Bsrgvty

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:

$$M_quote = e^−q_0Ttimes S_0(0)times N(d_+)−e^−q_1Ttimes S_1(0)times N(d_−)$$

where:

$$beginalign
& d_pm = fraclogfracS_0(0)S_1(0)+(q_1 − q_0 ±σ^2/2)TsigmasqrtT
\[4pt]
& sigma = sqrtsigma^2_0 + sigma^2_1 − 2rho_impsigma_0 sigma_1
endalign$$

Note that $d_− = d_+ − σsqrtT$.

Let $rhotriangleqrho_imp$. Note that:
$$fracpartial sigmapartial rho(rho)=-fracsigma_0sigma_1sigma(rho)<0$$
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:
$$tag1M_textquote=M(rho)$$
where:
$$M(rho)=e^−q_0TS_0(0)N(d_+)−e^−q_1TS_1(0)N(d_−)$$
and $d_pm$ as defined in your question, with $M_textquote$ the observed market price. In practice, this can be restated as:
$$beginalign
&min_rholeft(M(rho)-M_textquoteright)^2tag2
\
& texts.t. rho in [-1,1]
endalign$$
because $(M(rho)-M_textquote)^2geq0$. 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)=$113.5$, $sigma_0=45%$, $sigma_1=37%$, $T=1text year$ and $q_0=q_1=0$ and it has worked just fine.

We know that $-1lerho_imple 1$ 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.