General indifference pricing with small transaction costs

We study the utility indifference price of a European option in the context of small transaction costs. Considering the general setup allowing consumption and a general utility function at final time T, we obtain an asymptotic expansion of the utility indifference price as a function of the asymptotic expansions of the utility maximization problems with and without the European contingent claim. We use the tools developed in [54] and [48] based on homogenization and viscosity solutions to characterize these expansions. Finally we study more precisely the example of exponential utilities, in particular recovering under weaker assumptions the results of [6].Comment: 43 page

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 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

Option Pricing and Hedging with Small Transaction Costs

An investor with constant absolute risk aversion trades a risky asset with general It\^o-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.Comment: 20 pages, to appear in "Mathematical Finance

Utility indifference pricing with market incompleteness

Utility indifference pricing and hedging theory is presented, showing how it leads to linear or to non-linear pricing rules for contingent claims. Convex duality is first used to derive probabilistic representations for exponential utility-based prices, in a general setting with locally bounded semi-martingale price processes. The indifference price for a finite number of claims gives a non-linear pricing rule, which reduces to a linear pricing rule as the number of claims tends to zero, resulting in the so-called marginal utility-based price of the claim. Applications to basis risk models with lognormal price processes, under full and partial information scenarios are then worked out in detail. In the full information case, a claim on a non-traded asset is priced and hedged using a correlated traded asset. The resulting hedge requires knowledge of the drift parameters of the asset price processes, which are very difficult to estimate with any precision. This leads naturally to a further application, a partial information problem, with the drift parameters assumed to be random variables whose values are revealed to the hedger in a Bayesian fashion via a filtering algorithm. The indifference price is given by the solution to a non-linear PDE, reducing to a linear PDE for the marginal price when the number of claims becomes infinitesimally small

Utility based pricing and hedging of jump diffusion processes with a view to applications

We discuss utility based pricing and hedging of jump diffusion processes with emphasis on the practical applicability of the framework. We point out two difficulties that seem to limit this applicability, namely drift dependence and essential risk aversion independence. We suggest to solve these by a re-interpretation of the framework. This leads to the notion of an implied drift. We also present a heuristic derivation of the marginal indifference price and the marginal optimal hedge that might be useful in numerical computations.Comment: 23 pages, v2: publishe