Spikes. Shijie Deng 1. Georgia Institute of Technology. First draft: November 20, 1998

Transcription

1 Stochastic Models of Energy ommodity Prices and Their Applications: Mean-reversion with Jumps and Spikes Shijie Deng Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA Work Phone: è44è First draft: November 2, 998 Revised: February 5, 999 This version: October, 999 Iamvery grateful for many fruitful discussions with Darrell Duæe and his valuable comments. I thank Shmuel Oren for his valuable suggestions and comments. I have also beneæted from conversations with lake Johnson, Aram Sogomonian, and seminar participants at the University of alifornia Energy Institute èueiè, U erkeley, U Irvine, Georgia Institute of Technology, and the 999 erkeley POWER onference. All remaining errors are my own. This work was supported by a grant from UEI.

2 Stochastic Models of Energy ommodity Prices and Their Applications: Mean-reversion with Jumps and Spikes

3 Abstract I propose several mean-reversion jump-diæusion models to describe spot prices of energy commodities that may bevery costly to store. I incorporate multiple jumps, regime-switching and stochastic volatilityinto these models in order to capture the salient features of energy commodity prices due to physical characteristics of energy commodities. Prices of various energy commodity derivatives are derived under each model using the Fourier transform methods. In the context of deregulated electric power industry, I construct a real options approach tovalue physical assets such as generation and transmission facilities. The implications of modeling assumptions to the valuation of real assets are also examined.

4 I Introduction Most of the existing literatures on modeling commodity prices deal with commodities which are storable. This is mainly because, until a few years ago, there had not been any traded commodities which were diæcult to store. However, the presumption of all traded commodities being storable is no longer valid as electricity became a traded commodity in recent years. Energy commodity markets grow rapidly as the restructuring of electricity supply industries is spreading in the United States and around the world. The volume of trade for electricity reported by USpower marketers has increased almost æfty folds from 27 million MWh in 995 to,95 million MWh in 997 èsource: Edison Electric Instituteè. The global trend of electricity market reforms inevitably exposes the portfolios of generating assets and various supply contracts held by traditional electric power utility companies to market price risks. It has so far changed and will continue to change not only the way a utility company operates and manages its physical assets such aspower plants, but also the way a utility company values and selects potential investment projects. In general, risk management and asset valuation needs require in-depth understanding and sophisticated modeling of commodity spot prices. Primarily motivated by the surging demands for risk management and asset valuation due to deregulation in the $2 billion US electricity industry, Iinvestigate the modeling of energy commodity prices in the cases where the underlying commodities may bevery costly to store. Among all the energy commodities, electricity poses the biggest challenge for researchers and practitioners to model its price behaviors. A distinguishing characteristic of electricity is that it can not be stored or inventoried economically once generated. Moreover, electricity supply and demand in a bulk electric power network has to be balanced continuously so as to prevent the network from collapsing. Since the supply and demand shocks can not be smoothed byinventories, electricity spot prices are volatile. In the summer of 998, wholesale prices of electricity æuctuated between $èmwh and $7èMWh in the Midwest of US. It

5 is not uncommon to see a 5è implied volatility in traded electricity options. Figure èè 2% Implied Vol 9% 7% 5% 3% % 9% 7% 5% all Options (6//98) all Options (6/9/98) all Options (7/24/98) all Options (8/6/98) Moneyness (K/F) Figure : Implied Volatility of all èsept.è Options at inergy plots the implied volatility of electricity call options in Eastern US across diæerent strike prices at diæerent points in time. On top of the tremendous levels of volatility, the highly seasonal patterns of electricity prices also complicate the modeling issues. There have been few studies on modeling electricity prices since electricity markets only came into existence a few years ago in US. Schwartz è997è, and Miltersen and Schwartz è998è are two of the most recent papers which concern the modeling of commodityspot prices. They investigate several stochastic models for commodity spot prices and perform an empirical analysis based on copper, gold and crude oil price data. They ænd that stochastic convenience yields could explain the term structure of forward prices and demonstrate the implications to hedging and real asset valuation by diæerent models. Hilliard and Reis è998è consider the eæects of jumps and other factors in the spot price on the pricing of commodity futures, forwards, and futures options. One particular ænding of theirs is that the jump in the spot price does not aæect forward or futures prices. However, as I will illustrate 2

6 later, this may not be true if the underlying commodity is almost non-storable such as electricity. Kaminski è997è as well as arz and Johnson è998è are two papers on modeling electricity prices. Kaminski è997è point out the needs of introducing jumps and stochastic volatility in modeling electricity prices. The Monte-arlo simulation is used for electricity derivative pricing under the jump-diæusion price models. arz and Johnson è998è suggest the inadequacy of the Geometric rownian motion and mean-reverting process in modeling electricity spot prices. With the objective of reæecting the key characteristics of electricity prices, they oæer a price model which combined a mean-reverting process with a single jump process. However, they do not provide analytic results regarding derivative valuation under their proposed price model. In this paper, I examine a broader class of stochastic models which can be used to model behaviors in commodity prices including jump, stochastic volatility, aswell as stochastic convenience yield. I feel that models with jumps and stochastic volatility are particularly suitable for modeling the price processes of nearly non-storable commodities. While some energy commodities, such as crude oil, may be properly modeled as traded securities, the nonstorability of electricity makes such an approach inappropriate. However, we can always view the spot price of a commodity as a state variable or a function of several state variables. All the physical contractsèænancial derivatives on this commodity are therefore contingent claims on the state variables. I specify the spot price processes of energy commodities as aæne jump-diæusion processes which are introduced in Duæe and Kan è996è. Aæne jump-diæusion processes are æexible enough to allow me to capture the special characteristics of commodity prices such as mean-reversion, seasonality, and spikes 2. More importantly, I am able to compute the prices of various energy commodity derivatives under the assumed underlying aæne jump-diæusion price processes by applying the transform analysis developed in Duæe, Pan and Singleton è998è. I consider not only the usual aæne jump-diæusion models but also a regime-switching mean-reversion jump-diæusion model. The regime-switching model is used to capture the systematic alternations between ëabnormal" 3

