Variance Reduction Techniques in Simulation › Variance Reduction Techniques in Simulation

In the previous chapter, we explored Monte Carlo simulation as a powerful method for pricing derivatives that lack closed-form solutions. The technique's elegance lies in its simplicity: simulate many paths of the underlying asset, compute the payoff for each path, and average the discounted results. However, simplicity comes at a computational cost. To achieve accurate prices, we often need millions of simulations, and the standard error decreases only as $1/\sqrt{n}$, where $n$ is the number of simulations. This means reducing the error by half requires four times as many simulations.

For complex derivatives, path-dependent options, or real-time risk calculations, this computational burden becomes prohibitive. Variance reduction techniques address this challenge by modifying the simulation procedure to produce estimates with lower variance for the same number of simulations, or equivalently, achieving the same accuracy with far fewer simulations. These techniques exploit mathematical structure in the problem, such as symmetry, known analytical results, or strategic sampling, to squeeze more information from each simulation.

This chapter introduces three foundational variance reduction methods: antithetic variates, control variates, and stratified sampling. Each technique attacks the variance problem from a different angle. Efficient derivative pricing depends on choosing the method that fits the payoff and simulation structure. By the end of this chapter, you will be able to reduce simulation time by factors of 10 or more without sacrificing accuracy, a capability that separates practical quantitative work from textbook exercises.