The dynamics of stock share prices over time may be difficult to model accurately, but interest rates are much more complex. In earlier chapters, we have considered continuously compounded interest rates

 for an investment over the time interval . If we introduce randomness, we have to cope with a mathematical object like . This object is called a random field. If we fix , we obtain a stochastic process representing how an interest rate for a given time-to-maturity changes over time. If we fix , we obtain a function of time-to-maturity, describing the term structure of rates at time , for different scenarios ω. Clearly, there must be some connection between interest rates at different time instants, and rates for different maturities. Modeling interest rates in financially sensible, yet computationally tractable way is no easy task.

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r(t,t+T)