7 and ënormal" equilibrium states of supply and demand for a commodity. The remainder of this paper is organized as follows. In the next section, I propose three alternative models for energy commodity prices and compute the transform functions needed for contingent claim pricing. In section III, I present illustrative examples of the models speciæed in section II and derive the pricing formulae of several energy commodity derivatives. The comparisons of the prices of energy commodity derivatives under diæerent models are shown. I provide a heuristic method for estimating the model parameters by matching moment conditions using historical spot price data and calibrating the parameters to prices of traded options. In section IV, I construct a real options approach tovalue real assets such as generation and transmission facilities in the context of deregulated electricity industry. The implications of modeling assumptions to the valuation of real assets are also examined. Finally, I conclude in section V. II Mean-Reverting Jump-Diæusion Models The most noticeable price behavior of energy commodities is mean-reverting. When the price of a commodity is high, its supply tends to increase thus putting a downward pressure on the price; when the spot price is low, the supply of the commodity tends to decrease thus providing an upward lift to the price. Another salient feature of energy commodity prices is the presence of price jumps and spikes. This is particularly prominent in the case where massive storage of a commodity is not economically viable and demand exhibits low elasticity. A perfect example would be electricity which is almost non-storable. Figure è2è shows the historical on-peak electricity spot prices in Texas èerotè and at the alifornia and Oregon border èoè. The jumpy behavior in electricity spot prices is mainly attributed to the fact that a typical regional aggregate supply function of electricity almost always has a kink at certain capacity level and the supply curve has a steep upward slope beyond that capacity level. Figure è3è represents a snap shot of the marginal cost curve of the electricity supply resource stack inwestern US. 4

8 $6. Spot Price Electricity On-Peak Spot Price $8. $4. EROT on-peak O on-peak $7. $2. $6. $. $5. $8. $4. $6. $3. $4. $2. $2. $. $. 2//95 3//96 6/8/96 9/26/96 /4/97 4/4/97 7/23/97 /3/97 2/8/98 5/9/98 Time $. Figure 2: Electricity Historical Spot Prices In a competitive market, electricity prices are determined by the intersection points of the aggregate demand and supply functions èthe solid curves in Figure è3èè. A forced outage of a major power plant or sudden surging demand would either shift the supply curve to the left or lift up the demand curve èthe dashed curve in Figure è3èè therefore causing a price jump. When the contingency making the spot price to jump high is short-term in nature, the high price will quickly fall back down to the normal range as the contingency disappears therefore causing a spike in the commodity price process. In the summer of 998, we observed the spot price of electricity in Eastern and Midwestern US skyrocketing from $5èMWh to $7èMWh because of the unexpected unavailability of some power generation plants and congestion on key transmission lines. Within a couple of days the price fell back to the $5èMWh range as the lost generation and transmission capacity was restored. Electricity prices may also exhibit regime-switching jumps, caused by weather patterns and varying precipitation, in markets where the majority of installed electricity supply capacity ishydro power such as in the Nord Power Pool and the Victoria Power Pool. 5

9 Generation Resource Stack for the Western US $ $9 $8 $7 $6 Marginal ost ($/MWh) Resource stack includes: Hydro units; Nuclear plants; oal units; Natural gas units; Misc. Marginal ost $5 $4 Inverse Demand $3 $2 $ $ Demand (MW) Figure 3: Generation Stack for Electricity in a Region Although the mean reversion is well studied, there has been little work examining the implications by jumps and spikes to risk management and asset valuation. In this paper, I examine the following three types of mean-reverting jump-diæusion models for modeling energy commodity spot prices.. Mean-reverting jump-diæusion process with deterministic volatility. 2. Mean-reverting jump-diæusion process with regime-switching. 3. Mean-reverting jump-diæusion process with stochastic volatility. I consider two types of jumps in all of the above models. While our analytical approach could handle multiple types of jumps, I feel that, with properly chosen jump intensities, two types of jumps suæce in mimicking the jumps andèor spikes in the energy commodity price processes. The case of one type of jump is included as a special case when the intensity of type-2 jump is set to zero. In addition to the commodity price process under consideration, I also jointly specify another factor process which can be correlated with the underlying commodity price. This additional factor could be the price of another commodity, or something else, such as the 6

10 aggregate physical demand of the underlying commodity. In the case of electricity, the additional factor can be used to describe the spot price of the generating fuel such as natural gas. A jointly speciæed price process of the generating fuel is essential for risk management involving cross commodity risks between electricity and the fuel. There are empirical evidences demonstrating a positive correlation between electricity prices and the generating fuel prices in certain geographic regions during certain time periods of a year. In all models the risk free interest rate, r, is assumed to be deterministic. A Model : A mean-reverting deterministic volatility process with two types of jumps I start with specifying the spot price of an energy commodity as a mean-reverting jumpdiæusion process with two types of jumps. Let the factor process X t in èè denote ln S e t, where S e t is the price of the underlying energy commodity, e.g. electricity. Y t is the other factor process which, in the case of modeling electricity spot price, can be used to specify the logarithm of the spot price of a generating fuel, e:g: Y t =lns g t where S g t is the spot price of natural gas. In this formulation, I have type- jump representing the upward jumps and type-2 jump representing the downward jumps. y setting the intensity functions of the jump processes in a proper way, we can mimic the spikes in the price process of the underlying energy commodity. Suppose the state vector process èx t ;Y t è given by èè is under the true measure and the risk premia associated with all state variables are linear functions of state variables. Then the state vector process has the same form as that of èè under the risk-neutral measure, but with diæerent coeæcients. For the ease of pricing derivatives in section III, I choose to directly specify the state vector process under the riskneutral measure from here on with the assumption that the risk premia associated with all state variables are linear functions of state variables. Assume that, under regularity conditions, X t and Y t are strong solutions to the following 7

