3
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

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: $$E\left((X_T-Y_T)_+|\mathscr{F}_t\right)\ \not= \ E\left(E(X_T-Y_T)_+|Y_T)|\mathscr{F}_t\right) \ = \ E(C(X_t,Y_T)|\mathscr{F}_t)$$

$\endgroup$
  • 1
    $\begingroup$ 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? $\endgroup$ – ilovevolatility Jan 23 at 10:27
1
$\begingroup$

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 $\mathbb{Q}_Y$).

$\endgroup$
  • $\begingroup$ Thank you, yes I understand now. Don't know where I was with my thoughts that I didn't see it (that I had to orthogonalize I mean). $\endgroup$ – ilovevolatility Jan 23 at 13:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.