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...

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

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