11 stochastic diæerential equation èsdeè under the risk-neutral measure Q, X t Y t A ç ètèèç ètè, X t è A dt + ç 2 ètèèç 2 ètè, Y t è 2X + i= æz i ç ètè q çètèç 2 ètè, çètè 2 ç 2 ètè A dw t èè where ç ètè and ç 2 ètè are the mean-reverting coeæcients; ç ètè and ç 2 ètè are the long term means; ç ètè and ç 2 ètè are instantaneous volatility rates of X and Y ; W t is a F t -adapted standard rownian motion under Q in é 2 ; Z j is a compound Poisson process in é 2 with the Poisson arrival intensity being ç j ètè èj =; 2è. æz j denotes the random jump size in é 2 with distribution function v j èzè èj =; 2è. Let ç j Jèc ;c 2 ;tè ç R expèc æ R 2 zèdvj èzè bethe transform function of the jump size distribution of type-j jumps èj = ; 2è. The transform function Deæne the generalized transform function as 'èu; X t ;Y t ;t;tè ç E Q ëe,rèt,tè expèu X T + u 2 Y T è jf t ë for any æxed time T where u ç èu ;u 2 è 2 2. The transform function ' is well-deæned at a given u under technical regularity conditions on ç æ ètè, ç æ ètè, ç æ ètè, ç æ ètè, and ç æ Jèc ;c 2 ;tè. Under the regularity conditions èe.g. see Duæe, Pan and Singleton è998èè, ' æ e,rt is a martingale under the risk-neutral measure Q since it is a F t,conditional expectation of a single random variable expèu X T + u 2 Y T è. Therefore the drift term of ' æ e,rt is zero. Applying Ito's lemma for complex function, we observe that ' needs to satisfy the following fundamental partial diæerential equation èpdeè Df, rf = è2è 8

12 where Df t f + ç X X f + 2 trè@2 X fææt è+ 2X j= Z ç j ètè ëfèx t +æz j R 2 t ;tè, fèx t ;tèëdv j èzè The solution to the PDE è2è is given by èsee Appendix A. for detailsè 'èu; X t ;Y t ;t;tè = expèæèt; uè+æ èt; uèx t + æ 2 èt; uèy t è where æ èt; ëu ;u 2 ë è = u expè, R T t æ 2 èt; ëu ;u 2 ë è = u 2 expè, R T t æèt; uè = Z T è 2X t i= P,r + 2 ç èsèdsè ç 2 èsèdsè ëç i èsèç i èsèæ i ès; uè+ 2 ç2 i èsèæ 2 i ès; uèë + çèsèç èsèç 2 èsèæ ès; uèæ 2 ès; uè j= ç j èsèèç j Jèæ ès; uè;æ 2 ès; uè;sè, èèds è3è Model 2: A regime-switching mean-reverting process with two types of jumps To motivate this model, I consider modeling the electricity prices in which case the forced outages of generation plants or unexpected contingencies in transmission networks often result in abnormally high spot prices for a short time period and then a quick price fallback. In order to capture the phenomena of spot prices switching between ëhigh" and ënormal" states, I extend model to a Markov regime-switching model which I describe in detail below. Let U t be a continuous-time two-state Markov chain du t = Ut= æ æèu t èdn èè t + Ut= æ æèu t èdn èè t è4è 9

13 where N èiè t is a Poisson process with arrival intensity ç èiè èi =; è and æèè =,æèè=. I next deæne the corresponding compensated continuous-time Markov chain M ètè as dm t =,çèu t èæèu t èdt + du t è5è The joint speciæcation of electricity and the generating fuel price processes under the riskneutral measure Q is given by: X t Y t A ç ètèèç ètè, X t è A dt + ç 2 ètèèç 2 ètè, Y t è 2X + j= æz j t + çèu t, èdm ç ètè q çètèç 2 ètè, çètè 2 ç 2 ètè A dw t è6è where W t is a F t -adapted standard rownian motion under Q in é 2 ; fçèiè ç èç èiè;ç 2 èièè ; i = ; g denotes the sizes of the random jumps in state variables when regime-switching occurs. ç çèiè èc ;c 2 ;tè ç R expèc æ zèdv R çèièèzè is the transform function of the regime-jump size distribution v çèiè èi =; 2è. Z j,æz j and ç j Jèc ;c 2 ;tè are similarly deæned as those in Model. 2 Strong solutions to è6è exist under regularity conditions. The transform function Let F i èx; tè èi =; è denote Eëe,rèT,tè expèu X T + u 2 Y T èjx t = x; Y t = y; U t = ië where U t is the Markov regime state variable. The inænitesimal generator D of F i is given by DF èx; tè R =df èx;tè+ç èè R 2ëF èx + çèè;tè, F èx;tèëdç t èçèèè DF èx; tè R è7è =df èx;tè+ç èè R 2ëF èx + çèè;tè, F èx;tèëdç t èçèèè

14 where df i èx; tè = F i + F i æ t x çi èx;tè+ 2 trëf i xx çi èx;tèç i èx; tè T ë + 2X j= ç i jèx;tè èi = ; is the regime state variableè Z R 2 ëf i èx + z; tè, F i èx;tèëdç j;t èzè The fundamental PDEs satisæed by the transform functions F i èx; tè èi =; è are DF i èx; y; tè, rf i èx; y; tè=èi =; è è8è The solutions to è8è assume the following forms F èx; y; tè = expèæ ètè+æ ètèx + æ 2 ètèyè F èx; y; tè = expèæ ètè+æ ètèx + æ 2 ètèyè è9è where æ ètè, æ ètè, æ ètè, and æ ètè are solutions to a system of ordinary diæerential equations èodesè speciæed in Appendix A.2. Model 3: A mean-reverting stochastic volatility process with two types of jumps I consider a three-factor aæne jump-diæusion process with two types of jumps in this model èè. Once again, I motivate the model in the setting of modeling the electricity prices. onsider X t and Y t to be the logarithm of the spot prices of electricity and a generating fuel, e.g. natural gas, respectively. V t represents the stochastic volatility factor. There are empirical evidences alluding to the fact that the volatility of electricity price is high when the aggregate demand is high and vice versa. Therefore, V t can be thought as a factor which is proportional to the regional aggregate demand process for electricity. Jumps may

