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A Regression Start and Likelihood Setup for CIR Parameter Estimation

Article Quant Q&A · Author: SchnitzelRaver
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Summary

The document concerns estimating parameters for a stochastic volatility model in which variance follows a Cox–Ingersoll–Ross (CIR) square-root process. The question asks for estimation from time-series data rather than option prices, with quasi-maximum likelihood or generalized method of moments as possible approaches. The response provides a MATLAB routine for CIR estimation and describes inputs such as observations, time step, and choices for evaluating the likelihood.

The routine first transforms changes in the observed series and uses a linear regression to form initial estimates of mean-reversion speed, long-run level, and volatility. It then passes those starting values to a numerical optimizer, with alternative objective functions associated with a noncentral chi-square density or a Bessel-function approach. The example is only a scaffold: the objective functions are not included, and the stated input describes interest-rate observations, so it does not fully explain applying the routine to the correlated stochastic-volatility model in the question. No estimation results or validation are reported.

Key ideas

  • The example estimates CIR mean reversion, long-run level, and volatility from a time series.
  • A regression on transformed observations supplies starting values for numerical likelihood optimization.
  • The routine offers alternative likelihood calculations based on a noncentral chi-square density or a Bessel function.
  • The response omits the objective-function implementations and does not demonstrate validation on the requested stochastic-volatility model.

Tags

StatisticsVolatilityMachine learning
Full text
# Stochastic Volatility CIR estimation


# Stochastic Volatility CIR estimation












Would anyone have a code (pref. Matlab or R) for any type of estimation (QML, GMM) not using option prices of a stochastic volatility model driven by a CIR process described below?

\begin{equation} dS_t = \mu dt + \sqrt{v_t} dW_t^1 \end{equation} \begin{equation} dv_t = \kappa (\theta - v_t) dt + \sigma \sqrt{v_t} dW^2_t \end{equation} such that $dW_{t}^{1}\,dW_{t}^{2}=\rho dt$

I just need to cross-check my results.

Thanks!

## Answer by user16891 (score 1)

https://quant.stackexchange.com/a/19233



- Input Model.Data = Time series of interest rates observations. Model.TimeStep = Delta t; recommended: 1/250 for daily data and 1/12 for monthly data. Model.Disp = 'y' | 'n', (default: 'n'). Model.Method = 'ncx2pdf' | 'besseli' (default: 'besseli'). Model.MatlabDisp = 'off' | 'iter' | 'notify' |' final' (default: 'off').



```
function Results = CIRestimation(Model)
Nobs = length(Model.Data);
r = Model.Data(1:end-1);
dr = diff(Model.Data);           
dr = dr./r.^0.5;
regressors = [Model.TimeStep./r.^0.5, Model.TimeStep*r.^0.5];
drift = regressors\dr;
res = regressors*drift - dr;
alpha = -drift(2);
theta = -drift(1)/drift(2);
sigma = sqrt(var(res, 1)/Model.TimeStep);
InitialParams = [kappa theta sigma];
if ~isfield(Model, 'Disp'), Model.Disp = 'y'; 
end;
if strcmp(Model.Disp, 'y')
    fprintf('\n initial kappa=...%+3.6f\n initial theta=...%+3.6f\n initial sigma = %+3.6f\n',kappa, theta, sigma);
end
if ~isfield(Model, 'MatlabDisp'), Model.MatlabDisp = 'off';
end;
options = optimset('LargeScale', 'off', 'MaxIter', 300, 'MaxFunEvals',300, 'Display', Model.MatlabDisp, 'TolFun', 1e-4, 'TolX', 1e-4, 'TolCon', 1e-4); 
if ~isfield(Model, 'Method'), Model.Method = 'besseli'; 
end;
if strcmp(Model.Method, 'ncx2pdf')
    [Params, Fval, Exitflag] =  fminsearch(@(Params) CIRobjective2(Params, Model),InitialParams, options);   
else
    [Params, Fval, Exitflag] =  fminsearch(@(Params) CIRobjective1(Params, Model),InitialParams, options);   
end   
Results.Params = Params;
Results.Fval = -Fval/Nobs;
Results.Exitflag = Exitflag;

if strcmp(Model.Disp, 'y')
    fprintf('\n kappa = %+3.6f\n theta    = %+3.6f\n sigma = %+3.6f\n',...
        Params(1), Params(2), Params(3));
    fprintf(' log-likelihood = %+3.6f\n', -Fval/Nobs);
end
```

end

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