Methods of Mathematical Finance

Throughout this monograph we deal with a financial market consisting of N + 1 financial assets. One of these assets is instantaneously risk-free, and will be called a money market. Assets 1 through N are risky, and will be called stocks (although in applications of this model they are often commodities or currencies, rather than common stocks). These financial assets have continuous prices evolving continuously in time and driven by a D-dimensional Brownian motion. The continuity of the time parameter and the accompanying capacity for continuous trading permit an elegance of formulation and analysis not unlike that obtained when passing from difference to differential equations. If asset prices do not vary continuously, at least they vary frequently, and the model we propose to study has proved its usefulness as an approximation to reality. Our assumption that asset prices have no jumps is a significant one. It is tantamount to the assertion that there are no "surprises" in the market: the price of a stock at time t can be perfectly predicted from knowledge of its price at times strictly prior to t. We adopt this assumption in order to simplify the mathematics; the additional assumption that asset prices are driven by a Brownian motion is little more than a convenient way of phrasing this condition. Some literature on continuous-time markets with discontinuous asset prices is cited in the notes at the end of this chapter. The extent to which the results of this monograph can be extended to such models has not yet been fully explored.

A derivative security (also called contingent claim; cf. Definition 2.1 and discussion following it) is a financial contract whose value is derived from the value of another underlying, more basic, security, such as a stock or a bond. Common derivative securities are put options, call options, forward contracts, futures contracts, and swaps. These securities can be used for both speculation and hedging, but their creation and marketing are based much more on the latter use than the former. Some derivative securities are traded on exchanges, while others are arranged as private contracts between financial institutions and their clients. The world-wide market in derivative securities is in the trillions of dollars.

This chapter solves the problem of an agent who begins with an initial endowment and who can consume while also investing in a standard, complete market as set forth in Chapter 1. The objective of this agent is to maximize the expected utility of consumption over the planning horizon, or to maximize the expected utility of wealth at the end of the planning horizon, or to maximize some combination of these two quantities. Except for the completeness assumption, the market model is quite general, allowing the coefficient processes to be stochastic processes that are not even assumed to be Markovian. Specializations of this model to the case of deterministic and even constant coefficients are provided in Sections 3.8 and 3.9. The problem of this chapter is revisited in the context of incomplete markets in Chapter 6.

In the context of continuous-time financial markets, the equilibrium problem is to build a model in which security prices are determined by the law of supply and demand. The primitives in this model are the endowment processes and the utility functions of a finite number of agents. We shall assume in this chapter that all agents are endowed in units of the same perishable commodity, which arrives at some time-varying random rate. Agents may consume their endowment as it arrives, they may sell some portion of it to other agents, or they may buy extra endowment from other agents. The endowment, however, cannot be stored, and agents will wish to hedge the variability in their endowment processes by trading with one another. To facilitate the trading of endowment, there is a financial market consisting of a money market and of several stocks, in which agents may invest (positively or negatively).

As we saw in Chapter 5, when a financial market is incomplete due to portfolio constraints, it may no longer be possible to construct a perfect hedge for contingent claims. This led to the introduction in that chapter of superreplicating portfolios and upper-hedging prices for contingent claims. This is a conservative approach to pricing, since it begins from the assumption that agents trade only if their probability of loss is zero.