15 appear in both X t and V t since weather conditions such asunusual heat waves usually cause simultaneous jumps in both the electricity price and the aggregate load. The state vector process èx t ;V t ;Y t è is speciæed by èè. Under proper regularity conditions, there exists a Markov process which is the strong solution to the following SDEs under the risk-neutral measure Q. X t V t Y t A + ç ètèèç ètè, X t è ç V ètèèç V ètè, V t è ç 2 ètèèç 2 ètè, Y t è 2X i= æz i t A dt p Vt ç ètèç 2 ètè p V t q, ç 2 ètèç 2 ètè p V t ç 2 ètèç 3 ètè p V t ç 3 ètè where W is a F t -adapted standard rownian motion under Q in é 3 ; Z j Poisson process in é 3 A dw t is a compound with the Poisson arrival intensity being ç j èx t ;V t ;Y t ;tèèj =; 2è. I model the spiky behavior by assuming that the intensity function of type- jumps is only a function of time t, denoted by ç èè ètè, and the intensity oftype-2 jumps is a function of V t, i.e. ç è2è èv t ;tè=ç 2 ètèv t. Let ç j Jèc ;c 2 ;c 3 ;tè ç R R 2 expèc æ zèdvj èzè denote the transform function of the jump-size distribution of type-j jumps, v j èzè, èj =; 2è. èè The transform function that the transform function Following similar arguments to those used in Model, we know 'èu; X t ;V t ;Y t ;t;tè ç E Q ëe,rèt,tè expèu X T + u 2 V T + u 3 Y T è jf t ë is of form 'èu; X t ;V t ;Y t ;t;tè = expèæèt; uè+æ èt; uèx T + æ 2 èt; uèv T + æ 3 èt; uèy T è èè 2

16 where æèu; tè and æèu; tè ç èæ èu; tè;æ 2 èu; tè;æ 3 èu; tèè are solutions to a system of ODEs speciæed in Appendix A.3. III Pricing of Energy ommodity Derivatives Having speciæed the mean-reverting jump-diæusion price models and demonstrated how to compute the generalized transform functions of the state vector at any given time T, the prices of European-type contingent claims on the underlying energy commodity under the proposed models can then be obtained through the inversion of the transform functions. Suppose X t is a state vector in R n and u 2 n and the generalized transform function is given by 'èu; X t ;t;tè ç E Q ëe,rèt,tè expèu æ X T èjf t ë = expëæèt; uè+æèt; uè æ X t ë è2è Let Gèv; X t ;Y t ;t;t; a; bè denote the time-t price of a contingent claim with payoæ expèaæx T è when b æ X T ç v is true at time T where a, b are vectors in R n and v 2 R, then we have èsee Duæe, Pan and Singleton è998è for a formal proofè: Gèv; X t ;t;t; a; bè = E Q ëe,rèt,tè expèa æ X T è bæx T çv jf të = 'èa; X t;t;tè 2, ç Z Imë'èa + iwb; X t ;t;tèe,iwv ë dw w è3è For properly chosen v, a, and b, Gèv; X t ;Y t ;t;t; a; bè serves as building blocks in pricing contingent claims such as forwardsèfutures, callèput options, and cross-commodity spread options. To illustrate the points, I take some concrete examples of the models proposed in section II and compute the prices of several commonly traded energy commodity derivatives. Speciæcally, model Ia is a special case of model I èi =; 2; 3è. losed-form solutions of the derivative securities èup to the Fourier inversionè are provided whenever available. 3

17 A Illustrative Models The illustrative models presented here are obtained by setting the parameters to be constants in the three general models. The jumps appear in the primary commodity price and the volatility processes èmodel 3aè only. The jump sizes are distributed as independent exponential random variables in R n thus having the following transform function: ç j Jèc; tè ç ny k=, ç k j c k è4è The simulated price paths under the three illustrative models are shown in Figure è4è for $ $9 $8 2 regime (Regime switching) 2 factor (Determ. Vol. Jump-Diffusion) 3 factor (Stoch. Vol. Jump-Diffusion) $7 $6 $5 $4 $3 $2 $ $ Figure 4: Simulated Spot Prices under the Three Models parameters given in Table èè. The x-axis represents the simulation time horizon in years while the y-axis represents the commodity price level. A. Model a Model a è5è is a special case of èè with all parameters being constants. The jumps are in the logarithm of the primary commodity spot price, X t. The sizes of type-j jumps èj =, 2è are exponentially distributed with mean ç j J. The transform function of the jump-size 4

18 distribution is ç j èc J ;c 2 ;tè ç, ç j c èj =; 2è. J X t Y t A ç èç, X t è ç 2 èç 2, Y t è 2X + i= æz i t A dt ç q ç ç 2, ç 2 ç 2 A dw t è5è where W is a F t -adapted standard rownian motion in é 2. The transform function ' a written out explicitly for this model. The closed-form solution of the transform function can be ' a èu; X t ;Y t ;t;tè = expëæèçè+æ èçèx t + æ 2 èçèy t ë è6è where ç = T, t. y solving the ordinary diæerential equations in è35è with all parameters being constants, we get the following, æ èç;u è = u expè,ç çè æ 2 èç;u 2 è = u 2 expè,ç 2 çè P æèç;uè =,rç, 2 ç j J u ç j J, ln j= ç u ç j expè,ç J çè, + a ç 2 u2 + a 2ç 2 4ç 4ç 2 +u ç è, expè,ç çèè + u 2 ç 2 è, expè,ç 2 çèè + u u 2 ç ç ç 2 è, expè,èç + ç 2 èçèè ç + ç 2 2 u2 2 è7è with a =, expè,2ç çè and a 2 =, expè,2ç 2 çè. A.2 Model 2a Model 2a è9è is a regime-switching model with the regime-jumps appearing only in the primary commodity price process. In the electricity markets, this is suitable for modeling 5

19 the occasional price spikes in the electricity spot prices caused by forced outages of the major power generation plants or line contingency in transmission networks. For simplicity, I assume that there are no jumps within each regime. X t Y t A ç èç, X t è ç 2 èç 2, Y t è +çèu t, èdm t A dt ç q ç ç 2, ç 2 ç 2 A dw t è8è è9è where W is a F t -adapted standard rownian motion in é 2. U t is the regime state process as deæned in è4è. The sizes of regime-jumps are assumed to be distributed as independent exponential random variables and the transform functions of the regime-jump sizes are ç ç èc ;c 2 ;tè ç, ç ç c èç =; è where ç ç èupward jumpsè and ç ç èdownward jumpsè. The transform function ' 2a in closed-form completely. We have For this model the transform function ' 2a can not be solved ' 2aèx; y; tè = expèæ ètè+æ ètèx + æ 2 ètèyè ' 2aèx; y; tè = expèæ ètè+æ ètèx + æ 2 ètèyè è2è where æètè ç æèt; uè ç èæ èt; uè;æ 2 èt; uèè has the closed-form solution of æ èç;u è = u expè,ç çè æ 2 èç;u 2 è = u 2 expè,ç 2 çè but æètè ç æèt; uè ç èæ èt; uè;æ èt; uèè needs to be numerically computed from d æ ètè æ ètè A A èæètè;tè+ç èè ë expèæ ètè, æ ètèè, ç æ èt; u è A èæètè;tè+ç èè ë expèæ ètè, æ ètèè, ç æ èt; u è, ë, ë A 6

