Levy Model for Commodity Pricing

Transcription

1 Levy Model for Commodity Pricing V. Benedico, Undergraduate Student, ECE Paris School of Engineering, France. C. Anacleto, A. Bearzi, L. Brice, V. Delahaye Undergraduate Student, ECE Paris School of Engineering, France. Abstract The aim in present paper is to construct an affordable and reliable commodity pricer based on a recalculation of its cost through time which allows visualize the potential risks and thus, take more appropriate decisions regarding forecasts. Here attention has been focused on Levy model, more reliable and realistic than classical random Gaussian one as it takes into consideration observed abrupt jumps in case of sudden price variation. In application to Energy Trading sector where it has never been used before, equations corresponding to Levy model have been written for electricity pricing in European market. Parameters have been set in order to predict and simulate the price and its evolution through time to remarkable accuracy. As predicted by Levy model, the results show significant spikes which reach unconventional levels contrary to currently used Brownian model. Keywords: Commodity Pricing, Lévy Model, Price Spikes, Electricity Market 1

2 1. Introduction During the meeting organized by Europlace Institute in Paris in 2009, specialists of financial mathematics recognized that because mathematical models were too simple they could not predict the 2008 collapse of financial world. Usual models [1] are mainly based on Brownian motion which gives a simple and supposedly realistic vision of finance dynamics in that it provides reasonable approach to universally observed fluctuations [2,3]. But a Brownian motion based model is a continuous model which cannot simulate sharp price changes through time [4-6]. This may be a reason why this model has not been able to predict the subprime crisis and consequently has shown its limits. So the idea of present work is in a first step to find a way to simulate real historical data in order to define a model close enough to reality. Among all available markets, the study will be undertaken on commodity price, easier to handle as more global, and example of electricity price in Europe will be considered here [7,8]. The first point is to analyze data corresponding to electricity price evolution [9]. From to its observation information is collected on its fluctuations with jumps and spikes, ie on the way price oscillates around the same average. This evolution appears much messier than basic Gaussian one, so in a next step it is interesting to use instead a Levy process [10-12], here implemented in an Ornstein-Uhlenbeck model equation [13], to simulate data of observed price evolution. Rather than using a model with many adjustable parameters not adapted for optimal calibration, it has been chosen to directly introduce functions having the ability to simulate spikes and jumps [14-16]. So from different papers dealing with this subject [17-22] a model based on a Levy process has been calibrated and adapted to the electricity market in order to obtain a theoretical curve close enough to reality. Calculations with resulting system are showing remarkable agreement with real market data. 2. Real Data Observation Figures 1 and 2 give the current evolution of electricity prices over a one year period and a 27 days period respectively. Figure 1: Electricity Price Evolution over a 1 Year Period 2

3 Figure 2: Price Evolution for a Month Period It is observed that price is not varying in a continuous way as described in usual models. Indeed there exist important spikes, jumps and other discontinuities in the curves. So it is necessary to build a model able to simulate these discontinuities and these important spikes. It is also noted that the curves follow a regular oscillatory pattern evolution. Reason is that prices depend on external conditions due to weather, temperature and so on. These conditions are modeled through the seasonality which is a periodic function such as a sine or a cosine directly obtained by Fourier transforming price curve of Figure 2 for a onemonth period analysis developed here, see Figure 3, Figure 3: Frequency Spectrum of Electricity Price over a Month Period 3

