Description

the NFCP_MLE function performs parameter estimation of an n-factor model given observable term structure futures data through maximum likelihood estimation. NFCP_MLE allows for missing observations as well as constant or variable time to maturity of observed futures contracts.

Usage

Arguments

Details

NFCP_MLE is a wrapper function that uses the genetic algorithm optimization function genoud from the rgenoud package to optimize the log-likelihood score returned from the NFCP_Kalman_filter function. When Richardsons_extrapolation = TRUE, gradients are approximated numerically within the optimization algorithm through the grad function from the numDeriv package. NFCP_MLE is designed to perform parameter estimation as efficiently as possible, ensuring a global optimum is reached even with a large number of unknown parameters and state variables. Arguments passed to the genoud function can greatly influence estimated parameters and must be considered when performing parameter estimation. Recommended arguments to pass into the genoud function are included within the vignette of NFCP. All arguments of the genoud function may be passed through the NFCP_MLE function (except for gradient.check, which is hard set to false).

NFCP_MLE performs boundary constrained optimization of log-likelihood scores and does not allow does not allow for out-of-bounds evaluations within the genoud optimization process, preventing candidates from straying beyond the bounds provided by Domains. When Domains is not specified, the default bounds specified by the NFCP_domains function are used.

The N-factor model The N-factor model was first presented in the work of Cortazar and Naranjo (2006, equations 1-3). The N-factor framework describes the spot price process of a commodity as the correlated sum of \(N\) state variables \(x_t\).

When GBM = TRUE: \[log(S_{t}) = \sum_{i=1}^N x_{i,t}\] When GBM = FALSE: \[log(S_{t}) = E + \sum_{i=1}^N x_{i,t}\]

Additional factors within the spot-price process are designed to result in additional flexibility, and possibly fit to the observable term structure, in the spot price process of a commodity. The fit of different N-factor models, represented by the log-likelihood can be directly compared with statistical testing possible through a chi-squared test.

Flexibility in the spot price under the N-factor framework allows the first factor to follow a Brownian Motion or Ornstein-Uhlenbeck process to induce a unit root. In general, an N-factor model where GBM = T allows for non-reversible behaviour within the price of a commodity, whilst GBM = F assumes that there is a long-run equilibrium that the commodity price will revert to in the long-term.

State variables are thus assumed to follow the following processes:

When GBM = TRUE: \[dx_{1,t} = \mu^*dt + \sigma_{1} dw_{1}t\]

When GBM = FALSE: \[dx_{1,t} = - (\lambda_{1} + \kappa_{1}x_{1,t})dt + \sigma_{1} dw_{1}t\]

And: \[dx_{i,t} =_{i\neq 1} - (\lambda_{i} + \kappa_{i}x_{i,t})dt + \sigma_{i} dw_{i}t\]

where: \[E(w_{i})E(w_{j}) = \rho_{i,j}\]

The following constant parameters are defined as:

param \(\mu\): long-term growth rate of the Brownian Motion process.

param \(E\): Constant equilibrium level.

param \(\mu^*=\mu-\lambda_1\): Long-term risk-neutral growth rate

param \(\lambda_{i}\): Risk premium of state variable \(i\).

param \(\kappa_{i}\): Reversion rate of state variable \(i\).

param \(\sigma_{i}\): Instantaneous volatility of state variable \(i\).

param \(\rho_{i,j} \in [-1,1]\): Instantaneous correlation between state variables \(i\) and \(j\).

Disturbances - Measurement Error:

The Kalman filtering algorithm assumes a given measure of measurement error or disturbance in the measurement equation (ie. matrix \(H\)). Measurement errors can be interpreted as error in the model's fit to observed prices, or as errors in the reporting of prices (Schwartz and Smith, 2000). These disturbances are typically assumed independent.

var \(ME_i\) measurement error of contract \(i\).

where the measurement error of futures contracts \(ME_i\) is equal to 'ME_' [i] (i.e. 'ME_1', 'ME_2', ...) specified in arguments parameter_values and parameter_names.

There are three particular cases on how the measurement error of observations can be treated in the NFCP_Kalman_filter function:

Case 1: Only one ME is specified. The Kalman filter assumes that the measurement error of observations are independent and identical.

Case 2: One ME is specified for every observed futures contract. The Kalman filter assumes that the measurement error of observations are independent and unique.

Case 3: A series of ME's are specified for a given grouping of maturities of futures contracts. The Kalman filter assumes that the measurement error of observations are independent and unique to their respective time-to-maturity.

Grouping of maturities for case 3 is specified through the ME_TTM argument. This is a vector that specifies the maximum maturity to consider for each respective ME parameter argument.

in other words, ME_1 is considered for observations with TTM less than ME_TTM[1], ME_2 is considered for observations with TTM less than ME_TTM[2], ..., etc.

The first case is clearly the simplest to estimate, but can be a restrictive assumption. The second case is clearly the most difficult to estimate, but can be an infeasible assumption when considering all available futures contracts that make up the term structure of a commodity.

Case 3 thus serves to ease the restriction of case 1, and allow the user to make the modeling of measurement error as simple or complex as desired for a given set of maturities.

Diffuse Kalman Filtering

If estimate_initial_state = F, a 'diffuse' assumption is used within the Kalman filtering algorithm. Factors that follow an Ornstein-Uhlenbeck are assumed to equal zero. When estimate_initial_state = F and GBM = T, the initial value of the first state variable is assumed to equal the first element of log_futures. This is an assumption that the initial estimate of the spot price is equal to the closest to maturity observed futures price.

The initial covariance of the state vector for the Kalman Filtering algorithm assumed to be equal to matrix \(Q\)

Initial states of factors that follow an Ornstein-Uhlenbeck process are generally not estimated with a high level of precision, due to the transient effect of the initial state vector on future observations, however the initial value of a random walk variable persists across observations (see Schwartz and Smith (2000) for more details).

Value

NFCP_MLE returns a list with 10 objects. 9 objects are returned when the user has specified not to calculate the hessian matrix at solution.

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.

Mebane, W. R., and J. S. Sekhon, (2011). Genetic Optimization Using Derivatives: The rgenoud Package for R. Journal of Statistical Software, 42(11), 1-26. URL http://www.jstatsoft.org/v42/i11/.

Examples