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Exotic options in general exponential Levy models
 Prepublication 850, Universites Paris 6
, 2003
"... Recently, financial models of the type S(t) = exp(Xt), (1) where X is a Lévy process, have attracted a lot of interest, both among academics and the industry. Indeed, these models are a natural generalization of ..."
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Recently, financial models of the type S(t) = exp(Xt), (1) where X is a Lévy process, have attracted a lot of interest, both among academics and the industry. Indeed, these models are a natural generalization of
MATURITY RANDOMIZATION FOR STOCHASTIC CONTROL PROBLEMS
, 2006
"... We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the socalled Canadization procedure suggested by Carr [Revie ..."
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We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the socalled Canadization procedure suggested by Carr [Review of Financial Studies II (1998) 597–626] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the superreplication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly. 1. Introduction. It
Duality and Derivative Pricing with Lévy Processes
, 2003
"... The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Levy process. The main idea is to apply Girsanov's Theorem for Levy processes, in order to reduce the posed problem to the pricing of a one Levy driven sto ..."
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The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Levy process. The main idea is to apply Girsanov's Theorem for Levy processes, in order to reduce the posed problem to the pricing of a one Levy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu (1996) for two dimensional Brownian motion. Also we examine an existing relation between prices of put and call options, of both the European and the American type. This relation, based on a change of numeraire corresponding to a change of the probability measure through Girsanov's Theorem, is called putcall duality. It includes as a particular case, the relation known as putcall symmetry. Necessary and sufficient conditions for putcall symmetry to hold are obtained, in terms of the triplet of predictable characteristic of the Levy process.
Remarks on the American put option for jump diffusions
, 2007
"... We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasivariational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smoot ..."
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We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasivariational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each of which is a value function of an optimal stopping problem for a diffusion. This sequence, which converges to the value function of the American put option for jump diffusions, is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasivariational inequalities. This sequence converges to the value function of the American put option uniformly and exponentially fast, therefore it provides a good approximation scheme. In fact, the value of the American put option is the fixed point of the functional operator we use.
Remarks on the perpetual American put option for jump diffusions
, 2007
"... We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasivariational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smoot ..."
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We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasivariational inequality, that it is C 2 except at the stopping boundary and that it is C 1 everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each of which is a value function of an optimal stopping problem for a diffusion. This sequence, which converges to the value function of the American put option for jump diffusions, is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasivariational inequalities. This sequence converges to the value function of the American put option uniformly and exponentially fast, therefore it provides a good approximation scheme. In fact, the value of the American put option is the fixed point of the functional operator we use. 1
Perpetual options for Lévy processes in the Bachelier model
 Proceedings of the Steklov Mathematical Institute
, 2001
"... Solution to the optimal stopping problem V (x) = sup τ Ee−δτg(x+Xτ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K)+ or gp(x) = (K−x)+ Results, interpreted as option prices of perpetu ..."
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Solution to the optimal stopping problem V (x) = sup τ Ee−δτg(x+Xτ) is given, where X = {Xt}t≥0 is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form gc(x) = (x−K)+ or gp(x) = (K−x)+ Results, interpreted as option prices of perpetual options in Bachelier’s model are expressed in terms of the distribution of the overall supremum in case g = gc and overall infimum in case g = gp of the process X killed at rate δ. Closed form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for gc, and under arbitrary positive jumps and mixed exponentially distributed negative jumps for gp. In case g = gc a prophet inequality comparing prices of perpetual lookback call options and perpetual call options is obtained.
Optimal stopping and American options
 Daiwa Lecture Series, Kyoto
, 2008
"... 1.1 Essential supremum. Uniform integrability......................... 3 1.1.1 Essential supremum................................. 3 1.1.2 Uniform integrability................................. 4 ..."
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1.1 Essential supremum. Uniform integrability......................... 3 1.1.1 Essential supremum................................. 3 1.1.2 Uniform integrability................................. 4
Elementary Proofs on Optimal Stopping
, 2001
"... Elementary proofs of classical theorems on pricing perpetual call and put options in the standard BlackScholes model are given. The method presented does not rely on stochastic calculus and is also applied to give prices and optimal stopping rules for perpetual call options when the stock is driven ..."
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Elementary proofs of classical theorems on pricing perpetual call and put options in the standard BlackScholes model are given. The method presented does not rely on stochastic calculus and is also applied to give prices and optimal stopping rules for perpetual call options when the stock is driven by a Levy process with no positive jumps, and for perpetual put options for stocks driven by a Levy process with no negative jumps 1 This work was partially written at the Laboratoire de Statistique et Probabilites de l'Universite Paul Sabatier, Toulouse, and beneted from helpful discussion with Walter Moreira. Elementary Proofs on Optimal Stopping Ernesto Mordecki y Facultad de Ciencias. Centro de Matematica. Igua 4225. CP 11400. Montevideo. Uruguay. December 29, 2000 1 Introduction 1.1 Consider a model of a nancial market with two assets, a savings account B = fB t g t0 and a stock S = fS t g t0 . The evolution of B is deterministic, with B t = B 0 e rt ; B 0 = 1; r >...
Nomura International plc.
, 812
"... A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes ..."
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A transform approach to compute prices and greeks of barrier options driven by a class of Lévy processes
problems
, 2004
"... We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the socalled Canadization procedure suggested by P. Carr in ..."
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We study a maturity randomization technique for approximating optimal control problems. The algorithm is based on a sequence of control problems with random terminal horizon which converges to the original one. This is a generalization of the socalled Canadization procedure suggested by P. Carr in [2] for the fast computation of American put option prices. In addition to the original application of this technique to optimal stopping problems, we provide an application to another problem in finance, namely the superreplication problem under stochastic volatility, and we show that the approximating value functions can be computed explicitly.