# Price a SWAP contract in no arbitrage market

We have a single period, arbitrage free market with riskless rate of return $$r$$. A swap contract operates in the following way:

1. At $$t = 0$$ the buyer pays the seller amount $$q$$.
2. 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

$$$$q = \mathbb{E}_\pi(S^A_1 - S^B_1)e^{-r}.$$$$

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.