# Effect of correlation on a best-of rainbow option

EDIT 2: I found the problem(s) and the prices seem to behave as expected now. For anyone interested there was a bug when normalizing the dependant ranom normal variates used in the simulation, so while they had the correct correlation one of them had a standard deviation of 1 and the other a standard deviation much greater than 1. Causing the price to not drop (and even increase) even as correlation increased. The parity relation suggested by @ir7 seems to hold now which gives me confident that all is well.

I'm valuing a rainbow option numerically with a monte-carlo simulation and I'm getting some unexpected results. The price of a best-of call option is decreasing in price as correlation up to a certain point where it starts to increase, contrary to my intuition. The worst-of option case is much more well behaved with being an increasing function of the correlation as expected. Since I'm valuing them in much the same way (just taking min(...) instead of max(...) in my code) I'm very confused as to what could be wrong, or is it possible for the price to behave in this way? If it's completely unreasonable, does anyone want to hazard a guess as to why my calculations might be breaking down as the correlation increases? The program is written in C++ so if anyone fluent in C++ wants to have a look at my code for something erroneous I'd be more than happy to post it.

EDIT 1: After troubleshooting a bit with the help of @ir7 it seems there's something off with my Monte Carlo simulation for several assets (the single asset case works fine). Posting (some of) my C++ code below for the simpler problem of valuing an out-performance option which as a closed form solution for him (and anyone else who wants to of course) to check out and give help. The caclulations again seem to break down once the correlation goes above ~0.5, see below image. If there are some function calls used that you want me to explain or post the code for I'd be happy to do so, for now I'll try to keep it somewhat bare:

The class and function that does the actual valuation:

MonteCarloOutPerformanceOptionFunction::MonteCarloOutPerformanceOptionFunction(std::string uniqueIdentifier_, int nominal_, std::vector<double> S0_vect, std::vector<Wrapper<PayOff>> ThePayOffVect_, double r_, std::vector<double> d_vect_, std::vector<double> impvol_vect_, std::vector<std::vector<double>> covMatrix_, double TTM_, unsigned long numberOfPaths_)
: r(r_), S_vect(S0_vect), ThePayOffVect(ThePayOffVect_), d_vect(d_vect_), covMatrix(covMatrix_), valuationFunction(uniqueIdentifier_, TTM_, nominal_), numberOfPaths(numberOfPaths_), impvol_vect(impvol_vect_)
{
if (covMatrix.size() != S_vect.size())
throw("Missmatched Covariance matrix and initial spot values array sizes in OutPerformance Option");
if (2 != S_vect.size())
throw("More than two equities specified in OutPerformance Option");
}

void MonteCarloOutPerformanceOptionFunction::ValueInstrument()
{
std::vector<MJArray> correlatedNormVariates = GetArraysOfCorrelatedGauassiansByBoxMuller(numberOfPaths, covMatrix);
std::vector<StatisticAllPaths> thesePathGatherers;
for (unsigned long i = 0; i < S_vect.size(); i++)
{
StandardExcerciseOption thisOption(ThePayOffVect[i], TTM);
StatisticAllPaths onePathGatherer;
thesePathGatherers.push_back(onePathGatherer);
OneStepMonteCarloValuation(thisOption, S_vect[i], impvol_vect[i], r, d_vect[i], numberOfPaths, correlatedNormVariates[i], thesePathGatherers[i]);
}
f = 0;
for (unsigned long i = 0; i < numberOfPaths; i++)
{
std::vector<double> outcomes;
outcomes.reserve(S_vect.size());
for (unsigned long j = 0; j < S_vect.size(); j++)
{
outcomes.push_back(thesePathGatherers[j].GetOneValueFromResultsSoFar(i));
}
f += std::max(outcomes[0] - outcomes[1], 0.0);
}
f *= ((double)nominal / numberOfPaths);
return;
}


The Monte Carlo simulation function being called at OneStepMonteCarloValuation (this seems to work fine for single asset options like vanilla calls/puts)

void OneStepMonteCarloValuation(const StandardExcerciseOption& TheOption, double Spot, double Vol, double r, double d, unsigned long NumberOfPaths, MJArray normVariates, StatisticsMC& gatherer)
{
if (normVariates.size() != NumberOfPaths)
throw("mismatched number of paths and normal variates");
//Pre-calculate as much as possible
double Expiry = TheOption.GetExpiry();
double variance = Vol * Vol * Expiry;
double rootVariance = sqrt(variance);
double itoCorrection = -0.5 * variance;
double movedSpot = Spot * exp((r-d) * Expiry + itoCorrection);
double thisSpot;
double discounting = exp(-r * Expiry);
for (unsigned long i = 0; i < NumberOfPaths; i++)
{
thisSpot = movedSpot * exp(rootVariance * normVariates[i]);
double thisPayoff = TheOption.OptionPayOff(thisSpot);
gatherer.DumpOneResult(discounting * thisPayoff);
}
return;
}


The StatisticAllPaths class which is used as input in the simulation that collects all the final values of the simulation

StatisticAllPaths::StatisticAllPaths(const unsigned long minimumNumberOfPaths) : PathsDone(0)
{
ResultList.reserve(minimumNumberOfPaths);
}

void StatisticAllPaths::DumpOneResult(double result)
{
ResultList.push_back(result);
PathsDone++;
}

const double& StatisticAllPaths::GetOneValueFromResultsSoFar(unsigned long index) const
{
return ResultList[index];
}


