spot_price_forecast: Forecast spot prices of an N-factor model

In NFCP: N-Factor Commodity Pricing Through Term Structure Estimation

Forecast spot prices of an N-factor model

Description

\loadmathjax

Analytically forecast expected spot prices following the "true" process of a given n-factor stochastic model

Usage

spot_price_forecast(x_0, parameters, t, percentiles = NULL)

Arguments

x_0	vector. Initial values of the state variables, where the length must correspond to the number of factors specified in the parameters.
parameters	vector. A named vector of parameter values of a specified N-factor model. Function NFCP_parameters is recommended.
t	vector. Discrete time points, in years, to forecast spot prices
percentiles	vector. Optional. Probabilistic forecasting percentile intervals.

Details

Future expected spot prices under the N-factor model can be forecasted through the analytic expression of expected future prices under the "true" N-factor process.

Given that the log of the spot price is equal to the sum of the state variables (equation 1), the spot price is log-normally distributed with the expected prices given by:

\mjdeqn

E[S_t] = exp(E[ln(S_t)] + \frac12Var[ln(S_t)])exp(E[ln(S[t])] + 1/2 Var[ln(S[t])]) Where: \mjdeqnE[ln(S_t)] = season(t) + \sum_i=1^Ne^-(\kappa_it)x_i(0) + \mu tE[ln(S[t])] = season(t) + sum_i=1^N (e^(-(kappa[i] t)) x[i,0] + mu * t)

Where \mjeqn\kappa_i = 0kappa[i] = 0 when GBM=T and \mjeqn\mu = 0mu = 0 when GBM = F

\mjdeqn

Var[ln(S_t)] = \sigma_1^2t + \sum_i.j\neq1\sigma_i\sigma_j\rho_i,j\frac1-e^-(\kappa_i+\kappa_j)t\kappa_i+\kappa_j Var[ln(S[t])] = sigma[1]^2 * t + sum_i.j != 1 (sigma[i] sigma[j] rho[i,j] (1 - e^(-(kappa[i] + kappa[j])t)) / (kappa[i] + kappa[j]) )

and thus:

\mjdeqn

E[S_t] = exp(season(t) + \sum_i=1^N e^-\kappa_itx_i(0) + (\mu + \frac12\sigma_1^2)t + \frac12\sum_i.j\neq1 \sigma_i\sigma_j\rho_i,j\frac1-e^-(\kappa_i+\kappa_j)t\kappa_i+\kappa_j) E[S[t]] = exp(season(t) + sum_i=1^N e^(-kappa[i] t) x[i,0] + (mu + 1/2 sigma[1]^2)t + 1/2 (sum_i.j != 1( sigma[i] sigma[j] rho[i,j] (1 - e^(-(kappa[i] + kappa[j])t)) / (kappa[i] + kappa[j]))) )

Under the assumption that the first factor follows a Brownian Motion, in the long-run expected spot prices grow over time at a constant rate of \mjeqn\mu + \frac12\sigma_1^2mu + 1/2 sigma[1] as the \mjeqne^-\kappa_ite^(-kappa[i] * t) and \mjeqne^-(\kappa_i + \kappa_j)te^(-(kappa[i] + kappa[j])) terms approach zero.

An important consideration when forecasting spot prices using parameters estimated through maximum likelihood estimation is that the parameter estimation process takes the assumption of risk-neutrality and thus the true process growth rate \mjeqn\mumu is not estimated with a high level of precision. This can be shown from the higher standard error for \mjeqn\mumu than other estimated parameters, such as the risk-neutral growth rate \mjeqn\mu^*mu^*. See Schwartz and Smith (2000) for more details.

Value

spot_price_forecast returns a vector of expected future spot prices under a given N-factor model at specified discrete future time points. When percentiles are specified, the function returns a matrix with the corresponding confidence bands in each column of the matrix.

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

# Forecast the Schwartz and Smith (2000) two-factor oil model:

##Step 1 - Kalman filter of the two-factor oil model:
SS_2F_filtered <- NFCP_Kalman_filter(SS_oil$two_factor,
                                    names(SS_oil$two_factor),
                                    log(SS_oil$stitched_futures),
                                    SS_oil$dt,
                                    SS_oil$stitched_TTM,
                                    verbose = TRUE)

##Step 2 - Probabilistic forecast of N-factor stochastic differential equation (SDE):
spot_price_forecast(x_0 = SS_2F_filtered$x_t,
                   parameters = SS_oil$two_factor,
                   t = seq(0,9,1/12),
                   percentiles = c(0.1, 0.9))

NFCP documentation built on March 18, 2022, 5:06 p.m.