General indifference pricing with small transaction costs

We study the utility indifference price of a European option in the context of small transaction costs. Considering the general setup allowing consumption and a general utility function at final time T, we obtain an asymptotic expansion of the utility indifference price as a function of the asymptotic expansions of the utility maximization problems with and without the European contingent claim. We use the tools developed in [54] and [48] based on homogenization and viscosity solutions to characterize these expansions. Finally we study more precisely the example of exponential utilities, in particular recovering under weaker assumptions the results of [6].

[1]  Option pricing by large risk aversion utility¶under transaction costs , 2001 .

[2]  Jakša Cvitanić,et al.  HEDGING AND PORTFOLIO OPTIMIZATION UNDER TRANSACTION COSTS: A MARTINGALE APPROACH12 , 1996 .

[3]  H. Soner,et al.  Optimal Investment and Consumption with Transaction Costs , 1994 .

[4]  Bruno Bouchard,et al.  No-arbitrage in Discrete-time Markets with Proportional Transaction Costs and General Information structure , 2005, Finance Stochastics.

[5]  H. Soner,et al.  A free boundary problem related to singular stochastic control: the parabolic case , 1991 .

[6]  Mark H. A. Davis,et al.  A note on super-replicating strategies , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[7]  Dawn Hunter,et al.  Penalty methods for continuous-time portfolio selection with proportional transaction costs , 2010 .

[8]  Steven E. Shreve,et al.  Futures trading with transaction costs , 2010 .

[9]  B. Bouchard,et al.  Explicit solution to the multivariate super-replication problem under transaction costs , 2000 .

[10]  Y. Kabanov,et al.  Hedging of Contingent Claims under Transaction Costs , 2002 .

[11]  G. Constantinides,et al.  Portfolio selection with transactions costs , 1976 .

[12]  Daniel N. Ostrov,et al.  Balancing Small Transaction Costs with Loss of Optimal Allocation in Dynamic Stock Trading Strategies , 2010, SIAM J. Appl. Math..

[13]  Steven E. Shreve,et al.  Asymptotic analysis for optimal investment and consumption with transaction costs , 2004, Finance Stochastics.

[14]  Min Dai,et al.  Penalty Methods for Continuous-Time Portfolio Selection with Proportional Transaction Costs , 2008 .

[15]  G. Constantinides.  Capital Market Equilibrium with Transaction Costs , 1986, Journal of Political Economy.

[16]  D. Liu.  European Option Pricing with Transaction Costs , 2000 .

[17]  J. Kallsen,et al.  Portfolio Optimization under Small Transaction Costs: a Convex Duality Approach , 2013, 1309.3479.

[18]  Maxim Bichuch,et al.  Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment , 2011, Finance Stochastics.

[19]  Y. Kabanov,et al.  Hedging under Transaction Costs in Currency Markets: a Continuous‐Time Model , 2002 .

[20]  Steven E. Shreve,et al.  Utility Maximization Trading Two Futures with Transaction Costs , 2013, SIAM J. Financial Math..

[21]  Johannes Muhle-Karbe,et al.  Portfolio Selection with Small Transaction Costs and Binding Portfolio Constraints , 2012, SIAM J. Financial Math..

[22]  N. Touzi,et al.  On Super-Replication in Discrete Time under Transaction Costs , 2001 .

[23]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

[24]  P. Wilmott,et al.  An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs , 1997 .

[25]  Colin Atkinson,et al.  Multi‐asset portfolio optimization with transaction cost , 2004 .

[26]  M. Akian,et al.  On an Investment-Consumption Model With Transaction Costs , 1996 .

[27]  M. Akian,et al.  A finite horizon multidimensional portfolio selection problem with singular transactions , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[28]  H. Pham,et al.  Hedging in discrete time under transaction costs and continuous-time limit , 1999 .

[29]  Nizar Touzi,et al.  Large liquidity expansion of super-hedging costs , 2012, Asymptot. Anal..

[30]  Johannes Muhle-Karbe,et al.  Option Pricing and Hedging with Small Transaction Costs , 2012 .

[31]  H. Soner,et al.  There is no nontrivial hedging portfolio for option pricing with transaction costs , 1995 .

[32]  H. Soner,et al.  Regularity of the value function for a two-dimensional singular stochastic control problem , 1989 .

[33]  E. Jouini,et al.  Martingales and Arbitrage in Securities Markets with Transaction Costs , 1995 .

[34]  Min Dai,et al.  Finite Horizon Optimal Investment and Consumption with Transaction Costs , 2009, SIAM J. Control. Optim..

[35]  Nizar Touzi.  Super-replication under proportional transaction costs: From discrete to continuous-time models , 1999, Math. Methods Oper. Res..

[36]  H. Soner,et al.  TRADING WITH SMALL PRICE IMPACT , 2017 .

[37]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[38]  Min Dai,et al.  Characterization of Optimal Strategy for Multiasset Investment and Consumption with Transaction Costs , 2013, SIAM J. Financial Math..

[39]  Yuri Kabanov,et al.  Hedging and liquidation under transaction costs in currency markets , 1999, Finance Stochastics.

[40]  Nizar Touzi,et al.  Homogenization and Asymptotics for Small Transaction Costs , 2012, SIAM J. Control. Optim..

[41]  Steven E. Shreve,et al.  A free boundary problem related to singular stochastic control , 1990 .

[42]  Bruno Bouchard,et al.  Optimal Control under Stochastic Target Constraints , 2009, SIAM J. Control. Optim..

[43]  A. Skorohod,et al.  On the possibility of hedging options in the presence of transaction costs , 1997 .

[44]  Guy Barles,et al.  Option pricing with transaction costs and a nonlinear Black-Scholes equation , 1998, Finance Stochastics.

[45]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[46]  Bruno Bouchard.  Utility maximization on the real line under proportional transaction costs , 2002, Finance Stochastics.

[47]  Maxim Bichuch.  Asymptotic Analysis for Optimal Investment in Finite Time with Transaction Costs , 2011, SIAM J. Financial Math..

[48]  B. Bouchard.  Option Pricing via Utility Maximization in the presence of Transaction Costs: an Asymptotic Analysis , 2000 .

[49]  Bruno Bouchard,et al.  Hedging Under an Expected Loss Constraint with Small Transaction Costs , 2016, SIAM J. Financial Math..

[50]  B. Dumas,et al.  An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs , 1991 .

[51]  Walter Schachermayer,et al.  A super-replication theorem in Kabanov’s model of transaction costs , 2006, Finance Stochastics.

[52]  H. Soner,et al.  National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No . 845 Homogenization and Asymptotics for Small Transaction Costs : The Multidimensional Case , 2013 .

[53]  Huyên Pham,et al.  A closed-form solution to the problem of super-replication under transaction costs , 1999, Finance Stochastics.

[54]  J. Muhle‐Karbe,et al.  THE GENERAL STRUCTURE OF OPTIMAL INVESTMENT AND CONSUMPTION WITH SMALL TRANSACTION COSTS , 2013, 1303.3148.

[55]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[56]  Michael J. Klass,et al.  A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees , 1988, Math. Oper. Res..

[57]  Halil Mete Soner,et al.  Asymptotics for fixed transaction costs , 2015, Finance Stochastics.

[58]  W. Schachermayer,et al.  Asymptotics and duality for the Davis and Norman problem , 2010, 1010.0627.

[59]  M. Dai,et al.  Finite-Horizon Optimal Investment with Transaction Costs: A Parabolic Double Obstacle Problem , 2006 .

[60]  Y. Kabanov,et al.  Markets with transaction costs , 2010 .