Closed-Form Valuation of Forward-Starting Calls with Nonzero Strike

Summary

The document asks whether a forward-starting call on an asset’s price change has a closed-form valuation when its strike is nonzero. Its payoff is the positive part of the difference between the asset price at a later time and its price at an earlier time, minus the strike. The question compares this payoff with a call spread on two distinct assets, for which the author says a closed-form price is unavailable under log-normal asset prices.

The author points out that both cases involve a pair of log-normal prices with a correlation structure, but wonders whether the fact that the pair is observed at different times could make the forward-starting option easier to value. No pricing derivation, model assumptions beyond log-normality, numerical results, or answer are included. The document therefore frames a derivatives-pricing question rather than presenting a valuation method; any conclusion would depend on the joint price dynamics, discounting assumptions, and precise meaning of the log-normal model.

Key ideas

• The payoff is based on the difference between an asset’s prices at two different times, less a nonzero strike.
• The document compares the forward-starting call with a call spread on two assets.
• Both payoffs depend on pairs of log-normal prices with a correlation structure.
• The post asks whether different observation times permit a closed-form valuation but gives no answer.

Tags

# Valuation of forward-starting call with non-zero strike


# Valuation of forward-starting call with non-zero strike












We know prices for call spread options with strike $K\neq0$ that is an option whose payoff $\varphi(S_T^1,S_T^2)$ is given by: $$\varphi(S_T^1,S_T^2):=(S_T^1-S_T^2-K)^+$$ where $S^1,S^2$ are the prices of two distinct assets, do not admit a closed-form expression when the assets have a log-normal distribution.

Does the same hold for forward-starting calls on price returns? Namely letting $S$ be the price of some asset and $\Delta>0$, the option with payoff: $$\varphi(S_T,S_{T+\Delta}):=(S_{T+\Delta}-S_T-K)^+$$

Ultimately both $(S_T^1,S_T^2)$ and $(S_T,S_{T+\Delta})$ are two pairs of log-normal variables with some correlation structure, but I was wondering whether in the second case, the different evaluation times might allow for a close-form solution.

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