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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?

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

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  • $\begingroup$ Iff you have the flexibility to chose right up to expiry, then as you've written the value is the same as for a straddle. If you have to choose earlier, then it's a bit more complex. $\endgroup$ – will Oct 6 '18 at 0:15
  • $\begingroup$ Please have look of this question. $\endgroup$ – Gordon Oct 6 '18 at 0:35
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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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  • $\begingroup$ I didn't mention it but one of the conditions is that both call and put options should have the same strike and expire at the same time. $\endgroup$ – mirik Oct 6 '18 at 6:47
  • $\begingroup$ If I see it correct the options that consist the chooser option are actually virtual, they have the same strike and I can decide which one of them to take up to the specific time. On the other hand the real options in straddle have different strikes and I can decide which one of them to take up to the their expiration. $\endgroup$ – mirik Oct 6 '18 at 6:55
  • $\begingroup$ I think of it this way: In straddle you have both options until $\tau_2$. In chooser option, you have to give up one of them at $\tau_1$ and hence the difference. Glass half-full/half-empty kinda situation! $\endgroup$ – Magic is in the chain Oct 6 '18 at 13:00

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