20 @ æ è;uè æ è;uè A A with A èæètè;tè=,r + 2X i= ëç i ç i æ i + 2 ç2 i æ2 i ë, ç ç ç 2 æ æ 2 A.3 Model 3a Model 3a è2è is a stochastic volatility model in which the type- jumps are simultaneous jumps in the commodity spot price and volatility processes, and the type-2 jumps are in the commodity spot price only. All parameters are constants. X t V t Y t A ç èç, X t è ç V èç V, V t è A dt + ç 2 èç 2, Y t è p Vt p p ç ç 2 Vt q, ç 2 ç 2 Vt p A dw t ç 2 ç 3 Vt ç 3 + æz i t è2è i= where W is a F t -adapted standard rownian motion in é 3 ; Z i èi =; 2è is a compound Poisson process in é 3. The Poisson arrival intensity functions are ç èx t ;V t ;Y t ;tè=ç and ç 2 èx t ;V t ;Y t ;tè=ç 2 V t. The transform functions of the jump-size distributions are ç Jèc ;c 2 ;c 3 ;tè ç 2 J èc ;c 2 ;c 3 ;tè ç ç è, ç c èè, ç 2 c 2 è, ç 2c where ç k J is the mean size of the type-j èj =; 2è jump in factor k èk =; 2è. The transform function ' 3a From section II, we know that ' 3a is of form ' 3a èu; X t ;V t ;Y t ;t;tè = expèæèt; uè+æ èt; uèx T + æ 2 èt; uèv T + æ 3 èt; uèy T è 7

21 Similar to model 2a, the transform function ' 3a does not have a closed-form solution. I numerically solve for both æèt; uè and æèt; uè ç ëæ èt; uè;æ 2 èt; uè;æ 3 èt; uèë from the following ordinary diæerential equations èodesè with d æèt; uè+èæèt; uè;tè=; dt æè;uè=u d è22è dt æèt; uè+aèæèt; uè;tè=; æè;uè= P Aèæ;tè=,r + 3 ç i ç i æ i èt; uè+ 2 æ2 3ètèç ç ë i= è, ç æ èt; uèèè, ç 2 æ 2 èt; uèè, ë ç æ èt; uè+ç 2 è, ç 2æ èt; uè, è èæ;tè= ç 2 æ 2 èt; uè+ç 22 è, ç 2æ èt; uè, è + 2 ç 3 æ 3 èt; uè A è23è and èæ;tè = æ èt; uèèæ èt; uè+æ 2 èt; uèç ç 2 + æ 3 èt; uèç 2 ç 3 è +æ 2 èt; uèèæ èt; uèç ç 2 + æ 2 èt; uèç æ 3 èt; uèç ç 2 ç 2 ç 3 è +æ 3 èt; uèèæ èt; uèç 2 ç 3 + æ 2 èt; uèç ç 2 ç 2 ç 3 + æ 3 èt; uèç 2 2 ç2 3 è Energy ommodity Derivatives In this subsection, I derive the pricing formulae for the futuresèforwards, calls, spark spread options, and locational spread options. The derivative prices are calculated using the parameters given in Table èè. I show the comparisons of the derivative prices under diæerent models as well. 8

22 . FuturesèForward Price A futures èforwardè contract promising to deliver one unit of commodity S i at a future time T for a price of F has the following payoæ at time T Payoæ = S i T, F Since no initial payment is required to enter into a futures contract, the futures price F is given by F ès i t ;t;tè=eq ës i T jf t ë Rewrite the above expression as F ès i t ;t;tè = EQ ës i T jf të = e rç E Q ëe,rç æ expèx i T è jf të We therefore have F ès i t ;t;tè=erç æ 'èe T i ; X t;çè è24è where ç = T, t; 'èu; X t ;çè is the transform function given by è2è; e i is the vector with i th component being and all other components being. Futures price èmodel aè y setting u =ë; ë in ' a,we get Recall the transform function ' a is obtained in è6è and è7è. ' a èë; ë ;X t ;Y t ;çè = expëx t expè,ç çè, rç + a ç 2 4ç + ç è, expè,ç çèè, jèçèë where ç = T, t, X t =lnès t è, a =, expè,2ç çè and jèçè = 2 P Therefore by è24è, we have the following proposition. j= ç j J ç j J, ln ç ç j J expè,ç çè,. 9

23 Proposition In Model a, the futures price ofcommodity S t at time t with delivery time T is F ès t ;t;tè = e rç æ ' a èë; ë ;X t ;Y t ;çè = expëx t expè,ç çè+ a ç 2 4ç where ç = T, t, X t = lnès t è, a =, expè,2ç çè and jèçè = 2 P + ç è, expè,ç çèè, jèçèë j= è25è ç j J ç j J, ln ç ç j J expè,ç çè,. Note that the futures price in this model is simply the scaled-up futures price in the Ornstein- Uhlenbeck mean-reversion model with the scaling factor being expè,jèç èè. If we interpret the spikes in the commodity price process as upward jumps followed shortly by downward jumps of similar sizes, then over a long time horizon both the frequencies and the average sizes of the upward jumps and the downward jumps are roughly the same, i.e. ç J = ç2 J and ç J ç,ç 2 J. One might intuitively think that the up and down jumps would oæset each other's eæect in the futures price. What è25è tells us is that this intuition is not quite right and indeed, in the case where ç J = ç 2 J than that corresponding to the no-jump case. and ç J =,ç 2 J, the futures price is deænitely higher Futures price èmodel 2aè The futures price in model 2a is F ès t ;t;tè=e rç æ ' i 2a èë; ë ;X t ;Y t ;çè èi =; è where i is the Markov regime state variable; ç = T, t and the transform functions ' i 2a are computed in è2è. Futures price èmodel 3aè The futures price in model 3a is F ès t ;t;tè=e rç æ ' 3a èë; ; ë ;X t ;V t ;Y t ;çè where ç = T, t and the transform function ' 3a is given in èa.3è. 2

