跳-扩散模型下期权定价方法及参数校准

Abstract

Abstract:

In this paper, the pricing method and parameter calibration of jump-diffusion model are investigated. First, the risk-neutral characteristic function of jump-diffusion model is derived under the mean correctiong equivalent martingale measure. The option under jump-diffusion model is priced by using the COS pricing method. Then, the pricing error of the COS algorithm is analyzed and the effectiveness of the COS pricing method is verified through numerical experiment. Subsequently, the parameters of the jump-diffusion model are calibrated by the relative entropy regularization method. Numerical experiments demonstrate the accuracy and reliability of the proposed method. Finally, the calibration method is tested by analyzing the S&P500 market data. The results show that the values of calibrated parameter are qualitatively for each maturity. Moreover, the results indicate a better fitting to the market data for the Merton jump-diffusion model in comparison to the Black-Scholes model.

Key words: Jump-diffusion models, Option pricing, Parameter calibration, COS method, Regularization

CLC Number:

O211.6

Congcong Xu,Zuoliang Xu. Option Pricing Method and Parameter Calibration for Jump-Diffusion Model[J].Acta mathematica scientia,Series A, 2019, 39(3): 649-663.

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Figures/Tables 9

References 12

1	Black F , Scholes M . The pricing of option and corporate liabilities. J Political Econom, 1973, 81: 637- 659 doi: 10.1086/260062
2	Merton R . Option pricing when underlying stock returns are discontinuous. J Finan Econom, 1976, 3: 125- 144 doi: 10.1016/0304-405X(76)90022-2
3	Cont R , Tankov P , Voltchkova E . Option pricing models with jumps:integro-differential equations and inverse problems. European Congress on Computational Methods in Applied Sciences and Engineering, 2004, 1- 20
4	Delbaen F , Schachermayer W . The fundamental theorem of asset pricing for unbounded stochastic processes. Math Anna, 1996, 312: 215- 250
5	Yao L G , Yang G , Yang X Q . A note on the mean correcting martingale measure for geometric Lévy processes. Appl Math Lett, 2011, 24: 593- 597 doi: 10.1016/j.aml.2010.11.011
6	Carr P , Madan D . Option valution using the fast Fourier transform. J Comput Finance, 1999, 3: 463- 520
7	Fang F , Oosterlee C W . A novel option pricing method based on Fourier-cosine series expansions. SIAM J Sci Comput, 2008, 31: 826- 848
8	Cont R , Tankov P . Financial Modelling with Jump Processes. New York: Chapman & Hall/CRC Press, 2004
9	Cont R , Tankov P . Calibration of jump-diffusion option pricing models:A robust non-parametric approach. SSRN Electronic J, 2002 doi: 10.2139/ssrn.332400
10	Cont R , Tankov P . Non-parametric calibration of jump-diffusion option pricing models. J Comput Finance, 2004, 7: 1- 49
11	Schoutens W . Lévy Process in Finance:Pricing Financial Derivatives. New York: Wiley, 2003
12	Miyahara Y. Martingale measures for the geometric lévy process models. Discussion Papers in Economics, Nagoya City University, 2005, 431:1-14

	N	32	64	128	256	512
$K=90$	运行时间(s)	0.018	0.019	0.020	0.022	0.024
	绝对误差	13.486	1.007	2.378e-06	3.561e-9	3.561e-9
$K=100$	运行时间(s)	0.016	0.019	0.019	0.021	0.024
	绝对误差	13.758	1.009	2.400e-06	3.535e-9	3.535e-9
$K=110$	运行时间(s)	0.018	0.019	0.021	0.022	0.025
	绝对误差	13.770	1.013	2.380e-06	3.549e-9	3.549e-9

到期日	模型	$\sigma^Q$	$\lambda^Q$	$\mu^Q$	$\delta^Q$	RMSE
$T_1=0.038$	Merton模型	0.0656	8.2454	-0.0341	0.0440	0.8416
	Black-Scholes模型	0.1247				2.8718
$T_2=0.115$	Merton模型	0.0520	5.9989	-0.0388	0.0396	0.3094
	Black-Scholes模型	0.1136				4.9638
$T_3=0.211$	Merton模型	0.0648	1.6550	-0.0998	0.0176	0.2825
	Black-Scholes模型	0.1164				6.2999
$T_4=0.460$	Merton模型	0.0662	0.8905	-0.1412	0.0467	0.5395
	Black-Scholes模型	0.1252				11.9545

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Option Pricing Method and Parameter Calibration for Jump-Diffusion Model

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