Option pricing under the double exponential jump‐diffusion model with stochastic volatility and interest rate

This paper proposes an efficient option pricing model that incorporates stochastic interest rate (SIR), stochastic volatility (SV), and double exponential jump into the jump‐diffusion settings. The model comprehensively considers the leptokurtosis and heteroscedasticity of the underlying asset’s returns, rare events, and an SIR. Using the model, we deduce the pricing characteristic function and pricing formula of a European option. Then, we develop the Markov chain Monte Carlo method with latent variable to solve the problem of parameter estimation under the double exponential jump‐diffusion model with SIR and SV. For verification purposes, we conduct time efficiency analysis, goodness of fit analysis, and jump/drift term analysis of the proposed model. In addition, we compare the pricing accuracy of the proposed model with those of the Black–Scholes and the Kou (2002) models. The empirical results show that the proposed option pricing model has high time efficiency, and the goodness of fit and pricing accuracy are significantly higher than those of the other two models

A Fast Fourier Transform Technique for Pricing European Options with Stochastic Volatility and Jump Risk

We consider European options pricing with double jumps and stochastic volatility. We derived closed-form solutions for European call options in a double exponential jump-diffusion model with stochastic volatility (SVDEJD). We developed fast and accurate numerical solutions by using fast Fourier transform (FFT) technique. We compared the density of our model with those of other models, including the Black-Scholes model and the double exponential jump-diffusion model. At last, we analyzed several effects on option prices under the proposed model. Simulations show that the SVDEJD model is suitable for modelling the long-time real-market changes and stock returns are negatively correlated with volatility. The model and the proposed option pricing method are useful for empirical analysis of asset returns and managing the corporate credit risks

A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature, Volatility Smile, and Analytical Tractability

Brownian motion and normal distribution have been widely used, for example, in the Black-Scholes-Merton option pricing framework, to study the return of assets. However, two puzzles, emerged from many empirical investigations, have got much attention recently, namely (a) the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and (b) an empirical abnormity called ``volatility smile'' in option pricing. To incorporate both the leptokurtic feature and ``volatility smile'', this paper proposes, for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modeled by two parts, a continuous part driven by Brownian motion, and a jump part with the logarithm of the jump sizes having a double exponential distribution. In addition to the above two desirable properties, leptokurtic feature and ``volatility smile'', the model is simple enough to produce analytical solutions for a variety of option pricing problems, including options, future options, and interest rate derivatives, such as caps and floors, in terms of the $Hh$ function. Although there are many models can incorporate some of the three properties (the leptokurtic feature, ``volatility smile'', and analytical tractability), the current model can incorporate all three under a unified framework.

Quanto options under double exponential jump diffusion.

Lau, Ka Yung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 78-79).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Background --- p.5Chapter 2.1 --- Jump Diffusion Models --- p.6Chapter 2.2 --- Double Exponential Jump Diffusion Model --- p.8Chapter 3 --- Option Pricing with DEJD --- p.10Chapter 3.1 --- Laplace Transform --- p.10Chapter 3.2 --- European Option Pricing --- p.13Chapter 3.3 --- Barrier Option Pricing --- p.14Chapter 3.4 --- Lookback Options --- p.16Chapter 3.5 --- Turbo Warrant --- p.17Chapter 3.6 --- Numerical Examples --- p.26Chapter 4 --- Quanto Options under DEJD --- p.30Chapter 4.1 --- Domestic Risk-neutral Dynamics --- p.31Chapter 4.2 --- The Exponential Copula --- p.33Chapter 4.3 --- The moment generating function --- p.36Chapter 4.4 --- European Quanto Options --- p.38Chapter 4.4.1 --- Floating Exchange Rate Foreign Equity Call --- p.38Chapter 4.4.2 --- Fixed Exchange Rate Foreign Equity Call --- p.40Chapter 4.4.3 --- Domestic Foreign Equity Call --- p.42Chapter 4.4.4 --- Joint Quanto Call --- p.43Chapter 4.5 --- Numerical Examples --- p.45Chapter 5 --- Path-Dependent Quanto Options --- p.48Chapter 5.1 --- The Domestic Equivalent Asset --- p.48Chapter 5.1.1 --- Mathematical Results on the First Passage Time of the Mixture Exponential Jump Diffusion Model --- p.50Chapter 5.2 --- Quanto Lookback Option --- p.54Chapter 5.3 --- Quanto Barrier Option --- p.57Chapter 5.4 --- Numerical results --- p.61Chapter 6 --- Conclusion --- p.64Chapter A --- Numerical Laplace Inversion for Turbo Warrants --- p.66Chapter B --- The Relation Among Barrier Options --- p.69Chapter C --- Proof of Lemma 51 --- p.71Chapter D --- Proof of Theorem 5.4 and 5.5 --- p.74Bibliography --- p.7

Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes

[EN] This paper deals with the numerical analysis of PIDE option pricing models with CGMY process using double discretization schemes. This approach assumes weaker hypotheses of the solution on the numerical boundary domain than other relevant papers. Positivity, stability, and consistency are studied. An explicit scheme is proposed after a suitable change of variables. Advantages of the proposed schemes are illustrated with appropriate examples.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and by the Spanish M.E.Y.C. Grant DPI2010-20891-C02-01.Company Rossi, R.; Jódar Sánchez, LA.; El-Fakharany, M. (2013). Positive Solutions of European Option Pricing with CGMYProcess Models Using Double Discretization Difference Schemes. Abstract and Applied Analysis. 2013:1-12. https://doi.org/10.1155/2013/517480S1122013Kou, S. 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