Exponential Jump-diffusion Model Research Articles

Sequential Modeling of Dependent Jump Processes

Two multivariate models for asset returns are introduced. Both generalize popular univariate models without altering their marginal laws; a very convenient feature, e.g., for a sequential calibration of the model's parameters to market quotes. The first is a double exponential jump-diffusion model with constant volatility as presented in the univariate case by Kou (2002). The second is a generalization of the stochastic volatility model of Gamma-Ornstein-Uhlenbeck type as first presented by Barndorff-Nielsen and Shephard (2001). Univariate processes of these kinds are driven by a Brownian motion as well as a compound Poisson process. Unlike other existing multivariate extensions of these univariate models, one important advantage of the presented approach is to keep the parameters that govern the dependence structure between the assets separate from the parameters that determine the marginal distributions of the assets. This is achieved by introducing dependence to the univariate compound Poisson processes by means of a bespoke common stochastic time change.

Analytical Option Pricing Under a Displaced Double Exponential Jump-Diffusion Model

We generalize the Kou (2002) double exponential jump-diffusion model in two directions. First, we independently displace the two tails of the jump size distribution away from the origin. Second, we allow for each of the displaced tails to follow a gamma distribution with an integer-valued shape parameter. Both extensions introduce additional flexibility in the tails of the corresponding return distribution. Our model is supported by an equilibrium economy and we obtain closed-form solutions for European plain vanilla options. Our valuation function is computationally fast to evaluate and robust across the full parameter space. We estimate the physical model parameters through maximum likelihood and for a diverse sample of equities, commodities and exchange rates. For all assets under consideration, the original Kou (2002) model can be rejected in favor of our newly introduced asymmetrically displaced double gamma dynamics.

Fast Fourier transform option pricing with stochastic interest rate, stochastic volatility and double jumps

In this paper, we focus on pricing European options in a double exponential jump diffusion model with stochastic volatility and stochastic interest rate. Firstly, using fast Fourier transform (FFT) technique, we obtain numerical solutions for option prices. Then, we analyze several effects on option prices under the proposed model, including correlation between stock returns and volatility, stochastic interest rate. Simulations show that FFT is fast and efficient, stock returns are negatively correlated with volatility and the effect of stochastic interest rate over longer time horizons is significant.

Pricing discrete path-dependent options under a double exponential jump–diffusion model

We provide methodologies to price discretely monitored exotic options when the underlying evolves according to a double exponential jump diffusion process. We show that discrete barrier or lookback options can be approximately priced by their continuous counterparts’ pricing formulae with a simple continuity correction. The correction is justified theoretically via extending the corrected diffusion method of Siegmund (1985). We also discuss the jump effects on the performance of this continuity correction method. Numerical results show that this continuity correction performs very well especially when the proportion of jump volatility to total volatility is small. Therefore, our method is sufficiently of use for most of time.

이중 지수 점프확산 모형하에서의 마코브 체인을 이용한 아메리칸 옵션 가격 측정

This paper suggests a numerical method for valuation of American options under the Kou model (double exponential jump diffusion model). The method is based on approximation of underlying asset price using a finite-state, time-homogeneous Markov chain. We examine the effectiveness of the proposed method with simulation results, which are compared with those from the conventional numerical method, the finite difference method for PIDE (partial integro-differential equation).

European Option Pricing under Jump Diffusion with Proportional Transaction Costs

This paper considers the problem of European option pricing in the presence of proportional transaction costs when the price of the underlying follows a jump diffusion process. Using an approach that is based on maximization of the expected utility of terminal wealth, we transform the option pricing into stochastic optimal control problems, and argue that the value functions of these problems are the solutions of a free boundary problem, in particular, a partial integro-differential equation, under different boundary conditions. To solve the singular stochastic control problems associated with utility maximization and compute the value function and no-transaction boundaries, we develop a coupled backward induction algorithm that is based on the connection of the free boundary problem to an optimal stopping problem. Numerical examples of option pricing under a double exponential jump diffusion model are also provided.

