Monte Carlo pricing with & without antithetic variation

While generally speaking Aristotelian logic dictates that the whole is greater than the sum of its parts, in the world of finance, generally cannot be spoken that loosely, and that very dictum takes a back-seat when it comes to variance of a pay-off portfolio of two options written on two ‘perfectly’ anti-correlated assets. The variance of the pay-off of a portfolio consisting of two above mentioned options is less than the variance of the pay-off of each individual option. It is intuitive to understand this because when one option pays off the other one does not etc. This interesting property is taken advantage of in pricing any option using the beloved Monte Carlo simulation.

A comparison was done to assess the valuation of a simple European call option with and without utilizing the above mentioned trick which is also called as an antithetic variate. The following were the inputs to the simulation and the results are as below. 10000 simulations, 10 time steps, strike = 100, T=1, Vol=20%, dividend yield = 3%, interest rate = 6%.

summary(price_withoutav)
Min. 1st Qu. Median Mean 3rd Qu. Max.
4.418 8.184 9.074 9.111 9.994 16.210
> summary(price_withav)
Min. 1st Qu. Median Mean 3rd Qu. Max.
4.291 8.007 8.879 8.917 9.796 15.260

As can be seen above, the antithetic variate method reduces the deviation and brings more accuracy. Of course 100,000 simulations is definitely not an ‘ideal’ number and perhaps more runs are needed to be more efficient. But the point is that while dealing with very large portfolios even such minimal differences can make or break your competitiveness..

If you are interested in the R code for this do mail us..

Originally published at https://www.linkedin.com.