Discrete time models of portfolio optimisation and option pricing are studied under the
effects of proportional transaction costs. In a multi-period portfolio selection problem, an
investor maximises expected utility of terminal wealth by rebalancing the portfolio between
a risk-free 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 no-transaction 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 two-stage perturbation method is developed to derive approximate solutions
for the exponential and power utility functions. The first stage involves ignoring the no-transaction
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.