Abstract:

In this dissertation, we examine the properties of the Constant Elasticity of Variance model in the context of optimal investment problems. We begin by deriving the solution to the one stock problem, examining the properties of the solution by comparing it to the results from Merton [34] and those of a stochastic volatility model. The resulting solution for the CEV model has two parts: a moving Merton ratio and a correction factor. The correction factor depends only on time to maturity and is less than or equal to one. It implies that starting out, the optimal strategy is to invest a small amount of wealth in the risky asset, steadily increasing the amount invested as maturity nears. From examination of simulated paths, we note that the CEV model for long time periods is problematic given the strong trend of declining volatility.

Next, we extend the problem to the multi-asset case. In this case, we are unable to find explicit solutions and thus solve the resulting partial differential equation for the value function using numerical methods. We do this for the two stock case and compare the results with Merton's fixed-mix solution. In addition, we examine a two dimensional form of the stochastic volatility problem. However, we find that we must impose strict restrictions on the drift and diffusion coefficients in order to obtain explicit solutions of the kind proposed by Liu [28].

Finally, we establish some motivation for using a negative elasticity parameter by estimating the parameters of the CEV model using historical stock returns and options data. We find that both data sets imply a fairly negative elasticity parameter. We finish by proposing the Long-Term CEV model which corrects for the problem of declining volatility in the regular CEV model. We do so by making the parameter, ?, a separate, uncorrelated stochastic process which grows over time. This new model generates more realistic paths but also no longer allows for explicit solutions. Using asymptotic approximations we find the optimal portfolio is similar to that of the single stock CEV problem.