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?

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

• 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. – will Oct 6 '18 at 0:15
• Please have look of this question. – Gordon Oct 6 '18 at 0:35

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 $$K e^{-r(\tau_2-\tau_1)}$$.
• 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! – Magic is in the chain Oct 6 '18 at 13:00