Calibration of Stochastic
Convenience Yield Models For
Crude Oil Using the Kalman Filter.
Adriaan Krul
Delft – 22-02-08
www.ing.com
Contents
 Introduction
Convenience yield follows Ornstein-Uhlenbeck process
Analytical results
Convenience yield follows Cox-Ingersoll-Ross process
Analytical results
 Numerical results
Conclusion
Further research
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Introduction
A future contract is an agreement between two parties to
buy or sell an asset at a certain time in the future for a
certain price.
Convenience yield is the premium associated with
holding an underlying product or physical good, rather
than the contract of derivative product.
Commodities – Gold, Silver, Copper, Oil
We use futures of light crude oil ranging for a period from
01-02-2002 until 25-01-2008 on each friday to prevent
weekend effects.
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Stochastic convenience yield; first approach
We assume that the spot price of the commodity follows an
geometrical brownian motion and that the
convenience yield follows an Ornstein-Uhlenbeck process. I.e.,
we have the joint-stochastic process
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In combination with the transformation x = ln S we have
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Analytical results
Expectation of the convenience yield
Variance of the convenience yield
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Analytical results
PDE of the future prices
Closed form solution of the future prices
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Stochastic convenience yield; second approach
We assume that the spot price of the commodity follows an
geometrical brownian motion and that the convenience yield
follows a Cox-Ingersoll-Ross process. I.e., we have the jointstochastic process
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Together with the transformation x = ln S, we have
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Analytical results
Expectation of the convenience yield
Variance of the convenience yield
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Analytical results
PDE of the future prices
Closed form solution of the future prices
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Kalman filter
Since the spot price and convenience yield of commodities are
non-observable state-variables, the Kalman Filter is the
appropriate method to model these variables.
The main idea of the Kalman Filter is to use observable
variables to reconstitute the value of the non-observable
variables. Since the future prices are widely observed and
traded in the market, we consider these our observable
variables.
The aim of this thesis is to implement the Kalman Filter and test
both the approaches and compare them with the market data.
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Kalman Filter for approach one.
Recall that the closed form solution of the future price
was given by
From this the measurement equation immediately follows
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From
we can write
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Kalman filter for approach one
The difference between the closed form solution and the
measurement equation is the error term epsilon.
This error term is included to account for possible
errors. To get a feeling of the size of the error, suppose
that the OU process generates the yields perfectly and that
the state variables can be observed
form the market directly. The error term could then be
thought of as market data, bid-ask spreads etc.
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Kalman filter for approach one
Recall the join-stochastic process
the transition equation follows immediately
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For simplicity we write
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Kalman filter for approach two
From
it follows
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Kalman filter for approach two
Recall the join-stochastic process
the transition equation follows immediately
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For simplicity we write
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How does the Kalman Filter work?
We use weekly observations of the light crude oil market
from 01-02-2002 until 25-01-2008. At each observation we
consider 7 monthly contracts. The systems matrices consists of
the unknown parameter set. Choosing an initial set we can
calculate the transition and measurement equation and update
them via the Kalman Filter. Then the log-likelihood function is
maximized and the innovations (error between the market price
and the numerical price) is minimized.
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How to choose the initial state.
For the initial parameter set we randomly choose the value of the
parameters within a respectable bound.
For the initial spot price at time zero we retained it as the future
price with the first maturity and the convenience yield is initially
calculated via
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Numerical results for approach one
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Log future prices versus state variable x
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Implied convenience yield versus state variable
delta
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Innovation for F1
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Kalman forecasting applied on the log future
prices
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Kalman Forecasting applied on the state variables
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Conclusion
We implemented the Kalman Filter for the OU process.
Both the convenience yield as well as the state variable x (log
of the spot price) seems to follow the implied yield and the
market price (resp.) quite good. Also, different initial values for
the parameter set will eventually converge to the optimized
set with the same value of the log-likelihood. This is a good
result and tests the robustness of the method.
The main difference between the systems matrices of both
processes is the transition error covariance-variance matrix
Vt. In the CIR model, this matrix forbids negativity of the CY.
We simply replaced any negative element of the CY by zero,
but since it is negative for a large number of observations,
this will probably give rise to large standard errors in the
optimized parameter set.
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Conclusion
The Kalman Forecasting seems to work only if there is no
sudden drop in the data. To improve the Kalman Forecasting
we could update it every 10 observations.
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Further research
• Implement the Kalman Filter for the CIR model
• Inserting a jump constant in the convenience yield
• Compare both stochastic models
• Pricing of options on commodities, using the optimized
parameter set
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