Valuing a Simple Chooser Option as a Call Plus a Put

Summary

The document examines whether a simple chooser option can be valued as the sum of a call and a put. The proposed algebra uses put-call parity to express the maximum of call and put values as a call plus a put-like term. The answer confirms the decomposition but points out that the chooser's decision date and the options' expiration date must be distinguished.

At the decision date, the call component retains the later maturity, while the additional put has an earlier maturity and a strike discounted for the time remaining between decision and expiration. Thus the decomposition is not generally a call and put with identical maturities and strikes. The note provides a structural pricing identity, not numerical evidence or a full derivation. Its applicability depends on the chooser's stated exercise and maturity terms, and the simplified formula should not be applied without matching those dates and the corresponding strike.

Key ideas

• Put-call parity can express a simple chooser value as a call plus a put component.
• The chooser has a decision date distinct from the underlying options' expiration date.
• The call and put components in the decomposition have different maturities.
• The put component uses a strike adjusted for the time between decision and expiration.

Tags

# Value simple chooser option as a sum of call and put options


# Value simple chooser option as a sum of call and put options












There is a well known formula for valuating the chooser's option price: $H_{chooser}=max\{C(S_t, K, T-t), P(S_t, K, T-t)\}=max\{C(S_t, K, T-t), C(S_t, K, T-t)+Ke^{−r(T-t)}−S_t\}=C(S_t, K, T-t) + max\{0, Ke^{−r(T-t)}−S_t\}$

The max element of this formula resembles the regular European put option, so is it correct to rewrite the formula as a sum of a call and put options?

$H_{chooser}=C(S_t, K, T-t)+P(S_t, Ke^{−r(T-t)}, T-t)$

## Answer by Magic is in the chain (score 0, accepted)

https://quant.stackexchange.com/a/42070

Yes but you will need to account for two times: decision time and the option maturities, lets call them $\tau_1$ and $\tau_2$. The put call parity that you used relates prices of the options as at decision time $\tau_1$ for resdiual maturities $\tau_2 -\tau_1$. So when you take the call price out of the max, it has payoff at $\tau_2$. The other term becomes $ max \left( 0, -S+K e^{-r(\tau_2-\tau_1)}\right)$ which is a put option with maturity $\tau_1$ and strike $K e^{-r(\tau_2-\tau_1)}$.

So in summary you can write it as sum of a call and a put option but the options have different maturities and different strikes.

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