Preferences, Levy Jumps and Option Pricing
Dr Chenghu Ma, Department of AFM, University of Essex

Abstract:
This paper develops a class of equilibrium asset pricing models in continuous-time Lucas (1978) exchange economy. It distinguishes the existing literature into two parts: (a) the representative agent's preference over the life-time consumption programs is assumed to be represented by the so-called intertemporal recursive utility function formulated by Ma (2000), which generalizes that of Duffie and Epstein (1992a) by allowing non-expected utility specifications; (b) the uncertainty is Markovian with state variables driven by a Levy jump process that contains both Brownian and Poisson uncertainties. By introducing the so-called pseudo-state process, which is uniquely determined by the original state process and the representative agent's utility function, we were able to express the equilibrium price of a security as the expected net present value of its dividend streams under the original probability measure. This resolves the technical problem in identifying the risky neutral probability measure(s) in incomplete market economies as is the case in the presence of Levy jumps. The existence of equilibrium security prices as solutions to the Euler equation for the agent's optimal consumption and portfolio choices is proved. For a particular parameterisation of the economy, a closed-form formula for the European call options, as well as for other derivative securities on the aggregate equity, is derived and analyzed. Two applications are carried out.

The first application involves a derivation of an equilibrium option pricing formula, and a formula for pricing all other derivative securities on the aggregate equity. This is done for a particular parametric specifications of the agent's utility function and state variables. The set-up is the same as in Merton (1976) and Naik and Lee (1990) except that we allow general homothetic recursive utility function, and we do not restrict the jump sizes to follow a log-normal distribution as assumed by the others. In such a set-up, the utility specifications become relevant for option pricing, as well as for pricing other derivative securities. The relevant option pricing formula is expressed in terms of the Laplace inverse transformation of a complex function Phi. Preference parameters and other aggregate factors of the economy affecting the option prices are explicitly reflected through the Phi-function. The derived option pricing formula is found to be attractive not only because of its mathematical simplicity, but also because of its generality. For example, it is shown that this formula generalises those of Black-Scholes, Naik-Lee, Cox-Ross and Merton in two ways: First, the magnitude of the jump may follow any distribution with finite moments; second, the utility function is recursive but is not necessarily an intertemporal additive von-Neumann Morgenstern utility function. Nevertheless, for the special cases when the market becomes complete, the equilibrium conditions will lead to the same formulation for security prices as what is implied by imposing purely the no-arbitrage conditions.

As another application of the derived equilibrium asset pricing model, we re-examine Kocherlakota's (1990) observational equivalence between recursive utility functions and expected discounted utility functions in a continuous-time setting. Recursive utility function is of special interest because, in contrast to the intertemporal additive von Neumann Morgenstern utility, the recursive utility function of Epstein and Zin (1989) permits a degree of separation between risk aversion and intertemporal substitution, and agent's attitudes toward the timing of uncertainty resolution can be also captured by the recursive utility formulation. We ask: (1) how utility specifications may affect the equilibrium behaviour of security prices; and, conversely, (2) how equilibrium security prices can convey information about representative agent's preferences. For example, can we distinguish between the non-expected recursive utility function of Epstein and Zin (1989) and the traditional expected additive utility function from the security prices?

Progress has been made in the research towards some deeper understanding of these issues. Following Duffie and Epstein (1992b), Kocherlakota's observation equivalence between expected and non-expected utility functions prevails in the pure Brownian economy even when it is not i.i.d. This is because the M-function, which characterises the risk attitude towards uncertainty in continuous time, does not enter into their formulation of recursive utility. This is equivalent to assume expected utility certainty equivalent. This paper considers mixed Poisson-Brownian uncertainty. It is shown that the betweenness recursive utility function and expected utility functions are observational distinguishable even in an i.i.d. economy. Similar to Ma (1998 b) in discrete time, it is found that the prices of European call options contain the most relevant information on agent's utility function in continuous time. The observational non-equivalence finding reported here is based on the closed-form formula for pricing European call options on the aggregate equity in the presence of Levy jumps as mentioned above.