# Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities

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## Abstract

**:**

## 1. Introduction

## 2. Pricing Model

## 3. Dimension and Variance Reduction

#### 3.1. CMC Method

#### 3.2. Martingale Control Variate (CV)

**Theorem**

**1**(Martingale Representation Theorem)

**.**

**Proof.**

**Remark**

**1.**

## 4. Numerical Tests

#### 4.1. Exchange Options

#### 4.2. Basket Options

#### 4.3. Quanto Options with Real Data

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MC | Monte Carlo |

CV | control variate |

CMC | conditional Monte Carlo |

FFT | fast Fourier transformation |

FX | foreign exchange |

## References

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Reference | ${\mathit{\rho}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}\left(\mathit{y}\right)$ | ${\mathit{\mu}}_{\mathit{i}}(\mathit{t},\mathit{y})$ | ${\mathit{g}}_{\mathit{i}}(\mathit{t},\mathit{y})$ |
---|---|---|---|---|

Hull–White [8] | 0 | $\sqrt{y}$ | $\mu y$ | $\sigma y$ |

Scott [9] | 0 | ${e}^{y}$ | $a(\theta -y)$ | $\sigma $ |

Stein–Stein [10] | 0 | $\left|y\right|$ | $a(\theta -y)$ | $\sigma $ |

Ball–Roma [11] | 0 | $\sqrt{y}$ | $a(\theta -y)$ | $\sigma \sqrt{y}(2a\theta >{\sigma}^{2})$ |

Heston [12] | $\ne 0$ | $\sqrt{y}$ | $a(\theta -y)$ | $\sigma \sqrt{y}(2a\theta >{\sigma}^{2})$ |

Hagan et al. [14] | $\ne 0$ | y | 0 | $\sigma y$ |

${\mathit{\rho}}_{1}|{\mathit{\rho}}_{2}$ | −0.75 | −0.50 | −0.25 | 0.00 | 0.25 | 0.50 | 0.75 | |
---|---|---|---|---|---|---|---|---|

−0.75 | 2.8083 | 2.8218 | 2.8336 | 2.8436 | 2.8519 | 2.8587 | 2.8639 | |

−0.50 | 2.8169 | 2.8294 | 2.8402 | 2.8493 | 2.8567 | 2.8623 | 2.8663 | |

−0.25 | 2.8248 | 2.8362 | 2.8459 | 2.8539 | 2.8603 | 2.8649 | 2.8678 | |

0.00 | 2.8317 | 2.8420 | 2.8505 | 2.8573 | 2.8625 | 2.8661 | 2.8680 | ${\overline{V}}_{\mathrm{MC}}$ |

0.25 | 2.8372 | 2.8462 | 2.8536 | 2.8594 | 2.8635 | 2.8660 | 2.8667 | |

0.50 | 2.8413 | 2.8492 | 2.8554 | 2.8600 | 2.8630 | 2.8644 | 2.8641 | |

0.75 | 2.8440 | 2.8508 | 2.8558 | 2.8592 | 2.8612 | 2.8615 | 2.8601 | |

−0.75 | 2.8153 | 2.8294 | 2.8420 | 2.8528 | 2.8616 | 2.8684 | 2.8731 | |

−0.50 | 2.8253 | 2.8377 | 2.8486 | 2.8578 | 2.8651 | 2.8703 | 2.8735 | |

−0.25 | 2.8340 | 2.8446 | 2.8539 | 2.8615 | 2.8672 | 2.8709 | 2.8725 | |

0.00 | 2.8413 | 2.8503 | 2.8579 | 2.8639 | 2.8679 | 2.8700 | 2.8700 | ${\overline{V}}_{\mathrm{CMC}}$ |

0.25 | 2.8473 | 2.8546 | 2.8606 | 2.8649 | 2.8674 | 2.8678 | 2.8661 | |

0.50 | 2.8519 | 2.8576 | 2.8620 | 2.8647 | 2.8654 | 2.8641 | 2.8608 | |

0.75 | 2.8550 | 2.8592 | 2.8620 | 2.8630 | 2.8621 | 2.8591 | 2.8539 |

${\mathit{\rho}}_{1}|{\mathit{\rho}}_{2}$ | −0.75 | −0.50 | −0.25 | 0.00 | 0.25 | 0.50 | 0.75 | |
---|---|---|---|---|---|---|---|---|

−0.75 | 0.0132 | 0.0137 | 0.0143 | 0.0148 | 0.0153 | 0.0159 | 0.0164 | |

−0.50 | 0.0132 | 0.0137 | 0.0142 | 0.0148 | 0.0153 | 0.0158 | 0.0164 | |

−0.25 | 0.0131 | 0.0137 | 0.0142 | 0.0147 | 0.0153 | 0.0158 | 0.0164 | |

0.00 | 0.0131 | 0.0136 | 0.0141 | 0.0147 | 0.0152 | 0.0158 | 0.0163 | $St{d}_{\mathrm{MC}}$ |

