Quantitative Finance: The Demystification of Stochastic Calculus
In the world of finance, it is not uncommon to hear about stochastic calculus or stochastic processes. While the name may sound daunting, the concept and its application in finance is actually relatively straightforward.
A stochastic process, sometimes referred to as a random process, is simply a group (or “system”) of random variables and their evolution or changes over time. Stochastic calculus is the branch of mathematics used to model the behavior of these random systems. In the finance world, these systems are often stock prices or bond interest rates and the random variables are factors that influence them.
Take, for example, the price of Stock A. Stock A will rise and fall in value continuously during the open exchange period. Each time the price goes up or down, it can be thought of as a “step.” However, after each step, there is no way of knowing for certain in which “direction” (up or down) the price will go the next time it moves. The next step in the series is essentially random. A series of steps like this is sometimes referred to as a “random walk,” because with each “step” the “direction” of the walk can change randomly.
Modeling this behavior, while very useful as a finance tool, is very difficult due to its random nature. The Wiener process is one of the most widely used and best known examples of this sort of model. This process, in particular, is often used to represent the rise and fall of stock prices and/or interest rates over time.
The above example is a non-mathematical approximation of the concept behind the Wiener process. The process itself is a mathematical construct that can be used to translate the random motion of a non-mathematical concept (such as stock prices or the motion of molecular particles in a fluid) into something that can be calculated or processed as numbers using stochastic calculus. Essentially, it takes the random variables of reality and turns them into a mathematical formula so that they can be quantified and used in calculating financial models.
This is useful for a number of reasons, not the least of which is the relative ease of understanding that a model can provide. Rather than tracking each individual variable (an impossible task, even with the best of computing technology), a model boils down the highly complex, real-life situation into a set of simple (and therefore easier to understand) components. This approach is not without its flaws but it is still a powerful tool in understanding and tracking economic progress and forces over time.
The Wiener process, and stochastic processes and calculus in general, are important tools in quantitative analysis as it is applied to finance. Stochastic calculus is used for the valuation of stock options and derivatives, assessment of financial risk, and many other financial purposes. The models it produces provide insight and aid in a plethora of financial endeavors.
However complex the mathematics behind stochastic calculus may be, the concept behind stochastic processes is relatively straightforward and a powerful tool in understanding financial markets and forces.
Guest post contributed by Charles Ronson for www.easyfinance.com. Charles is a freelance business writer. He has extensive experience in consulting with small to medium sized businesses. His articles appear on various business blogs.