t)). Use an appropriate martingale condition, and the fact that 1/S_T is also lognormal, to find γ in terms of r and σ.

(f) Hence use the fundamental theorem to prove that, assuming the distribution in (e) holds,