Two stochastic control problems in finance: American options and illiquid investments

2011-07-20Stochastic control problems are ubiquitous in modern finance. However, explicit solutions and policies of such problems faced by investors receive disproportionately little attention. This dissertation focuses on characterizing and solving the policies for two stochastic control problems that buy-side investors face in the market, exercising American options and optimal redemption of illiquid investments such as hedge funds. ❧ The return an investor realizes from his investment in an American or Bermudan style derivative is highly dependent on the exercise policy he employs. Despite the fact that the exercise policy is as crucial to the option buyer as the price, constructing these policies has not received as much attention vis-à-vis the pricing problem. The existing research on the optimal exercise policies is complex, unpractical and conducted to the extent it is utilized to reach accurate prices. Here we present a simple and practical new heuristic to develop exercise policies for American and Bermudan style derivatives, which are immensely valuable to buy-side entities in the market. Our heuristic relies on a simple look-ahead algorithm, which yields comparatively good exercise policies for Bermudan options with few exercise opportunities. We resort to policy improvement to construct more accurate exercise frontiers for American options with frequent exercise opportunities. This exercise frontier is in turn used to estimate the price of the derivative via simulation. Numerical examples yield prices comparable to the existing sophisticated simulation methods in terms of accuracy. Chapter 1 introduces the problem and lays out the valuation framework, Chapter 2 defines and describes our heuristic approach, chapter 3 provides algorithms for implementation with numerical examples provided in Chapter 4. ❧ Optimal redemption policies for illiquid investments are studied in Chapter 5, where we consider a risk-averse investor whose investable assets are held in a perfectly liquid asset (a portfolio of cash and liquid assets or a mutual fund) and another investment that has liquidity restrictions. The illiquidity could be due to restrictions on the investments (such as hedge funds) or due to nature of the asset held (such as real estate). The investor’s objective is to maximize the utility he derives from his terminal wealth at a future end date of his investment horizon. Furthermore the investor wants to hold his liquid wealth above a certain subsistence level, below which he incurs hefty borrowing costs or shortfall penalty. We consider the optimal conditions under which the investor must liquidate his illiquid assets. The redemption notification problem for hedge fund investors has certain affinity with the optimal control methods used in widely-studied inventory management problems. The optimal policy has a monotone structure similar in nature to inventory management problems. ❧ Chapter 6 concludes the study and suggests possible extensions