Why the Proposed Commodity Risk-Factor Differentials Conflict
Summary
The document examines a proposed differential for the second risk factor in the two-factor Gibson–Schwartz commodity model. The factor is first defined as the time integral of an Ornstein–Uhlenbeck process, then rewritten as a stochastic integral using Fubini’s theorem. The question asks why a different expression involving time to product maturity should follow from that representation.
The accepted response points out that the two stated dynamics are incompatible: the time-integrated process is not a martingale, whereas the proposed differential describes a martingale. It therefore does not provide a derivation of the maturity-dependent expression. The response says the starting point must be clarified, and suggests that a related quantity in the model may instead follow a standard Ornstein–Uhlenbeck specification used in short-rate models. This is a conceptual warning about identifying the correct process before deriving its differential; the document does not establish a corrected model or full derivation.
Key ideas
- The stated integral definition and proposed differential do not describe the same process.
- The time integral of the Ornstein–Uhlenbeck factor is not a martingale.
- A maturity-dependent stochastic differential requires a clearly specified starting process.
- A related model quantity may follow a standard Ornstein–Uhlenbeck specification.
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Full text
# Gibson-Schwartz 2-factor model: derivation proof of the differential of the risk factor $x_{2,t}$
# Gibson-Schwartz 2-factor model: derivation proof of the differential of the risk factor $x_{2,t}$
In the Gibson-Schwartz 2-factor model for commodity products, the dynamics of one of the two risk factors is defined as:
$$x_{2,t} = \int_{0}^{t}z_{u}du \;\;\;\;\text{ where } \;\;\;\; z_{u} = \int_{0}^{u} \sigma_{2} e^{-\alpha_{2}(u-s)}dW_{2,s}$$
$dW_{2,s}$ is a Brownian Motion increment.
By plugging in $z_u$, and by using Fubini's theorem the above equation can be re-written as:
$$x_{2,t} = \int_{0}^{t}\Big(\int_{s}^{t}\sigma_{2} e^{-\alpha_{2}(u-s)}du\Big)dW_{2,s}= \frac{\sigma_{2}}{\alpha_{2}}\int_{0}^{t}(1-e^{-\alpha_{2}(t-s)})dW_{2,s}$$
My question is about the following differential $dx_{2,t}$ where, as you can see in the exponential function, the time difference is between $t$ and $T$ (maturity of a commodity product):
$$dx_{2,t} = \frac{\sigma_{2}}{\alpha_{2}}(1-e^{-\alpha_{2}(T-t)})dW_{2,t}$$
Apparently the derivation of the latter total differential is based on the final equation of $x_{2,t}$.Nevertheless, it is not clear to me. Can someone provide the derivation proof?
## Answer by Andrea (score 1, accepted)
https://quant.stackexchange.com/a/81809
These 2 differentials are incompatible
$$x_{2,t} = \int_{0}^{t}z_{u}du$$
and
$$dx_{2,t} = \frac{\sigma_{2}}{\alpha_{2}}(1-e^{-\alpha_{2}(T-t)})dW_{2,t}$$
The latter is a martingale, the former isn't.
You will need to formulate the question more precisely. What is the real starting point? and what are your calculations?
EDIT: if you are asking the same as Gibson & Schwartz two factor model: mathematical derivation of the total expected return of a commodity contingent claim, then $d \delta$ is the usual OU process used in many short rate models (e.g. Vasicek) for which there will be tons of posts about the derivation.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.