Pricing a Margrabe Option and the Role of Correlation
Summary
The document compares two proposed ways to price a Margrabe option on two correlated assets in a Black–Scholes setting. The first changes numeraire to express the payoff using the asset used as numeraire; under that measure, the relative asset price has volatility that depends on both assets’ volatilities and their correlation. The answer notes that this relative-price process can be derived by orthogonalizing the Brownian drivers and applying Itô’s formula.
The second proposal integrates a single-asset call price over the terminal distribution of the other asset. The accepted response identifies a conditioning error: conditioning only on the path of one asset does not capture the full information needed to derive the stated conditional option price. The excerpt offers no complete integration formula or numerical comparison. It illustrates why dependence between the assets must be handled in the joint model, rather than inferred from the marginal distribution of one asset alone.
Key ideas
- A change of numeraire expresses the option payoff in terms of the ratio of the two asset prices.
- The relative asset’s volatility under the chosen measure depends on both volatilities and their correlation.
- Integrating a call price over the other asset’s marginal distribution can fail if the conditioning information is incomplete.
- The excerpt explains the conceptual issue but does not provide a full numerical integration method or comparison.
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# Margrabe option: change of numeraire versus conditioning and numerical integration
# 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!
## Answer by Daneel Olivaw (score 2, accepted)
https://quant.stackexchange.com/a/50804
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: $$E\left((X_T-Y_T)_+|\mathscr{F}_t\right)\ \not= \ E\left(E(X_T-Y_T)_+|Y_T)|\mathscr{F}_t\right) \ = \ E(C(X_t,Y_T)|\mathscr{F}_t)$$
## Answer by ir7 (score 0)
https://quant.stackexchange.com/a/50808
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 $\mathbb{Q}_Y$).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.