Excerpt from Nonnegative Wealth, Absence of Arbitrage, and Feasible Consumption Plans

While much of the intuition in option pricing and portfolio choice can be exhibited in discrete time models, continuous time models using Ito calculus have been dominant in these areas of finance.l One reason is that it is generally easier to derive a closed - form solution to a differential equation than to a difference equation. Early development of continuous-time finance using Ito calculus tended to be intuitively based and assumed sufficiency of the natural first order conditions (see, for example, Merton [1971] and Black and Scholes The intuitive appeal of the results was reassuring, as was consistency with limiting versions of discrete-time results (as in Cox, Ross, and Rubinstein but at that time no attempt was made to make sure that the mathematical analysis was rigorously correct.

Harrison and Kreps [1979] set out to give the continuous time analysis a rigorous foundation. They showed that this task is not straightforward, since arbitrage profits can be obtained using seemingly reasonable strategies called doubling strategies (after the strategy of doubling one's bet at roulette). Having continuous trading allows one to do in any finite time interval what would take infinitely many turns at the roulette wheel. Presence of the doubling strategies strikes at the core of the continuous time model, rendering it vacuous. Having arbitrage opportunities precludes having a solution to the optimal investment problem (for strictly monotone preferences) and, of course, invalidates option pricing theory based on the assumption that there is no arbitrage Opportunity. Harrison and Kreps removed arbitrage possibilities by restricting trading strategies to simple trading strategies that allow trade only at finitely many times chosen in advance. This restriction allowed them to use and formalize the risk - neutral pricing approach of Cox and Ross Cox and Ross argued that in the absence of arbitrage, one could always reassign the probabilities to give all assets the same expected returns. Harrison and Kreps called this approach the martingale approach because of its relation to martingale theory.

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