To capture previously described behaviour of realised historical correlations, NYU Professor Roza Galeeva proposes a simple model and hereby shares the model’s applications.
In the previous introduction to our research, we described the importance of correlations for energy derivatives, and the challenges in their modelling. Here, we give further details of our approach and its applications.
To capture previously described behaviour of realised historical correlations, we start with the assumptions on the dynamics of the instantaneous, or local, correlation ρ(t, Ti, Tj) between two contracts Ti, Tj, with time moving closer to the earlier contract. The form of decay of correlations with distance between contracts, or even seasonality can be embedded in the local correlations at time t=Ti.
Using a simple form for Samuelson decay of volatilities, we show that the realised correlations between two contracts, calculated on the interval [t1, t2] can be obtained by transformations of the local correlations at the expiry of earlier contract, t=Ti, namely realised correlations are amplified versions of the those local correlations. The growth factor will depend on the length of the observation period t2-t1, the time to earliest contract Ti-t2 and the parameter α, which controls the dynamics (growth with time) of local correlations.
Based on this model, and using different forms of decay, we arrive to our models of correlation matrices, which present the desirable properties of typical calendar commodity correlations. The models are controlled by two parameters of growth and decay. We performed a detailed investigation of the surfaces of the minimum eigenvalue of the correlation matrix in the parameter space.
We fit one of the models to the historically realised oil correlations (WTI and Brent), using settlement futures price data for more than 10 years. The quality of the fit is excellent, with the average error of historical vs model correlations being less than 0.7% percent over the total correlation matrix. We also analysed Henry Hub natural gas historical calendar correlations, which present a bigger challenge due to seasonality. We had to introduce a parameter of scaling correlation between two different storage periods.
In addition to the physical commodities, we apply our model to interest rate futures, specifically the Eurodollar futures. Our model performs exceptionally well for these futures as well. As with oil correlations, we fit matrices within the average statistical error.
We apply our model to price a typical commodity derivative, subject to correlations, such as swaption. We investigate in detail how swaption prices depend on the “non-transparent” inputs, as Samuelson and correlations. Our correlation framework provides a simple way to parameterise correlation matrices and calibrate the parameters of the model to market swaption quotes, therefore obtain market implied parameters, similar to implied vols.
Our model provides a way to construct large correlation matrices that are simply constructed, carefully controlled, and positive definite. Furthermore, the framework is consistent with the realised historical correlations of physical and financial futures. The two parameters of growth and decay are straightforward parameters that capture the well-known phenomenon in oil and gas correlations. The methods can be also generalised to a correlation of any dimension with any delivery frequency. These features make for an effective model for derivatives pricing correlation products on assets like oil, gas, power and beyond.
We are excited to present our new innovative model to model commodity correlations at QuantMinds International and we are looking very much forward to the event.