Abstract
Pricing derivatives with Monte-Carlo simulations involve standard errors that typically decrease at a rate proportional to where N is the sample size. Several approaches have been discussed to reduce the empirical variance for a given sample size. This article analyzes the joint application of the put-call-parity approach and importance sampling to variance reduced option pricing. For this purpose, we examine non-path-dependent and path-dependent options. For European options, we observe dramatic variance reduction, especially for in-the-money options. Also for arithmetic Asian options, a significant variance reduction is achieved.
Highlights
Monte-Carlo simulations are frequently used to estimate prices of financial products for which no analytical formulae exist, e.g., several path-dependent options
An alternative approach to variance reduction at least for in-the-money options is the application of the put-call-parity: instead of simulating an in-the-money call price, the corresponding out-of-the-money put price can be simulated
Foundations of Importance Sampling This section introduces the concept of importance sampling, which serves to reduce the empirical variance of estimators
Summary
Monte-Carlo simulations are frequently used to estimate prices of financial products for which no analytical formulae exist, e.g., several path-dependent options. A difficulty related to this approach is that the empirical variance of estimators decreases slowly, typically at a rate ∝ 1 N where N is the sample size. Several techniques have been discussed to reduce the empirical variance for a given sample size N [1]. Importance sampling turns out to be a effective variance reduction technique [2]-[5]. The call price can be calculated from the put-call-parity yielding a variance reduced estimator [6] [7]. (2016) Improved Variance Reduced Monte-Carlo Simulation of in-the-Money Options. A. Müller lead to significant variance reduction in the simulation of Monte-Carlo estimators.
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