The Schwartz-Smith 2-factor model decomposes the log spot price of a commodity into two components:
- Long-term equilibrium level (non-stationary, modelled as Brownian motion with drift)
- Short-term deviations (mean-reverting component)
The original code uses a custom Kalman filter for parameter estimation and filtering. To adapt this into MATLAB's SSM (State-Space Model) framework, the model needs to be expressed in the standard linear Gaussian state-space form:
SSM Formulation Overview
State Equation:
Let the state vector be:
xt=[χt
ξt]
- χt : long-term (non-stationary)
- ξt: short-term (mean-reverting)
The dynamics are:
χt+1=χt+μ+ηt
ξt+1=χt+μ+ηt=ϕξt+ϵt
where ηt∼N(0,ση2) and ϵt∼N(0,σϵ2)
Observation Equation:
yt=χt+ξt+εt , εt∼N(0,σε2)
Please refer the MATLAB pseudo-code for the same:
% Define state-space model for Schwartz-Smith
% Parameters: mu, phi, sigma_eta, sigma_eps, sigma_obs
A = @(params) [1 0; 0 params(2)]; % phi
B = @(params) [1; 0]; % mu only affects chi_t
C = @(params) [1 1]; % observation: chi + xi
D = @(params) eye(2) .* [params(3); params(4)]; % process noise
E = @(params) params(5); % observation noise
% Define ssm object
ssmModel = ssm(...
@(params) deal(A(params), B(params) * params(1), C(params), E(params), D(params) * D(params)'));
% Initial parameter guess: [mu, phi, sigma_eta, sigma_eps, sigma_obs]
param0 = [0.1, 0.9, 0.1, 0.1, 0.1];
% Load your observed log spot price time series
% y = logPrices;
% Estimate parameters
[estModel, estParams, estSE] = estimate(ssmModel, logPrices, param0);
Please refer these MATLAB documentations for more details:
- State-Space Models - https://www.mathworks.com/help/control/ug/state-space-models.html
- Estimate State-Space Model - https://www.mathworks.com/help/ident/ref/estimatestatespacemodel.html
- Kalman Filtering - https://www.mathworks.com/help/control/ug/kalman-filtering.html
Hope this is beneficial!
