TSfit_volatility {NFCP}

Calculate the volatility term structure of futures returns

Description

Estimate the theoretical and empirical volatility term structure of futures returns

Usage

TSfit_volatility(parameters, futures, futures_TTM, dt)

Arguments

parameters

vector. A named vector of parameters of an N-factor model. Function NFCP_parameters is recommended.

futures

matrix. Historical observes futures price data. Each column must correspond to a listed futures contract and each row must correspond to a discrete observation of futures contracts. NA's are permitted.

futures_TTM

vector. Each element of 'futures_TTM' must correspond to the time-to-maturity from the current observation point of futures contracts listed in object 'futures'.

dt

numeric. Constant, discrete time step of observations, in years.

Details

The fit of an N-factor models theoretical volatility term structure of futures returns to those obtained directly from observed futures prices can be used as a measure of robustness for the models ability to explain the behaviour of a commodities term structure.

The theoretical model volatility term structure of futures returns is given by the following equation:

\[\sigma_F(\tau) = \sum_{i=1}^N \sum_{j=1}^N \sigma_i \sigma_j \rho_{i,j} e^{-(\kappa_i + \kappa_j)\tau}\]

Under the case that \(\kappa_1 = 0\), the model volatility term structure converges to \(\sigma_1^2\) as \(\tau\) grows large.

The empirical volatility term structure of futures returns is given by:

\[\hat\sigma_F^2(\tau) = \frac{1}{\Delta t}\sum_{i=1}^N(log(F(t_i,\tau)/F(t_i-\Delta t,\tau)) - \bar\mu)^2\]

According to Cortazar and Naranjo (2006): "A larger number of factors gives more flexibility to adjust first and second moments simultaneously, hence explaining why (a) four-factor (may) outperform (a) three-factor one in fitting the volatility term structure."

Value

TSfit_volatility returns a matrix with the theoretical and empirical volatility term structure of futures returns, with the number of columns of this matrix coinciding with the number of input futures contracts.

References

Schwartz, E. S., and J. E. Smith, (2000). Short-Term Variations and Long-Term Dynamics in Commodity Prices. Manage. Sci., 46, 893-911.

Cortazar, G., and L. Naranjo, (2006). An N-factor Gaussian model of oil futures prices. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 26(3), 243-268.

Examples

# Test the volatility term structure fit of the Schwartz-Smith two-factor model on crude oil:
V_TSFit <- TSfit_volatility(
parameters = SS_oil$two_factor,
futures = SS_oil$stitched_futures,
futures_TTM = SS_oil$stitched_TTM,
dt = SS_oil$dt)