# Black-Scholes market and payoff with integrals

I am struggling with the following exercise:

Prove that on Black-Scholes market, with some parameters $$r, \mu, \sigma >0$$, a payoff $$X=\int_{0}^{T}\ln \frac{S_t}{S_0}\mathrm{d}t+\frac{1}{\sigma}\Big(\ln^2\frac{S_T}{S_0} + (\sigma^2-2\mu)T\ln\frac{S_T}{S_0}\Big),$$

is replicable. Find the appropriate strategy.

I know that if $$\ \mathbb{E}_{Q}X^2<\infty, \$$ then $$X$$ is replicable. I guess that this can be easily checked here. But I do not know how to look for replication strategy. This $$X$$ is not in the form of $$X=f(S_T)$$, which I have seen in some materials. I will be grateful for any help.

• Compute the discounted expectation: this is the price of the payoff today. Then work out the SDE governing its evolution. Then neutralize the risky terms. This is the strategy. – Ivan Jun 2 at 17:17
• Ok, computing $\Pi_0(X)$ is easy. By SDE governing "its" evolution You mean evolution of $\Pi$? What do You understand by neutralizing risky terms? Could You point me towards a simillar problem? – John Tuwim Jun 2 at 19:12
• The evolution of $P_i$, correct. Using Ito, you will find that there is a risky (stochastic) term i.e. $dS_t$ term. The replication strategy is that which neutralizes this term by means of a suitably chosen dynamic holding of the stock. – Ivan Jun 2 at 19:55
• Thank You! I do understand now. – John Tuwim Jun 3 at 10:42
• This is a very nice example for calculations. All risky terms can be nicely written as integral with respect to $\hat{W}_t$! – John Tuwim Jun 3 at 11:40