### Abstract

The dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi-Monte Carlo (QMC) method. The first factor implies that the QMC method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large. The second factor suggests that the QMC method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes. Motivated by these findings, methods such as the linear transformation (LT) method of [J. Imai and K. S. Tan, J. Comput. Finance, 10 (2006), pp. 129-155] and the orthogonal transformation (OT) method of [X. Wang and K. S. Tan, Management Sci., 59 (2013), pp. 376-389] have been proposed to increase the efficiency of QMC. However, both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the LT method enhances QMC by effective dimension reduction while the OT method accomplishes the same objective by discontinuity realignment. On the other hand there are many problems in finance that are both of high dimensionality and discontinuous. The main objective of this paper is to propose an efficient QMC method for handling problems of this kind. We first prove that the OT method is a special case of the LT method. We then show that by integrating the LT and OT methods, the proposed method has the advantage of addressing both dimensionality and discontinuity concurrently. The numerical examples indicate that relative to the standard QMC, the proposed hybrid method is extremely effective and could attain a variance reduction as high as several thousand times. We further discuss a possible extension of the proposed method to the case under a Lévy process and demonstrate its competitive efficiency using some additional numerical examples.

Original language | English |
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Pages (from-to) | A2101-A2121 |

Journal | SIAM Journal on Scientific Computing |

Volume | 36 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Digital options
- Dimension reduction
- Linear transformation
- Lévy process
- Orthogonal transformation
- Quasi-Monte Carlo method

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

**Pricing derivative securities using integrated Quasi-Monte Carlo methods with dimension reduction and discontinuity realignment.** / Imai, Junichi; Tan, Ken Seng.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 36, no. 5, pp. A2101-A2121. https://doi.org/10.1137/130926286

}

TY - JOUR

T1 - Pricing derivative securities using integrated Quasi-Monte Carlo methods with dimension reduction and discontinuity realignment

AU - Imai, Junichi

AU - Tan, Ken Seng

PY - 2014

Y1 - 2014

N2 - The dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi-Monte Carlo (QMC) method. The first factor implies that the QMC method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large. The second factor suggests that the QMC method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes. Motivated by these findings, methods such as the linear transformation (LT) method of [J. Imai and K. S. Tan, J. Comput. Finance, 10 (2006), pp. 129-155] and the orthogonal transformation (OT) method of [X. Wang and K. S. Tan, Management Sci., 59 (2013), pp. 376-389] have been proposed to increase the efficiency of QMC. However, both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the LT method enhances QMC by effective dimension reduction while the OT method accomplishes the same objective by discontinuity realignment. On the other hand there are many problems in finance that are both of high dimensionality and discontinuous. The main objective of this paper is to propose an efficient QMC method for handling problems of this kind. We first prove that the OT method is a special case of the LT method. We then show that by integrating the LT and OT methods, the proposed method has the advantage of addressing both dimensionality and discontinuity concurrently. The numerical examples indicate that relative to the standard QMC, the proposed hybrid method is extremely effective and could attain a variance reduction as high as several thousand times. We further discuss a possible extension of the proposed method to the case under a Lévy process and demonstrate its competitive efficiency using some additional numerical examples.

AB - The dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi-Monte Carlo (QMC) method. The first factor implies that the QMC method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large. The second factor suggests that the QMC method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes. Motivated by these findings, methods such as the linear transformation (LT) method of [J. Imai and K. S. Tan, J. Comput. Finance, 10 (2006), pp. 129-155] and the orthogonal transformation (OT) method of [X. Wang and K. S. Tan, Management Sci., 59 (2013), pp. 376-389] have been proposed to increase the efficiency of QMC. However, both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the LT method enhances QMC by effective dimension reduction while the OT method accomplishes the same objective by discontinuity realignment. On the other hand there are many problems in finance that are both of high dimensionality and discontinuous. The main objective of this paper is to propose an efficient QMC method for handling problems of this kind. We first prove that the OT method is a special case of the LT method. We then show that by integrating the LT and OT methods, the proposed method has the advantage of addressing both dimensionality and discontinuity concurrently. The numerical examples indicate that relative to the standard QMC, the proposed hybrid method is extremely effective and could attain a variance reduction as high as several thousand times. We further discuss a possible extension of the proposed method to the case under a Lévy process and demonstrate its competitive efficiency using some additional numerical examples.

KW - Digital options

KW - Dimension reduction

KW - Linear transformation

KW - Lévy process

KW - Orthogonal transformation

KW - Quasi-Monte Carlo method

UR - http://www.scopus.com/inward/record.url?scp=84911420620&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84911420620&partnerID=8YFLogxK

U2 - 10.1137/130926286

DO - 10.1137/130926286

M3 - Article

VL - 36

SP - A2101-A2121

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -