# Black Scholes modified boundary conditions

Compute the price of the payoff $$(2\log(S(T))-K)^+$$. Before I do any algebra, I want to make sure I understand. To solve this problem, I need to solve the Black Scholes PDE with boundary condition $$C(S,T)=(2\log(S(T))-K)^+$$ instead of $$C(S,T)=(S-K)^+$$. Then I will be done. Is there another way to do it?

• If you assume log-normality for $S(T)$, then $\log S(T)$ is normal. This will become a simple exercise. – Gordon Feb 6 at 20:36
• It’s called Bachelier model, there is explicit formula – Ezy Feb 6 at 23:44
• Since you haven't accepted this, is ZRH's answer sufficient or do you need more clarification? – Sanjay Feb 10 at 2:36

Assuming the underlying follows GBM price dynamics, I would do the following to avoid solving the PDE: $$2 log(S(T))-K$$ is positive for $$S>e^{K/2}$$. So if you take $$g_{T}(\xi)$$ to be the lognormal distribution of the underlying at time $$T$$, given an initial underlying price $$S(t)$$, then you should be able to obtain the solution as:
$$C(S(t),t)=e^{-r(T-t)}\int_{e^{K/2}}^{\infty}g_{T}(\xi)(2log(\xi)-K)d\xi$$
• As I understand, the problem is to compute the price of the payoff at time $t$ in terms of $S(t)$, i.e. the expected value of $\max(2\log(S(T))-K,0)$ at time $t$. – L Wheels Feb 6 at 20:21