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95
Stochastic Volatility for Lévy Processes
, 2001
"... Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include NonGaussian models that are so ..."
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Cited by 209 (12 self)
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Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include NonGaussian models that are solutions to OU (OrnsteinUhlenbeck) equations driven by one sided discontinuous L¶evy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean corrected exponential model is not a martingale in the ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1
Optimal stopping and perpetual options for Lévy processes
, 2000
"... Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this sol ..."
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Cited by 67 (8 self)
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Solution to the optimal stopping problem for a L'evy process and reward functions (e x \Gamma K) + and (K \Gamma e x ) + , discounted at a constant rate is given in terms of the distribution of the overall supremum and infimum of the process killed at this rate. Closed forms of this solutions are obtained under the condition of positive jumps mixedexponentially distributed. Results are interpreted as admissible pricing of perpetual American call and put options on a stock driven by a L'evy process, and a BlackScholes type formula is obtained. Keywords and Phrases: Optimal stopping, L'evy process, mixtures of exponential distributions, American options, Derivative pricing. JEL Classification Number: G12 Mathematics Subject Classification (1991): 60G40, 60J30, 90A09. 1 Introduction and general results 1.1 L'evy processes Let X = fX t g t0 be a real valued stochastic process defined on a stochastic basis(\Omega ; F ; F = (F t ) t0 ; P ) that satisfy the usual conditions. A...
Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation
 Analysis and Computation, SIAM Books
, 2007
"... Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions usi ..."
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Cited by 33 (7 self)
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Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jumpdiffusions. Next, markedjumpdiffusions are treated using compound Poisson processes to include random marked jumpamplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measuretheoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding nonsmooth and jump discontinuities to the model. Finally, the stochastic optimal control of markedjumpdiffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jumpdiffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There
Numerical valuation of options with jumps in the underlying
, 2005
"... A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in tim ..."
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Cited by 19 (3 self)
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A jumpdiffusion model for a singleasset market is considered. Under this assumption the value of a European contingency claim satisfies a general partial integrodifferential equation (PIDE). The equation is localized and discretized in space using finite differences and finite elements and in time by the second order backward differentiation formula (BDF2). The resulting system is solved by an iterative method based on a simple splitting of the matrix. Using the fast Fourier transform, the amount of work per iteration may be reduced to O(n log 2 n) and only O(n) entries need to be stored for each time level. Numerical results showing the quadratic convergence of the methods are given for Merton’s model and Kou’s model.
Time Changed Markov Processes in Unified CreditEquity Modeling ∗
, 2008
"... This paper develops a novel class of hybrid creditequity models with statedependent jumps, localstochastic volatility and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time changed Markov diffusion process with statede ..."
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Cited by 17 (4 self)
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This paper develops a novel class of hybrid creditequity models with statedependent jumps, localstochastic volatility and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time changed Markov diffusion process with statedependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with statedependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has localstochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has statedependent jumps, localstochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the
Exotic Options under Lévy Models: An Overview
, 2004
"... In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models. ..."
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Cited by 17 (0 self)
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In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models.
Pricing Options in JumpDiffusion Models: An Extrapolation Approach
, 2008
"... We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization sc ..."
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Cited by 17 (3 self)
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We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is firstorder accurate in time. Secondorder accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new highorder time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton’s and Kou’s jumpdiffusion models, the diffusionextended variance gamma model, as well as the twodimensional DuffiePanSingleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a oneyear doublebarrier option in Kou’s jumpdiffusion model, our scheme attains accuracy of 10−5 in 72 time steps (in 0.05 seconds). In contrast, it takes the firstorder IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the secondorder IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive
Implied volatility: Statics, dynamics, and probabilistic interpretation
 Recent Advances in Applied Probability
, 2005
"... Given the price of a call or put option, the BlackScholes implied volatility is the unique volatility parameter for which the BulackScholes formula recovers the option price. This article surveys research activity relating to three theoretical questions: First, does implied volatility admit a pro ..."
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Cited by 15 (0 self)
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Given the price of a call or put option, the BlackScholes implied volatility is the unique volatility parameter for which the BulackScholes formula recovers the option price. This article surveys research activity relating to three theoretical questions: First, does implied volatility admit a probabilistic interpretation? Second, how does implied volatility behave as a function of strike and expiry? Here one seeks to characterize the shapes of the implied volatility skew (or smile) and term structure, which together constitute what can be termed the statics of the implied volatility surface. Third, how does implied volatility evolve as time rolls forward? Here one seeks to characterize the dynamics of implied volatility. 1
Option pricing under a mixedexponential jump diffusion model
 Management Science
, 2011
"... This paper aims to extend the analytical tractability of the Black–Scholes model to alternative models witharbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixedexponential distribution, which is a weighted average of expo ..."
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Cited by 12 (2 self)
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This paper aims to extend the analytical tractability of the Black–Scholes model to alternative models witharbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixedexponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and doubleexponential jump diffusion models, because the mixedexponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavytailed distributions. The mixedexponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for pathdependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a highorder integrodifferential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day. Key words: jump diffusion; mixedexponential distributions; lookback options; barrier options; Merton’s normal jump diffusion model; first passage times
Temporary versus Permanent Shocks: Explaining Corporate Financial Policies,” Review of Financial Studies
, 2010
"... We investigate corporate financial policies in the presence of both temporary and permanent shocks to firms ’ cash flows. In our framework firms can experience negative cash flows, changes in and levels of cash flows are imperfectly correlated with firm value, and earnings volatility differs from a ..."
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Cited by 12 (1 self)
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We investigate corporate financial policies in the presence of both temporary and permanent shocks to firms ’ cash flows. In our framework firms can experience negative cash flows, changes in and levels of cash flows are imperfectly correlated with firm value, and earnings volatility differs from asset volatility. These results are consistent with empirical stylized facts and are contrary to the implications of existing dynamic capital structure models that allow only for permanent shocks to cash flows. Temporary shocks increase the importance of financial flexibility and may provide an intuitively simple and realistic explanation of empirically observed financial conservatism and low leverage phenomena. The theoretical framework developed in this paper is general enough to be used in various corporate finance applications.