Kasper Larsen

No arbitrage and the growth optimal portfolio

Abstract Recently, several papers have expressed an interest in applying the Growth Optimal Portfolio (GOP) for pricing derivatives. We show that the existence of a GOP is equivalent to the existence of a strictly positive martingale density. Our approach circumvents two assumptions usually set forth in the literature: 1) infinite expected growth rates are permitted and 2) the market does not need to admit an equivalent martingale measure. In particular, our approach shows that models featuring credit constrained arbitrage may still allow a GOP to exist because this type of arbitrage can be removed by a change of numéraire. However, if the GOP exists the market admits an equivalent martingale measure under some numéraire and hence derivatives can be priced. The structure of martingale densities is used to provide a new characterization of the GOP which emphasizes the relation to other methods of pricing in incomplete markets. The case where GOP denominated asset prices are strict supermartingales is analyzed in the case of pure jump driven uncertainty.

Continuity of Utility-Maximization with Respect to Preferences

This paper provides an easily verifiable regularity condition under which the investors utility maximizer depends continuously on the description of her preferences in a general incomplete financial setting. Specifically, we extend the setting of Jouini and Napp to include noise generated by a general continuous semi-martingale and to the case where the market price of risk process is allowed to be a general adapted process satisfying a mild integrability condition. This extension allows us to obtain positive results for both the mean-reversion model of Kim and Omberg and the stochastic volatility model of Heston. Finally, we provide an example set in Samuelsons complete financial model illustrating that without imposing additional regularity, the continuity property of the investors optimizer can fail.

On utility maximization under convex portfolio constraints

We consider a utility-maximization problem in a general semimartingale financial model, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin; that is, it may be inadmissible for an investor to hold no risky investment at all. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present. Our main result establishes the existence of optimal trading strategies in such models under no smoothness requirements on the utility function. The result also shows that, up to attainment, the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.

Equilibrium in securities markets with heterogeneous investors and unspanned income risk

In a finite time horizon, incomplete market, continuous-time setting with dividends and investor incomes governed by arithmetic Brownian motions, we derive closed-form solutions for the equilibrium risk-free rate and stock price for an economy with finitely many heterogeneous CARA investors and unspanned income risk. In equilibrium, the Sharpe ratio is the same as in an otherwise identical complete market economy, whereas the risk-free rate is lower and, consequently, the stock price is higher. The reduction in the risk-free rate is highest when the more risk-averse investors face the largest unspanned income risk.

Optimal portfolio delegation when parties have different coefficients of risk aversion

We consider the problem of delegated portfolio management when the involved parties are risk-averse. The agent invests the principals money in the financial market, and in return he receives a compensation which depends on the value that he generates over some period of time. We use a dual approach to explicitly solve the agents problem analytically and subsequently we use this solution to solve the principals problem numerically. The interaction between the principals and the agents risk aversion and the optimal compensation scheme is studied and, for example, in the case of the more risk averse agent according to common folklore the principal should optimally choose a fee schedule such that the agents derived risk aversion decreases. We illustrate that this is not always the case.

Taylor approximation of incomplete Radner equilibrium models

In the setting of exponential investors and uncertainty governed by Brownian motions, we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential–quadratic models and prove that the corresponding incomplete equilibrium is characterized by a coupled set of Riccati equations. Finally, we prove that these exponential–quadratic models can be used to approximate the incomplete models we studied in the first part.

Horizon dependence of utility optimizers in incomplete models

This paper studies the utility maximization problem with changing time horizons in the incomplete Brownian setting. We first show that the primal value function and the optimal terminal wealth are continuous with respect to the time horizon T. Secondly, we exemplify that the expected utility stemming from applying the T-horizon optimizer on a shorter time horizon S<T may fail to converge to the T-horizon value as S↑T. Finally, we provide necessary and sufficient conditions preventing the existence of this phenomenon.

A note on the existence of the power investor's optimizer

Karatzas et al. (SIAM J. Control Optim. 29:707–730, 1991) ensure the existence of the expected utility maximizer for investors with constant relative risk aversion coefficients less than one. In this note, we explain a simple trick that allows us to use this result to provide the existence of utility maximizers for arbitrary coefficients of relative risk aversion. The simplicity of our approach is to be contrasted with the general existence result provided in Kramkov and Schachermayer (Ann. Appl. Probab. 9:904–950, 1999).