European Option Pricing with Transaction Costs

Abstract

This paper is dedicated to Wendell Fleming on the occasion of his 65 th birthday.

Singular Stochastic Control in Option Hedging with Transaction Costs

“…Davis et al [14] derived the Hamilton-Jacobi-Bellman (HJB) equations associated with the control problems V(t, S, x, y) and H(t, S, x)=1 − V(t, S, x,0) in the same way as we did in Section 2.1. By applying the transformation h(s, z, x)=H(t, S, x) to their HJB equations, we obtain the following free boundary problem (FBP) for h(s, z, x): Optimal buy (lower) and sell (upper) boundaries from negative exponential (CARA) utility maximization for a short asset-settled call with strike price K = 20 (solid black lines) and for the case of no option (solid red lines), with proportional transaction costs incurred at the rate of λ = μ = 0.5%, absolute risk aversion γ = 2.0, risk-free rate r = 8.5%, asset return rate α = 10% and asset volatility σ = 5%, at 1.5, 0.5, 0.25 and 0.1 period(s) from expiration T. For each pair of boundaries, the "buy asset" region is below the buy boundary and the "sell asset" region is above the sell boundary; the no-transaction region is between the two boundaries.…”

“…Associated with each of the problems V i (t, S, x, y), i = 0 given by (14) corresponding to an investor with no option position and i = s, b being analogs of (14) corresponding to an investor with a short or long call position, respectively, is an optimal trading strategy x i t (i = 0, s, b) of the form…”

“…If the investor is presented with an opportunity to enter into a position in a European call option written on the given asset, with strike price K and expiration date T, the problem can be formulated in the same way as (14) but with Ω 0 T replaced by Ω i T with i = s indicating a short call position and i = b indicating a long call position. The corresponding value functions V i (t, S, x, y) also admit reductions in dimensionality via H i (t, S, x)=1 − V i (t, S, x,0), but with Z 0 (S T , x T ) in (15) replaced by Z i (S T , x T ).…”

“…The nature of the optimal hedge is that an investor with an option position should rebalance his portfolio only when the number of units of the asset falls "too far" out of line. For the negative exponential utility function, Davis et al [14], Clewlow & Hodges [11], and Zakamouline [20] have developed numerical methods to compute the optimal hedge by making use of discrete-time dynamic programming on an approximating binomial tree for the asset price; see Kushner & Dupuis [3] for the general theory of Markov chain approximations for continuous-time processes and their use in the numerical solution of optimal stopping and control problems. More recently, Lai & Lim [18] introduced a new numerical method for solving the singular stochastic control problems associated with utility maximization, yielding a much simpler algorithm to compute the buy and sell boundaries and value functions in the utility-based approach.…”

Utility valuation of multi-name credit derivatives and application to CDOs

“…We refer to [22,23] and [7] as classical references in this area. The indifference approach was initiated for European claims by Hodges and Neuberger [19] and further extended by Davis et al [9].…”