Title:

Portfolio optimisation and option pricing in discrete time with transaction costs

Discrete time models of portfolio optimisation and option pricing are studied under the effects of proportional transaction costs. In a multiperiod portfolio selection problem, an investor maximises expected utility of terminal wealth by rebalancing the portfolio between a riskfree and risky asset at the start of each time period. A general class of probability distributions is assumed for the returns of the risky asset. The optimal strategy involves trading to reach the boundaries of a notransaction region if the investor’s holdings in the risky asset fall outside this region. Dynamic programming is applied to determine the optimal strategy, but it can be computationally intensive. In the limit of small transaction costs, a twostage perturbation method is developed to derive approximate solutions for the exponential and power utility functions. The first stage involves ignoring the notransaction region and transacting to the optimal point corresponding to the zero transaction costs case. Approximations of the resulting suboptimal value functions are obtained. In the second stage, these suboptimal value functions are corrected to obtain approximations of the optimal value functions and optimal boundaries at all time steps. A discrete time option pricing model is developed based on the utility maximisation approach. This model reduces to the binomial model in the special case where the risky asset follows a binomial price process without transaction costs. Incorporating transaction costs, the utility indifference price and marginal utility indifference price of the option are observed to depend on the price of the underlying risky asset and the investor’s holdings in the risky asset. The regions where these option prices do not vary with the investor’s holdings in the risky asset are identified. An example illustrates how utility indifference pricing or marginal utility indifference pricing enables one to determine the bid and ask price of a European call option.