4 showing a main highest peak at lowest frequency corresponding to seasonality (around.5hz) and higher much smaller frequencies which will be neglected in the sequel. Lastly, price tends to vary around an average. This condition can be simulated through a mean reversion process which makes the curve oscillate around the same average. Collected data are showing totally unpredictable and messy evolution. Usually, to represent such dynamics, the choice is between either a model which already has most of these abilities and consequently implies a large number of parameters, or a simpler model in which it is possible to implement functions and to add proper parameters. First option is usually too complicated because of the amount of parameters generally difficult to evaluate for correct model calibration. The second more interesting one will be used here, because it provides a much more direct control of implemented functions in the model. 3. Sources and Discussion Along this line, a model based on Ornstein-Uhlenbeck equation with a Levy process has been implemented, because of Levy process ability to be discontinuous and to simulate important spikes. A reason is that Levy process fixes some negative points of the Brownian motion classically used in Black-Scholes (BS) model [ ]. Considering the properties of Brownian motion, the yield only depends on the period on which it is calculated. According to yields properties, when calculated for a market share, they have to be independent and identically distributed according to the same normal distribution. Moreover, Brownian motion path is continuous and so it is for the underlying. As empirically shown, yields are not identically distributed on the market and all but Gaussian [ ]. More specifically, the underlying path brings up some jumps at some high volatility periods. Through this point, BS model can be easily criticized in practical applications. This is where Levy process can step in. Indeed, jumps increase is independent and identically distributed. In addition, trajectories are continuous on the right and limited on the left. This is why advantages of Levy processes allow them to have discontinuous paths as jumps and spikes. However, the same issue still exists concerning the yields. Figure 4: Path of Levy Process (Brownian Motion + Poisson Process) Figure 4 shows Brownian motion continuity associated with jumps due to Poisson 4

5 process, illustrating the fundamental property of a Levy process to have a continuous path on the right and a limited one on the left as indicated before. Consequently the prevailing advantage of Levy process lies in the fact that it allows simulate spikes in high volatility case. 4. Model used: To model system dynamics, the following expression of spot price function S t will be S t = exp [f(t) + X t + Y t ] (1) With St the Spot price function, f(t) the Seasonality function, Xt a mean-reverting process making the price oscillate around an average, and Yt a mean-reverting process with a jump component. The spot price is defined as the price paid in a fixed spot market for immediate delivery. The seasonality corresponds to the part of model defining seasonal or periodic particularity (year, trimester, month, day ) of spot price variation, as for instance analyzed in Part II for a month period. This function undergoes a predictable cyclic variation represented here by f(t) = f(0) + A cos( t (2) with amplitude, frequency and phase A, and F(0) is adjustable shift to match with observations. The mean reversion process of Ornstein-Uhlenbeck, allowing spot price to be constantly attracted to the overall average, writes dx t = X t dt + dl t (3) X t+1 = (1 h)x ti + (L t,i+1 L t,i ) (4) With Levy process L = a.brownian + b.poisson, a, b positive constants, the mean reversion rate, the (constant) price volatility. Finally, as electricity price undergoes large and abrupt variations, the added mean-reverting process with a jump component Y t is obeying the system dy t = Y t dt + J n dn t (5) Y t+1 = Y t,i 1 h + Y t,i + J t.(n t,i+1 N t,i ) (6) With Nt: Poisson process of intensity, β : mean-reversion rate, Jn: the jumps length, and J t = exp(1/ ) where is a constant. Equations (1-6) represent system state dynamics of present spot price model. Next step is to calibrate the various coefficients in their expression. Starting from data published in [X], and adapting them, one finally gets the Table Table 1: Numerical Values of System Coefficients Constant a b f(0) A Value /MWh 35 /MWh 4rad/s.2rad 5. Results and Comparison 5.1 Calculations with system (1-6) and coefficients values in Table I give the following 5

6 results Figure 5: Model Simulation over a Month Period (Left) and a Year Period (Right) As expected, the curve exhibits strong discontinuities with high jumps and spikes. Because of the exponential in its expression (1), the price oscillates between very small and very large values, and the contribution of each term in spot price expression can be analyzed. -The mean-reverting process makes the curve oscillate around the same average which is around for this example. -The jump function gives the model the ability to undergo large spikes. -The seasonality shapes the curve around a cosine oscillation, see Figure 6 for a year period analysis. Figure 6: Electricity Price Seasonality over a Year Period To visualize the difference between present model and Black-Scholes (BS) model commonly used in finance, same calculation with the later gives instead the curve displayed on Figure 7 Figure 7: Price Calculation over a Year Period with BS Model 6