0.25 | 0.0130 | 0.0135 | 0.0141 | 0.0146 | 0.0152 | 0.0157 | 0.0163 | |

0.50 | 0.0130 | 0.0135 | 0.0140 | 0.0146 | 0.0151 | 0.0157 | 0.0163 | |

0.75 | 0.0129 | 0.0134 | 0.0140 | 0.0145 | 0.0151 | 0.0156 | 0.0162 | |

−0.75 | 0.0088 | 0.0065 | 0.0049 | 0.0045 | 0.0058 | 0.0084 | 0.0119 | |

−0.50 | 0.0081 | 0.0055 | 0.0036 | 0.0032 | 0.0049 | 0.0077 | 0.0115 | |

−0.25 | 0.0076 | 0.0048 | 0.0025 | 0.0020 | 0.0042 | 0.0073 | 0.0112 | |

0.00 | 0.0074 | 0.0045 | 0.0019 | 0.0011 | 0.0039 | 0.0071 | 0.0110 | $St{d}_{\mathrm{CMC}}$ |

0.25 | 0.0075 | 0.0046 | 0.0020 | 0.0012 | 0.0039 | 0.0072 | 0.0111 | |

0.50 | 0.0078 | 0.0050 | 0.0027 | 0.0021 | 0.0043 | 0.0074 | 0.0113 | |

0.75 | 0.0084 | 0.0057 | 0.0036 | 0.0031 | 0.0048 | 0.0078 | 0.0116 | |

−0.75 | 1.5014 | 2.1268 | 2.9394 | 3.2711 | 2.6283 | 1.8898 | 1.3762 | |

−0.50 | 1.6372 | 2.4911 | 3.9677 | 4.6222 | 3.1339 | 2.0450 | 1.4323 | |

−0.25 | 1.7336 | 2.8222 | 5.5991 | 7.3682 | 3.6300 | 2.1571 | 1.4677 | |

0.00 | 1.7692 | 2.9943 | 7.3944 | 13.0340 | 3.9300 | 2.2070 | 1.4805 | $R=\frac{St{d}_{\mathrm{MC}}}{St{d}_{\mathrm{CMC}}}$ |

0.25 | 1.7385 | 2.9276 | 7.0000 | 11.8935 | 3.8805 | 2.1900 | 1.4714 | |

0.50 | 1.6551 | 2.6807 | 5.2352 | 7.0184 | 3.5503 | 2.1170 | 1.4433 | |

0.75 | 1.5402 | 2.3701 | 3.9081 | 4.7380 | 3.1223 | 2.0066 | 1.4001 |

${\mathit{\rho}}_{12}$ | ${\mathit{\rho}}_{1}={\mathit{\rho}}_{2}$ | ${\mathit{Std}}_{\mathbf{MC}}$ | ${\mathit{Std}}_{\mathbf{Mar}}$ | ${\mathit{Std}}_{\mathbf{Fun}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ |
---|---|---|---|---|---|---|