24 $5. Forward Price Electricity Forward urves under Different Models (ontango) $45. $4. $35. $3. $25. $2. $ Deliverary Date (yrs) 2 factor (Determ. Vol. Jump-Diffusion) 2 regime (Regime switching) 3 factor (Stoch. Vol. Jump-Diffusion) GM Figure 5: Forward urves under Diæerent Models èontangoè Forward curves Using the parameters in Table èè for modeling the electricity spot price in the Eastern region of US èinergy to be speciæcè, I obtain forward curves at inergy under each of the three illustrative models. The jointly speciæed factor process is the spot price of natural gas at Henry Hub. For the initial values of S e = $24:63; S g =$2:5; V =:5; U= and r = 4è, Figure è5è illustrates the three forward curves of electricity which are all in contango form since the initial value S e is lower than the long-term mean value. Figure è6è plots three electricity forward curves in backwardation when the initial electricity price S e is set to be $4 which is higher than the long-term mean value. The forward curves under the Geometric rownian motion ègmè price model are also shown in the two ægures. Under the GM price model, the forward prices always exhibit a æxed rate of growth..2 all Option A ëplain vanilla" European call option on commodity S i with strike price K has the payoæ of ès i ;K;T;Tè = T maxèsi T, K; è 2

25 Electricity Forward urves under Different Models (ackwardation) $5. Forward Prices $45. $4. $35. $3. $25. $2. $5. $. $5. 2 factor (Determ. Vol. Jump-Diffusion) 2 regime (Regime switching) 3 factor (Stoch. Vol. Jump-Diffusion) GM $ Deliverary Date (yrs) Figure 6: Forward urves under Diæerent Models èackwardationè at maturity time T. The price of the call option is given by ès i t ;K;t;Tè = EQ ëe,rèt,tè maxès i T, K; è jf t ë = E Q ëe,rç expèx i T è X i T çln K jf t ë, K æ E Q ëe,rç X i T çln K jf t ë = G, K æ G 2 è26è where ç = T, t and G, G 2 fa =,b =,e i, v =, ln Kg in è3è, respectively. are obtained by setting fa = e i, b =,e i, v =, ln Kg and G = E Q ëe,rç expèx T è XT çln K jf t ë = F i t e,rç, Z Imë'èë, w æ i; ë; X t ;çè expèi æ w ln Kèë dw 2 ç w = F i t e,rç è 2, Z Imë'èë, w æ i; ë; X t ;çè expèrç + i æ w ln Kèë dwè ç wft i è27è where F i t T. = e rç æ 'èe T i ; X t;çè is the time-t forward price of commodity S i with delivery time G 2 = E Q ëe,rç X i T çln K jf t ë 22

26 Z = 'è; X t;t;tè, Imë'èi æ we i ; X t ;t;tèe iæw ln K ë dw 2 ç è w = e,rç 2, Z! Imë'èi æ we i ; X t ;t;tèe ræç +iæw ln K ë dw ç w è28è all option price Substituting ' a, ' 2a, and ' 3a into è27è and è28è we have the call option price given by è26è under Model a, 2a and 3a, respectively. $25. all Price all Options Price under Different Models (Strike K = $35, rf = 4%) $2. $5. $. $5. Model a Model 2a Model 3a GM $ Maturity Time (yrs) Figure 7: all Options Price under Diæerent Models Volatility smile Figure è7è plots the call option values with diæerent maturity time under diæerent models. The call values under a Geometric rownian motion ègmè model are also plotted for comparison purpose. Note that, as maturity time increases, the mean-reversion eæects in all three models cause the value of a call option to converge to a long-term value which is diæerent from the spot price of the underlying. Figure è8è illustrates the implied volatility curves under the three illustrative models with the set of parameters given in Table èè. 23

27 4% Volatility Implied Volatility urves of all Options (Maturity:.25yrs; Se =$24.63 ) 3% 2% % % 9% 8% 7% 6% 5% Model a Model 2a Model 3a GM 4% $28 $38 $48 $58 $68 $78 $88 $98 Strike Price Figure 8: Volatility-Smile under Diæerent Models.3 ross ommodity Spread Option In energy commodity markets, cross commodity derivatives play crucial roles in risk management. rack spread options in crude oil markets as well as the spark spread and locational spread options in electricity markets are good examples. Deng, Johnson and Sogomonian è998è illustrated how the spark spread options, which are derivatives on electricity and the fossil fuels used to generate electricity, can havevarious applications in risk management for utility companies and power marketers. Moreover, such options are essential in asset valuation for fossil fuel electricity generation plants. A European spark spread call èssè option pays oæ the positive part of the diæerence between the electricity spot price and the generating fuel cost at the time of maturity. Its payoæ function is: SSèS e T ;Sg T ;H;Tè = maxèse T, H æ Sg T ; è 24

28 where S e and T Sg T are the prices of electricity and the generating fuel, respectively; the constant H, termed strike heat rate, represents the number of units of generating fuel contracted to generate one unit of electricity. Another type of cross commodity option called locational spread option was also introduced in Deng, Johnson and Sogomonian è998è. A locational spread option pays oæ the positive part of the price diæerence between the prices of the underlying commodityattwo diæerent delivery points. In the context of electricity markets, locational spread options serve the purposes of hedging the transmission risks and they can also be used to value transmission expansion projects as shown in Deng, Johnson and Sogomonian è998è. The time T payoæ of a European locational spread call option is LSèS a T ;Sb T ;L;Tè = maxèsb T, L æ S a T ; è where S a T and S b T are the time-t commodity prices at location a and b. The constant L is a loss factor reæecting the transportationètransmission losses or costs associated with shipping one unit of the commodity from location a to b. Observing the similar payoæ structures of the abovetwo spread options, I deæne a general cross-commodity spread call option as an option with the following payoæ at maturity time T, where S i T SèS T ;S2 T ;K;Tè = maxès T, K æ S 2 T ; è is the spot price of commodity i èi =; 2è and K is a scaling constant associated with the spot price of commodity two. The interpretation of K is diæerent in diæerent examples. For instance, K represents the strike heat rate H in a spark spread option, and it represents the loss factor L in a locational spread option. The time-t value of a European cross-commodity spread call option on two commodities 25