The double exponential jump diffusion model for pricing European options under fuzzy environments

The interest rate and volatility may have different values in the different commercial banks and financial institutions. Moreover, the fluctuations of the underlying assets are rare events, and there are not enough historical data to estimate the jump intensity in a precise sense. This paper considers European option pricing problems with the fuzzy interest rate, fuzzy drift, fuzzy volatility and fuzzy jump intensity. We present the fuzzy pricing formula of European options based on the Kou's double exponential jump diffusion model. We also obtain the crisp possibilistic mean option pricing formula in fuzzy double exponential jump diffusion model by using the crisp possibilistic mean values of the fuzzy numbers. Comparing with B-S formula, numerical analysis and empirical results show that the fuzzy double exponential jump diffusion formula and the crisp possibilistic mean option pricing formula are reasonable and can be taken as reference pricing tools for the financial investors.

Fast Fourier transform technique for the European option pricing with double jumps

In this paper, we provided a fast algorithm for pricing European options under a double exponential jump-diffusion model based on Fourier transform. We derived a closed-form (CF) representation of the characteristic function of the model. By using fast Fourier transform (FFT) technique, we obtained an approximation numerical solution for the prices of European call options. Our numerical results show that our method is fast, accurate and easy to implement. The proposed option pricing method is useful for empirical analysis of asset returns and managing the corporate credit risks.

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.

Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

Pricing and Hedging of Quantile Options in a Flexible Jump Diffusion Model

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

Catastrophe Bond and Risk Modeling: A Review and Calibration Using Chinese Earthquake Loss Data

ABSTRACT Earthquake risks are attracting increased attention as a result of recent catastrophic events such as the Wenchuan earthquake in China. This article aims to select, tailor, and develop loss modeling methods for catastrophic insurance. We review the state-of-the-art approaches in modeling catastrophe losses for catastrophe bonds’ modeling and pricing. The methods are applied to the 1966–2008 losses resulted from the earthquakes in China. Various error measures are proposed for validating catastrophe modeling. Results suggest that the double exponential jump-diffusion model fits the data well.

Occupation Times of Jump-Diffusion Processes with Double Exponential Jumps and the Pricing of Options

In this paper, we provide Laplace transform-based analytical solutions to pricing problems of various occupation-time-related derivatives such as step options, corridor options, and quantile options under Kou's double exponential jump diffusion model. These transforms can be inverted numerically via the Euler Laplace inversion algorithm, and the numerical results illustrate that our pricing methods are accurate and efficient. The analytical solutions can be obtained primarily because we derive the closed-form Laplace transform of the joint distribution of the occupation time and the terminal value of the double exponential jump diffusion process. Beyond financial applications, the mathematical results about occupation times of a jump diffusion process are of more general interest in applied probability.

Optimal design of profit sharing rates by FFT

This paper addresses the calculation of a fair profit sharing rate for participating policies with a minimum interest rate guaranteed. The bonus credited to policies depends on the performance of a basket of two assets: a stock and a zero coupon bond and on the guarantee. The dynamics of the instantaneous short rates are driven by a Hull and White model, whereas the stocks follow a double exponential jump–diffusion model. The participation level is determined such that the return retained by the insurer is sufficient to hedge the interest rate guaranteed. Given that the return of the total asset is not lognormal, we rely on a Fast Fourier Transform to compute the fair value of bonus and guarantee options.

Analytical binomial lookback options with double-exponential jumps

We study the problem of the convergence of the adjusted binomial lookback option in double-exponential jump diffusion models. By using the results of [Dai, M., (2000). A modified binomial tree method for currency lookback options. Acta Mathematica Sinica, 16, 445–454; Kou, S., & Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management Science, 50, 1178–1192] and [Park, H.S., Kim, K.I., & Qian, X. (2009). A Mathematical modeling for the Lookback option with jump diffusion using Binomial tree method. Journal of Computational and Applied Mathematics, preprint], we show the equivalence between the adjusted binomial tree method and the explicit difference scheme. The convergence is also theoretically proved through the notion of viscosity solution. Numerical results coincide with the theoretical results.

