Financial Simulation Modeling in Excel

STOCHASTIC DIFFERENTIAL EQUATIONS

In the prior section, we discussed partial differential equations and we have seen previously that we can use random walks to model financial processes. Here we will try to bring both concepts together by introducing stochastic differential equations. Much like with regular ODEs, we will not discuss the mathematical peculiarities of stochastic calculus or the complex methodologies by which these equations are solved. Instead we will try to attain a conceptual understanding of a differential stochastic process by understanding their numerical properties. (We saw these equations in action when exploring the Black-Scholes and Hull-White models in Chapter 4.)

In the financial world of derivatives and option pricing, the interest rate is king. Trading strategies typically pair stocks with bonds in an arbitrage-free market, and as such, the rate on the bond becomes the driving force of how stocks are traded. Hence, stochastic differential equations are commonly used to model interest rates that are then applied toward a specific trading strategy. The most simplistic interest rate model, and the basis for the Ho-Lee Model, is given by equation A.18.

Conceptually speaking, it is pretty clear what “dt” means. Most everyone who has a watch or a calendar can imagine incremental changes in time. However, what does “dW” mean? What does it mean to take the derivative ...