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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...

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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.

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  • $\begingroup$ I understand the calculus you wrote but not the "philosophy". For me the 1st line is not the payoff the holder of such contract can receive but only the maximum between the price of a Call and a Put both with maturity T (?) $\endgroup$
    – glork
    Nov 29 '15 at 10:11
  • $\begingroup$ Agreed. It should be the value at time $t$ instead of the payoff. As you need to decide whether you want the call or put, the value is the maximum. $\endgroup$
    – Gordon
    Nov 29 '15 at 12:43
  • $\begingroup$ Ok and for the value at time 0 we just have to discount it one more time. Just a last question, I don't completely understand that from the holder point of view, the price of his contract is the $maximum$ between the price of a call and the price of a put. If a person A sells the contract to a person B, then the premium that B gives to A should be the maximum between the price of a call and the price of a put in order that A can hedge his position for all the situations, right ? $\endgroup$
    – glork
    Nov 29 '15 at 14:15
  • $\begingroup$ Yes. That is correct. $\endgroup$
    – Gordon
    Nov 29 '15 at 14:22
  • $\begingroup$ How do you get from line 1 to line 2? Both values should be positive because of the extrinsic value right? $\endgroup$
    – SRKX
    Nov 30 '15 at 1:29
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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 $K' = e^{-r(T_2-T_1)}K$.

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