I could find any reference restricting the sign of the volatilities in the multi-factor HJM framework.

Can someone please confirm if $\sigma_i(t,T)$ can assume negative values for some $i,t$ and $T$?

$$ df(t,T) = \left(\sum_i \sigma_i(t,T)\int_t^T \sigma_i(t,u) du \right) dt + \sum_i \sigma_i(t,T) dW_i(t) $$


1 Answer 1


At first, I also could not find a single source that formally restricts $\sigma_i(t,T)$. However, the formally very precise article [1] explicitly states that the $\sigma_i(t,T)$ are assumed to be non-negative:

[...] the volatilities $\sigma_i(\cdot,\cdot,\omega_t)$ belong to $\mathcal{F}$, the set of all functions defined from ${(t, T ) : t \in [0, T ]}\times\Omega$ onto $\mathbb{R}$, that are $\mathbb{P}$-almost everywhere non-negative, bounded, square integrable on any finite time horizon, and Lipschitz continuous with respect to the second variable. Further the $\sigma_i(\cdot,\cdot,\omega_t)$ are jointly measurable from $\mathcal{B} {(t, T ) : t \in [0, T]} \times \mathcal{F}_T → \mathcal{B}$, where $\mathcal{B}$ is the Borel $\sigma$-algebra restricted to $[0, T]$.

Most references quietly assume this by stating that the $\sigma_i(t,T)$ are volatility functions.

Additionally, the volatility functions are usually modeled as a non-negative process, for example when showing its correspondence to a Hull-White model [2].

[1] Tchuindjo, L. (2009). An extended Heath–Jarrow–Morton risk-neutral drift. Applied Mathematics Letters, 22(3), 396–400.

[2] Hull-White model: match between HJM framework and short model formulation


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