−0.5 | −0.6 | 0.0160 | 0.0015 | 0.0039 | 10.3761 | 4.1166 |

−0.4 | 0.0163 | 0.0013 | 0.0040 | 12.2233 | 4.0626 | |

−0.2 | 0.0166 | 0.0012 | 0.0041 | 13.9129 | 4.0540 | |

0.0 | 0.0170 | 0.0011 | 0.0042 | 15.0719 | 4.0821 | |

0.2 | 0.0173 | 0.0013 | 0.0042 | 13.2504 | 4.1410 | |

0.4 | 0.0176 | 0.0016 | 0.0042 | 11.1433 | 4.2271 | |

0.6 | 0.0180 | 0.0019 | 0.0041 | 9.5880 | 4.3348 | |

0.0 | −0.75 | 0.0132 | 0.0014 | 0.0039 | 9.6978 | 3.4035 |

−0.50 | 0.0137 | 0.0012 | 0.0040 | 11.2209 | 3.4318 | |

−0.25 | 0.0142 | 0.0010 | 0.0041 | 14.6458 | 3.4829 | |

0.00 | 0.0147 | 0.0009 | 0.0041 | 16.2942 | 3.5500 | |

0.25 | 0.0152 | 0.0011 | 0.0042 | 13.3596 | 3.6291 | |

0.50 | 0.0157 | 0.0015 | 0.0042 | 10.2764 | 3.7178 | |

0.75 | 0.0162 | 0.0017 | 0.0043 | 9.6550 | 3.8097 | |

0.5 | −0.6 | 0.0106 | 0.0010 | 0.0038 | 10.9374 | 2.8023 |

−0.4 | 0.0110 | 0.0009 | 0.0040 | 12.9115 | 2.7755 | |

−0.2 | 0.0115 | 0.0006 | 0.0041 | 17.7294 | 2.7910 | |

0.0 | 0.0119 | 0.0006 | 0.0042 | 19.8106 | 2.8424 | |

0.2 | 0.0123 | 0.0008 | 0.0042 | 14.7523 | 2.9285 | |

0.4 | 0.0127 | 0.0012 | 0.0042 | 10.7868 | 3.0531 | |

0.6 | 0.0131 | 0.0012 | 0.0041 | 10.6183 | 3.2264 |

**Table 5.**Exchange option: Numerical results for CVs with different volatilities of the stochastic volatilities.

${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{Std}}_{\mathbf{MC}}$ | ${\mathit{Std}}_{\mathbf{Mar}}$ | ${\mathit{Std}}_{\mathbf{Fun}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ |
---|---|---|---|---|---|---|

0.2 | 0.1 | 0.0151 | 0.0013 | 0.0030 | 11.7123 | 4.9767 |

0.2 | 0.0157 | 0.0015 | 0.0042 | 10.2764 | 3.7178 | |

0.3 | 0.0163 | 0.0018 | 0.0057 | 9.0825 | 2.8678 | |

0.4 | 0.0169 | 0.0021 | 0.0073 | 8.1624 | 2.3279 | |

0.05 | 0.2 | 0.0157 | 0.0014 | 0.0034 | 11.2239 | 4.5840 |

0.10 | 0.0157 | 0.0014 | 0.0036 | 10.9406 | 4.3545 | |

0.15 | 0.0157 | 0.0015 | 0.0039 | 10.6170 | 4.0447 | |

0.20 | 0.0157 | 0.0015 | 0.0042 | 10.2764 | 3.7178 |

**Table 6.**Geometric average basket option: Numerical results for CVs with different correlation coefficients.

n | ${\mathit{\rho}}_{\mathit{i}}$ | ${\mathit{Std}}_{\mathbf{MC}}$ | ${\mathit{Std}}_{\mathbf{Mar}}$ | ${\mathit{Std}}_{\mathbf{Fun}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ |
---|---|---|---|---|---|---|

2 | −0.75 | 0.0110 | 0.0012 | 0.0021 | 9.2740 | 5.2542 |

−0.50 | 0.0112 | 0.0009 | 0.0021 | 12.8705 | 5.2650 | |

−0.25 | 0.0115 | 0.0004 | 0.0022 | 27.4754 | 5.2935 | |

0.00 | 0.0117 | 0.0004 | 0.0022 | 30.5615 | 5.3375 | |

0.25 | 0.0120 | 0.0005 | 0.0022 | 23.5176 | 5.3968 | |

0.50 | 0.0122 | 0.0010 | 0.0022 | 11.9347 | 5.4719 | |

0.75 | 0.0124 | 0.0013 | 0.0022 | 9.2236 | 5.5629 | |

5 | −0.75 | 0.0068 | 0.0009 | 0.0013 | 7.6660 | 5.1178 |

−0.50 | 0.0069 | 0.0006 | 0.0013 | 11.7653 | 5.1320 | |

−0.25 | 0.0069 | 0.0002 | 0.0013 | 27.9824 | 5.1434 | |

0.00 | 0.0070 | 0.0001 | 0.0014 | 110.3248 | 5.1544 | |

0.25 | 0.0071 | 0.0003 | 0.0014 | 25.1755 | 5.1663 | |

0.50 | 0.0071 | 0.0006 | 0.0014 | 11.1250 | 5.1785 | |

0.75 | 0.0072 | 0.0010 | 0.0014 | 7.4879 | 5.1952 | |

10 | −0.75 | 0.0049 | 0.0008 | 0.0010 | 6.3077 | 5.0690 |

−0.50 | 0.0049 | 0.0005 | 0.0010 | 10.4984 | 5.0820 | |

−0.25 | 0.0049 | 0.0002 | 0.0010 | 26.5073 | 5.0928 | |

0.00 | 0.0050 | 1.7 × 10^{−5} | 0.0010 | 291.4214 | 5.1023 | |

0.25 | 0.0050 | 0.0002 | 0.0010 | 24.2257 | 5.1104 | |

0.50 | 0.0050 | 0.0005 | 0.0010 | 10.0253 | 5.1192 | |

0.75 | 0.0050 | 0.0008 | 0.0010 | 6.1413 | 5.1310 |

**Table 7.**Geometric basket option: Numerical results for CVs with different volatilities of the stochastic volatility.

n | ${\mathit{\sigma}}_{\mathit{i}}$ | ${\mathit{Std}}_{\mathbf{MC}}$ | ${\mathit{Std}}_{\mathbf{Mar}}$ | ${\mathit{Std}}_{\mathbf{Fun}}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ |
---|---|---|---|---|---|---|

