Option pricing with transaction costs using a Markov chain approximation

Abstract

An e cient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, UK). A fair option price is determined by requiring that an inÿnitesimal diversion of funds into the purchase or sale of options has a neutral e ect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payo into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely speciÿed option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general deÿnition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed. ?

On Transaction-Cost Models in Continuous-Time Markets

“…Initially, proportional costs were added, and later costs that had a fixed and a proportional part (still linear though) were considered. In relation to the addition of proportional costs, we can find among others the work initially of Merton [1] and later of Monoyios [2]. Whalley and Willmot [3] included small transaction costs, while Damgaard [4,5] and Kocinski [6], followed a more global approach.…”

“…α is the instantaneous conditional expected change in price per unit time, 2 σ is the instantaneous conditional variance per unit time and W is a Brownian Motion. Moreover we assume the existence of a riskless asset that has a rate of return …”

Time-Consistent Actuarial Valuations

“…This method is frequently used to price path dependent derivatives such as American options, barrier options, etc. See for example Duan et al (2003) and Monoyios (2004).…”

Dynamic Portfolio Optimization with Transaction Costs and State-Dependent Drift

“…where H(t, s, y) is defined by H(t, s, y) := V (t, s, 0, y), see, e.g., Davis et al (1993) or Monoyios (2004). This representation allows reducing the dimension of the optimization problem by one.…”

“…Davis et al (1993) proposes a backward recursive method which has seen a number of improvements in the past 20 years. For instance, Monoyios (2004) provides an approximation to the optimal decision in the final period which allows searching over a smaller range of stock holdings. Zakamouline (2005) proposes bounds on stock holdings.…”