Calibration of Forward Price, Volatility, and Correlations across multiple assets

Calibration of multiple Fwd Prices and Vol Curves

This paper outlines a methodology for calibrating the Schwartz-Smith two factor commodity pricing model across multiple commodities such that on the valuation date:

1. The calibration produces simulations that are consistent with the current forward curve term structure
2. The calibration produces simulations that are consistent with the current ATM (at-the-money) volatility term structure
3. The calibration produces simulations that are consistent with the correlation inputs between the lognormal movements of different commodities

The primary difference between the approach presented here and the calibration methodologies typically used by other papers implementing a derivative of the Schwartz-Smith model is that this approach does not consider the historical forward price evolution to calibrate the spot dynamics. Instead, it relies on the use of available market data, more specifically the forward term structure and implied volatility curves, as inputs.

In this example, I use four assets: 5x16,2x16,7x8 PJM forward prices and ATM volatilities along with natural gas forward prices and ATM volatilities

Calibration of the parameters is done in Excel. By inputing the vectors of Forward and ATM volatilites for each commodity, I can compute the theoretical Schwartz Smith Forward Prices and Standard Deviations for a given maturity. I then use Excel solver to minimize the difference between the observed market values and their theoretical values to obtain the Schwartz-Smith parameters for each commodity. Note this methodology is drastically different from the one described in "Short Term Variation and Long-Term Dynamics in Commodity Prices" in which Schwartz and Smith calibrate their model to historical futures prices. Here I calibrate the model to the current forward and volatility curve as well as adjust for seasonality. This procedure is much more practical pricing methodology.

The more difficult step was insuring that the correlation between and asset(i) and asset(j) at maturity (t) was consistent with the implied correlation between the forward price vectors. This was done by adjusting the correlation between the short-term factors of commodities at each maturity. The matlab code factors in this adjustment and simulates the 4 commodities in this example to show that the theoretical prices, volatilities, and correlations match up with the observed market data.

There is not, to my knowledge, a commodities methodology that incoporates so many market factors across multible commodities into one simulation. The advantages of such a model allows for more accurate modeling of spark spreads and pricing of deals that are dependent on multiple commodities prices. I have included all files, including excel, associated with this calibration and simulation.