Separability and Markovianity in Stochastic Volatility Term Structures

Summary

The document asks how the forward-rate volatility specification in a general stochastic-volatility term-structure model can produce a Markovian model. It contrasts the HJM condition that Markovian forward-rate volatility be separable into a time-dependent factor and a maturity-dependent factor with a specification that depends on time to maturity through a linear term multiplied by an exponential.

The author attempts to rearrange the expression into separable form but finds that calendar time remains inside a term that cannot be factored in the proposed way. The document gives no answer or derivation, so it does not establish whether the model is Markovian, whether a broader notion of separability applies, or whether the expression needs to be rewritten using additional state variables. It is useful as a focused question about the link between volatility structure and finite-dimensional Markov representations, but readers need another source to resolve the issue.

Key ideas

• The document asks how a stochastic-volatility term-structure specification can yield Markovian dynamics.
• It cites separability of forward-rate volatility as a condition associated with Markovian HJM models.
• The displayed specification depends on time to maturity through both a linear and an exponential term.
• The attempted rearrangement leaves calendar time inside a factor, and the document provides no resolution.

Tags

# Separability of Stochastic Volatility Model


# Separability of Stochastic Volatility Model












After having read the article of Trolle & Schwartz regarding their general stochastic volatility term structure model (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=966364), it is not clear how the model becomes Markovian. It is well-known that if and only if HJM model is Markovian, then the forward rate volatility is separable in time i.e. $$\sigma_f(t,T)=h(t)g(T),$$ where g and h is a deterministic vector function and matrix process respectively. However the specification in the article is $$\sigma_f(t,T)=(\alpha_0 + \alpha_1(T-t))e^{\gamma(T-t)},$$ which I don't see satisfies the representation from above. I'm getting stuck rewriting here $$\sigma_f(t,T) = (\alpha_0 + \alpha_1)e^{-\gamma T}e^{\gamma t} - \alpha_1 e^{-\gamma T}e^{\gamma t} t,$$ as it not possible to take the last term on the RHS underneath the parenthesis due to the "t".

Hope you can help clarify.

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