We have a single period, arbitrage free market with riskless rate of return $r$. A swap contract operates in the following way:
- At $ t = 0$ the buyer pays the seller amount $q$.
- The seller agrees to give the buyer a unit of asset $A$ in exchange for a unit of asset $B$ at time $t = 1$
The question is to find the fair market value of the SWAP contract, q. Below is my solution which I would like verified.
Solution:
Since the market is arbitrage free, by the fundamental theorem of arbitrage pricing, there exists a risk-neutral measure $\pi$ such that for any asset $C$ the price of $C$ at $t = 0$ is given by
$$S_0^C = e^{-r}\mathbb{E}_\pi S_1^C, $$
where $S_i^C$ is the price of $C$ at time $i$. The expected value of the SWAP contract at maturity is simply $\mathbb{E}_\pi(S^A_1 - S^B_1)$, so my guess for the fair value at $t = 0$ is
\begin{equation} q = \mathbb{E}_\pi(S^A_1 - S^B_1)e^{-r}. \end{equation}
We verify this by showing that there is an arbitrage if it does not hold. Suppose $ q > \mathbb{E}_\pi(S^A_1 - S^B_1)e^{-r}$. Then
at t=0: Sell a SWAP contract for $q$. Sell a stock of $B$ short. Use the proceeds
$$ q + S_0^B = q + e^{-r}\mathbb{E}_\pi S_1^B > S_0^A,$$
to buy a stock of $A$. Invest the remained in the riskless asset.
at $t = 1$: Execute the swap of B with A. Return shorted B stock. You get to keep
$$(q + S^B_0 - S^A_0)e^{r} > 0.$$
So this is an arbitrage.
On the other hand if $q < \mathbb{E}_\pi(S^A_1 - S^B_1)e^{-r}$, then
At $t = 0$: Sell a stock of $A$ short. Buy a swap contract with the proceeds. Then you have
$$S_0^A - q > S_0^B$$
left. Buy a share of B. Invest the rest in the riskless asset.
At $t =1$: Swap your share of $B$ for $A$, then return the shorted $A$ stock. You made $$(S_0^A - q - S_0^B)e^r > 0,$$ hence there is an arbitrage. This complete the proof.
