Numerical approximations of optimal portfolios in mispriced asymmetric Lévy markets

Highlights

• Optimal portfolios in mispriced asymmetric Levy markets are presented.

• The model is applied to the Kou jump-diffusion and VG markets.

• Analytic formulas are derived for derivatives of the portfolio objective function.

• Optimal portfolios are simulated via Newton’s method and incomplete Beta functions.

• Optimal portfolios are presented with levels of information asymmetry and horizon.

Abstract

We present numerical approximations of optimal portfolios in mispriced Lévy markets under asymmetric information for informed and uninformed investors having logarithmic preference. We apply our numerical scheme to Kou (2002) jump-diffusion markets by deriving analytic formulas for the first two derivatives of the underlying portfolio objective function which depend only on the Lévy measure of the jump-generating process. Optimal portfolios are then simulated using the Box–Muller algorithm, Newton’s method and incomplete Beta functions. Convergence dynamics and trajectories of sample paths of optimal portfolios for both investors are presented at different levels of information asymmetry, mispricing, horizon, asymmetry in the Kou density, jump intensity, volatility, mean-reversion speed, and Sharpe ratios. We also apply the proposed Newton’s algorithm to compute optimal portfolios for investors in Variance Gamma markets via instantaneous centralized moments of returns.

Introduction

Asymmetric information models assume that there are two types of investors in the market—informed and uninformed. The informed investor trades because he or she has non-public informational advantage, while the uninformed is a liquidity trader or hedger without an information advantage. The informed investor partially reveals information to the uninformed investor through trades. Kelly and Ljungqvist (2012) and Easley and O’Hara (2004) provide evidence of the importance of information asymmetry in asset pricing. They find that prices and demand for the risky asset by uninformed investors fall as information asymmetry increases. Their results confirm that information asymmetry is priced, and imply that liquidity is a primary channel that links asymmetry to prices.

Mispricing¹ is the difference between the asset price and its fundamental value. It is modeled by a mean-reverting Orstein–Uhlenbeck process. Because of the exponential decay associated with mean-reverting processes, it has an associated half-life which is inversely proportional to its mean-reversion rate. Half-life is a measure of the slowness of a mean-reverting process, and is measured in trading days. It may be defined as the average time it takes the mean-reverting process to get pulled half-way back to its long-term mean. In this context, half-life gives an estimate of how long we should expect the mispricing to remain far from zero. Thus, a half-life of 10 days means that it takes 20 trading days on average for the mispricing to revert to zero. We define mean-reversion time as the inverse of the mean-reversion rate, and use it as an equivalent measure of mispricing. In other words, mean-reversion speed or mean-reversion time is a proxy for mispricing. In this paper, we combine the concepts of asymmetric information, mispricing, and mean-reversion, and apply them to the Kou (2002) jump-diffusion and Variance Gamma (VG) markets to study the impact on the optimal demand for the risky asset by investors having logarithmic preference.

Portfolio allocation problems² have been extensively studied in various settings since the seminal work of Markowitz (1952). Recently, Buckley et al. (2014) show that optimal portfolios satisfy some nonlinear stochastic equations for both informed and uninformed investors in the asymmetric and mispriced Lévy markets. Usually there are no analytic solutions to these nonlinear equations. Numerical and analytic approximations are therefore critical for portfolio allocation problems. We solve our problem by presenting a simple but highly efficient approximation to our asset allocation problem in the asymmetric and mispriced Kou and VG market setting. In particular, we apply the theory presented in the recent studies by Buckley et al. (2012), Buckley et al. (2014) and Buckley et al. (2015) to compute numerical approximations of the optimal portfolios with the aid of Newton’s method, the Box–Muller algorithm, and the incomplete Beta functions.

Kou (2002) jump-diffusion model deals with a finite activity process in the sense that only a small number of jumps occur per unit of time. Empirical tests in Ramenzani and Zeng (2007) demonstrate that Kou’s double exponential jump-diffusion model fits stock data better than Merton’s normal jump-diffusion model. It is also very tractable, and therefore, is a natural candidate to apply our general numerical scheme. For the Kou markets, we derive analytic formulas for the first two derivatives of the underlying portfolio objective function presented in Buckley et al. (2014) which depends only on the Lévy measure of the process generating the jumps. Using the parameters in Kou (2002) and Ramenzani and Zeng (2007), we then implement our model by simulating paths of optimal portfolios for each investor at various levels of mispricing, information asymmetry, investment horizon, volatility, Sharpe ratio, jump probability, and jump size.

The Variance Gamma (VG) model introduced by Madan and Seneta (1990) and Madan, Carr, and Chang (1998) has found extensive applications in modeling stock returns, option pricing and structural credit risk models. It is of infinite activity, that is, there are infinitely many arrivals of small jumps per unit of time. We also employ our theory to the Variance Gamma (VG) Lévy markets to examine how asset mispricing and information asymmetry affect the optimal portfolio and maximum expected utilities of investors in this market. Cvitanic et al. (2008) report that jumps in asset prices lead to higher moments that must be captured when modeling returns and allocating portfolios. Following Cvitanic et al. (2008) and Buckley et al. (2014), we employ instantaneous centralized moments of returns (ICMR) in our analysis of the VG markets and then combine Newton’s method with the ICMR to produce numerical approximations of the optimal portfolios of the investors at different levels of mispricing, asymmetric information, jump intensity, volatility, and horizon.

To the best of our knowledge, this paper is the first to report on the impact of mispricing and asymmetric information in the Kou (2002) and Variance Gamma markets, and therefore, contributes to the literature on mispricing and asymmetric information asset pricing models.

The rest of the paper is organized as follows: the model is briefly reviewed in Section 2. In Section 3, Newton’s method is proposed to give numerical computation of optimal portfolios in the asymmetric Lévy markets. The Kou jump-diffusion market is introduced in Section 4. Newton’s algorithm depends on the first two derivatives of a non-trivial objective function G(.), which is a moment generating function. In this section, we also develop analytic formulas for said derivatives in terms of the cumulative distribution functions of incomplete Beta random variables. The mispriced VG market is introduced in Section 5, where an explicit formula is given for the optimal portfolio. We also report details of the VG market, including ICMR and their effects on mean, variance, skewness and kurtosis of the return process. Concluding remarks are given in Section 6. Numerical results for the Kou and VG markets under various levels of asymmetric information, volatility, mean-reversion speed, Poisson arrival rates and asymmetric distributions are reported in the Appendices.