Pricing a Logarithmic Call Payoff with a Lognormal Expectation

Summary

The document considers a European-style payoff that is positive when twice the log of the terminal asset price exceeds a strike parameter. Under geometric Brownian motion, the answer avoids solving a modified Black–Scholes partial differential equation directly. It identifies the asset-price threshold where the payoff becomes positive and prices the claim as the discounted expectation of that payoff over the lognormal terminal-price distribution.

This expectation can be evaluated by integration, with a change of variables suggested as a way to simplify the calculation. The method relies on the stated GBM assumption and a risk-neutral pricing setup, including discounting at the risk-free rate. The note gives no numerical example or comparison with a PDE solution, and it does not work through the integral or discuss alternative dynamics, so it serves as a concise valuation setup rather than a complete derivation.

Key ideas

• Under geometric Brownian motion, the terminal asset price has a lognormal distribution.
• The payoff is positive only above the asset-price threshold implied by its logarithmic expression.
• The claim can be valued as a discounted expectation over terminal prices above that threshold.
• A variable substitution may help evaluate the resulting integral.

Tags

# Black Scholes modified boundary conditions


# 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?

## Answer by ZRH (score 1, accepted)

https://quant.stackexchange.com/a/43932

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:

$C(S(t),t)=e^{-r(T-t)}\int_{e^{K/2}}^{\infty}g_{T}(\xi)(2log(\xi)-K)d\xi$

wrt Gordon's comment, you would apply variable substitution to solve this

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.