The PayOffVect used is used here to take the payoff of each path in the MC valuation function, but since we're just collecting all the paths here and processing them later (in the last part of the main valuation class) it doesn't really do anything here. It it used in this case just to make the outperformance relative values with this inherited class:

PayOffRelPerformance::PayOffRelPerformance(double startValue_) : startValue(startValue_)
{
}

double PayOffRelPerformance::operator()(double spot) const
{
return spot / startValue;
}


The GetArraysOfCorrelatedGauassiansByBoxMuller does the job of generating the vectors of normal variates that will be used in the simulation. I have checked that the Cholezky Matrix is correct for real cases, and I have also checked that the outputted normal variates are in fact dependent with the correlation implied by the covariance Matrix.

std::vector<MJArray> GetArraysOfCorrelatedGauassiansByBoxMuller(unsigned long numberOfVariates, std::vector<std::vector<double>> covMatrix)
{
//Calculate the cholezky Matrix
std::vector<std::vector<double>> cholezkyMatrix = Cholesky_Decomposition(covMatrix);
//Fix the size of the arrays to contain correlated normal variates
std::vector<MJArray> corrNormVariatesVector(cholezkyMatrix.size());
for (unsigned long j = 0; j < corrNormVariatesVector.size(); j++) {
corrNormVariatesVector[j].resize(numberOfVariates);
corrNormVariatesVector[j] = 0;
}
//calculate correlated normal variates and fill the arrays with values
MJArray NormVariates(cholezkyMatrix.size());
for (unsigned long k = 0; k < numberOfVariates; k++) {
for (unsigned long i = 0; i < cholezkyMatrix.size(); i++)
{
NormVariates[i] = GetOneGaussianByBoxMuller();
for (unsigned long j = 0; j < cholezkyMatrix[i].size(); j++) {
corrNormVariatesVector[i][k] += cholezkyMatrix[i][j] * NormVariates[j];
}
corrNormVariatesVector[i][k] /= cholezkyMatrix[i][i]; //normalize the random variates
}
}
return corrNormVariatesVector;
}

• I agree that best of call price should be lower as correlation increase. It seems like that there is something hapenning when around $\rho = 0.5$. Can you check similar pattern happens when the correlation is below 0? – spar7453 Jul 4 at 9:37

Intuitively, they should both be short correlation, that is the less correlated the assets are the higher the value of the worst of/best of option.

The best of option payoff is sandwiched by an exchange option payoff (plus other vanilla forward/option payoffs on single stock, insensitive to correlation):

$$X_T -K + (Y_T-X_T)^+ \leq \max(X_T - K ,Y_T - K,0) \leq (X_T-K)^+ + (Y_T-X_T)^+$$

It is clear intuitively that the exchange option is short correlation (also explicitly seen in Margrabe's world).

Edit: For worst of option we have a similar relation:

$$K-X_T + (X_T-Y_T)^+ \leq \max(K-X_T,K-Y_T,0) \leq (K-X_T)^+ + (X_T-Y_T)^+$$

Edit2: You can look at the behaviour of the product packages I claim sandwich the rainbow, for various correlations. This could expose something quickly. Remember that those product packages need both MC prices (from the same loops you use for rainbows) and also have closed-form solutions (of course, vols are flat etc. in this debugging phase), so insert both versions in your comparisons.

• Is there a typo in your equation? The LHS and RHS are the same. I think I do see the intuition for the bets-of option still, but what about the worst-of? If you have 2 assets currently at-the-money and a correlation of -1, then wouldn't that mean your option will never have a positive payoff (since if one goes up the other goes down). Same situation for best-of will obviously be very much in your favor however since you're guaranteed a payoff, so how can both of them be sensitive to correlation in the same way? – Oscar Jul 4 at 6:19
• @Oscar I subtracted a strike everywhere to allow for negative payoffs. In the worst of option contract the investor receives the highest decline (as opposed to increase). In both cases they should benefit from low correlation. In practice, they use percentage returns (against respective original stock prices) in the payoffs. I made some edits. – ir7 Jul 4 at 11:47
• @Oscar Btw, you might want to inquire (maybe here on SE) if there is an analytic formula (even if just approximation) to implement and compare against your MC implementation (of course with standard assumptions like flat vols etc.). – ir7 Jul 4 at 12:02
• @Oscar Added one more edit. Might help with your debugging. – ir7 Jul 4 at 12:16
• Hey, thank you. In what I'm trying to model the worst-of option would pay the smallest increase rather than highest decline so I think your example is different from mine there. For your debugging advice, to clarify, you advice I price and compare: (vanilla call + outperformance_X-Y) ; (best-of call) ; (vanilla call + outperformance_Y-X) and compare for both analytical and MC valuations? Is that right? – Oscar Jul 4 at 12:37

Best-of + Worst-of = Call1 + Call2

The right hand side is independent of correlation (and you can check it in your model).

Therefore if Best-of is short correlation, worst-of must be long correlation.

Increasing correlation makes the two assets more similar and therefore makes the best-of more like a vanilla. This is why a best-of is short correlation.

(Hope I understood the question right!)