Historical VaR for Commodity Forwards Without Full Revaluation
Summary
The document discusses estimating value at risk for a commodity forward exposed to commodity price, foreign exchange, interest rate, and commodity spread movements. It explains that full revaluation can be practical when the forward pricing formula is simple: sample joint risk-factor scenarios, reprice the forward, and take the desired loss percentile. Historical observations can provide scenarios when no joint model is available.
For faster approximation, it proposes focusing on the dominant underlying price risk or using a first-order expansion of the pricing formula. This expresses P&L as a weighted sum of changes in the risk drivers, with sensitivities computed before sampling. The example claims scenario repricing is fast for the stated formula, but does not provide a validation study or quantify approximation error. Linearization may be less reliable for large moves or nonlinear exposures, and the answer does not specify how to model commodity spreads or dependence among all listed factors.
Key ideas
- A commodity forward can be repriced under sampled joint scenarios, with VaR estimated from the resulting P&L distribution.
- Historical risk-factor observations can serve as scenarios when a joint probability model is unavailable.
- A first-order pricing expansion approximates P&L as a weighted sum of risk-factor changes.
- Focusing on the underlying commodity may simplify analysis when it dominates the position's risk.
- Linear approximations can lose accuracy when price changes are large or exposure is nonlinear.
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# Historical VaR on Commodity Physical Forward
# Historical VaR on Commodity Physical Forward
Recently came across building Histroical VaR for commodity forward position. Understood from quants guru the best way to calculate VaR is using full re-valuation, Full reval is computationally intensive . Any other alternative approach will be appreciated . On a different note, my forward position has following risk drivers
- Commodity Price
- Fx Rate
- IR Rate (To discount the forward cash flows)
- Commodity Spread.
I need to find a model that can accommodate all the risk drivers without using full reval and parametric approach.
## Answer by Chris Taylor (score 2, accepted)
https://quant.stackexchange.com/a/25418
Why is it so expensive to use the full revaluation method? The commodity forward price is
$$ F = (S + U)e^{rT} $$
where $S$ is the current spot price, $U$ is the cost of storage between $0$ and $T$ and $r$ is the risk-free rate (you may also have an FX rate if the forward is priced in a different currency from the underlying).
If you have a joint model for the distribution of $(S, U, r)$ you can sample from this distribution (say 10,000 times), compute the forward price (which is fast) and find the 5th percentile (which is also fast). It takes ~2 milliseconds in MATLAB.
If you don't have a joint distribution, you can sample from the historical distribution instead (e.g. over the last 252 days).
If you want to speed it up, notice that for most commodities the primary risk driver is the price of the underlying, so you can either just sample from the distribution of the underlying, or if you have a model for the price moves (e.g. lognormal) you can calculate the VaR exactly by applying the appropriate transofmration to the VaR of the underlying.
Another approach, which will approximate the VaR, is to expand the pricing formula to first order. For example, for the pricing formula above,
$$ dF = FTdr + e^{rT}dS + e^{rT}dU $$
which expresses the change in the futures value as a linear combination of the risk drivers $dr$, $dS$ and $dU$ (you can do the same for the specific risk drivers for your contracts).
You can compute the coefficients in front of the risk drivers before historical sampling, and now just compute VaR as the 5th percentile of your approximate P&L $dF$.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.