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105
Option pricing under a double exponential jump diffusion model
 Management Science
, 2004
"... Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with ..."
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Cited by 95 (4 self)
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Analytical tractability is one of the challenges faced by many alternative models that try to generalize the BlackScholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the BlackScholes model to alternative models with jumps. We demonstrate a double exponential jump diffusion model can lead to an analytic approximation for Þnite horizon American options (by extending the BaroneAdesi and Whaley method) and analytical solutions for popular pathdependent options (such as lookback, barrier, and perpetual American options). Numerical examples indicate that the formulae are easy to be implemented and accurate.
Russian and American put options under exponential phasetype Lévy models
, 2002
"... Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage ..."
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Cited by 89 (4 self)
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Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phasetype jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phasetype case via martingale stopping and WienerHopf factorisation. Also the first passage time problem is studied for a regime switching Lávy process with phasetype jumps. This is achieved by an embedding into a a semiMarkovian regime switching Brownian motion.
Optimal portfolio choice and the valuation of illiquid securities
 The Review of Financial Studies
, 2001
"... Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type o ..."
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Cited by 82 (13 self)
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Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type of illiquidity behaves very differently from an unconstrained investor. A liquidityconstrained investor endogenously acts as if facing borrowing and shortselling constraints, and one may take riskier positions than in liquid markets. I solve for the shadow cost of illiquidity and show that large price discounts can be sustained in a rational model. The brass assembled at headquarters at 7 a.m. that Sunday. One after another, LTCM's partners, calling in from Tokyo and London, reported that their markets had dried up. There were no buyers, no sellers. It was all but impossible to maneuver out of large trading bets.Wall Street Journal, November 16, 1998. 1.
Exit Problems for Spectrally Negative Lévy Processes and Applications to Russian, American and Canadized Options
 Ann. Appl. Probab
"... this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for thi ..."
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Cited by 72 (21 self)
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this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for this class of processes. Such processes are often considered in the context of the theories of dams, queues, insurance risk and continuous branching processes; see for example [6, 4, 5, 19]. Following the exposition on two sided exit problems in Bertoin [3] we study first exit from an interval containing the origin for spectrally negative L'evy processes reflected in their supremum (equivalently spectrally positive L'evy processes reflected in their infimum). In particular we derive the joint Laplace transform of the time to first exit and the overshoot. The aforementioned stopping time can be identified in the literature of fluid models as the time to buffer overflow (see for example [1, 13]). Together Universit'e de Pau, email: Florin.Avram@univpau.fr y Utrecht University, email: kyprianou@math.uu.nl z Utrecht University, email: pistorius@math.uu.nl 1 with existing results on exit problems we apply our results to certain optimal stopping problems that are now classically associated with mathematical finance. In sections 2 and 3 we introduce notation and discuss and develop existing results concerning exit problems of spectrally negative L'evy processes. In section 4 an expression is derived for the joint Laplace transform of the exit time and exit position of the reflected process from an interval containing the origin. This Laplace transform can be written in terms of scale functions that already appear in the solution to the two sided exit problem. In Section 5 we ou...
Pricing and hedging pathdependent options under the CEV process
 MANAGEMENT SCIENCE
, 2001
"... Much of the work on pathdependent options assumes that the underlying asset price follows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the socalled constant ..."
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Cited by 31 (0 self)
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Much of the work on pathdependent options assumes that the underlying asset price follows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the socalled constant elasticity of variance (CEV) diffusion model where the volatility is a function of the underlying asset price. We derive analytical formulae for the prices of important types of pathdependent options under this assumption. We demonstrate that the prices of options, which depend on extrema, such as barrier and lookback options, can be much more sensitive to the specification of the underlying price process than standard call and put options and show that a financial institution that uses the standard geometric Brownian motion assumption is exposed to significant pricing and hedging errors when dealing in pathdependent options.
An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems
, 2006
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Some calculations for Israeli options
 Finance and Stoch
"... Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general Americantype option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pri ..."
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Cited by 27 (2 self)
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Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general Americantype option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating an optimal stopping problem assocaited with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.
Optimal switching with application to energy tolling agreements
, 2005
"... We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimens ..."
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Cited by 24 (4 self)
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We consider the problem of optimal switching with finite horizon. This special case of stochastic impulse control naturally arises during analysis of operational flexibility of exotic energy derivatives. The current practice for such problems relies on Markov decision processes that have poor dimensionscaling properties, or on strips of spark spread options that ignore the operational constraints of the asset. To overcome both of these limitations, we propose a new framework based on recursive optimal stopping. Our model demonstrates that the optimal dispatch policies can be described with the aid of ‘switching boundaries’, similar to standard American options. In turn, this provides new insight regarding the qualitative properties of the value function. Our main contribution is a new method of numerical solution based on Monte Carlo regressions. The scheme uses dynamic programming to simultaneously approximate the optimal switching times along all the simulated paths. Convergence analysis is carried out and numerical results are illustrated with a variety of concrete
A Wiener– Hopf Monte Carlo simulation technique for Lévy processes
 Ann. Appl. Probab
, 2011
"... We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s socalled “Can ..."
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Cited by 22 (8 self)
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We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Lévy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s socalled “Canadization ” technique as well as Doney’s method of stochastic bounds for
Lifetime consumption and investment: retirement and constrained borrowing
 Journal of Economic Theory
, 2010
"... Abstract Retirement exibility and inability to borrow against future labor income can signi cantly affect optimal consumption and investment. With voluntary retirement, there exists an optimal wealthtowage ratio threshold for retirement and human capital correlates negatively with the stock market ..."
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Cited by 21 (4 self)
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Abstract Retirement exibility and inability to borrow against future labor income can signi cantly affect optimal consumption and investment. With voluntary retirement, there exists an optimal wealthtowage ratio threshold for retirement and human capital correlates negatively with the stock market even when wages have zero or slightly positive market risk exposure. Consequently, investors optimally invest more in the stock market than without retirement exibility. Both consumption and portfolio choice jump at the endogenous retirement date. The inability to borrow limits hedging and reduces the value of labor income, the wealthtowage ratio threshold for retirement, and the stock investment. Journal of Economic Literature Classi cation Numbers: D91, D92, G11, G12, C61.