7 By comparison, BS model is simple and produces continuous curve. There are no violent jumps, but only a succession of small jumps. It allows minimizing the risks when it comes to options such as Swing options, contrary to a Levy process based model which tends to maximize the risks. 5.2 In order to check if present model correctly represents real price, it has been compared to real historical graphs corresponding to price evolution during May 2014, see Figure 8 Figure 8: European Electricity Price Evolution during May 2014 Rather than usual direct evaluation [23,24],The idea is to subtract each calculated function in (1) from the real price evolution in order to show that after all the subtractions the mean of remaining curve tends toward zero. This method would confirm the fact that model calibration respects reality. A. The first step is to subtract seasonality, see Figure Figure 9: Electricity Price with and without Seasonality Price with seaso nality -100 Price without seasonality is more hectic as expectable B. The second step is to subtract mean-reversion process. To calibrate this function it is necessary to compare the graphs on different periods or moments because of random nature of the model. Figure 10: Prices with and without Mean Reverting Process 7

8 Price with the mean reverting process Mean reverting process Prices without the mean reverting process Figure 10 exhibits an important element for the next step. It can be observed that some spikes tend to repeat themselves. The process developed for the jumps precisely includes this feature. C. Last step consists in calibrating the jumps according to results shown on Figure 10 (green curve) Figure 11: Prices after Jumps Substraction Price with Jumps Jump Function Price without Jumps Figure 11 describes the remaining behavior of real electricity price after every function in model representation (1) has been subtracted. It is not possible to obtain exactly zero because of model (unpredictable) random nature and only moments of observed curves can be compared to calculations. For lowest average one, exact mean value of remaining curve is equal to, much smaller than average fluctuation of initial observed price. So already within model approximation where higher order oscillations of seasonality function have been neglected, present approach provides a much more convenient setting for representing dynamical commodity evolution than usual BS model. 6. Conclusion A difficulty in price evaluation of economic items such as commodities and stocks is their inherent unpredictable dynamical behavior. Models trying to include this component in their structure by random Gaussian type terms have been developed but they are still not adapted to handle very steep jumps exhibited by recorded market prices and usually offer a relatively 8

9 poor predictive potential typically too weak for correct risk evaluation. To overcome this difficulty, a more complete model including an added Lévy component to handle the jumps has been worked out and application has been made to electricity price on European market. On a one month base analysis, it is shown that even with a relatively rough lowest order approximation, new proposed model correctly covers price evolution with a very small residual compared to observed fluctuations, validating the approach which will be refined in a next step elsewhere. Acknowledgments The authors are very much indebted to ECE Paris School of Engineering for having provided the necessary environment where the project has been developed and to Pr M. Cotsaftis for help in preparation of the manuscript. References A. Papapantoleon : An Introduction to Lévy Processes with Applications in Finance, PhD Thesis, Financial and Actuarial Math. Dept., TU Vienna, A.E. Kyprianou : An introduction to the Theory of Lévy Processes, PhD Thesis, Math. Sciences Dept., Bath Univ., C. Mancini : Disentangling the Jumps from the Diffusion in a Geometric Jumping Brownian Motion, Giornale dell'istituto Italiano degli Attuari, Vol.LXIV, pp.19-47, Cambridge Univ. Press, Cambridge, UK, D. Bates : Jumps and Stochastic Volatility: the Exchange Rate Processes Implicit in Deutschemark Options, Rev. Fin. Studies, Vol.9, pp , F. Black, M. Scholes : The Pricing of Options and Corporate Liabilities. J. Political Economy, Vol.81(3), pp , 1973; R. Merton : Option Pricing when Underlying Stock Returns are Discontinuous, J. Financial Economics, Vol.3, pp , H. Geman : Pure Jump Levy Processes for Asset Price Modeling, J. of Banking and Finance, Vol.26, pp , ICIS : European Electricity Market Reports, Quadrant House, The Quadrant, Sutton, Surrey, SM2 5AS, UK. J. Bertoin : Levy Processes, Cambridge Univ. Press, Cambridge, J.P. Bouchaud, M. Potter : Theory of Financial Risk, Cambridge Univ. Press, Cambridge, UK, M. Kjaer : Pricing of Swing Options in a Mean Reverting Model with Jumps, Applied Math. Finance, Vol.15(56), pp , O. Barndorff-Nielsen, T. Mikosch, S. Resnick : eds., Levy Processes, Theory and Applications, Birkhauser, Boston,

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