How to calculate implied correlation via observed market price (Margrabe option)

Question

I can't seem to figure out how to do the following: compute the implied correlation $ρ_{imp}$ by using the observed market price $M_{quote}$ of a Margrabe option, and solving the non-linear equation shown below:

$$M_{quote} = e^{−q_0T}\times S_0(0)\times N(d_+)−e^{−q_1T}\times S_1(0)\times N(d_−)$$

where:

$$\begin{align} & d_\pm = \frac{\log\frac{S_0(0)}{S_1(0)}+(q_1 − q_0 ±σ^2/2)T}{\sigma\sqrt{T}} \\[4pt] & \sigma = \sqrt{\sigma^2_0 + \sigma^2_1 − 2\rho_{imp}\sigma_0 \sigma_1} \end{align}$$

Note that $d_− = d_+ − σ\sqrt{T}$.

Answer 1

Let $\rho\triangleq\rho_{imp}$. Note that: $$\frac{\partial \sigma}{\partial \rho}(\rho)=-\frac{\sigma_0\sigma_1}{\sigma(\rho)}<0$$ Therefore $\sigma$ is monotonic in implied correlation. In addition, the Margrabe pricing function $M(\cdot)$ is also monotonic in volatility $\sigma$ thus you can find an unique solution to the equation: $$\tag{1}M_{\text{quote}}=M(\rho)$$ where: $$M(\rho)=e^{−q_0T}S_0(0)N(d_+)−e^{−q_1T}S_1(0)N(d_−)$$ and $d_\pm$ as defined in your question, with $M_{\text{quote}}$ the observed market price. In practice, this can be restated as: $$\begin{align} &\min_\rho\left(M(\rho)-M_{\text{quote}}\right)^2\tag{2} \\ &\ \text{s.t. } \rho \in [-1,1] \end{align}$$ because $(M(\rho)-M_{\text{quote}})^2\geq0$. This is an optimization problem which can be solved through traditional techniques:

• The solution suggested by @Alex C will give you a quick, approximate answer;
• If you want arbitrary precision, you can use a simple Newton algorithm on either $(1)$ or $(2)$ with root value $\rho=0$, this is quick to program in Excel VBA, or you can maybe even find an online tool that does it. This PDF explains the method for a vanilla call in a Black-Scholes framework to find the implied volatility, but the set-up is very similar. Another alternative is gradient descent but this would probably take longer to program and you have to do it on $(2)$;
• You can also use Excel's Solver to find a solution to $(1)$ directly. I have tried with $S_0(0)=\$101$, $S_1(0)=\$113.5$, $\sigma_0=45\%$, $\sigma_1=37\%$, $T=1\text{ year}$ and $q_0=q_1=0$ and it has worked just fine.

Answer 2

We know that $-1\le\rho_{imp}\le 1$ so perhaps the simplest approach is to try the possible values $\rho_{imp}=\{-1,-0.9,-0.8,\cdots,0.8,0.9,+1\}$, to calculate resulting $\sigma$ values, d± values, and $M_{quote}$ values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed correlations to pin it down more precisely. It is a manual but relatively simple method.