What does the formula 'dr = k x (theta - r) x dt + sigma x dw' represent?

The formula 'dr = k x (theta - r) x dt + sigma x dw' is known as the Ornstein-Uhlenbeck process, which is commonly used to model the behavior of interest rates over time. In this context, 'dr' represents the change in interest rates, while 'k', 'theta', and 'sigma' are parameters that capture the rate of mean reversion, the long-term mean level of the interest rate, and the volatility of the interest rate process, respectively.

The term '(theta - r)' indicates how far the current interest rate (r) is from its long-term mean (theta), and 'dt' represents a small increment of time. The inclusion of 'sigma x dw' introduces a stochastic component to reflect the randomness in interest rate movements, where 'dw' denotes the increment of a Wiener process.

This formulation adequately captures the dynamics of interest rates, particularly noting that they tend to revert to a long-term average over time while also incorporating the element of volatility. Thus, the formula effectively describes the change in interest rates over time, making that choice the correct interpretation of the equation.