An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs

Abstract

Davis, Panas, and Zariphopoulou (1993) and Hodges and Neuberger (1989) have presented a very appealing model for pricing European options in the presence of rehedging transaction costs. In their papers the 'maximization of utility' leads to a hedging strategy and an option value. The latter is different from the Black-Scholes fair value and is given by the solution of a three-dimensional free boundary problem. This problem is computationally very time-consuming. In this paper we analyze this problem in the realistic case of small transaction costs, applying simple ideas of asymptotic analysis. The problem is then reduced to an inhomogeneous diffusion equation in only two independent variables, the asset price and time. The advantages of this approach are to increase the speed at which the optimal hedging strategy is calculated and to add insight generally. Indeed, we find a very simple analytical expression for the hedging strategy involving the option's gamma. Copyright Blackwell Publishers Inc. 1997.

Option pricing with transaction costs using a Markov chain approximation

“…The dashed curves indicate the region in which the hedging portfolio is not rebalanced, whilst the solid curve is the BlackScholes delta hedging strategy. The replacement of the unique Black-Scholes delta by a hedging bandwidth is in accordance with intuition and with previous results on optimal hedging under transaction costs, such as Hodges and Neuberger (1989) and (for the limiting case of small transaction costs) Whalley and Wilmott (1997).…”

Analytical approximation of the transition density in a local volatility model

“…The application of asymptotic expansions and singular perturbation methods in mathematical finance has been studied by several authors: one of the first results was obtained by Whalley and Wilmott [31] in the study of transaction costs; in the seminal paper [18] Hagan and Woodward derived an asymptotic expansion formula for the implied volatility in a local volatility (LV) model where the volatility function can be written as the product of two independent functions of time and underlying, the prototype example being the constant elasticity of variance (CEV) model; the approximation of more general one-dimensional LV models, using different deterministic and probabilistic techniques, was studied among others by Howison [22], Widdicks, Duck, Andricopoulos and Newton [32], Capriotti [5], Gatheral, Hsu, Laurence, Ouyang and Wang [15], Benhamou, Gobet and Miri [3]. A further different approach based on heat kernel expansion and the parametrix method was proposed by Corielli, Foschi and Pascucci [7], Cheng, Constantinescu, Costanzino, Mazzucato and Nistor [6], Kristensen and Mele [24].…”

(S,s)-adjustment Strategies and Hedging under Markovian Dynamics

“…There are two main approaches in the literature taking the effects of transaction costs into account: the first considers discrete portfolio adjustments, where the time step of portfolio rebalancing is exogenously given, while the second considers traders as continuously monitoring the price of the underlying asset, although adjusting their portfolio only if the gain from adjustment is larger than the cost of adjustment. This latter approach can be subdivided into two further approaches: the first is called local in time (Leland, 1985;Hoggard et al, 1994), whereas the second is called global in time (Davis et al, 1993;Whalley and Wilmott, 1997;Zariphopoulou, 1999, 2001). The former is a nonoptimizing approach, where rehedging is based on minimizing the variance of the hedged portfolio, while the latter is an optimizing one and is based on utility maximization and stochastic optimal control.…”

“…In this framework, Davis et al (1993) consider European option pricing with proportional transaction costs charged on sales and purchases of stock. Whalley and Wilmott (1997) provide an asymptotic analysis of Davis et al (1993) in the limit of small transaction costs. Zariphopoulou (1999, 2001) derive in closed form bounds to the reservation price of a call option for a large class of utility functions.…”