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52
Optimal Replication of Contingent Claims Under Portfolio Constraints
 Rev. of Financial Studies
, 1998
"... We study the problem of determining the minimum cost of superreplicating a nonnegative contingent claim when there are convex constraints on the portfolio weights. It is shown that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a ..."
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Cited by 49 (3 self)
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We study the problem of determining the minimum cost of superreplicating a nonnegative contingent claim when there are convex constraints on the portfolio weights. It is shown that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a dominating claim, i.e., a claim whose payoffs are increased in an appropriate way relative to the original claim. The results hold for a wide variety of options, including standard European and American calls and puts, multiasset options, and some pathdependent options. We also provide a somewhat similar analysis when there are constraints on the gamma of the replicating portfolio. Constraints on portfolio amounts and constraints on number of shares of assets are also considered. Optimal Replication of Contingent Claims Under Portfolio Constraints 2 Since the pioneering option pricing work of Black and Scholes (1973) and Merton (1973), much research has focused on relaxing the assumptio...
Asymptotic Analysis for Optimal Investment and Consumption with Transaction Costs
 Finance Stoch
, 2004
"... We consider an agent who invests in a stock and a money market and consumes in order to maximize the utility of consumption over an infinite planning horizon in the presence of a proportional transaction cost λ>. The utility function is of the form U(c) = c 1−p /(1 − p) for p> 0, p � = 1. We ..."
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Cited by 46 (3 self)
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We consider an agent who invests in a stock and a money market and consumes in order to maximize the utility of consumption over an infinite planning horizon in the presence of a proportional transaction cost λ>. The utility function is of the form U(c) = c 1−p /(1 − p) for p> 0, p � = 1. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of λ 1/3, and we also obtain asymptotic results on the boundary of the “notrade” region.
A ClosedForm Solution to the Problem of SuperReplication Under Transaction Costs
, 1997
"... We study the problem of finding the minimal price needed to dominate Europeantype contingent claims under proportional transaction costs in a continuoustime diffusion model. The result we prove has already been known in special cases  the minimal superreplicating strategy is the least expensive b ..."
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Cited by 42 (2 self)
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We study the problem of finding the minimal price needed to dominate Europeantype contingent claims under proportional transaction costs in a continuoustime diffusion model. The result we prove has already been known in special cases  the minimal superreplicating strategy is the least expensive buyandhold strategy. Our contribution consists in showing that this result remains valid for general pathindependent claims, and in providing a shorter and more intuitive, financial mathematicstype proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation. Key words: transaction costs, superreplicating strategies, viscosity solutions. JEL classification: G11, G12. AMS 1991 subject classifications: Primary 90A09, 93E20, 60H30; secondary 60G44, 90A16. Research of the first author partially supported by NSF grant #DMS9503582. 1 Introductio...
Homogenization and asymptotics for small transaction costs: Multi dimensions,
, 2012
"... Abstract We consider the classical Merton problem of lifetime consumptionportfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form ..."
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Cited by 20 (6 self)
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Abstract We consider the classical Merton problem of lifetime consumptionportfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the onedimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from the homogenization theory and use the convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our accompanying paper [31] using the same approach.
Market Illiquidity as a Source of Model Risk in Dynamic Hedging
, 2000
"... In the present paper we study market illiquidity as a particular source of model risk in the hedging of derivatives. We depart from the usual BlackScholes framework, where it is assumed that option hedgers are small traders, and consider a model where the implementation of a hedging strategy aec ..."
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Cited by 15 (3 self)
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In the present paper we study market illiquidity as a particular source of model risk in the hedging of derivatives. We depart from the usual BlackScholes framework, where it is assumed that option hedgers are small traders, and consider a model where the implementation of a hedging strategy aects the price of the underlying security. We derive a formula for the feedbackeect of dynamic hedging on market volatility and present a formula for the hedging error due to market illiquidity. We go on and characterize perfect hedging strategies by a nonlinear version of the BlackScholes PDE. We relate this PDE to other models for the riskmanagement of derivatives under market frictions and present some simulations. Key words: Option hedging, illiquid markets, large trader models, feedbackeects, nonlinear BlackScholes equation 1 Introduction The recent turbulences on nancial markets and in particular the events surrounding the LTCMdebacle have made market liquidity an issue...
Option hedging for small investors under liquidity costs, Finance and Stochastics
, 2010
"... Abstract Following the framework of Ç etin, Jarrow and Protter ..."
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Cited by 15 (3 self)
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Abstract Following the framework of Ç etin, Jarrow and Protter
High order compact finite difference schemes for a nonlinear BlackScholes equation
, 2001
"... A nonlinear BlackScholes equation which models transaction costs arising in the hedging of portfolios is discretized semiimplicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite ..."
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Cited by 11 (0 self)
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A nonlinear BlackScholes equation which models transaction costs arising in the hedging of portfolios is discretized semiimplicitly using high order compact finite difference schemes. In particular, the compact schemes of Rigal are generalized. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and nonoscillatory properties and are generally more efficient than the considered classical schemes.
The dual optimizer for the growthoptimal portfolio under transaction costs. Finance Stoch
, 2011
"... Abstract. We consider the maximization of the longterm growth rate in the BlackScholes model under proportional transaction costs as in Taksar, Klass and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and MuhleKarbe [Ann. Appl. Probab., 20, 2010] for optimal consumption over an infini ..."
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Cited by 9 (4 self)
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Abstract. We consider the maximization of the longterm growth rate in the BlackScholes model under proportional transaction costs as in Taksar, Klass and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and MuhleKarbe [Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite horizon, we tackle this problem by determining a shadow price, which is the solution of the dual problem. It can be calculated explicitly up to determining the root of a deterministic function. This in turn allows to explicitly compute fractional Taylor expansions, both for the notrade region of the optimal strategy and for the optimal growth rate. 1.
Liquidity Risk and Option Pricing Theory
 in Handbook in Operation Research and Management Science: Financial Engineering
, 2007
"... This paper summarizes the recent advances of Çetin[6],Çetin,Jarrow and Protter [7], Çetin, Jarrow, Protter and Warachka [8], Blais [4], and Blais and Protter [5] on the inclusion of liquidity risk into option pricing theory. This research provides new insights into the relevance of the classical tec ..."
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Cited by 9 (3 self)
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This paper summarizes the recent advances of Çetin[6],Çetin,Jarrow and Protter [7], Çetin, Jarrow, Protter and Warachka [8], Blais [4], and Blais and Protter [5] on the inclusion of liquidity risk into option pricing theory. This research provides new insights into the relevance of the classical techniques used in continuous time finance for practical risk management. 1