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.

  • 2
    $\begingroup$ 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. $\endgroup$ – Ivan Jun 2 at 17:17
  • $\begingroup$ 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? $\endgroup$ – John Tuwim Jun 2 at 19:12
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    $\begingroup$ 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. $\endgroup$ – Ivan Jun 2 at 19:55
  • $\begingroup$ Thank You! I do understand now. $\endgroup$ – John Tuwim Jun 3 at 10:42
  • $\begingroup$ This is a very nice example for calculations. All risky terms can be nicely written as integral with respect to $\hat{W}_t$! $\endgroup$ – John Tuwim Jun 3 at 11:40

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