Limiting connection between discrete and continuous time forward interest rate curve models
The authors study limiting connections between discrete time forward interest rate models introduced by themselves and corresponding generalized well-known continuous time models. The main aim is to derive a functional limit theorem for a sequence of discrete time forward interest rate models driven by random fields. To explain the sense of generalization let \(f(t,x)\) denote the instantaneous forward rate at time \(t\) with time to maturity \(x\) and in the famous Heath-Jarrow-Morton model be assumed to follow the dynamics \(df(t,x)=\alpha(t,x)\,dt+\sigma(t,x)\,dW(t),\) where \(W(t)_{t\in R_+}\) is a standard Wiener process. Then the price of zero coupon bond \(P(t,s)\) at time \(t\) with maturity date \(s\) may be defined by \[ P(t,s)=\exp\left\{-\int_0^{s-t}f(t,u)\,du\right\}, \quad 0\leq t\leq s. \] And having this equation one can set up and solve different problems in finance in general and in derivative securities in particular, for example. In this classical HJM model the forward rate process \(\{f(t,x)\}_{t\in R_+}\) is driven by the same Wiener process for any value \(x\geq 0.\) Therefore it is natural to generalize the model by incorporating a random driving field instead of a single driving process as follows. Let \(\{Z(t, s)\}_{t, s\in R_+}\) be a random field and suppose that for each fixed \(x\in R_+\) the forward rate dynamics is given by \[ df(t,x)=\alpha(t,x)\,dt+\sigma(t,x)\,dZ(dt,x), \] where \(\{Z(t,s)\}_{t\in R_+}\) is a semimartingale for any \(s\geq 0.\) In this way forward rates with different time to maturity can be driven by different processes and one can clearly get a much larger range of forward rate families than in the classical setup. Such generalization of the continuous time model has been proposed by Kennedy, and the corresponding discrete time forward rate models driven by random fields were proposed by the authors.
scientific article; zbMATH DE number 2015154