Double Exponential Jump Diffusion Process Research Articles

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.

Real Option with Uncertain Expected Return

We consider an irreversible investment in a project, which generates cash flow following a double exponential jump-diffusion process and its expected return is governed by a continuous-time two-state Markov chain. If the expected return is observable, we present explicit expressions for the pricing and timing of the option to invest. With partial information, i.e. if the expected return is unobservable, we provide an explicit project value and an integral-differential equation for the pricing and timing of the option. We show a method to measure the information value, i.e. the difference between the values of the option to invest under the two cases. We present numerical solutions by finite difference methods if jumps are absent. By numerical analysis, we find that: (i) The value of the option to invest increases with the belief on economic boom; (ii) If investors are more uncertain about the state of the economy, information is more valuable; (iii) The more likely the transition from boom to recession, the less the value of the option; (iv) The bigger the dispersion of the expected return, the higher the information value; (v) A higher cash flow volatility induces a less information value.

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 Valuation of Turbo Warrants under Double Exponential Jump Diffusion

A turbo call warrant is a kind of barrier option, which pays off like a European call if the barrier is not reached before expiration, but crossing the barrier causes the original call to be replaced by a lookback call with a different strike. This complicated instrument is strangely popular in the market. But valuation clearly presents some serious challenges. In this article, Wong and Lau dive into the problem and manage to obtain a closed-form pricing equation for the case in which the underlying asset follows a double exponential jump diffusion process. <b>TOPICS:</b>Options, statistical methods

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Levy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models.

Valuing credit derivatives in a jump-diffusion model

This paper presents a simple framework for valuing single-name credit derivatives in jump-diffusion models. The Gaver–Stehfest algorithm is used to calculate the CDS spread when the value of the reference entity is assumed to the double exponential jump-diffusion process. We model directly the credit spread using a geometric Ornstein–Uhlenbeck process with a jump and derive the pricing formula for credit spread option.

ANALYTICAL PRICING OF DOUBLE-BARRIER OPTIONS UNDER A DOUBLE-EXPONENTIAL JUMP DIFFUSION PROCESS: APPLICATIONS OF LAPLACE TRANSFORM

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

Capacity expansion with exponential jump diffusion processes

This paper studies an irreversible capacity expansion problem where the industry demand is described by a double exponential jump diffusion process. Formally, we consider the following optimization problem over all adapted, non-decreasing process Here we understand X t as the cumulative capital investment up to time t (it is non-decreasing since the investment is assumed to be irreversible). is the exogenous industry demand process, which is modeled by a double exponential jump diffusion. The profit flow is assumed to have rate for some concave function H, while k denotes the cost of unit capital. The goal is choosing a capital investment strategy X so as to maximize the discounted overall profit, net the cost of investing. We explicitly solve the problem when H is assumed to be the Cobb–Douglas production function, that is, for some Throughout the paper, a slightly different (but essentially the same) optimization criterion is adopted, for the sake of technical convenience.

First passage times of a jump diffusion process

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

First passage times of a jump diffusion process

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.