Double Exponential Jumps Research Articles

Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

ABSTRACTThis article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.

American-style options in jump-diffusion models: estimation and evaluation

We propose dynamic programming coupled with finite elements for valuing American-style options under Gaussian and double exponential jumps à la Merton [J. Financ. Econ., 1976, 3, 125–144] and Kou [Manage. Sci., 2002, 48, 1086–1101], and we provide a proof of uniform convergence. Our numerical experiments confirm this convergence result and show the efficiency of the proposed methodology. We also address the estimation problem and report an empirical investigation based on Home Depot. Jump-diffusion models outperform their pure-diffusion counterparts.

Asymptotic Analysis of Stock Price Densities and Implied Volatilities in Mixed Stochastic Models

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions defined on $(0,\infty)$ and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps and perform a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the normal inverse Gaussian law.

A Structural Model with Jump Diffusion Processes

In this paper, we extend the framework of Leland’s 94 by examining corporate debt, equity and firm values with jump-difffusion processes. We choose two kinds of jumps such as the uniform and double exponential jumps to model the distribution of the log jump sizes. By this choice, we are able to derive closed-form results in both models for equity, debt and firm values. Analysis of credit spread, debt value and firm value has been done for three proposed models: diffusion process diffusion with uniform jump and diffusion with double exponential jumps. Our results have the same forms as those of Leland’s 94. However, in both of our models, the spreads are modified significantly in comparison with those of Leland due to jumps’ assumption.

European Option Pricing under the Double Exponential Jump Model with Stochastic Interest Rate, Stochastic Volatility and Stochastic Intensity

This paper explores the valuation of European options when the underlying asset follows the double exponential jump process with stochastic rate, stochastic volatility and stochastic intensity. This model better describes market characteristics, such as the volatility smile, and jump behavior. By using FFT (Fast Fourier Transform) approach, a closed form representation of the characteristic function of the process is derived for the valuation of European options. Numerical results show that the FFT method is effective and competent.

Anticipated Default Stopping Time with Jump Diffusion Process

This study extends the framework of Merton (1976) by examining the corporate defaultable debt, the equity and the firm values with jump diffusion processes. We look at the correlation effect of cascade falls in firm values on the value of the debt and its credit spreads. Two kinds of jumps have been chosen, including the double exponential jumps and the normal jumps, to model the distribution of the jump sizes. The described method has closed-form results in both models for equity, debt and firm values. The results of this study have the same forms as those of Merton. However, in both models, the credit spreads are modified significantly in comparison with those of Merton due to correlation jumps’ assumptions that have been used in this model.

Barrier Caps and Floors under the LIBOR Market Model with Double Exponential Jumps

The LIBOR market model (LMM) has become the standard model for many kinds of interest rate derivatives, such as cap contracts. It assumes that the distribution of the one-period interest rate at each future repricing date is lognormal, so that every caplet can be easily priced using the Black model. Volatilities at different dates are tied together by assumptions that restrict the number of parameters to a manageable set. But interest rates are strongly affected by things like monetary policy decisions, which can lead to sharp, nondiffusive jumps when policy changes. The effect may be seen in the leptokurtosis of the empirical distributions of interest rates and in volatility smiles for caplets. The low probability of extreme rate moves in the standard LMM is especially problematical for barrier options. In this article, the authors introduce a new jump-diffusion process for the LLM that can capture the effects of jumps on interest rate barrier contracts. The authors allow exponential jumps, both up and down, so the model is called the LMM with a double-jump process, or LMMDJ. Explicit valuation equations are provided for the four classes of barrier contracts and are shown to be quite accurate in Monte Carlo simulations. <b>TOPICS:</b>Derivatives, quantitative methods

A Family of Mortality Jump Models Applied to US Data

Abstract Mortality models are fundamental to quantify mortality/longevity risks and provide the basis of pricing and reserving. In this article, we consider a family of mortality jump models and propose a new generalized Lee–Carter model with asymmetric double exponential jumps. It is asymmetric in terms of both time periods of impact and frequency/severity profiles between adverse mortality jumps and longevity jumps. It is mathematically tractable and economically intuitive. It degenerates to a transitory exponential jump model when fitting the US mortality data and is the best fit compared with other jump models.

A Family of Mortality Jump Models with Parameter Uncertainty: Application to Hedging Longevity Risk in Life Settlements

Mortality models are fundamental to quantify mortality/longevity risks and provide the basis of pricing and reserving. In this paper, we consider a family of mortality jump models and propose a new generalized Lee-Carter model with asymmetric double exponential jumps. It is asymmetric in terms of both time periods of impact and frequency/severity profiles between adverse mortality jumps and longevity jumps. It is mathematically tractable and economically intuitive. It degenerates to a transitory exponential jump model when fitting the U.S. mortality data and is the best fit compared with other jump models.

Longevity/Mortality Risk Modeling and Securities Pricing

Securitization of longevity/mortality risk provides insurers and pension funds an effective, low-cost approach to transferring the longevity/mortality risk from their balance sheets to capital markets. The modeling and forecasting of the mortality rate is the key point in pricing mortality-linked securities that facilitates the emergence of liquid markets. The catastrophic longevity jumps and mortality jumps are significant in historical data and have a critical effect on securities pricing. This paper introduces a stochastic diffusion model with a Double Exponential Jump Diffusion (DEJD) process for mortality time-series, which is the first to capture both asymmetric jump features and cohort effect as the underlying reason for the mortality trend. The DEJD model has the advantage of easy calibration and mathematical tractability. The form of the DEJD model is neat, concise and practical. Compared with previous stochastic models with or without jumps, the DEJD model fits the actual data better. To apply the model, the implied risk premium is calculated based on the Swiss Re mortality bond price. The DEJD model is the first to provide a close-form solution to price the q-forward, which is the standard product contingent on the LifeMetrics index for hedging longevity or mortality risk.

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 option...

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.

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.

Modeling Greek equity prices using jump diffusion processes

This study compares two popular jump diffusion models using equity data from the Greek market. The models considered are those proposed by Merton (1974) and Kou (2002) and differ on the specification of the jump component. In the former model jumps follow a lognormal distribution whereby in the latter the jump component is drawn from a double exponential distribution. Maximum Likelihood estimation provides evidence of jumps in both basket indices and individual stock returns. Moreover, the empirical comparison shows that double exponential jumps are more consistent with the empirical data.