Calibration of Stochastic Convenience Yield Models For Crude Oil Using the Kalman Filter.

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Calibration of Stochastic Convenience Yield
Models For Crude Oil Using the Kalman Filter.

• Adriaan Krul

Delft 22-02-08 www.ing.com
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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
joint-stochastic 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 marketfrom 01-02-2002 until 25-01-2008. At
  each observation weconsider 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