# Pricing exchange options

I am really puzzled about the mechanism of pricing of exchange options using a change in numeraire:

Suppose that $$S^{(1)}$$ and $$S^{(2)}$$ are stocks satisfying SDEs

$$dS^{(1)}_t = \mu_1 S^{(1)}_t \,dt + \sigma_1 S^{(1)}_t \, dW^{(1)}_t , \quad \quad dS^{(2)}_t = \mu_2 S^{(2)}_t \,dt + \sigma_2 S^{(2)}_t \, dW^{(2)}_t ,$$ where $$W^{(1)}$$ and $$W^{(2)}$$ are Brownian motions with correlation $$\rho$$.

We want to price a European exchange call option that pays $$\max \{ S^{(2)}_T-S^{(1)}_T ,0 \}$$ at maturity $$T$$.

The approach in my book is to set $$S^{(1)}$$ as the numeraire and note that $$\max \{ S^{(2)}_T-S^{(1)}_T ,0 \} = S^{(1)}_T \max \bigg\{ \frac{S^{(2)}_T}{S^{(1)}_T} - 1,0 \bigg\}.$$ Then, by Ito's formula, one can show that $$d \bigg( \frac{S^{(2)}_t}{S^{(1)}_t} \bigg) = \hat{\mu}\frac{S^{(2)}_t}{S^{(1)}_t}\,dt + \hat{\sigma} \frac{S^{(2)}_t}{S^{(1)}_t} \,dW_t,$$ for some Brownian motion $$W$$, where $$\hat{\mu}:= \mu_2 - \mu_1 + \sigma^2_1 - \rho \sigma_1 \sigma_2, \quad \quad \quad \hat{\sigma}:= \sqrt{\sigma^2_1 -2 \rho \sigma_1 \sigma_2 + \sigma^2_2} .$$ Therefore, by defining measure $$\mathbb{Q}$$ as $$\frac{d \mathbb{Q}}{d \mathbb{P}} := \exp \bigg\{ - (\hat{\mu}- \hat{\sigma})W_T - \frac{1}{2}(\hat{\mu}- \hat{\sigma})^2 T \bigg\},$$ it follows by Girsanov's theorem that $$\{W_t + (\hat{\mu}- \hat{\sigma})t \}_{t \in [0,T]}$$ is a $$\mathbb{Q}$$-Brownian motion and hence $$\frac{S^{(2)}_t}{S^{(1)}_t}$$ is a $$\mathbb{Q}$$-martingale.

How to proceed from here?

In the standard Black-Scholes setting of a stock (with dynamics $$S_t$$) and cash (with dynamics $$B_t$$), one constructs a self-financing portfolio with value $$\Pi_t$$. After showing that $$\{ \frac{\Pi_t}{B_t} \}$$ is a $$\mathbb{Q}$$-martingale, for some equivalent measure $$\mathbb{Q}$$, for $$\Pi$$ to be a replicating strategy of a contingent claim $$X$$ at time $$T$$, we set $$\Pi_T = X$$, which implies that $$\text{Value of claim at time } t = \mathbb{E}^{\mathbb{Q}} \big[ \frac{B_t}{B_T} X \big| \mathcal{F}_t \big] = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}} \big[ X \big| \mathcal{F}_t \big] .$$

However, this approach does not work in this case. We simply cannot set $$S^{(2)}_T$$ to be equal to $$\max \big\{ \frac{S^{(2)}_T}{S^{(1)}_T} - 1,0 \big\}$$ just as the case in the Black-Scholes formula. Am I missing something? I am lost...

• Not clear what you are asking for. Is it for the option pricing formula or the PDE satisfied by the option price? – Gordon Oct 3 '19 at 19:53