Estimate the theoretical and empirical volatility term structure of futures returns
TSfit_volatility(parameters, futures, futures_TTM, dt)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.
vector. A named vector of parameters of an N-factor model. Function NFCP_parameters is recommended.
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.
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'.
numeric. Constant, discrete time step of observations, in years.
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:
_F() = _i=1^N _j=1^N _i _j _i,j e^-(_i + _j)sigma_F(tau) = sum_i = 1, j = 1^N sigma[i] sigma[j] rho[i,j] e^(-(kappa[i] + kappa[j]) tau)
Under the case that _1 = 0kappa[1] = 0, the model volatility term structure converges to _1^2sigma[1]^2 as tau grows large.
The empirical volatility term structure of futures returns is given by:
_F^2() = 1 t_i=1^N(log(F(t_i,)/F(t_i- t,)) - )^2hat(sigma)[F^2](tau) = 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."
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.
# 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)
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