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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.

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

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. Then, the call price can be calculated from the put-call-parity yielding a variance reduced estimator [

This analysis will investigate how the joint application of the put-call-parity and importance sampling can lead to significant variance reduction in the simulation of Monte-Carlo estimators. The article is organized as follows: in Section 2 the method of importance sampling is introduced. Section 3 explains how put-call-parities can be employed for variance reduction. Next, the combination of the two approaches is introduced in Section 4. Numerical results of the achieved variance reductions are presented in Section 5 for both non-path-dependent and path-dependent options. A conclusion follows in Section 6.

Foundations of Importance Sampling This section introduces the concept of importance sampling, which serves to reduce the empirical variance of estimators.

The expectation value of a non-negative function

An estimator of

With any other probability density

The fraction

An unbiased estimator of this expression is

Its variance

vanishes if we choose

To obtain a a new probability density this product must be normalized (note that

For this purpose, the integral

must be calculated, which is the original problem to be solved in Equation (1). However, an approximation of the optimal density can already lead to significant variance reduction [

Monte-Carlo simulation of the Feynman-Kac formula Option prices can be calculated as the expectation value of the derivative’s discounted pay-off function. In the following, a variance reduced estimator of the resulting Feynman-Kac formula

will be presented. Note that the Feynman-Kac formula also covers the case of path-dependent options, i.e. the case where h does not only depend on the terminal value of the underlying’s price

For simplicity, a constant risk-free interest rate r is assumed. For the purpose of numerical calculations, the Feynman-Kac formula can be expressed as

using a finite-dimensional approximation on a grid

In accordance with Equation (2), a simple Monte-Carlo estimator of this expression is

Importance sampling by adding an additional drift term In 2014, Singer presented a variance reduced Monte-Carlo estimator for Equation (9) which relies on importance sampling by adding an additional drift term to the stochastic differential equation

where

The optimal importance sampling density

In the Black-Scholes model with

using the abbreviation

for the option price elasticity.

Here, C is the Feynman-Kac formula from Equation (9). Again, the problem of Equations (7) and (8) materializes: To describe the optimal stochastic differential equation, knowledge of the unknown quantity C is required.

However, variance reduction can also be achieved by approximating C. For this purpose, in our analysis we use the Black-Scholes formula for European options [

In the univariate Black-Scholes case that is considered in this article, the Radon-Nikodým derivative for one step on the discretized grid yields

Ultimately, for the variance reduced Monte-Carlo estimator with sample size N, n discretization steps and sampling from

As mentioned,

European options Simple non-arbitrage arguments require that for a European call with pay-off function

the following put-call-parity holds true where

Arithmetic Asian options Also for arithmetic Asian call options with the pay-off function

a put-call-parity holds.

With

the following relation is required to avoid arbitrage [

Variance reduction Reider (1994) [

Put-call-parities allow to split a stochastic quantity into deterministic components and a residual stochastic quantity. The deterministic components’ standard errors vanish by definition. If the residual stochastic quantity is reduced in absolute size compared to the stochastic quantity of interest, the residual stochastic quantity’s standard error will generally tend to be lower as well, all else being equal. According to Gaussian propagation of uncertainty, variance reduction is achieved [

In a previous article, it is suggested that a combined application of the put-call-parity and importance sampling might be particularly attractive [

1) To obtain a benchmark against which variance reductions can be analyzed, a simple Monte-Carlo simulation of the estimator (11) is conducted. Both the estimator and its empirical variance are calculated.

2) To simulate the option price of a call option with a given set of parameters, the corresponding put option with the same set of parameters is simulated using the variance reduced Monte-Carlo estimator (17).

3) The call price is calculated from the put-call-parity (20) resp. (23). The empirical variance of this estimated call price is the same as the empirical variance of the previously simulated put price. This is the case as all other terms in Equation (20) resp. (23) have zero variance.

4) The variance reduction ratio (“VarRatio”) is calculated as the ratio between the empirical variance of the benchmark estimator and the empirical variance of the variance reduced estimator calculated employing importance sampling and the put-call-parity.

