Introduction

I'm constantly reminded of the value of choice; the ability to select between, say - two potential future outcomes must be valuable! Under rationality, one selects the outcome with the highest perceived value - and this certainly makes someone at least as better of as if they had no choice at all!

This reminder stemmed from an interesting client request - to have the option to enter into one of two swaps, on two different assets, at a fixed future date.

While the value of a vanilla option is trivial - I was curious on how the pricing for this request would deviate.

In this piece - I satiate this curiosity.

The Final Value

Let $V(t)$ be the value of the client's option at some time $t$, and let $X(t)$ and $Y(t)$ represent the value of two assets at the same time. Recalling the two swaps - set $K_{X}$ and $K_{Y}$ to be fixed prices on $X$ and $Y$ respectively. Without loss of generality, we consider the case where the underlying volume is a single unit.

Take $t=T>0$ to be the expiry date of the option. The value of the client's position at $T$ is

\begin{align*} V(T) &= \max(X(T)-K_{X}, Y(T)-K_{Y}) \end{align*}

The familiar $\max$ operator has appeared, although its form can be amended as such

\begin{align} V(T) &= \max(X(T)-K_{X}, Y(T)-K_{Y}) \newline &= \max(X(T)-K_{X} - (Y(T)-K_{Y}), 0) + (Y(T)-K_{Y}) \newline V(T) &= \max((X(T)-Y(T)) - (K_{X}-K_{Y}), 0) + (Y(T)-K_{Y}) \newline \end{align}

By manipulating the initial payoff - we conclude that the value of the client's option is the addition of two derivatives: first - the option on the spread between the assets; and two - an outright swap on a single commodity.

If you're not convinced let's consider some cases. If the the $X$-swap is worth than the $Y$-swap, then from the equation above, the client will exercise the option and it holds that.

\begin{align*} V(T)|\text{Exer.} &= X(T)-Y(T) - (K_{X}-K_{Y}) + Y(T) - K_{Y} \newline V(T)|\text{Exer.} &= X(T)-K_{X} \end{align*}

If the $Y$-swap is worth more - then the option is not exercised. Therefore

\begin{align*} V(T)|\overline{\text{Exer.}} &= 0 + Y(T) - K_{Y} \newline V(T)|\overline{\text{Exer.}} &= Y(T) - K_{Y} \end{align*}

Both of these results are expected for the client. Recall that the result was at $t=T$. At $t$, in a Black-Scholes world, we may express the results more elegantly

\begin{align*} V(t) = C_{t}(S, K_{S}, T-t) + (Y(t) - K_{Y})\text{DF}(t) \end{align*}

where $C_{t}$ is the value of a call option on the spread $S(t) = X(t) - Y(t)$, with strike $K_{S} = K_{X} - K_{Y}$, and time to maturity $T-t$. $\text{DF}(t)$ is the discount factor

In conclusion - to record the client's request - we would, as the market maker, book a short call option on a basket of assets, and a short outright swap on the negative asset.

This is the worth of the client's option! What's yours?