## A purview of the Gabillon commodity price model widely used in the financial sector

Investment dollars into commodities have increased dramatically from 2004-2008 with the growth of exchange traded funds and over the counter structured derivatives. However, the global financial crisis in 2008 and the present Euro crisis have crimped the growth in derivatives. Further, the legal tussle over the Ceylon Petrochemical Company (CPC) investment into target redemption notes has impeded its growth^{1}, at least in Asia.

But commodity derivatives continue to be important – more so for vanilla products especially Asian swaps/ options, which are actively traded on the Platts window in Asian trade. An academic survey of stochastic commodity models reveals commodity prices modelled as either spot prices/ convenience yield or futures/ forward curves. Spot price/ convenience yields are modelled in the Schwartz model, but has not gained industry-wide acceptance due to the unobservability of the convenience yield.

Instead futures prices used for hedging are modelled in the pricing of commodity derivatives. This is done via the Gabillon model which is the most popular model used in the financial industry. The Gabillon stochastic differential equation (SDE) for each futures F(t,T_{i}) observed at t for delivery in T_{i} is:

Here the W_{s} and W_{l} are short and long term innovations respectively. The short term factor generally refers to short term shocks like inventory, production disruptions or demand changes, whilst the long term factors are technological innovations or discovery of new production fields. The other parameters in the equation are:

Name |
Symbol |
Significance |

Base vol | σ_{l} |
Affects the long end of the volatility term structure more. |

Short vol | σ_{s} |
Affects the short end of the volatility term structure relatively more due to short term shock. |

Correlation between base and short vols | ρ | Not calibrated but fixed normally. A value closer to 1.0 implies more serial correlation. The reason is it is fixed normally ~0.9-0.97 is it tends to be very unstable in calibration. |

Mean reversion | κ | Steepness of the volatility term structure. A larger κ makes it steeper. A larger value means more serial de-correlation between futures further apart. |

Potentially there are as many SDEs as each delivery date for a commodity. Unlike interest rate models, there are no arbitrage opportunities between contract months, since these are essentially different contracts with different delivery dates. A pertinent feature of commodities is that futures volatilities tend to mean revert to a long term mean. This is observed through the ATM volatility term structure where the ‘long end’ reverts to a ‘base volatility’ – a Samuelson effect. The reversion is regulated through κ. The parameters above are calibrated to the ATM volatility term structure observed in the market:

The Gabillon model means that each futures contract has a common early expiry profile. In this profile, each futures volatility increases closer to maturity (a common characteristic in commodity markets as short term demand/ supply factors dominate) in an ‘exponential-like’ manner:

The common evolution of the futures volatility for σ_{s} = 40%, σ_{l} = 20%, ρ=0.95 and κ=0.3 as the futures moves closer to expiry can be seen graphically in:

This common early expiry profile and the modelling of only ATM volatilities are the shortcomings of the Gabillon model. In most investment banks, the model is modified by adding a term α(F,t) in the SDE. This term α(F,t) is calibrated to instruments that are sensitive to early expiry. To account for fat tails in the commodities return, the SDE can be mapped from a normal distribution to a skewed distribution in a local volatility-like model;

*Simulated* *Normal returns —>** Skewed returns*

The Gabillon model allows for closed form pricing for asian swaps/ options, swaptions and European / American options. The model is in turn used for pricing more exotic products like target redemption notes, window barriers and volume options in the market.

Footnotes:

*1 – CPC invested into target redemption notes for ‘hedging purposes’. These notes are basically a series of monthly long/ short forwards/ options that extinguish on reaching a certain target sum. The sudden plummet of oil prices in the 2 ^{nd} half of 2008 turned their positions red. More information on the legal tussle can be found on:*

http://investing.businessweek.com/research/stocks/private/snapshot.asp?privcapId=21865991

*2 – Serial correlation refers to the correlation among the different futures contract months. *