On portfolio management and derivatives hedging

Abstract

In this thesis, efforts are devoted to modeling and addressing practical problems arising in portfolio management, derivatives pricing, and quantitative trading. Optimal portfolio selection problems with multiple risk constraints, vulnerable European options pricing, and pairs trading strategies under a mean-variance framework are investigated.

Expected utility maximization problems with initial-time and intermediate-time Value-at-Risk (VaR) constraints on terminal wealth are studied. The closed-form solutions to the two-VaR problem are first derived, which are optimal among all feasible controls at the initial time, i.e., precommitted strategies. Moreover, we find that the precommitted strategies are optimal at the intermediate time for "bad" market states as well. A contingent claim on Merton's portfolio is constructed to replicate the optimal portfolio. Risk management with intermediate-time risk constraints turns out to be prudent in hedging "bad" intermediate market states and outperforms the single terminal-wealth risk constrained solutions under the relative loss ratio measure. We then investigated an extended problem with joint initial-time and intermediate-time VaR regulations and a portfolio insurance constraint on terminal wealth. It is a real-world problem faced by defined-contribution pension fund managers, whose objective is to maximize the expected utility of terminal wealth over the minimum guaranteed annuity by allocating assets in a risk-free bond, an indexed bond, and a stock. We apply the martingale method to transform the dynamic optimization problem into a static pointwise optimization problem and derive the closed-form precommitted strategies. The numerical results show that an intermediate-time VaR constraint can significantly change the tail distribution of optimal solutions and greatly reduce the risk of loss in "bad" market conditions.

A pricing model is proposed for vulnerable European options, where the price dynamics of the underlying and counterparty assets follow two jump-diffusion processes with fast mean-reverting stochastic volatility. The pricing problem is transformed into solving a partial differential equation by the Feynman-Kac formula. We then approximate the solution by pricing formulas with constant volatility obtained from a two-dimensional Laplace transform via a multiscale asymptotic analysis. Thus, an analytic approximation formula for vulnerable European options is derived in our setting.

This thesis also studied the static and dynamic optimal pairs trading strategies under the mean-variance criterion. The spread of the entity pairs is assumed to be mean-reverting and follows an Ornstein-Uhlenbeck process. Considering the constrained optimal control problem, we adopt the Lagrange multiplier technique to transform the primal problem into a family of linear-quadratic optimal control problems that can be solved by the classical dynamic programming principle. Solutions to static and dynamic optimal pairs trading problems are derived and discussed. We show that the "static and dynamic optimality" is a viable approach to the time-inconsistent control problem. Furthermore, numerical experiments of optimal pair trading strategies are presented.