Thaleia Zariphopoulou

European option pricing with transaction costs

The authors consider the problem of pricing European options in a market model similar to the Black–Scholes one, except that proportional transaction charges are levied on all sales and purchases o...

A solution approach to valuation with unhedgeable risks

Abstract. We study a class of stochastic optimization models of expected utility in markets with stochastically changing investment opportunities. The prices of the primitive assets are modelled as diffusion processes whose coefficients evolve according to correlated diffusion factors. Under certain assumptions on the individual preferences, we are able to produce reduced form solutions. Employing a power transformation, we express the value function in terms of the solution of a linear parabolic equation, with the power exponent depending only on the coefficients of correlation and risk aversion. This reduction facilitates considerably the study of the value function and the characterization of the optimal hedging demand. The new results demonstrate an interesting connection with valuation techniques using stochastic differential utilities and also, with distorted measures in a dynamic setting.

An example of indifference prices under exponential preferences

Abstract.The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivative’s payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy.

Hedging in incomplete markets with HARA utility

In the context of Merton’s original problem of optimal consumption and portfolio choice in continuous time, this paper solves an extension in which the investor is endowed with a stochastic income that cannot be replicated by trading the available securities. The problem is treated by demonstrating, using analytic and, in particular, ‘viscosity solutions’ techniques, that the value function of the stochastic control problem is a smooth solution of the associated Hamilton-Jacobi-Bellman (HJB) equation. The optimal policy is shown to exist and given in a feedback form from the optimality conditions in the HJB equation. At zero wealth, a fixed fraction of income is consumed. For ‘large’ wealth, the original Merton policy is approached. We also give a sufficient condition for wealth, under the optimal policy, to remain strictly positive.

Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences

Abstract. Analytic bounds on the reservation write price of European-style contingent claims are derived in the presence of proportional transaction costs in a model which allows for intermediate trading. The option prices are obtained via a utility maximization approach by comparing the maximized utilities with and without the contingent claim. The mathematical tools come mainly from the theories of singular stochastic control and viscosity solutions of nonlinear partial differential equations.

Consumption-Investment Models with Constraints

The paper examines a general investment and consumption problem for a single agent who consumes and invests in a riskless asset and a risky one. The objective is to maximize the total expected discounted utility of consumption. Trading constraints, limited borrowing, and no bankruptcy are binding, and the optimization problem is formulated as a stochastic control problem with state and control constraints. It is shown that the value function is the unique smooth the associated Hamilton-Jacobi-Bellman equation and the optimal consumption and portfolios are provided in feedback form.

Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

We introduce an expected utility approach to price insurance risks in a dynamic financial market setting. The valuation method is based on comparing the maximal expected utility functions with and without incorporating the insurance product, as in the classical principle of equivalent utility. The pricing mechanism relies heavily on risk preferences and yields two reservation prices - one each for the underwriter and buyer of the contract. The framework is rather general and applies to a number of applications that we extensively analyze.

Investment-consumption models with transaction fees and Markov-chain parameters

This paper considers an infinite horizon investment-consumption model in which a single agent consumes and distributes his wealth in two assets, a bond and a stock. The problem of maximization of the total utility from consumption is treated. State (amount allocated in assets) and control (consumption, rates of trading) constraints are present. It is shown that the value function is the unique viscosity solution of a system of variational inequalities with gradient constraints.

An optimal investment/consumption model with borrowing

This paper considers a consumption and investment decision problem for a single agent. Wealth is divided between a riskless asset and a risky asset with logarithmic Brownian motion price fluctuations. Short-selling is not allowed, but borrowing is allowed at rate exceeding the rate of return on the riskless asset. An explicit solution of the dynamic programming differential equation for the maximum total discounted expected utility function U is available only in the HARA case. However, using viscosity solution methods the asymptotic behavior of the value function v ( x ) is found for small wealth x and for large wealth x .

Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random

This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics. The derivatives to be priced are of European type with the payoff depending on both the stock and the volatility. The valuation approach uses utility-based criteria under the assumption of exponential risk preferences. This methodology yields the indifference prices as solutions to second order quasilinear PDEs. Two sets of price bounds are derived that highlight the important ingredients of the utility approach, namely, nonlinear pricing rules with dynamic certainty equivalent characteristics, and pricing measures depending on correlation and the Sharpe ratio of the traded asset. The problem is further analyzed by asymptotic methods in the limit of the volatility being a fast mean-reverting process. The analysis relates the traditional market-selected volatility risk premium approach and the preference-based valuation techniques.