Pricing a European Chooser Option with an Adjusted-Strike Put

# How is the Chooser Option's value computed in this example?


# How is the Chooser Option's value computed in this example?












In preparation for my finals, I am attempting a question on chooser options. One question asks

> A European chooser option on an index ETF paying a yield of 3.0% with strike \$64 has a maturity of T2 = 21 months and a choice regarding the type of the option must be made after T1 = 12 months. The risk-free rate is 7%, stock return volatility is assumed to be 33% per year and currently a share costs $61. What is the value of the option?

The answers are

> One T2-Call = \$10.4641; 0.9778 times T1-Put (\$7.0157) with adjusted strike 62.1085 for a total cost of \$17.32.

Here is my interpretation (most likely incorrect but necessary to illustrate my problem) :

First and foremost, I do not understand how we can value such an option today given the information.

What I do know is that at time t = 1 (yrs), the value of the option is $$V(1) = \mathrm{max}(c,p).$$

At t = 1, both options have the same strike price (\$64) and remaining maturity (0.75 yrs.). It can be shown through the put-call parity that $$V(T1) = \mathrm{max}(c,c+e^{-r\cdot 0.75}K-S_{1}e^{-q\cdot 0.75}) \\ = c+e^{-q\cdot 0.75}\mathrm{max}(0,Ke^{-(r-q)\cdot 0.75}-S_{1}).$$

The constants are given by

- $q$ = dividend yield = 3%

- $r$ = risk-free rate = 7%

- $K$ = strike price = 64

- $S_{1}$ = spot price at time 1 = unknown

Now in order to calculate the value of the call, I require the spot price at t = 1. This is my first problem since I have been given no such information.

How can I move forward from here, find the value of the chooser option at t = 1, and furthermore its value at t = 0 (if that's what the original question requires)?

## Answer by Quantuple (score 5, accepted)

https://quant.stackexchange.com/a/27646

Although the answer of @SRKX is right on spot, I was already writing a solution along the lines of how you had specifically approached the problem. I think it might still be useful to you, so here it goes

The price of the chooser option, as seen of today $t=0$ is by definition \begin{align} V_0 &= \underbrace{e^{-r T_2}}_{\text{Payoff dicount factor}} \underbrace{\mathbb{E}\left[\ \ \underbrace{\max\left( \mathbb{E}[(S_{T_2}-K)^+ \vert \mathcal{F}_{T_1}], \mathbb{E}[(K-S_{T_2})^+ \vert \mathcal{F}_{T_1} ] \right)}_{\text{Expected payoff at $T_2$ as seen of $T_1$}} \ \ \vert \mathcal{F}_0 \ \ \right]}_{\text{Expected payoff at $T_2$ as seen of $t=0$}} \\ &= e^{-r T_2} \mathbb{E}_0\left[ \max\left( \mathbb{E}_{T_1}[(S_{T_2}-K)^+], \mathbb{E}_{T_1}[(K-S_{T_2})^+] \right) \right] \end{align} If you're not familiar with the notation $\mathcal{F}_t$ used for filtrations, you can interpret it as "all the information we know at time $t$". The notation $\mathbb{E}_t[.]$ simply figures that the expectation is taken conditionally on the knowledge of $\mathcal{F}_t$. Naturally all of these expectations are taken under the risk-neutral measure $\mathbb{Q}$.

By definition, we also have that the price of European call/put options is given by $$ C(T_1,S_{T_1};K,(T_2-T_1)) = e^{-r(T_2-T_1)} \mathbb{E}_{T_1}[(S_{T_2}-K)^+] \tag{def 1} := C_{12} $$ $$ P(T_1,S_{T_1};K,(T_2-T_1)) = e^{-r(T_2-T_1)}\mathbb{E}_{T_1}[(K-S_{T_2})^+] \tag{def 2} := P_{12} $$ where $C(t,S_t;K,\tau)$ (resp. $P(t,S_t;K,\tau)$) denotes the price of a European call (resp. put) option as seen of time $t$, given the spot value $S_t$, the strike price $K$ and the time to expiry $\tau$.

Therefore, $$ V_0 = e^{-r T_2} \mathbb{E}_0\left[ \max\left( \frac{C_{12}}{e^{-r(T_2-T_1)}}, \frac{P_{12}}{e^{-r(T_2-T_1)}} \right) \right]$$ Yet by call-put parity: $$ C_{12} - P_{12} = e^{-r(T_2-T_1)}( S_1 e^{(r-q)(T_2-T_1)} - K ) $$ so that we can further write (similarly to what you did) \begin{align} V_0 &= e^{-r T_2} \mathbb{E}_0\left[ \max\left( \frac{C_{12}}{e^{-r(T_2-T_1)}}, \frac{C_{12}}{e^{-r(T_2-T_1)}} - (S_1e^{(r-q)(T_2-T_1)} - K) \right) \right] \\ &= e^{-r T_2} \mathbb{E}_0\left[ \left( \frac{C_{12}}{e^{-r(T_2-T_1)}} + \max\left( 0, K - S_1e^{(r-q)(T_2-T_1)} \right) \right) \right] \\ &= \mathbb{E}_0\left[ e^{-rT_1} C_{12} \right] + \mathbb{E}_0\left[ e^{-rT_2} \max\left( 0, K - S_1e^{(r-q)(T_2-T_1)} \right) \right] \tag{1} \end{align}

Now using $(\text{def } 1)$ the first term of $(1)$ becomes: $$ \mathbb{E}_0 \left[ e^{-rT_1} C_{12} \right] = \mathbb{E}_0 \left[ e^{-rT_2} \mathbb{E}_{T_1}[(S_{T_2}-K)^+] \right] = C(0,S_0;K,T_2)$$ by the tower property of conditional expectations.

Similarly, the second term of $(1)$ can on the other hand be expressed as: \begin{align} \mathbb{E}_0\left[ e^{-rT_2} \max\left( 0, K - S_1e^{(r-q)(T_2-T_1)} \right) \right] &= \mathbb{E}_0\left[ \max\left( 0, Ke^{-rT_2} - S_1e^{-rT_1-q(T_2-T_1)} \right) \right] \\ &= e^{-q(T_2-T_1)} \mathbb{E}_0\left[ e^{-r{T_1}} \max\left( 0, Ke^{-(r-q)(T_2-T_1)} - S_1 \right) \right] \\ &= e^{-q(T_2-T_1)} P(0,S_0; Ke^{-(r-q)(T_2-T_1)}, T_1) \end{align} So that $(1)$ becomes $$ V_0 = C(0,S_0;K,T_2) + \underbrace{e^{-q(T_2-T_1)}}_{= 0.9778} P(0,S_0; \underbrace{Ke^{-(r-q)(T_2-T_1)}}_{= 62.1085}, T_1) $$ hence a $T_2$ call struck at $K$ + 0.9778 units of a $T_1$ put with adjusted strike 62.1085.

## Answer by SRKX (score 3)

https://quant.stackexchange.com/a/27645

You can refer to one of my previous answers here for a detailed development.

There are actually two ways you can price this: - the price of a call plus a put with adjusted strike (like above) - a put plus the price of a call with an adjusted strike (like in my answer).

The only difference is whether you do $\max( a, b ) = b + ( a - b )^+$, or $\max( a, b ) = a + ( b - a )^+$.