Modeling commodity futures including probability of negative prices
- Origin of negative prices of commodity futures contracts;
- Modeling shift-log normal futures do not require further implementations.
We are living a very rare and particular time period. Profound changes have been speculated about the future of the mankind in the next ten years after this global pandemic. The future of the financial market is also uncertain. What are the new instruments that will come up in the derivative markets? What about the insurance market? In this post, I will focus on the energy market, particularly on discussing modeling of the significant drop of future prices of crude oil for contracts expiring on May/2020.
To be honest, I think just a few risk models were taking such a movement into account. Nevertheless, it is worth mentioning that other commodity futures went negative in the past, such as natural gas in 2016.
 |
| Photo by Patrick Hendry on Unsplash |
Unlike equities, (most of the) commodities require storage to be taken into account as a significant component of the price. Let's illustrate the future price composition, starting simple and getting more complex during the process. In an arbitrage free world, future price relates to the spot price $S$ as follows:
$F(t, T) = (S+U) e^{r(T-t)}$
where $F(t, T)$ denotes a future contract expiring at $T$, $r$ is a deterministic risk free rate, and $U$ is the present value of the storage cost. Equation above is usually rewritten in terms of a convenience yield $\delta$ that encompass storage cost for thee counterpart that owns the commodity acquired at the spot price. Therefore:
$F(t, T) = S e^{(r-\delta)(T-t)}$
If one assumes a stochastic spot price and interest rate, future price would become:
$F(t, T)=E[\beta (t) \frac{S(T)}{\beta (T)} | F(t)] + E[\beta (t) \frac{U}{\beta (T)} | F(t)]$
where $\beta (t)$ denotes the risk neutral numeraire and $E[...|F(t)]$ is the expectation under risk neutral measure with filtration at $t$. Once we assume storage cost is deterministic, we obtain:
$F(t, T)=E[\beta (t) \frac{S(T)}{\beta (T)} | F(t)] + U P(t, T)$
where $P(t, T)$ is the bond price paying unit at maturity $T$.
The rational behind formulas above is quite simple. In order to prevent arbitrage between spot and future prices, one need to take into account money costs (interest rate) and storage costs. However, those formulas do not account negative prices for futures. The reason behind that is because it is assumed that once the future contract expires, the holder of the long position is able to immediately trade the received commodity (supposing physical settlement). For low demand situations (such as the current one for crude oil), the holder of the long position must take into account her/his own storage costs until closing off the physical position. This might be easier to understand if we consider that the holder of the long position is a distributor that resells for air companies or gas stations.
Under such low demand condition, we rewrite formulas above as:
$F(t, T) + U_d(t, T_t)=E[\beta (t) \frac{S(T)}{\beta (T)} | F(t)] + U P(t, T)$
where $U_d(t, T_t)$ is the storage cost payed by the holder of the long position of the future contract from the expiration of the contract $T$ until sell off the whole amount received by contract at a expected time $T_t$. If $T~T_t$, storage costs are part of the business as usual. But for low demand $T_t >> T$ this cost is largely significant. Rearranging formula above makes clear a situation where emerges negative prices for future contracts:
$F(t, T)=E[\beta (t) \frac{S(T)}{\beta (T)} | F(t)] + U P(t, T) - U_d(t, T_t)$
Worth mentioning that this is a model for the current situation. Other approaches might also be possible. I will be glad to hear different ideas.
Pricing derivatives including probability of negative future prices
Futures contracts are largely traded as well as derivatives whose the underlying is a future or basket of futures. My goal here is certainly not develop a innovative model that can be used for pricing (although one might think on improve this modeling for risk purposes). This would require much more research and formalism than the one proposed on this blog. Instead, I will discuss fundamentals for including a probability of negative prices.
Consider the following dynamics for futures:
$dF(t, T) = \lambda \phi(F(t, T)) dW$
where $\lambda$ is a constant. In the must-read book written by Piterbarg (here), there is a whole description of the alternatives for function $\phi(F)$ in chapter 7, although Piterbarg is focusing on swap rates, not commodity futures. Anyway, consider a shift-log normal dynamics:
$dF(t, T) = \sigma (\alpha F(t) + (1-\alpha)L) dW$
where $L$ is a constant that might be set as $L \approx F(t_0)$, Parameter $\alpha$ describe how much Gaussian the dynamics is. If $\alpha =0$ the model is essentially Gaussian and allows negative prices, while with $\alpha = 1$ the prices are higher than zero. For completeness, for $\alpha > 1$ the model is over Gaussian.
In order to price derivatives with the model above, one can recover a geometric Brownian motion by making the substitution $F' = \alpha F + (1-\alpha)L$, resulting in $dF' = \sigma \frac{F'}{\alpha}dW$. For an European call option, for instance, one obtains:
$C(t) = E[ (F-K)^+ |F(t)] = \alpha E[ (F' - K')^+ |F(t)]$
where $K' = (1-\alpha)L + \alpha K$.
The most amazing conclusion, for me at least, is that one can include the probability of negative prices for futures simply rewriting the input of the Black formula. As you may have noticed, futures are martingales under risk neutral measure. Here is some discussion on this.
Calibration of this model can be done by optimizing simultaneously $\alpha$ and $\sigma$. For example, one can write the following objective function:
$OF = \sum_{i}(C_{mkt}(t, T_i) - C(\sigma(t, T_i), t, T_i, \alpha))^2$
It is worth mentioning that this is not a well-posed problem. Meaning that additional conditions are required to recover implied $\alpha$.
I hope you enjoyed the post, leave your comments and share!
May the Force be with you!
#=================================
Diogo Pedroso
LinkedIn: www.linkedin.com/in/diogomourapedroso
Comments
Post a Comment