1. Do not put content on the brand signature area Calibration of Stochastic Convenience Yield Models For Crude Oil Using the Kalman Filter. Delft – 22-02-08 www.ing.com Adriaan Krul

2. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING2 Contents  Introduction  Convenience yield follows Ornstein-Uhlenbeck process  Analytical results  Convenience yield follows Cox-Ingersoll-Ross process  Analytical results  Numerical results  Conclusion  Further research

3. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING3 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.

4. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING4 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

5. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING5 In combination with the transformation x = ln S we have

6. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING6 Analytical results Expectation of the convenience yield Variance of the convenience yield

7. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING7 Analytical results PDE of the future prices Closed form solution of the future prices

8. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING8 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

9. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING9 Together with the transformation x = ln S, we have

10. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING10 Analytical results Expectation of the convenience yield Variance of the convenience yield

11. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING11 Analytical results PDE of the future prices Closed form solution of the future prices

12. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING12 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.

13. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING13 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

14. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING14 From we can write

15. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING15 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.

16. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING16 Kalman filter for approach one Recall the join-stochastic process the transition equation follows immediately

17. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING17 For simplicity we write

18. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING18 Kalman filter for approach two From it follows

19. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING19 Kalman filter for approach two Recall the join-stochastic process the transition equation follows immediately

20. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING20 For simplicity we write

21. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING21 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.

22. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING22 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

23. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING23 Numerical results for approach one

24. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING24 Log future prices versus state variable x

25. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING25 Implied convenience yield versus state variable delta

26. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING26 Innovation for F1

27. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING27 Kalman forecasting applied on the log future prices

28. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING28 Kalman Forecasting applied on the state variables

29. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING29 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.

30. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING30 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.

31. Do not put content on the brand signature area Orange RGB= 255,102,000 Light blue RGB= 180,195,225 Dark blue RGB= 000,000,102 Grey RGB= 150,150,150 ING colour balance Guideline www.ing-presentations.intranet ING31 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