Sign Constraints on Volatility in the Multi-Factor HJM Model
Summary
The document asks whether factor volatilities in the multi-factor Heath–Jarrow–Morton framework may be negative. It notes that the drift expression uses products of volatility functions, then reports a reference that explicitly assumes these functions are non-negative, bounded, square-integrable over finite horizons, and regular in maturity. The answer also says many sources leave the sign restriction implicit in their use of the term volatility.
This is a report of conventions in references, not a derivation that negative volatility is mathematically impossible. The cited discussion gives no comparison of equivalent model representations or explanation of how a sign change would affect the Brownian drivers. Its evidence is limited to one formal article and a typical modeling choice illustrated by a Hull–White connection. Readers should distinguish the cited non-negativity assumption from a universal requirement of HJM theory.
Key ideas
- Some HJM references explicitly assume volatility functions are non-negative and satisfy regularity and integrability conditions.
- The answer describes non-negativity as a common modeling convention.
- The document does not establish that negative volatility is impossible in every HJM formulation.
- The cited evidence is a reference and a modeling example rather than a general proof.
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Full text
# Can volatility assume negative values under multi-factor HJM framework?
# Can volatility assume negative values under multi-factor HJM framework?
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 $T$?
$$ df(t,T) = \left(\sum_i \sigma_i(t,T)\int_t^T \sigma_i(t,u) du \right) dt + \sum_i \sigma_i(t,T) dW_i(t) $$
## Answer by Hans-Peter Schrei (score 1)
https://quant.stackexchange.com/a/74983
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 formulationShown 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.