Probability and Finance: Flows of Conditional Prices, Liquidity Issues, and Impulse Control AMC-SS

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In this proposal we propose to study the mathematical framework of incomplete markets in Mathematical Finance Theory, and in particular in the equity markets (and markets with the same mathematical structure). Incomplete markets are characterized by there being an infinite number of risk neutral measures, and therefore there is not a unique price, for the typical contingent claim. Various methods have been proposed over the last 20 years or so to give a reasonable method to choose one risk neutral measure over the others, but they have always been unsatisfying, and ultimately arbitrary. A current popular approach is to use "indifference pricing," which involves economics arguments when choosing (still arbitrarily) risk neutral measure. We propose instead to try to "complete" incomplete markets, using a modification of the idea of Heath, Jarrow and Morton, who did an analogous feat for the bond market. The idea is to use the continuous time pricing of financial derivatives, such as options, which are traded in the markets, and which (by the theory) have to be local martingales. Therefore we need to show that there exist risk neutral measures that simultaneously make the underlying and its traded derivatives all martingales (or local martingales, or even sigma martingales); this can get quite technical, especially in cases where the maturity of the derivative is before the trading horizon.In the trading of equities, such as stocks, an element which has become of fundamental importance are financial derivatives, such as options. An option allows one to transfer risk, such as (for example) portfolio exposure, or foreign currency risk, from one party to another, for a fee. In this way, it is like fire insurance on a home, where the home owner transfers the financial risk of losing his home to an insurance company willing to bear the risk, for of course, a fee. However unlike fire insurance or life insurance, it is quite complicated mathematically to calculate a fair price for a financial derivative. (It is also difficult to price some types of home insurance too, at times, such as storm insurance for homes in Florida and along the Gulf Coast of the US, and insurance companies are increasingly relying on "re-insurance," which can often be modeled in ways analogous to option pricing.) The models of Black, Scholes, and Merton, for which the latter two were awarded a Nobel prize, explain how to calculate a fair price in simple situations. We now know how to calculate fair prices in slightly more sophisticated models, known as "complete markets", however it is widely believed that the world is more complicated than the complete market case, and is in fact "incomplete." All of the many methods proposed to date to calculate a fair price in incomplete markets have been arbitrary, and in general not accepted either by academics, nor practitioners. This proposal might go a long way towards solving that problem, by using the market prices of the options themselves, together with the underlying market stock prices, essentially to make an incomplete market into a complete one.