Valuing an Option to Switch from a Call to a Put

# How to price an option allowing to change a call into a put?


# How to price an option allowing to change a call into a put?












A recruiter asked me this question:

Suppose you have the following contract:

- a call option with maturity $T$ = 2 years

- the possibility to change this call into a put at $t$ = 1 year

What is the price of such contract ?

I begin with $E[\left((-1)^{\tau}(S_{T}-K)\right)^{+}e^{-rT}]$ with $\tau$ a random variable that equals $+1$ if we change the call into a put and $-1$ if we don't change it, but i'm stuck with this...

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/21978

Let $t=1$ and $T=2$. The value at time $t$ is given by \begin{align*} &\ e^{-r(T-t)}\max\left(E\left((S_T-K)^+\mid \mathcal{F}_{t}\right), \, E\left((K-S_T)^+\mid \mathcal{F}_{t}\right)\right) \\ =&\ e^{-r(T-t)}E\left((K-S_T)^+\mid \mathcal{F}_{t}\right) +e^{-r(T-t)}\max\left(E\left((S_T-K)\mid \mathcal{F}_{t}\right), \, 0\right)\\ =&\ e^{-r(T-t)}E\left((K-S_T)^+\mid \mathcal{F}_{t}\right) +\max\left(S_{t}-Ke^{-r(T-t)}, \, 0\right). \end{align*} That is, the value is for a portfolio with a put at $T$ and a call at $t$, and can be computed using formulas of Black-Scholes type.

## Answer by SRKX (score 5)

https://quant.stackexchange.com/a/22016

Let's define $t=0$, $T_1 = 1$ and $T_2 = 2$.

I believe the interviewer is looking for the price of the "global" option $V_t$ for $t \leq T_1 \leq T_2 $.

Let's define the payoff at time $T_1$: it is the maximum between the value of a call or a put on the same underlying with maturity at $T_2$.

$$\text{Payoff}_{T_1} = \max( c_{T_1}, p_{T_1} )$$

where $c_{T_1}$ and $p_{T_1}$ are respectively the price at time $T_1$ of a call and a put under the same underlying with expiry at time $T_2$.

As noted by Gordon, we know that $\max( a, b ) = b + ( a - b )^+$, hence:

$$\text{Payoff}_{T_1} = p_{T_1} + ( c_{T_1} - p_{T_1} )^+$$

By the put call parity, we know that $c_{T_1} - p_{T_1} = S_{T_1} - e^{-r(T_2-T_1)}K$ and therefore we get:

$$\text{Payoff}_{T_1} = p_{T_1} + ( S_{T_1} - e^{-r(T_2-T_1)}K )^+$$.

Let's write the general statement: $$V_t = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} \text{Payoff}_{T_1} | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} \left( p_{T_1} + ( S_{T_1} - e^{-r(T_2-T_1)}K )^+ \right) | \mathcal{F}_t \right]$$

We can split the value into two main terms:

$$V_t = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} p_{T_1} | \mathcal{F}_t \right] + \mathbb{E}_Q \left[ e^{-r(T_1 - t)} ( S_{T_1} - e^{-r(T_2-T_1)}K )^+ | \mathcal{F}_t \right]$$

We see that the second term is simply the price at time $t$ of a call option on $S$ expiring at $T_1$ with strike $K' = e^{-r(T_2-T_1)}K$.

Furthermore, we know that:

$$ p_{T_1} = \mathbb{E}_Q \left[ e^{-r(T_2 - T_1)} (K - S_{T_2}) | \mathcal{F}_{T_1} \right]$$

So the first term can be seen as:

$$\mathbb{E}_Q \left[ e^{-r(T_1 - t)} \mathbb{E}_Q \left[ e^{-r(T_2 - T_1)} (K - S_{T_2})^+ | \mathcal{F}_{T_1} \right] | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} e^{-r(T_2 - T_1)} (K - S_{T_2})^+ | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_2 - t)} (K - S_{T_2})^+ | \mathcal{F}_t \right] $$

because of the iterated conditoning rule and since $\mathcal{F}_t \subseteq \mathcal{F}_{T_1}$.

This second term is the value at time $t$ of a put expiring at time $T_2$ with strike $K$.

So the answer is that at time $t$ the value of the option is the value of the put plus the value of a call expiring at $T_1$ with strike $K' = e^{-r(T_2-T_1)}K$.