In this section, we apply the introduced combined approach of importance sampling and employing the put- call-parity to simulate variance reduced Monte-Carlo estimators. To make results comparable, we present two examples that have served previously in the analysis of other approaches discussed in literature [

According to the Black-Scholes model, the underlying

We use a discretized time-grid

For benchmark simulations without importance sampling

European options First, we consider the example of a European call option with pay-off function

with

Column “PCP-IS method” of

The approach presented in this article achieves higher variance reductions for all considered parameter constellations compared to the other two approaches. Especially for (deep-)in-the-money options (

Two further aspects should be noted: First, in the case of out-of-the money options a direct application of the introduced importance sampling procedure would be more appropriate than the joint application employing the put-call-parity. E.g., when foregoing the put-call-parity for

Parameters | GHS method | ZLG method | PCP-IS method | ||||
---|---|---|---|---|---|---|---|

s | K | Price | VarRatio | Price | VarRatio | Price | VarRatio |

0.1 | 30 | 21.463 | 103.3 | 21.463 | 931.2 | 21.463 | 8.37E+17 |

45 | 7.317 | 8.2 | 7.316 | 15.9 | 7.314 | 9.76E+03 | |

50 | 3.404 | 7.2 | 3.405 | 12.1 | 3.402 | 201.7 | |

55 | 1.087 | 11.2 | 1.088 | 12.5 | 1.086 | 14.5 | |

0.3 | 30 | 21.602 | 14.9 | 21.601 | 30.0 | 21.598 | 6.45E+04 |

45 | 9.869 | 9.5 | 9.855 | 15.9 | 9.848 | 283.1 | |

50 | 7.118 | 10.3 | 7.121 | 15.8 | 7.115 | 96.6 | |

55 | 5.010 | 11.8 | 5.014 | 5.9 | 5.013 | 40.4 |

for

Second, the combination of the put-call-parity approach with importance sampling outperforms the two standalone approaches (put-call-parity and importance sampling) for in-the-money options. The standalone importance sampling approach in the case of deep-in-the money calls only leads to limited variance reduction. For the standalone put-call-parity approach applied to the low-volatility case

Arithmetic Asian options The second example examines an arithmetic Asian option with pay-off function

Again, we set

In the same structure as before,

For European options, we achieved higher variance reduction than for arithmetic Asian options. This is not surprising, as we calculated the option price elasticity

Again, the numerical results for the GHS method and the ZLG method are taken from [

Parameters | GHS method | ZLG method | PCP-IS method | ||||
---|---|---|---|---|---|---|---|

s | K | Price | VarRatio | Price | VarRatio | Price | VarRatio |

0.05 | 45 | 6.042 | 39.8 | 6.042 | 931.2 | 6.041 | 2.10E+07 |

50 | 1.437 | 6.4 | 1.438 | 11.5 | 1.438 | 11.5 | |

55 | 0.007 | 138.1 | 0.007 | 35.9 | 0.007 | 0.0 | |

0.1 | 45 | 6.055 | 10.8 | 6.055 | 24.5 | 6.055 | 754.6 |

50 | 1.919 | 7 | 1.919 | 11.9 | 1.919 | 5.3 | |

55 | 0.202 | 21.2 | 0.202 | 13.3 | 0.200 | 0.2 | |

0.2 | 45 | 6.418 | 7.7 | 6.419 | 14.5 | 6.419 | 51.2 |

50 | 3.028 | 8.2 | 3.028 | 13.4 | 3.030 | 5.4 | |

55 | 1.106 | 13 | 1.106 | 14 | 1.107 | 0.9 | |

0.3 | 45 | 7.15 | 8.3 | 7.151 | 15 | 7.150 | 31.2 |

50 | 4.169 | 9.2 | 4.169 | 14.9 | 4.168 | 7.2 | |

55 | 2.211 | 12.2 | 2.21 | 15.4 | 2.215 | 2.1 | |

0.5 | 45 | 8.996 | 10.6 | 8.996 | 18.6 | 9.001 | 29.3 |

50 | 6.459 | 11.6 | 6.457 | 18.7 | 6.461 | 10.6 | |

55 | 4.455 | 13.5 | 4.543 | 18.9 | 4.541 | 4.3 |

We conducted variance reduced Monte-Carlo simulations for European and arithmetic Asian options. For in- the-money options, we achieve significant variance reduction by combining importance sampling with the put- call-parity approach. The presented approach delivers best results in low-volatility environments. For European options, it outperforms previously published results for all parameter constellations. Also for arithmetic Asian options, we obtain higher variance reduction ratios for in-the-money options.

This analysis does not consider the combination with other variance reduction techniques, e.g., antithetic variates or stratification, which could lead to further increased variance reduction ratios.

I thank the Editor and the referee for their commitment. I am very grateful to my supervisor Prof. Hermann Singer for the supervision of my research and the related discussions. This work was supported by the Konrad Adenauer Foundation.

Armin Müller, (2016) Improved Variance Reduced Monte-Carlo Simulation of in-the-Money Options. Journal of Mathematical Finance,06,361-367. doi: 10.4236/jmf.2016.63029