MODELING COMMODITY PRICES (COPPER)

Transcription

1 MODELING COMMODITY PRICES (COPPER) U. (WITH ROGER J-B WETS) UNIVERSIDAD DE CHILE & UNIVERSITY OF CALIFORNIA, DAVIS

2 OUTLINE INTRODUCTION MODELS XIII ISCP, Bergamo

3 INTRODUCTION XIII ISCP, Bergamo

4 INTRODUCTION Motivation Optimization problems related with commodities (copper) Mining Extraction planning Capacity and equipments allocation Others Finance Valuation of derivatives Investments They don t put much attention on the distribution! XIII ISCP, Bergamo

5 INTRODUCTION Nevertheless Prices are highly volatile XIII ISCP, Bergamo

6 XIII ISCP, Bergamo

7 MODEL Mean Reverting Process where: is the value of index at time is the speed of mean reversion for index captures the relation between indexes are independent Wiener processes is the long term price of index XIII ISCP, Bergamo

8 MODEL Solution where: Moments XIII ISCP, Bergamo

9 MODEL Approximation Considering and replacing by its expectation Why? Error is small Estimating the parameters is nearly impossible XIII ISCP, Bergamo

10 MODEL Approximation Considering and replacing by its expectation slightly displaced log-gaussian process predominantly volatility XIII ISCP, Bergamo

11 XIII ISCP, Bergamo

12 MODEL Geometric Brownian Motion where: is the value of index at time is the drift term of index captures the relation between indexes are independent Wiener processes XIII ISCP, Bergamo

13 MODEL Solution, for Moments, for XIII ISCP, Bergamo

14 EXPLOITING MARKET INFORMATION Existing contracts Stocks: producers, users, traders Exploration activities, recent discoveries Economic forecasts (future demand) Many other factors, Information summarized in future prices XIII ISCP, Bergamo

15 FROM FUTURES TO SPOT PRICES Convert futures into appropriate spot prices Derive a discount factor curve Collection of instruments Delivery times Delivery values Properties Nonnegative, decreasing and NPV of cash-flows must be as close as possible to zero Forward and spot rate curves must be smooth Find a such that for all : (in ) XIII ISCP, Bergamo

16 FROM FUTURES TO SPOT PRICES Finite dimensional problem: Epi-spline Type of constrained spline Obtained from a set of piecewise continuous functions and a set of constants Example: 2 nd order epi-spline XIII ISCP, Bergamo

17 FROM FUTURES TO SPOT PRICES Finite dimensional problem: Epi-spline In practice, we can split into sub-intervals of length and let the function be constant ad each sub-interval Then, it can be shown that XIII ISCP, Bergamo

18 FROM FUTURES TO SPOT PRICES Finite dimensional problem: Epi-spline For example, choosing the norm SPOT prices: XIII ISCP, Bergamo

19 DRIFT ESTIMATION Considering the set of spot prices 1. Fit a curve to our spot prices using an epi-spline, 2. Derive the drift recalling that 3. Finally, we can assume that rather than XIII ISCP, Bergamo

20 THE PROCESSES XIII ISCP, Bergamo

21 MID TERM MODEL If is the transient process and the stationary process, then the blended process can be defined as where XIII ISCP, Bergamo

22 MID TERM MODEL This is still and open question Estimation of the parameters and Functional form of the transition process We have solved it only experimentally Transient process: 1 year Decreasing influence at a rapid rate Mid term: between years 1 and 4 Long term: after year 4 XIII ISCP, Bergamo

23 XIII ISCP, Bergamo

24 DATA Historical information Monthly average LME spot copper prices Source: COCHILCO From 01/2000 to 10/2010 Market information Future contracts settlement price First 12 contracts Source: Bloomberg From 01/2000 to 10/2010 XIII ISCP, Bergamo

25 Drift comparison *Example estimated considering t = 01/2001 XIII ISCP, Bergamo

26 Including historical information Including historical and market information *Example estimated considering t = 01/2001 XIII ISCP, Bergamo

27 Drift comparison *Example estimated considering t = 10/2010 XIII ISCP, Bergamo

28 Including historical information Including historical and market information *Example estimated considering t = 10/2010 XIII ISCP, Bergamo

29 Distributions t=12 t=120 t=24 t=60 *Example estimated considering t = 10/2010 XIII ISCP, Bergamo

30 Distributions t=12 t=120 t=24 t=60 *Example estimated considering t = 10/2010 XIII ISCP, Bergamo

31 CONCLUSIONS We depart from previous work in several ways Split short and long term regimes Short term -Transient process Long term - Stationary process Incorporate market information Convert future contracts to spot prices Incorporate them in the process estimation Multi-dimensional extension Relevance of modeling the underlying probability distribution XIII ISCP, Bergamo

32 MODELING COMMODITY PRICES (COPPER) U. (WITH ROGER J-B WETS) UNIVERSIDAD DE CHILE & UNIVERSITY OF CALIFORNIA, DAVIS