NFCP (version 1.2.1)

futures_price_forecast: Forecast the futures prices of an N-factor model

Description

Analytically forecast future expected Futures prices under the risk-neutral version of a specified N-factor model.

Usage

futures_price_forecast(
  x_0,
  parameters,
  t = 0,
  futures_TTM = 1:10,
  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

numeric. The time point, in years, at which to forecast futures prices.

futures_TTM

vector. the time-to-maturity, in years, of futures contracts to forecast.

percentiles

vector. Optional. Probabilistic forecasting percentile intervals.

Value

futures_price_forecast returns a vector of expected Futures prices under a given N-factor model with specified time to maturities at time tt. When percentiles are specified, the function returns a matrix with the corresponding confidence bands in each column of the matrix.

Details

Under the assumption or risk-neutrality, futures prices are equal to the expected future spot price. Additionally, under deterministic interest rates, forward prices are equal to futures prices. Let F_T,tF[T,t] denote the market price of a futures contract at time tt with time TT until maturity. let * denote the risk-neutral expectation and variance of futures prices. The following equations assume that the first factor follows a Brownian Motion.

E^*[ln(F_T,t)] = season(T) + _i=1^Ne^-_iTx_i(0) + ^*t + A(T-t)E^*[ln(F[T,t])] = season(T) + sum_i=1^N (e^(-kappa[i] T) x[i,0] + mu * t + A(T-t))

Where: A(T-t) = ^*(T-t)-_i=1^N - 1-e^-_i (T-t)_i_i+12(_1^2(T-t) + _i.j 1 _i _j _i,j 1-e^-(_i+_j)(T-t)_i+_j) A(T-t) = mu^* (T-t) - sum_i=1^N ( - (1 - e^(-kappa[i] (T-t))lambda[i]) / kappa[i]) + 1/2 sigma[1]^2 (T-t) + sum_i.j != 1 sigma[i] sigma[j] rho[i,j] (1 - e^(-(kappa[i] + kappa[j])(T-t))) / (kappa[i] + kappa[j]) The variance is given by: Var^*[ln(F_T,t)]= _1^2t + _i.j1 e^-(_i + _j)(T-t)_i_j_i,j1-e^-(_i+_j)t_i+_j Var^*[ln(F[T,t])] = sigma[1]^2 * t + sum_i.j != 1 e^(-(kappa[i] + kappa[j])(T-t)) sigma[i] sigma[j] rho[i,j] (1 - e^(-(kappa[i] + kappa[j])t))/(kappa[i] + kappa[j])

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

# NOT RUN {
# Forecast futures prices of the Schwartz and Smith (2000) two-factor oil model:
## Step 1 - Run the Kalman filter for the two-factor oil model:
SS_2F_filtered <- NFCP_Kalman_filter(parameter_values = SS_oil$two_factor,
                                    parameter_names = names(SS_oil$two_factor),
                                    log_futures = log(SS_oil$stitched_futures),
                                    dt = SS_oil$dt,
                                    futures_TTM = SS_oil$stitched_TTM,
                                    verbose = TRUE)

## Step 2 - Probabilistic forecast of the risk-neutral two-factor
## stochastic differential equation (SDE):
futures_price_forecast(x_0 = SS_2F_filtered$x_t,
                      parameters = SS_oil$two_factor,
                      t = 0,
                      futures_TTM = seq(0,9,1/12),
                      percentiles = c(0.1, 0.9))
# }