29 is given by SèS t ;S2 t ;K;tè = EQ ëe,rèt,tè maxès T, K æ S2 T ; è jf të = E Q ëe,rç expèx T è S T,KæS2 T ç jf t ë, E Q ëe,rç K expèx 2 T è S T,KæS2 T ç jf t ë = G, G 2 è29è where G = Gè; ln S t ; lnèk æ S2 t è;t;t;ë; ; æææ; ë ; ë,; ; ; æææ; ë è G 2 = Gè; ln S t ; lnèk æ S2 t è;t;t;ë; ; ; æææ; ë ; ë,; ; ; æææ; ë è è3è and recall that Gèv; X t ;t;t; a; bè = 'èa; X t;t;tè 2, ç Z Imë'èa + iwb; X t ;t;tèe,iwv ë dw w The cross-commodity spread call option pricing formula è29è generalizes the exchange option pricing result of Margrabe è978è to cases where the underlying state variables follow the proposed mean-reversion jump-diæusion processes. $25. Spark Spread all Spark Spread all under Different Models (Heat Rate: 9.5MMtu/MWh, rf = 4%) $2. $5. $. $5. Model a Model 2a Model 3a GM $ Maturity Time (yrs) Figure 9: Spark Spread all Price under Diæerent Models 26

30 ross-commodity spread call option price by substituting ' a, ' 2a, and ' 3a into è29è and è3è. under each of the three models are obtained The spark spread call option value with strike heat rate H =9:5 for diæerent maturity time is shown in Figure è9è. Again, the spark spread call option value converges to the current spot price under the GM price model. However, under the mean-reversion jumpdiæusion price models, it converges to a long-term value which is most likely to be depending on fundamental characteristics of supply and demand. Model a Model 2a Model 3a ç ç ç 3 NèA NèA.8 ç ç ç 3 NèA NèA.87 ç.74.8 NèA ç ç 3 NèA NèA.54 ç ç 2 NèA NèA.2 ç ç ç 2 NèA NèA.22 ç ç Table : Parameters for the Illustrative Models Parameter Estimation In this subsection, I provide a heuristic method for estimating the model parameters using the electricity price data. As an illustration, I pick Model a and derive the moment conditions from the transform function of the unconditional distribution of the underlying price return. I assume that the risk premium associated with the factor X is proportional to X, i.e. the risk premium is of form ç X æ X. For simplicity, I further assume the risk 27

31 premia associated with the jumps are zero. As noted earlier, the price processes under the true measure are of the same forms as è5è. In particular, ç i = ç æ i + ç i ç i = çæ i æ ç æ i ç æ i + ç i ç j = ç æ j ç j = ç æ j I then use the electricity and natural gas spot and futures price series to get the estimates for the model parameters under the true measure and the risk premia by matching moment conditions as well as the futures prices. The following proposition provides the unconditional mean, variance and skewness of the logarithm of the electricity price in Model a. Proposition 2 In Model a, let X ç lim t! X t denote the unconditional distribution of X t where X t =lns e t.ifeëjx j n ë é, then the mean, variance and skewness of X are mean = ç + ç ç + ç 2 ç 2 ç variance = ç2 + ç ç 2 + ç 2ç 2 2 2ç ç skewness = 4p 2ç èç ç 3 + ç 2 ç 3 2è èç 2 +2ç ç 2 +2ç 2 ç 2 2è 3 2 Proof. In the transform function ' a as deæned by è6è and è7è, let u = iæw and t!, I obtain the characteristic function of X to be æ Xèwè = expëi æ wç, w2 ç 2 4ç, 2X j= ç j J ç lnè, i æ wç j Jèë If EëjX j n ë é, then the n th moment ofx is given by EëX n ë=è,ièn dn dw n æ Xèwè æ æ w= 28

32 In particular, EëX ë = è,iè d dw æ Xèwè æ æ æ w= = ç + ç ç + ç 2 ç 2 ç The formulae for variance and skewness are obtained in the same fashion. As shown in the proof of the Proposition è2è, we can derive as many moment conditions as we desire for X from the characteristic function of X for estimating the model parameters of Model a. The parameters in Model 2a and Model 3a are obtained by minimizing the mean squared errors of the traded options prices. IV Real Options Valuation of apacity In this section, I construct a real options approach for the valuation of installed capacity or real assets. For the ease of exposition, I take the deregulated electric power industry as an example to illustrate how to use derivative securities to value generation and transmission facilities. This section follows the methodology proposed in Deng, Johnson and Sogomonian è998è but here the valuation is based on assumptions about the electricity spot price processes instead of the futures price processes. I also examine the implications of modeling assumptions to capacityvaluation. For an electric power generating asset which is used to transform some fuel into electricity, its economic value is determined by the spread between the price of electricity and the fuel that is used to generate electricity. The amount of fuel that a particular generation asset requires to generate a given amount of electricity depends on the asset's eæciency. This eæciency is summarized by the asset's operating heat rate, which is deæned as the number of ritish thermal units ètusè of the input fuel èmeasured in millionsè required to generate one megawatt hour èmwhè of electricity. Thus the lower the operating heat rate, the more 29

