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73
Option pricing when underlying stock returns are discontinuous
 Journal of Financial Economics
, 1976
"... The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying ..."
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Cited by 507 (1 self)
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The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the moregeneral cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular,
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 73 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
Brownian Motion in a Weyl Chamber, NonColliding Particles, and Random Matrices
, 1997
"... . Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting posit ..."
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Cited by 66 (2 self)
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. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...
Free martingale polynomials
 Journal of Functional Analysis
"... ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. ..."
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Cited by 28 (3 self)
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ABSTRACT. In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota’s finite operator calculus can be modified for the free context.
Exact simulation of diffusions
 Annals of Applied Probability
, 2005
"... We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without ..."
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Cited by 28 (12 self)
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We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process. 1. Introduction. Exact
Dynamic Hedging Portfolios For Derivative Securities In The Presence Of Large Transaction Costs
 Appl. Math. Finance
, 1994
"... : We introduce a new class of strategies for hedging derivative securities in the presence of transaction costs assuming lognormal continuoustime prices for the underlying asset. We do not assume necessarily that the payoff is convex as in Leland [11] or that transaction costs are small compared to ..."
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Cited by 18 (1 self)
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: We introduce a new class of strategies for hedging derivative securities in the presence of transaction costs assuming lognormal continuoustime prices for the underlying asset. We do not assume necessarily that the payoff is convex as in Leland [11] or that transaction costs are small compared to the price changes between portfolio adjustments, as in Hoggard, Whalley and Wilmott [8]. The type of hedging strategy to be used depends on the value of the Leland number A = q 2 ß k oe p ffit , where k is the roundtrip transaction cost, oe is the volatility of the underlying asset, and ffit is the timelag between transactions. If A ! 1 it is possible to implement modified BlackScholes deltahedging strategies, but not otherwise. We propose new hedging strategies that can be used with A 1 to control effectively hedging risk and transaction costs. These strategies are associated with the solution of a nonlinear obstacle problem for a diffusion equation with volatility oe A = oe p...
Inertial Range Scaling Of Laminar Shear Flow As A Model Of Turbulent Transport
 Commun. Math. Phys
, 1991
"... Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport. Key words: Turbulent transport, random field, scaling exponent, anomalous diffusion. I. INTRODUCTION The determination of the statistical behav ..."
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Cited by 13 (5 self)
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Asymptotic scaling behavior, characteristic of the inertial range, is obtained for a fractal stochastic system proposed as a model for turbulent transport. Key words: Turbulent transport, random field, scaling exponent, anomalous diffusion. I. INTRODUCTION The determination of the statistical behavior of a fluid from the statistical properties of a random velocity field is important in the study of tracer flow in heterogeneous porous media, ground water ecology, and fully developed turbulence. In earlier work [4,9], the asymptotic scaling exponents were obtained for the motion of a fluid determined by a convectiondiffusion equation T t + v ® ( x ® , t ) . ÑT = µDT, T (0,x ® ) = T 0 ( x ® ), (1.1) with µ = 0. Here T is a physical quantity, v ® is a random velocity field, and µ is the molecular diffusion coefficient. The purpose of this paper is to relate these exponents to the similar but in some cases distinct exponents of [1], obtained for the equation T t + v(x,t)T y ...
Unitary Brownian motions are linearizable
"... Brownian motions in the infinitedimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent onedimensional Brownian motions. The proof involves continuous ..."
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Cited by 11 (3 self)
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Brownian motions in the infinitedimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent onedimensional Brownian motions. The proof involves continuous tensor products and continuous quantum measurements. A byproduct: a Brownian motion in a separable Fspace (not locally convex) is a Gaussian process.
Equivalent and absolutely continuous measure changes for jumpdiffusion processes” to appear in the Annals of Applied Probability
"... We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential. ..."
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Cited by 11 (2 self)
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We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential.