Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps

51 Pages Posted: 16 Aug 2021 Last revised: 13 Sep 2021

See all articles by Pingping Zeng

Pingping Zeng

Hong Kong University of Science & Technology (HKUST) - Department of Mathematics

Ziqing Xu

affiliation not provided to SSRN

Pingping Jiang

Nankai University - School of Mathematical Sciences

Yue Kuen Kwok

Hong Kong University of Science & Technology - Department of Mathematics

Date Written: August 13, 2021

Abstract

We perform a thorough investigation on the analytical solvability of general stochastic volatility (SV) models with Levy jumps and propose a unified, accurate, and efficient almost exact simulation method to price various financial derivatives. Our theoretical results lay a foundation for a range of valuation, calibration, and econometric problems. Our almost exact simulation method is applicable to a broad class of models and enables effective pricing of path-dependent financial derivatives, whereas the traditional exact simulation method is always tailor-made for some specific models and is generally time-consuming, which limits its use in the case of path-dependent financial derivatives. More specifically, by combining a decomposition technique with a change of measure approach, we first develop a simple probabilistic method to derive a unified formula for the conditional characteristic function of the log-asset price under general SV models with Levy jumps and show under which conditions this new formula admits a closed-form expression. The conditional and unconditional joint characteristic functions of the log-asset price and the integrated variance can be easily obtained as byproducts. Second, we take advantage of our main theoretical result, the Hilbert transform method, the interpolation technique, and the dimension reduction technique to construct unified and efficient almost exact simulation schemes. Finally, we apply our almost exact simulation method to price European options, discretely monitored weighted variance swaps, and discretely monitored variance options under a wide variety of SV models with Levy jumps. Extensive numerical examples demonstrate the high level of accuracy and efficiency of our almost exact simulation method in terms of bias, root-mean-squared error (RMS error), and CPU time.

Keywords: stochastic volatility models with Levy jumps; conditional characteristic function; Hilbert transform method; interpolation; enhanced Hilbert interpolation method; almost exact simulation; weighted variance swaps; variance options

JEL Classification: G

Suggested Citation

Zeng, Pingping and Xu, Ziqing and Jiang, Pingping and Kwok, Yue Kuen, Analytical Solvability and Exact Simulation of Stochastic Volatility Models with Jumps (August 13, 2021). Available at SSRN: https://ssrn.com/abstract=3904498 or http://dx.doi.org/10.2139/ssrn.3904498

Pingping Zeng

Hong Kong University of Science & Technology (HKUST) - Department of Mathematics ( email )

Rm. 3461, Lift 25-26
Clear Water Bay
Kowloon
Hong Kong

Ziqing Xu

affiliation not provided to SSRN

Pingping Jiang (Contact Author)

Nankai University - School of Mathematical Sciences ( email )

Weijin Road #94
Tianjin, 300071
China

Yue Kuen Kwok

Hong Kong University of Science & Technology - Department of Mathematics ( email )

Clearwater Bay
Kowloon, 999999
Hong Kong

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