Modeling Energy Futures Curves with Shared Risk Factors

# Modelling energy derivatives from a purist perspective


# Modelling energy derivatives from a purist perspective












I continue to attempt wrap my head around the mathematical foundations of modelling in energy trading, but I get stuck, or rather, cannot find any definitive references.

From a purist perspective, I want to find a very clean derivation from the real-world measure through to the pricing measure (not sure if it is the risk-neutral measure or the forward measure) and its calibration across futures -- similar to what is done in the pure equities case (i.e., over there we start from the spot price directly, and progress by using no-abitrage arguments, getting the risk-neutral drift, etc.).

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My thoughts:

Naturally in energy, we do not really observe the spot price $(S_{t})_{t}$ but rather the futures prices $(\operatorname{Fut}(t,T))_{t}$ in the market, where $T$ represents a delivery month (e.g., AUG-26) and its predefined expiry. Naturally, the idea would be then to model the futures path as follows for pricing:

\begin{align} \frac{d\operatorname{Fut}(t,T)}{\operatorname{Fut}(t,T)} = \sigma(t,T)dW(t),\; (*) \end{align} but would the above Brownian motion above, $W(t)$ be the same as the Brownian motion $W_{T_{2}}(t)$ for a similar futures but of different delivery period, e.g., \begin{align} \frac{d\operatorname{Fut}(t,T_{2})}{\operatorname{Fut}(t,T_{2})} = \sigma(t,T_{2})dW_{T_{2}}(t)\; ? \end{align}

So I guess we are viewing all futures under the same measure, i.e., the risk-neutral measure?

I also know that from standard arbitrage theory: \begin{align} \operatorname{Fut}(t,T) = \mathbb E_{t}[S_{T}],\; (**) \end{align}

but I am not sure if that helps since we do not actually model the spot prices.

Furthermore, I am aware that sometimes forward and futures are not the same thing, so is the clean approach to model futures or forwards in commodities (even though we do not observe forwards in the market really).

My reading:

Pilz, Schlögl; A Hybrid Commodity and Interest Rate Market Model

Nastasi, Pallavicini, Sartorelli; Smile Modelling in Commodity Markets

Final Thoughts: Apologies if my thoughts seem a bit all over the place but I am indeed looking for a definitive reference on how pricing methods in commodities (particularly oil) are derived from mathematical first principles, similar to what is seen in the standard Black-Scholes case.

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## Answer by João (score 2)

https://quant.stackexchange.com/a/85208

Do we have spot prices in commodities?

Yes.

So that means we have a shipment of 145MW Liquified Natural Gas (e.g., TTF) instantly?

No.

The word spot in commodities opposed to the general description in Equity and FX markets have some differences and somewhat depend on the product we are speaking and even the desk that is referring.

E.g.,

A energy desk(not accounting for the use of dark spreads and spark spreads) could refer to spot as the day ahead so, intuitively:

$$ S_{\text{spot}} = S_{t+1\text{ day}} $$

Another desk with the same characteristics could refer as intra day prices with 15min granularity.

To model using spot prices or futures prices remains only on the objective of what you want/need.

We can also have e.g., a platinum seller company and the buyer, in general they would never meet and if even they did they would not agree on prices, timing, specifications* and so on. So there are intermediaries who play the role of go-between, are prepared to take delivery of goods that may not resell immediately and organize the storage and shipping.

Another example:

If I search for WTI - Crude Prices, some will say the continuous price "spot" is in fact a CFD - OTC proxy, others will say it's the closest Futures contract, so DEC25

IMO I would avoid referring spot as the closest delivery but if the environment allows for that, so it is.

> but would the above Brownian motion above, 𝑊(𝑡) be the same as the Brownian motion 𝑊𝑇2(𝑡) for a similar futures but of different delivery periods

The shocks for different maturities are not literally the same Brownian motion. What we do in term-structure models (HJM e.g.) is to assume that all futures are driven by the same underlying source of randomness, typically a vector of Brownian motions, and each maturity loads on that vector with its own volatility function. Being:

$$ \frac{dF(t,T)}{F(t,T)} = \sigma(t,T)\, dW_t^{Q}, $$

where $dW_{t}^Q$ is not just one scalar Brownian motion but a vector

$$ dW_t^{Q} = \big(dW_t^{(1)}, dW_t^{(2)}, \ldots \big), \qquad dW_t^{(i)}\, dW_t^{(j)} = \rho_{ij}\, dt. $$

So for two specific maturities $T_1$ and $T_2$ we should think of their shocks as different linear combinations of the same factors.

$$ dW_t^{T_1} \neq dW_t^{T_2}, \ \text{but} \ \mathrm{Corr}\!\left(dW_t^{T_1}, dW_t^{T_2}\right) > 0. $$

In essence they are not the same Brownian motion, but they are normally taken to be very positively correlated because they come from the same underlying market factors. Allowing to model the whole futures curve in a consistent, no-arbitrage way.

Since spot is not well defined as before. You can instead, invert this relation and define the implied spot as

$$ S_t := \operatorname{Fut}(t,T), $$

and then model the entire family of futures prices $Fut(t,T)$ directly, ensuring that they remain arbitrage-free and consistent with some (latent) spot process.

> sometimes forward and futures are not the same thing

Futures -> Cleared by House clearing

Forwards -> Not Cleared, thus more credit risk since it's not mark to market

And you're correct, Futures are observable and measurable. Forwards on the other hand are not.

*Specifications in Commodities markets are important, if you need to buy heavy-duty water resistant grease, that will not oxidize you need to buy that. Same goes for Oil, wanting to buy WTI sweet and light is not the same as Brent, and changing these characteristics is very expensive from infrastructures and time consuming.

Crude oil is a mixture of hydrocarbons whose composition varies by source, with about 400 different grades being traded. It is classified mainly by density (API) and by sulfur content. Light crude oils, typically between 35 and 45 API, are fluid, rich in gasoline components, and the most valuable. Heavy crudes, below 25 API such as those from Venezuela’s are thick, dark, and contain large amounts of asphalt, requiring refining or blending to be marketable. Crudes are also described as sweet or sour depending on their sulfur content: sweet crudes contain less than 1% sulfur, are easier to refine, and usually earn a price premium, while sour crudes contain more sulfur and are less desirable because burning them produces sulfur dioxide, a pollutant.

There's also some cases of specifications frauds e.g., Allied Crude Vegetable Oil

> Ships supposedly full of salad oil for Allied would dock, and inspectors would certify the cargo, allowing Allied to post the oil as collateral and obtain millions of dollars in bank loans. In reality, the ships tanks contained only water, with a few feet of salad oil floating on top to trick inspectors. When inspectors audited Allied's facilities, the company would transfer the same oil stock from tank to tank to fool the inspectors while entertaining them during lunch.