Consumption and Portfolio Choice with Option-Implied State Prices

We propose an empirical implementation of the consumption-investment problem using the martingale representation alternative to dynamic programming. Our method is based on the direct observation of state prices from options data. This greatly simpli…es the investor’s task of specifying the investment opportunity set and inherits the computational convenience of the martingale representation. Our method also makes explicit the economic trade-o¤ between exploiting di¤erences in state prices and probabilities, which generate variation in consumption, and the consumption smoothing induced by risk aversion. Using options-implied information, we …nd quantitatively di¤erent optimal consumption and portfolio policies than those implied by standard return dynamics. We thank seminar participants at the CIRANO Conference on Portfolio Choice for their comments and suggestions. Financial support from the NSF under grant SBR-0350772, the Bendheim Center for Finance at Princeton University is gratefully acknowledged. yPrinceton, NJ 08544. Phone: (609) 258-4015. E-mail: [email protected]. zDurham, NC 27708. Phone: (919) 660-1948. E-mail: [email protected]. 1. Introduction Intertemporal consumption and portfolio choice is a daunting problem, requiring as input a complete characterization of the joint distribution of returns across all states of the world from the current date until the end of the investment horizon. Furthermore, professional investment advice is often of limited help because it delivers mostly predictions about mean returns at di¤erent horizons. For example, an analyst may give a stock a “near-term hold" or a “long-term buy" recommendation. How can an investor make portfolio and consumption decisions based on such a terse description of the investment opportunity set? We propose a new empirical approach to address this question. We decompose the portfolio and consumption choice into two separate problems and use di¤erent sources of information to get a handle on each. Consider an economy in which the uncertainty is driven by the stochastic movements of a stock and bond index that, in addition to a riskless money market asset, jointly determine the investment opportunity set. At the most abstract level, the investor’s problem consists of choosing how to allocate scarce resources to the di¤erent states of the world at all future dates. We use the martingale representation theory of Cox and Huang (1989), Cox and Huang (1991), Karatzas, Lehoczky, and Shreve (1987) and Pliska (1986) to turn this inherently dynamic optimization problem into a static one. This static solution to the portfolio and consumption problem requires two pieces of information. First, the investor has to …gure out how expensive one unit of consumption will be in each future state of the world. Second, the investor needs to determine how likely each state is to be realized. To obtain the …rst piece of information, the price of a unit of consumption in each future state, we use the market prices of traded options to infer the joint state price density q of stocks and bonds.1 The resulting option-implied prices of state-contingent consumption bundles allows the investor to determine in which states consumption is relatively cheap or expensive. The investor then allocates consumption to each state in order to maximize expected utility under the budget constraint. The solution to the static optimization over state-contingent consumption bundles depends 1Our approach of evaluating the cost of prospective consumption bundle using option-implied information di¤ers from that of Cox and Huang (1989), Cox and Huang (1991) and related theory papers that link the optimal portfolio and consumption choices to the growth-optimal policies under log utility.