When HJM Short Rates Are Markov and Why Separability Matters
Summary
The document compares the Markov property of short rates in Heath–Jarrow–Morton (HJM) and Vasicek models. In HJM, the short rate is obtained by evaluating the forward rate at the current date, and its expression contains a stochastic integral whose integrand also depends on that date. The question is whether this dependence prevents the short rate from being Markov, since the Vasicek solution likewise has time inside an integral as well as at an integration boundary.
The cited model discussion says that a general HJM short rate is not Markov, while certain volatility specifications can make it Markov. In particular, separable volatility, written as a product of deterministic functions of the integration time and maturity, gives the short-rate expression additional structure. The post asks what feature of that factorization matters but does not provide a resolution. It therefore identifies a useful modeling distinction without proving the Markov result or spelling out the state variables required; the formulas and claims are posed for explanation rather than established by evidence in the thread.
Key ideas
- A general HJM forward-rate specification does not necessarily produce a Markov short-rate process.
- The short rate in HJM depends on a stochastic integral whose integrand varies with maturity.
- Separable volatility, expressed as a product of functions of time and maturity, is identified as a condition under which Markov behavior can arise.
- The presence of time in an integral is also found in the Vasicek representation, so that feature alone does not settle whether a process is Markov.
- The discussion poses the state-reduction question but does not supply a proof or full derivation.
Tags
Full text
# Markov property in HJM model and Vasicek model
# Markov property in HJM model and Vasicek model
I am reading Section 5.1 `The HJM Forward-Rate Dynamics` in Brigo Mercurio.
Starting from the dynamics of the the instantaneous forward rate $f(t,T)$: \begin{align} df(t,T) = \alpha(t, T)dt + \sigma(t, T)dW(t), \end{align} \begin{align} f(0,T) = f^M(0,T), \end{align} the authors derive the following expression for the instantaneous short rate $r(t)$: \begin{align} r(t) &= f(t,t) = f(0,t) + \int_0^t \sigma(u,t) \left( \int_u^t \sigma(u,s) ds \right) du + \int_0^t \sigma(s,t) dW(s) \\ &= f(0,t) + \sum_{i=1}^{N} \int_0^t \sigma_i(u,t) \left( \int_u^t \sigma_i(u,s) ds \right) du + \sum_{i=1}^{N} \int_0^t \sigma_i(s,t) dW_i(s). \end{align} They then explain that the short-rate process is not a Markov process in general. Notice the time $t$ appears in the stochastic integral both as extreme of integration and inside the integrand function. However, there are suitable specifications of $\sigma$ for which $r$ is indeed a Markov process. This happens, for example, if we can write, for each $i = 1, \ldots, N$, \begin{align} \sigma_i(t, T) = \xi_i(t) \psi_i(T), \end{align} with $\xi_i$ and $\psi_i$ strictly positive and deterministic functions of time. Under such a separable specification, the short-rate process becomes \begin{align} r(t) = f(0,t) + \sum_{i=1}^{N} \int_0^t \xi_i(u) \psi_i(t) \left( \int_u^t \xi_i(u) \psi_i(s) ds \right) du + \sum_{i=1}^{N} \int_0^t \xi_i(s) \psi_i(t) dW_i(s), \end{align}
Questions:
- Could you help me understand why the fact that $t$ appears in the stochastic integral both as extreme of integration and inside the integrand function makes the process for $r$ non-Markov? I understand the Markovian property, but I am confused about what is different in this case compared to the specification of $r$ in the Vasicek model: \begin{align} r(t) = r(s)e^{-k(t-s)} + \theta \left( 1 - e^{-k(t-s)} \right) + \sigma \int_s^t e^{-k(t-u)} dW(u). \end{align} Also in this case $t$ appears as extreme of integration and inside the integrand function.
- What is peculiar about the specification $\sigma_i(t, T) = \xi_i(t) \psi_i(T)$ that makes the process for $r$ Markov?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.