Markov Properties of Two-Factor Hull–White and HJM Short-Rate Models
Summary
The document poses questions about Markovianity in the two-factor Hull–White interest-rate model and in short-rate dynamics derived from the Heath–Jarrow–Morton framework. It describes a calibration setting in which the factor correlation is close to minus one, raising the question of why a degenerate, effectively one-factor short rate may still be non-Markovian. The model is specified as the sum of two mean-reverting stochastic factors and a deterministic shift used to fit the initial forward curve.
The discussion also asks whether a time integral signals loss of the Markov property when the current time appears inside its integration range, and why non-Markov dynamics might improve on a Markovian model. The document supplies no answers, derivations, calibration evidence, or criteria for determining Markovianity. It is therefore useful as a focused statement of modeling questions and the equations under discussion, rather than as an explanation or a guide to resolving them.
Key ideas
- The document presents a two-factor Hull–White short rate as the sum of two mean-reverting factors and a deterministic curve-fitting shift.
- It asks whether near-perfect negative correlation can produce a one-factor short rate that remains non-Markovian.
- It raises a proposed rule about how the placement of current time in an integral affects Markovianity.
- It asks why non-Markov dynamics could improve model performance, but provides no explanation or evidence.
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Full text
# Hull White 2 factors and non Markov interest rates # Hull White 2 factors and non Markov interest rates I am studying the calibration of the 2 factors Hull White model on Brigo and Mercurio's book. They point out that, using cap volatilities, the value of $\rho$ is almost minus one and this means that the HW 2 factors model tends to degenerate into a one-factor model. So far, so good. Now, the authors claim that this one-factor model is non-Markov. Could you explain to me why? I know (more or less) what a Markov process is, but, still, I don't fully understand their statement. They also go further, stating the following: "the degenerate process for the short rate is still non-Markovian (if $a \ne b$), which explains what really makes the G2++ model outperform its one-factor version" There are many things I don't get about this sentence. First, why are they using the word "still"? Does it mean that even the non-degenerate Hull White 2 factors is non-Markovian? How can I say quickly whether an interest rate model is Markovian or not? Second, why on earth (at least in financial modelling) should a non-Markovian process outperfom a Markovian one? Their statement looks very strong... Since we are talking about Markov/non-Markov interest rates, I would also like to ask a question concerning the HJM framework. One of the main problems with this approach is that the dynamics of the short rate is often messy and non-Markovian. Again, by simply looking at the dynamics, how can we say that? The drift of the short rate involves some (deterministic and stochastic) integrals and I read somewhere that if the variable $t$ appears only as one of the extremes of the integral, then the short rate is Markovian; if, on the other hand, $t$ appears also inside the integral, the Markovianity is lost. Is it true? If so, could you give me an intuitive and/or rigorous explanation? Thank you very, very much for any help you can provide!!! P.S. Here is the model :) $r_t=x_t+y_t+\phi_t$ with $d x_t=-a x_t\ dt+\sigma dW^1_t $ $d y_t=-b y_t\ dt+\eta d W^2_t $ $d W^1_t d W^2_t = \rho dt$ where $\phi_t$ is the deterministic shift chosen so as to fit any initial forward curve
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