Stochatics Modeling


QR Analytics Server offers state-of-the-art volatility modeling capabilities within its Stochastic Modeling Module.

Modeling Capabilities
  • This Module models the volatility of historic data. These could be load or prices at various pricing point. The models used are SDEs or stochastic differential equations. These models are dynamic, that is they evolve in time, in all aspects such as mean or average values including trend and seasonal features, as well as randomness. In this regard they are different from and more powerful than simple time series models used in econometrics.
  • SDEs have parameters which must be estimated or calibrated based on spot historic data. This is mathematically carried out in a way that a calibrated SDE has the highest likelihood of simulating time series such as the historic data that was used to train it. We implement quasi-maximum likelihood techniques.
  • The Stochastic Modeling Module allows you to select a pricing point / market node and attach to any such driver a stochastic differential equation from a pull down list.
  • The models used include 1 and 2-factor stochastic differential equations (SDE). The drift of the models is time variable to allow modeling seasonality and trend for energy and commodities. We offer several mean reverting models. These are well suited for modeling the term structure of forward markets. The models offer proportional noise or volatility for better modeling commodity and energy prices.
  • Jump processes can be added. You can switch jumps off and on for any model chosen.
  • The 2 factor models with jumps are appropriate for electricity prices and can build realistic (non-flat) term structure of energy forward markets.

Calibration Process
  • Select a pricing point / market node, open the modeling dialog and assign to it one of the stochastic models.
  • Select the period of historic data you wish to use to calibrate the SDE model.
  • Once you assign an SDE to a pricing point, this choice determines the Monte Carlo engine, which is thus configured one pricing point at a time. You name this selection and save it.
  • Note that you can assign different SDE models to the same pricing point, a 1 factor model, a 2 factor model, etc, and you can save them under different names and use them in different Monte Carlo simulations.

  • Calibration or Parameter Estimation of the SDE model attached to a pricing point in your portfolio is a far reaching generalization of estimating volatility or correlations. It is a lot more sophisticated than distribution fitting, it is a dynamic and non-stationary process. After you assign an SDE model to a Pricing Point to model its random dynamics, this Module automatically estimates its parameters:
    • The strength or speed of mean reversion.
    • The reversion level or market equilibrium.
    • Seasonal Shape.
    • Volatility, etc.

Back Testing
  • After calibration, the module immediately runs a series of Monte Carlo simulations using the parameters thus calibrated. Then it creates a risk curve around the actual historic data to backtest the calibration process to visually prove goodness of fit.
  • The volatility parameter of the model estimated during calibration, can be adjusted by the user after backtesting to achieve a better fit (narrow or widen the risk cone).
  • The system has the ability to diagnose a wrong stochastic model for a given data type and make appropriate suggestions.

Sample calibration and backtesting of a 2-factor mean-reverting model applied to electricity prices (see image). The model is highly successful for following reasons:


Daily Nodal Price Modeling
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1/2 Hourly Nodal Price Modeling
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  • The black curve is the historical data points used for to calibrate or estimate the parameters of the selected stochastic model.
  • The red band is the 90% confidence or risk band, containing 90% of the scenarios simulated with the calibrated model. This particular calibration is quite good as approximately 90% of historical black data points are within the red band.

  • The green line appears to be the moving average of the historical black data curve. This is good because the green line is the expected value of the simulations.

  • The simulated blue sample path appears to have the same stochastic behavior as the black historical data. In other words it has the same amount of jaggedness.

  • The jumps of the blue sample path have approximately same heights (on average) as the black historical data.

  • The 2-factor model has a mean or average that is a "hidden" or is the unobservable factor. This is represented by the (purple line). Note that the Monte Carlo simulated blue sample path reverts to the purple line.

3 Months
out of the box Implementation $80 K per Month
Video Demo
Sample Videos
Stochastic Modeling