2 | 0.1 | 0.0177 | 0.0013 | 0.0011 | 13.7479 | 15.6830 |

0.2 | 0.0179 | 0.0013 | 0.0023 | 13.3158 | 7.9626 | |

0.3 | 0.0182 | 0.0014 | 0.0034 | 12.8095 | 5.4211 | |

0.4 | 0.0184 | 0.0015 | 0.0044 | 12.2580 | 4.1894 | |

5 | 0.1 | 0.0093 | 0.0007 | 0.0006 | 12.7129 | 14.3708 |

0.2 | 0.0093 | 0.0007 | 0.0013 | 12.4417 | 7.2688 | |

0.3 | 0.0094 | 0.0008 | 0.0019 | 12.1388 | 4.9147 | |

0.4 | 0.0094 | 0.0008 | 0.0025 | 11.8127 | 3.7571 | |

10 | 0.1 | 0.0060 | 0.0005 | 0.0004 | 11.8384 | 13.8295 |

0.2 | 0.0060 | 0.0005 | 0.0009 | 11.6365 | 6.9848 | |

0.3 | 0.0060 | 0.0005 | 0.0013 | 11.4192 | 4.7179 | |

0.4 | 0.0060 | 0.0005 | 0.0017 | 11.1910 | 3.5979 |

View Date | 13 October 2010 |
---|---|

S&P500 | 1169.77 |

FX Rate (KRW/USD) | 1127 |

Volatility of S&P500 | 18.58% |

Volatility of FX Rate | 11.83% |

Correlation between S&P500 and FX Rate | −0.2297 |

Correlation between S&P500 and its volatility | −0.55 |

Volatility of volatility of S&P500 | 11.72% |

Volatility of volatility of FX Rate | 16.8% |

USD LIBOR(1Y) | 0.77% |

KRW Treasury Rate(1Y) | 2.91% |

${\mathit{\rho}}_{12}$ | ${\mathit{V}}_{\mathrm{Appro}}$ (10^{6})
| ${\mathit{V}}_{\mathrm{MC}}$ (10^{6})
| ${\mathit{Std}}_{\mathrm{MC}}$ (10^{6})
| ${\mathit{V}}_{\mathrm{Mar}}$ (10^{6})
| ${\mathit{Std}}_{\mathrm{Mar}}$ (10^{6})
| ${\mathit{V}}_{\mathbf{Fun}}$ (10^{6})
| ${\mathit{Std}}_{\mathbf{Fun}}$ (10^{6})
| ${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|

−0.6 | 9.1393 | 9.1082 | 0.0462 | 9.1223 | 0.0038 | 9.1162 | 0.0047 | 12.1373 | 9.7825 |

−0.4 | 8.8325 | 8.8187 | 0.0454 | 8.8316 | 0.0037 | 8.8258 | 0.0046 | 12.3813 | 9.8215 |

−0.2 | 8.5311 | 8.5357 | 0.0447 | 8.5473 | 0.0035 | 8.5416 | 0.0045 | 12.6122 | 9.8533 |

0.0 | 8.2352 | 8.2592 | 0.0439 | 8.2696 | 0.0034 | 8.2638 | 0.0044 | 12.8264 | 9.8768 |

0.2 | 7.9448 | 7.9890 | 0.0432 | 7.9984 | 0.0033 | 7.9924 | 0.0044 | 13.0198 | 9.8906 |

0.4 | 7.6599 | 7.7253 | 0.0424 | 7.7335 | 0.0032 | 7.7276 | 0.0043 | 13.1884 | 9.8948 |

0.6 | 7.3807 | 7.4679 | 0.0417 | 7.4750 | 0.0031 | 7.4691 | 0.0042 | 13.3285 | 9.8900 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liang, Y.; Xu, X.
Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities. *Sustainability* **2019**, *11*, 815.
https://doi.org/10.3390/su11030815

**AMA Style**

Liang Y, Xu X.
Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities. *Sustainability*. 2019; 11(3):815.
https://doi.org/10.3390/su11030815

**Chicago/Turabian Style**

Liang, Yijuan, and Xiuchuan Xu.
2019. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities" *Sustainability* 11, no. 3: 815.
https://doi.org/10.3390/su11030815