مدل های گسستگی در برابر مدل های پیوسته زمان: فرآیندهای منحصر به فرد در تئوری قیمت گذاری دارایی

In economic theory, both discrete and continuous time models are commonly believed to be equivalent in the sense that one can always be used to approximate the other, or equivalently, any phenomena present in one is also present in the other. This common belief is misguided. Both (strict) local martingales and singular processes exist in continuous time, but not in discrete time models. More importantly, their existence reflects real economic phenomena related to arbitrage opportunities, large traders, asset price bubbles, and market efficiency. And as an approximation to trading opportunities in real markets, continuous trading provides a better fit and should be the preferred modeling approach for asset pricing theory.

It is commonly believed that in economic theory both discrete and continuous time models are equivalent in the sense that one can always be used to approximate the other, or equivalently, any economic phenomena present in one is also present in the other. Unfortunately, this common belief is misguided. Both (strict) local martingales and increasing singular processes exist in continuous time models, but not in discrete time models. These processes arise naturally in the mathematics of stochastic integration theory. For example, Itô integrals with respect to Brownian motion may not be martingales, but they are always local martingales. Such integrals arise when calculating the results of a hedging strategy, integrated against a price process. Local times for Brownian motion are examples of naturally occurring increasing processes that have paths that are singular with respect to Lebesgue measure. An example of how local times can arise is when certain kinds of convex functions (including the function f(x) = (x − K)+) are composed with a price process, as happens when describing certain kinds of options. The reason this is important is that the absence of arbitrage is essentially equivalent to the existence of a risk neutral measure that makes the price process a local martingale (for processes with continuous paths), and in the Brownian paradigm the presence of a local times makes this procedure impossible. 2 This would not be a concern except for the fact that these processes characterize real and important economic phenomena related to arbitrage opportunities, large traders, asset price bubbles, and market efficiency. Discrete time models will not exhibit these phenomena. In addition, as an approximation to current security markets, continuous trading provides a better fit. This is true because a trade can take place at any time during a day, and not on a fixed and predetermined grid. Consequently, at least for asset pricing theory, continuous time is the preferred modeling approach. The purpose of this paper is to clarify and to justify these previous assertions. An outline for this paper is as follows. Section 2 quickly explains how local martingales and singular processes are generated in continuous time models, and it characterizes the economic phenomena they represent. Section 3 discusses why continuous time models provide the better approximation to actual security markets, while Section 4 concludes.

Before concluding we discuss one remaining issue related to the testing of continuous time models using discretely sampled price data. Given a continuous time setting, it is sometimes falsely believed that estimating using discretely sampled observations destroys the relevant continuous time phenomena. This is not true. Such phenomena still exist independent of the sampling methodology, and such continuous time phenomena can still be seen using advanced statistical techniques (e.g. see Jarrow et al., 2011 and Jarrow et al., 2011). In addition, one can also test the nature of whether or not a local martingale exists indirectly by testing some consequences of the model using discretely sampled data.