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24
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 182 (21 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
A Simple Option Formula for General JumpDiffusion and Other Exponential Levy Processes
 Other Exponential Lévy Processes,” Environ Financial Systems and OptionCity.net
, 2001
"... Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very ..."
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Cited by 86 (2 self)
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Option values are wellknown to be the integral of a discounted transition density times a payoff function; this is just martingale pricing. It's usually done in 'Sspace', where S is the terminal security price. But, for L6vy processes the Sspace transition densities are often very complicated, involving many special functions and infinite summations. Instead, we show that it's much easier to compute the option value as an integral in Fourier space  and interpret this as a Parseval identity. The formula is especially simple because (i) it's a single integration for any payoff and (ii) the integrand is typically a compact expressions with just elementary functions. Our approach clarifies and generalizes previous work using characteristic functions and Fourier inversions. For example, we show how the residue calculus leads to several variation formulas, such as a wellknown, but less numerically efficient, 'BlackScholes style' formula for call options. The result applies to any Europeanstyle, simple or exotic option (without pathdependence) under any L6vy process with a known characteristic function.
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 68 (7 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...
The Term Structure of Simple Forward Rates with Jump Risk
 Jump Risk.” Mathematical Finance
, 2002
"... This paper characterizes the arbitragefree dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instan ..."
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Cited by 42 (5 self)
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This paper characterizes the arbitragefree dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent research on "market models." We consider very general types of jump processes, modeled through marked point processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates.
Numerical solution of jumpdiffusion LIBOR market models
 Finance and Stochastics
"... This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in particular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this for ..."
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Cited by 26 (3 self)
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This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in particular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this formulation offers some attractive modeling features, it presents a challenge for computational work. As a first step, we therefore show how to reformulate a term structure model driven by marked point processes with suitably bounded statedependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes because the Poisson random measure can be simulated without discretization error. Jumps in LIBOR rates are then thinned from the Poisson random measure using statedependent thinning probabilities. Because of discontinuities inherent to the thinning process, this procedure falls outside the scope of existing convergence results; we provide some theoretical support for our method through a result establishing first and second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments.
Pricing swaps and options on quadratic variation under stochastic time change models
 Columbia University
, 2007
"... We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As ..."
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Cited by 18 (1 self)
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We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of Lévy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closedform expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar logcontract approach is provided. ∗We thank Arthur Sepp and an anonymous referee for useful comments. We assume full responsibility for any remaining errors.
2003): ”Cap and Swaption Approximations in LIBOR Market Models with Jumps
 Journal of Computational Finance
, 2003
"... This paper develops formulas for pricing caps and swaptions in Libor market models with jumps. The arbitragefree dynamics of this class of models were characterized in Glasserman and Kou (2003) in a framework allowing for very general jump processes. For computational purposes, it is convenient to ..."
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Cited by 14 (0 self)
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This paper develops formulas for pricing caps and swaptions in Libor market models with jumps. The arbitragefree dynamics of this class of models were characterized in Glasserman and Kou (2003) in a framework allowing for very general jump processes. For computational purposes, it is convenient to model jump times as Poisson processes; however, the Poisson property is not preserved under the changes of measure commonly used to derive prices in the Libor market model framework. In particular, jumps cannot be Poisson under both a forward measure and the spot measure, and this complicates pricing. To develop pricing formulas, we approximate the dynamics of a forward rate or swap rate using a scalar jumpdiffusion process with timevarying parameters. We develop an exact formula for the price of an option on this jumpdiffusion through explicit inversion of a Fourier transform. We then use this formula to price caps and swaptions by choosing the parameters of the scalar diffusion to approximate the arbitragefree dynamics of the underlying forward or swap rate. We apply this method to two classes of models: one in which the jumps in all forward rates are Poisson under the spot measure, and one in which the jumps in each forward rate are Poisson under its associated forward measure. Numerical examples demonstrate the accuracy of the approximations. 1
Portfolio Optimization with JumpDiffusions: Estimation of TimeDependent Parameters and Application
, 2002
"... This paper treats jumpdiffusion processes in continuous time, with emphasis on the jumpamplitude distributions, developing more appropriate models using parameter estimation for the market in one phase and then applying the resulting model to a stochastic optimal portfolio application in a second ..."
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Cited by 9 (5 self)
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This paper treats jumpdiffusion processes in continuous time, with emphasis on the jumpamplitude distributions, developing more appropriate models using parameter estimation for the market in one phase and then applying the resulting model to a stochastic optimal portfolio application in a second phase. The new developments are the use of uniform jumpamplitude distributions and timevarying market parameters, introducing more realism into the application model, a LogNormalDiffusion, LogUniformJump model.
Stochastic Analysis of Jump–Diffusions for Financial Log–Return
 Processes,” Proceedings of Stochastic Theory and Control Workshop
, 2002
"... Abstract. A jumpdiffusion logreturn process with lognormal jump amplitudes is presented. The probability density and other properties of the theoretical model are rigorously derived. This theoretical density is fit to empirical logreturns of Standard & Poor’s 500 stock index data. The model ..."
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Cited by 9 (7 self)
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Abstract. A jumpdiffusion logreturn process with lognormal jump amplitudes is presented. The probability density and other properties of the theoretical model are rigorously derived. This theoretical density is fit to empirical logreturns of Standard & Poor’s 500 stock index data. The model repairs some failures found from the lognormal distribution of geometric Brownian motion to model features of realistic financial instruments: (1) No large jumps or extreme outliers, (2) Not negatively skewed such that the negative tail is thicker than the positive tail, and (3) Nonleptokurtic due to the lack of thicker tails and higher mode. This is the corrected version of the published paper. 1