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Delft – 22-02-08 www.ing.com Adriaan Krul Contents: Contents Introduction Convenience yield follows Ornstein-Uhlenbeck process Analytical results Convenience yield follows Cox-Ingersoll-Ross process Analytical results Numerical results Conclusion Further researchIntroduction: 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. Stochastic convenience yield; first approach: 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 processSlide5: In combination with the transformation x = ln S we haveAnalytical results: Analytical results Expectation of the convenience yield Variance of the convenience yieldAnalytical results : Analytical results PDE of the future prices Closed form solution of the future pricesStochastic convenience yield; second approach: 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 processSlide9: Together with the transformation x = ln S, we haveAnalytical results: Analytical results Variance of the convenience yieldAnalytical results: Analytical results PDE of the future prices Closed form solution of the future pricesKalman filter: 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.Kalman Filter for approach one.: Kalman Filter for approach one. Recall that the closed form solution of the future price was given by From this the measurement equation immediately followsSlide14: From we can writeKalman filter for approach one: 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.Kalman filter for approach one: Kalman filter for approach one Recall the join-stochastic process the transition equation follows immediatelySlide17: For simplicity we writeKalman filter for approach two: Kalman filter for approach two From it followsKalman filter for approach two: Kalman filter for approach two Recall the join-stochastic process the transition equation follows immediately Slide20: For simplicity we writeHow does the Kalman Filter work?: 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.How to choose the initial state.: 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 Numerical results for approach one: Numerical results for approach oneLog future prices versus state variable x: Log future prices versus state variable xImplied convenience yield versus state variable delta: Implied convenience yield versus state variable deltaInnovation for F1: Innovation for F1Kalman forecasting applied on the log future prices: Kalman forecasting applied on the log future pricesKalman Forecasting applied on the state variables: Kalman Forecasting applied on the state variablesConclusion: 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. Conclusion: 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.Further research: 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 You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.