Quasi-Monte Carlo for finance beyond Black--Scholes

Quasi-Monte Carlo methods are used to approximate integrals of high dimensionality. However, if the problem under consideration is of unbounded dimensionality, it is not obvious if one can apply quasi-Monte Carlo methods at all. We introduce a hybrid approach combining quasi-Monte Carlo and Monte Carlo methods and apply it to a finance problem of unbounded dimensionality. We find that this hybrid approach improves on a Monte Carlo approach. References A. Kyprianou, W. Schoutens, and P. Wilmott. {Exotic option pricing and advanced {L}evy models}. Wiley, Chichester, 2005. P. L'Ecuyer and C. Lemieux. Recent advances in randomized quasi-{M}onte {C}arlo methods. In {Modeling uncertainty}, volume 46 of {Internat. Ser. Oper. Res. Management Sci.}, pages 419--474. Kluwer Acad. Publ., Boston, MA, 2002. {http://www.iro.umontreal.ca/ lecuyer/myftp/papers/survey01.ps}. S. M. Ross. {Introduction to probability models}. Harcourt/Academic Press, Burlington, MA, sixth edition, 1997. R. E. Caflisch, W. J. Morokoff, and A. B. Owen. Valuation of mortgage backed securities using {B}rownian {Br}idges to reduce effective dimension. {J. Comp. Finance}, 1:27--46, 1997. {http://www-stat.stanford.edu/ owen/reports/cmo.ps}. R. Cont and P. Tankov. {Financial {M}odelling with {J}ump {P}rocesses}. Chapman and Hall/CRC, Boca Raton, London, New York, Washington, D. C., 2004. P. Glasserman. {Monte {C}arlo {M}ethods in {F}inancial {E}ngineering}. Springer, New York, Berlin, Heidelberg, Hong Kong, London, Milan, Paris, Tokyo, 2004. S.G. Kou and H. Wang. Option pricing under a double exponential jump diffusion model. {Management Science}, 50:1178--1192, 2004.

A canonical optimal stopping problem for American options under a double exponential jump-diffusion model

This paper presents a simple numerical approach to compute accurately the values and optimal exercise boundaries for American options when the underlying process is a double exponential jump-diffusion model that prices jump risk. The present work extends the canonical representation for American options initially developed in the Brownian motion set-up. Here, too, jump-diffusion pricing models can be reduced to a single optimal stopping problem, indexed by one more parameter, and linear spline approximations of the stopping boundary in the canonical scale with only a few knots are supported through numerical evidence. These approximations can then be exploited to solve the integral equation defining the early exercise boundary of an American option efficiently and accurately, thus leading to its efficient and accurate pricing and hedging.

Pricing European option in a double exponential jump-diffusion model with two market structure risks and its comparisons

Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closed-form solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure risks that there exist CIR stochastic volatility of stock return and Vasicek or CIR stochastic interest rate in the market. In the end, the result of the model in the paper is compared with those in other models, including BS model with numerical experiment. These results show that the double exponential jump-diffusion model with CIR-market structure risks is suitable for modelling the real-market changes and very useful.

Maximum likelihood estimation of the double exponential jump-diffusion process

The double exponential jump-diffusion (DEJD) model, recently proposed by Kou (Manage Sci 48(8), 1086–1101, 2002) and Ramezani and Zeng (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=606361, 1998), generates a highly skewed and leptokurtic distribution and is capable of matching key features of stock and index returns. Moreover, DEJD leads to tractable pricing formulas for exotic and path dependent options (Kou and Wang Manage Sci 50(9), 1178–1192, 2004). Accordingly, the double exponential representation has gained wide acceptance. However, estimation and empirical assessment of this model has received little attention to date. The primary objective of this paper is to fill this gap. We use daily returns for the S&P-500 and the NASDAQ indexes and individual stocks, in conjunction with maximum likelihood estimation (MLE) to fit the DEJD model. We utilize the BIC criterion to assess the performance of DEJD relative to log-normally distributed jump-diffusion (LJD) and the geometric brownian motion (GBM). We find that DEJD performs better than these alternatives for both indexes and individual stocks.

PRICING OF FIRST TOUCH DIGITALS UNDER NORMAL INVERSE GAUSSIAN PROCESSES

We calculate prices of first touch digitals under normal inverse Gaussian (NIG) processes, and compare them to prices in the Brownian model and double exponential jump-diffusion model. Numerical results are produced to show that for typical parameters values, the relative error of the Brownian motion approximation to NIG price can be 2–3 dozen percent if the spot price is at the distance 0.05–0.2 from the barrier (normalized to one). A similar effect is observed for approximations by the double exponential jump-diffusion model, if the jump component of the approximation is significant. We show that two jump-diffusion processes can give approximately the same results for European options but essentially different results for first touch digitals and barrier options. A fast approximate pricing formula under NIG is derived.