# Margrabe option: change of numeraire versus conditioning and numerical integration

I am having a slight brain meltdown because I do not seem to be able to understand the following basic thing.

Consider a BS economy, and two assets $$X$$ and $$Y$$ $$dX = \sigma X dW$$ $$dY = \nu Y dZ$$ $$dWdZ = \rho dt$$

I would like to price a Margrabe option $$(X_T - Y_T)_+$$.

The first and most straightforward method is a change of numeraire approach. In other words $$E_t(X_T - Y_T)_+ = Y_t E^{Q_Y}( X_T/Y_T -1 )_+$$ where $$Q_Y$$ is the measure with $$Y$$ as numeraire. Now if you evaluate the above expression under this measure you get a relatively simple option price expression, and where the correlation $$\rho$$ will appear in the formula. Agree?

The second approach is to use conditioning. Does everyone agree that I can also price the options as follows: $$E_t(X_T - Y_T)_+ = \int_0^\infty C(X_t, y) q(y) dy$$ where $$C(X_t, y)$$ is the single asset BS option price with strike $$y$$, and $$q(y)$$ is the lognormal distribution of $$Y$$.

I can always calculate using the numerical integration above right? If so, here is where I am confused: how does the correlation parameter $$\rho$$ appear in the numerical integration? I cannot see it, but it must somehow play a role.

Help!

Might you be using the tower law in a wrong way? I have the impression you derive your second equation by conditioning by the $$\sigma$$-algebra generated by $$(Y_t)_{t\geq0}$$, however note that: $$\mathscr{F}_t\nsubseteq\sigma(Y_t)_{0\leq t\leq T}$$ Hence: $$E\left((X_T-Y_T)_+|\mathscr{F}_t\right)\ \not= \ E\left(E(X_T-Y_T)_+|Y_T)|\mathscr{F}_t\right) \ = \ E(C(X_t,Y_T)|\mathscr{F}_t)$$
• Yes I think you are right. What I have to do is split the Brownian motions $W$ and $Z$ into orthogonal components, and then I can use the tower law. So if I write $dZ = \rho dW + \sqrt{1-\rho^2} dW^\bot$, then I can condition on $W^\bot$, agree? Jan 23 '20 at 10:27
Agreeing to your first observation: After orthogonalization, with independent W and W’, and using self explanatory notation for the new diffusion coefficients, which obviously depend on $$\rho$$, we can show that, under $$\mathbb{Q}_Y$$, we have:
$$dR = R[(\sigma_{XW} - \sigma_{YW})dW + (\sigma_{XW’} - \sigma_{YW’})dW’],$$
where $$R=XY^{-1}$$ (used only Ito calculations and the fact that R is a martingale under $$\mathbb{Q}_Y$$).