We consider the problem faced by an investor who must liquidate a given basket of assets over a finite time horizon. The investor's goal is to maximize the expected utility of the sales revenues over a class of adaptive strategies. We assume that the investor's utility has constant absolute risk aversion (CARA) and that the asset prices are given by a very general continuous-time, multiasset price impact model. Our main result is that (perhaps surprisingly) the investor does no worse if he narrows his search to deterministic strategies. In the case where the asset prices are given by an extension of the nonlinear price impact model of Almgren [(2003) Applied Mathematical Finance, 10, pp. 1-18], we characterize the unique optimal strategy via the solution of a Hamilton equation and the value function via a nonlinear partial differential equation with singular initial condition.Market impact modelling, illiquid markets, optimal liquidation, optimal trade execution, algorithmic trading, utility maximization, Hamilton-Jacobi-Bellman equation, finite fuel control,
In this note we prove H older-type inequalities for products of certain functionals of correlated Brownian motions. These estimates are applied to the study of optimal portfolio choice in incomplete markets when the investor's utility is of the form U (X; Y ) = g(X )h(Y ), where X is the investor's wealth and Y is a random factor not perfectly correlated with the market. Explicit solutions are found when g is the exponential, power, or logarithmic utility function.
This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case when this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions.These results allow us to construct a large family of the aforementioned optimality criteria, including some closed form examples in relevant financial models.
This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.
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