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:
E[S_t] = exp(E[ln(S_t)] + 12Var[ln(S_t)])exp(E[ln(S[t])] + 1/2 Var[ln(S[t])])
Where:
E[ln(S_t)] = _i=1^Ne^-(_it)x_i(0) + tE[ln(S[t])] = sum_i=1^N (e^(-(kappa[i] t)) x[i,0] + mu * t)
Where _i = 0kappa[i] = 0 when GBM=T
and = 0mu = 0 when GBM = F
Var[ln(S_t)] = _1^2t + _i.j1_i_j_i,j1-e^-(_i+_j)t_i+_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:
E[S_t] = exp(_i=1^N e^-_itx_i(0) + ( + 12_1^2)t + 12_i.j1 _i_j_i,j1-e^-(_i+_j)t_i+_j)
E[S[t]] = exp( 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 + 12_1^2mu + 1/2 sigma[1] as the e^-_ite^(-kappa[i] * t) and e^-(_i + _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 mu is not estimated with a high level of precision. This can be shown from the higher standard error for mu than other estimated parameters, such as the risk-neutral growth rate ^*mu^*. See Schwartz and Smith (2000) for more details.