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Stochastic Processes

Risk often refers to the distribution of random asset values at some future horizon. In many instances, risk models are based on simulations of asset values or of risk factors over time. The dynamics of returns and of asset prices are used in this text with the main purpose of assessing such distributions.

Specific applications to risk models include the distribution of asset values when asset returns follow a normal distribution, the simulation of defaults when defaults follow a "rare event" process, or the simulation of interest rates. There is a considerable amount of publications on asset pricing and stochastic processes[1]. This chapter is no substitute to such texts. Rather it covers the minimum pre-requisites of common stochastic processes that serve for risk modeling and simulations, and relate risk modeling to the rest of the literature.

The starting point is a review of the common properties of the time process followed by random variables. The next step is to explain the Ito process that applies to all variables having normal instantaneous returns, such as stock prices, the mean-reverting processes that apply to variables that tend towards long-term value, such as interest rates, and the "rare event process" because it relates directly to default models. Simpler processes are described before because they help understanding these main processes. Two sections provide typical examples of applications, for stock prices and interest rates. The last section briefly introduces the Ito lemma widely used in the literature, because it allows deriving the process of any function of any underlying process. It serves, notably, for deriving the process followed by a derivative from the process of the underlying asset.

  • [1] See Hull [36] for an overview of common stochastic processes. Neftci [55] provides more details on the underlying concepts and properties. The book from Grimmett G. R, Stirzaker, D. R. [35] is also a useful introduction.
 
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