Replicating a European Chooser Option with Calls and Puts

Summary

The document derives a replication for a European chooser option, which lets its holder decide at an intermediate date whether to receive a call or a put, both expiring later. At the choice date, put–call parity expresses the put payoff as the call payoff plus the value of a put with a discounted strike and expiry at the choice date. Taking the greater of the call and put values therefore yields a portfolio consisting of a call expiring at the later maturity and a shorter-dated put with strike equal to the original strike discounted to the choice date.

The two component options can be priced with an option model, and their values added to obtain the chooser value. The explanation gives an algebraic payoff identity rather than numerical examples. It assumes European exercise and the stated parity setup, including a rate used to discount the strike; practical valuation must use market-consistent inputs and conventions. When the choice date coincides with final expiry, the payoff reduces to the combined call and put payoffs, with only one typically having positive intrinsic value.

Key ideas

• A chooser option allows selection between a call and a put at an intermediate date.
• Put–call parity rewrites the put value at the choice date using a call and a discounted strike.
• The chooser payoff can be replicated by a longer-dated call and a shorter-dated put.
• The shorter-dated put uses a strike discounted from the original strike to the choice date.
• The replication relies on European option assumptions and consistent pricing inputs.

Tags

# Replication of the payoff of a chooser option


# Replication of the payoff of a chooser option












With numerical examples, how can the payoff of a chooser option be replicated with European call and put options?

## Answer by Kevin (score 3, accepted)

https://quant.stackexchange.com/a/77386

Consider a European chooser option which allows you to choose at time $\tau$ if you want to receive a put option and call option with maturity $T>\tau$.

At time $\tau$, using the put-call parity, the payoff is $$\max\{C,P\}=\max\{C,C-S+Ke^{-r(T-\tau)}\}=C+\max\{Ke^{-r(T-\tau)}-S,0\}.$$

Thus, owning the chooser option is identical to owning a call option with strike price $K$ and maturity $T$ and a put option with strike price $Ke^{-r(T-\tau)}$ and maturity $\tau$.

You can now use any option pricing model (eg Black-Scholes option) to determine the value of these two option and the combined value will be the value of the chooser option.

Interestingly, if $\tau=T$, then $\max\{C,P\}=C+P$. Note that one of the options will be worthless though (expire OTM).

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.