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.
NFCP.MLE(
log.futures,
dt,
TTM,
N.factors,
GBM = TRUE,
S.Constant = TRUE,
Estimate.Initial.State = FALSE,
Richardsons.Extrapolation = TRUE,
cluster = FALSE,
Domains = NULL,
...
)Object of class matrix corresponding to the natural logarithm of observable futures prices.
NA's are allowed within the matrix. Every column of the matrix must correspond to a particular futures contract,
with each row corresponding to a quoted price on a given date.
Constant, discrete time step of observations
Object of class vector or matrix that specifies the time to maturity of observed futures contracts.
time to maturity can be either constant (i.e. class vector) or time dependent (i.e. class matrix).
When the time to maturity of observed futures contracts is time dependent, the dimensions of
TTM must be identical to that of log.futures. Every element of TTM
corresponds to the time to maturity, in years, of a futures contract at a given observation date.
numeric. Number of state variables in the spot price process.
logical. When TRUE, factor 1 of the model is assumed to follow a Brownian Motion, inducing a unit-root in the spot price process.
logical. When TRUE, the white noise of observable contracts are assumed identical and independent.
logical. When TRUE, the initial state vector is specified as unknown parameters of the commodity pricing model. When FALSE,
a "diffuse" assumption is taken instead (see details)
logical. When TRUE, the grad function from the numDeriv package is called to
approximate the gradient within the genoud optimization algorithm.
an optional object of the 'cluster' class returned by one of the makeCluster commands in the parallel package to allow for parameter estimation
to be performed across multiple cluster nodes.
an optional matrix of the lower and upper bounds for the parameter estimation process. The NFCP.bounds function is highly recommended.
When Domains is not specified, the standard bounds specified within the NFCP.bounds function are used.
additional arguments to be passed into the genoud function. See help(genoud)
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.
|
numeric The Maximum-Likelihood-Estimate of the solution |
|
vector. The estimated parameters |
|
vector. Standard error of the estimated parameters. Returned only when hessian = T is specified |
|
vector. The final observation of the state vector |
|
matrix. All observations of the state vector, after the updating equation has been applied |
|
matrix. Estimated futures prices at each observation |
|
matrix. Estimation error of each futures contracts at each observation |
|
matrix. The Mean Error (Bias), Mean Absolute Error, Standard Deviation of Error and Root Mean Squared Error (RMSE) of each
observed contract, matching the column names of log.futures |
|
matrix. The theoretical and empirical volatility of futures returns for each observed contract as returned from the TSFit.Volatility function |
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.bounds 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 NN state variables x_tx[t].
When GBM = TRUE:
log(S_t) = _i=1^N x_i,tlog(S[t]) = sum_i=1^n x[i,t]
When GBM = FALSE:
log(S_t) = E + _i=1^N x_i,tlog(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.
State variables are thus assumed to follow the following processes:
When GBM = TRUE:
dx_1,t = ^*dt + _1 dw_1tdx[1,t] = mu^* dt + sigma[1] dw[1]t
When GBM = FALSE:
dx_1,t = - (_1 + _1x_1,t)dt + _1 dw_1tdx[1,t] = - (lambda[1] + kappa[1] x[1,t]) dt + sigma[1] dw[t]t
And: dx_i,t =_i 1 - (_i + _ix_i,t)dt + _i dw_itdx[i,t] =_(i != 1) - (lambda[i] + kappa[i] x[i,t]dt + sigma[i] dw[i]t)
where: E(w_i)E(w_j) = _i,jE(w[i])E(w[j])
The following constant parameters are defined as:
var mu: long-term growth rate of the brownian motion process.
var EE: Constant equilibrium level.
var ^*=-_1mu^* = mu-lambda[1]: Long-term risk-neutral growth rate
var _ilambda[i]: Risk premium of state variable ii.
var _ikappa[i]: Reversion rate of state variable ii.
var _isigma[i]: Instantaneous volatility of state variable ii.
var _i,j [-1,1]rho[i,j] in [-1,1]: Instantaneous correlation between state variables ii and jj.
Measurement Error:
The Kalman filtering algorithm assumes a given measure of measurement error (ie. matrix HH). 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) and are assumed independent.
var s_is[i] Observation error of contract ii.
When S.Constant = T, the values of parameter s_is[i] are assumed constant and equal to parameter object 'sigma.contracts'. When S.Constant = F, observation errors are assumed unique, where the error of
futures contracts s_is[i] is equal to parameter object 'sigma.contract_' [i] (i.e. 'sigma.contract_1', 'sigma.contract_2', ...).
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 QQ
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).
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/.
# NOT RUN {
##Example 1 - Perform One Generation of Maximum Likelihood Estimation on the
##first 20 weekly observations of the Schwartz and Smith (2000) Crude Oil Data:
##Example 1 - Complete, Stitched Data:
Schwartz.Smith.Two.Factor.Stitched <- NFCP.MLE(
####Arguments
log.futures = log(SS.Oil$Stitched.Futures)[1:5,1],
dt = SS.Oil$dt,
TTM= SS.Oil$Stitched.TTM[1],
S.Constant = FALSE, N.factors = 2, GBM = TRUE,
cluster = NULL,
hessian = TRUE,
####Genoud arguments:
Richardsons.Extrapolation = FALSE,
pop.size = 4, optim.method = "L-BFGS-B", print.level = 0,
max.generations = 0, solution.tolerance = 0.1)
##Example 2 - Incomplete, Contract Data:
Schwartz.Smith.Two.Factor.Contract <- NFCP.MLE(
####Arguments
log.futures = log(SS.Oil$Contracts)[1:20,1:5],
dt = SS.Oil$dt,
TTM= SS.Oil$Contract.Maturities[1:20,1:5],
S.Constant = TRUE,
N.factors = 2, GBM = TRUE,
cluster = NULL,
hessian = TRUE,
####Genoud arguments:
Richardsons.Extrapolation = FALSE,
pop.size = 4, optim.method = "L-BFGS-B", print.level = 0,
max.generations = 0, solution.tolerance = 0.1)
# }
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