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Monte Carlo Simulation

Stochastic Modeling of Spot and Forward Prices. Monte Carlo Simulation.

QR Monte Carlo module offers a unique and powerful easy to use environment to model the volatility of financial spot and forward market prices and volume at various pricing points.

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    QR Monte Carlo simulations apply to any security type: equity, FX, interest rates, energy, commodity or any data for that matter.
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    A complete range of mathematical models (stochastic differential equations or SDEs) are offered by QR Monte Carlo Simulation module to choose from. These dynamic models 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 time series models used in econometrics, or simple volatility matrix.
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    Build via Monte Carlo simulation of 2-factor models very realistic (non-flat) term structure of forward markets. The Monte Carlo engine automatically builds or simulates the spot and forward curves together in a coherent term structure, where nearby forwards are as volatile as the spot and far out forwards have decaying volatility. A Market Price of Risk is estimated for every forward position.
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    When performing Monte Carlo simulation, you can turn jumps on and off for any chosen model. The 2 factor models with jumps are appropriate for spot electricity price modeling.
    QR Monte Carlo Module
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    Automatic correlation computation across multiple spot and forward curves.
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    All you need to provide to calibrate a Monte Carlo simulation model is minimal historic data. The System performs all tasks automatically.

Stochastic Models

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    The stochastic differential equations (SDEs) models we use for Monte Carlo simulation are 1 and 2-factor models, with time variable mean reverting drifts to allow the modeling of seasonality or cyclical shapes as well as trend for energy, commodities, and various interest and FX rates and all market indices.
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    We offer about 10 different SDEs all adapted to financial modeling. These stochastic models have proportional noise or volatility to perfectly capture and model market volatility via Monte Carlo simulation. We offer several mean reverting 2-factor models. These are well suited for modeling the term structure of forward markets

Model Calibration

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    Any pricing point/market node can be pointed to a stochastic differential equation (SDE) for modeling. SDEs have parameters such as strength or speed of mean reversion, the reversion level or market equilibrium, the seasonal shape, volatility, etc.
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    The SDEs parameters must be estimated or calibrated based on historic data. Parameter estimation 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; and for 2 factor models, the second or unobservable variable is unfolded via advanced simulation filtering followed by noise reduction.
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    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 used for calibration to visually back test and validate the calibration process.
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    All parameters including volatility parameter that are estimated during calibration are displayed and can be adjusted to achieve a better fit (narrow or widen the risk cone).
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    The system has the ability to diagnose a wrong stochastic model for a given data type and make appropriate suggestions.

Monte Carlo Simulations

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    The stochastic differential equation (SDEs) models are solved numerically by simulating many sample paths and forming a probability distribution. This is called Monte Carlo simulation. We implement 1st and 2nd order methods for faster convergence and dynamic time-step mesh for robustness and preventing divergence.
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    The Monte Carlo engine automatically builds or simulates the spot and forward curves together in a coherent term structure. The differential equations linking the spot and forwards have been solved for every SDE Model provided. A Market Price of Risk is estimated for every forward position.
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    For each day full statistics are computed on the probability distribution of the simulated paths predicting the future: mean, standard deviation, skewness, Kurtosis, risk tale computed at 95-99 percentile. These results are displayed in tabular formats and plotted as multi-colors curves, displaying the full risk cone as it expands in time.
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    You can set the parameters of a Monte Carlo simulation run such as number of simulations, the time horizon in days, months or years, and the timescale, such as hourly, 1/2 hourly and 1/4 hourly.
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    The output of a Monte Carlo simulation are daily statistics computed on the probability distribution of the simulated paths predicting the future. First and foremost, the tales are computed at 95 percentile: the highest or best case (meaning future predicted values will only exceed this value 5 times in 100) and the risk or worst case (meaning future predicted values will only fall below this value 5 times in 100). But also typical moments from the distribution are computed: the mean, the standard deviation, skewness or 3rd moment and Kurtosis or 4th moment.