33 eæcient the facility. The right to operate a generation asset with operating heat rate H that uses generating fuel g is clearly given by the value of a spark spread option with ëstrike" heat rate H written on generating fuel g because they yield the same payoæ. Similarly, the value of a transmission asset that connects location to location 2 is equal to the sum of the value of the locational spread options to buy electricity at location and sell it at location 2aswell as the value of the options to buy electricity at location 2 and sell it a location èin both cases, less the appropriate transmission costsè. This equivalence between the value of appropriately deæned spark and locational spread options and the right to operate a generation or a transmission asset can be readily used to value such assets. I demonstrate this approach by developing a simple spark spread based model for valuing a fossil fuel generation asset. Once established, I æt the model and use it to generate estimates of the value of several gas-æred plants that have recently been sold. The accuracy of the model is then evaluated by comparing the estimates constructed to the prices at which the assets were actually sold. In the analysis I make the following simplifying assumptions about the operating characteristics of the generation asset under consideration: Assumption Ramp-ups and ramp-downs of the generation facility can be done with very little advance notice. Assumption 2 The facility's operation èe.g. start-upèshutdown costsè and maintenance costs are constants. These assumptions are reasonable, since for a typical combined cycle gas turbine cogeneration plant the response time èramp upèdownè is several hours and the variable costs èe.g. operation and maintenanceè are generally stable over time. To construct a spark spread based estimate of the value of a generation asset, I estimate the value of the right to operate the asset over its remaining useful life. This value can be found by integrating the value of the spark spread options over the remaining life of the asset. Speciæcally, 3

34 Deænition Let one unit of the time-t capacity right of a fossil fuel æred electric power plant represent the right to convert K H units of generating fuel into one unit of electricity by using the plant at time t, where K H is the plant's operating heat rate. The payoæ of one unit of time-t capacity right ismaxès e, K t HS g t ; è, where S e and t Sg t are the spot prices of electricity and the generating fuel at time t, respectively. Denote the value of one unit of the time-t capacity right by uètè. For a natural gas æred power plant, the value of uètè is given by the corresponding spark spread call option on electricity and natural gas with a strike heat rate of K H.However, for a coal-æred power plant, it often has long-term coal supply contracts which guarantee the supply of coal at a predetermined price c. Therefore the payoæ of one unit of time-t capacity right for a coal plant degenerates to that of a call option with strike price K H æ c. In this case uètè is equal to the value of a call option. Deænition 2 Denote the virtual value of one unit of generating capacity of a fossil fuel power plant by V. Then V is obtained through integrating the value of one unit of the plant's time-t capacity right over the remaining life ë;të of the power plant, i.e. V = R T uètèdt. The virtual value of one unit of transmission capacity connecting locations i and j èi.e. nodes i and jè in a radial electricity transmission network 3 can be similarly deæned as that of generating capacity. Deænition 3 Let V i,j denote the virtual value of one unit of transmission capacity connecting locations i and j. Then V i,j consists of two parts: one part corresponds to the value of transmitting electricity from location i to location j and the other part corresponds to the value of transmitting electricity from location j to location i. Namely, V i,j = Z T u ètèdt + Z T u 2 ètèdt è3è 3

35 where u ètè is the value of a locational spread option from i to j with maturity time t èi.e. time- t payoæ is maxès j t,l ij S i t ; èè;4 and u 2 ètè is the value of a locational spread option from j to i with maturity time t èi.e. time- t payoæ is maxès i t, L ij S j t ; èè. Under Assumption and 2, the virtual value less the present value of the future O&M costs is very close to the value of one unit capacity. Since the O&M costs are assumed to be constants and they do not vary with model parameters, I set them to be zero. For numerical computations, the virtual values in Deænition 2 and 3 are approximated by the following discrete formula: V ç nx i= uèt i èèt i, t i, è where t ;t 2 ; æææ;t n are n maturity times satisfying ç t ét é æææ ét n, ét n ç T. The larger the number n is, the more accurate the approximation is in è32è. In what follows I will investigate the sensitivity of capacity value with respect to model parameters under the setting of Model a. è32è I calculate the virtual capacity value for a hypothetical gas-æred power plant using spark spread valuation with parameters given in Table è2è. The operating life-time of the power plant is assumed to be 5 years. ç 4 ç 2 3 ç 3.5 ç ç.748 ç ç.3 ç 2 ç 2 2 ç.934 ç S 2.7 S Table 2: Parameters èmodel aè for apacity Valuation I ærst examine how the presence of jumps in commodity prices aæects the capacity value. Figure èè plots the capacity value of a æctitious natural gas æred power plant, for the operating heat rate ranging from 7, MMtuèMWh to 5, MMtuèMWh, both with and without jumps èç = ç 2 = è in the electricity price process. The axis on the left is for the capacity value and the axis on the right is for the diæerence in capacity value of the 32

36 4 apacity Value in Model I apacity Value $/% - loss apa Value (ase) apa Value (No jump) %loss (apa Value) (No jump) $loss (apa Value) (No jump) Heat Rate Figure : apacity Value withèwithout Jumps èmodel aè two cases. The solid curve with diamonds represents the capacity value using parameters in Table è2è which is the base for the comparison. Figure èè illustrates the change in capacity value in absolute dollar term and in percentage term on the right axis as a result of setting the jump intensities to zero in Model a. We can see that the less eæcient apower plant the more portion of its capacity value attributed to the jumps in the spot price process. The presence of jumps makes up as much as 35è of the capacity value of the very ineæcient power plants. We note that the capacity values with jumps èthe solid curve with diamondsè computed here will also serve as the basis for the subsequent comparative static analysis on capacity value with respect to the changes in other model parameters. In Figure èè, I plot the changes in capacity value due to a è increase or a è decrease in the electricity price volatility parameter ç, respectively. The solid curves represent the dollar value change with respect to the base case in Figure èè on the left-side axis and the dotted curves show the percentage change with respect to the base case on the right-side axis. It is clear that the change in volatility causes greater dollar value changes for eæcient plants than for ineæcient plants. The sensitivity of capacity value with respect to the correlation coeæcient between the 33

37 $5. apacity Value hange v.s. Volatility hange (ase case: σ = 75%) apacity Value hange 5.% $..% $5. 5.% $..% Heat Rate -$5. -5.% -$. -$5. d$(apacity) (sig_e+7.5%) d$(apacity) (sig_e-7.5%) d%(apacity) (sig_e+7.5%) d%(apacity) (sig_e-7.5%) -.% -5.% Figure : Sensitivity of apacity Value to Electricity Spot Volatility èmodel aè generating fuel price and the electricity price is illustrated in Figure è2è. While it is still true that the percentage change of capacity value varies monotonically with respect to the eæciency measure èoperating heat rateè of a power plant, the largest dollar value change does not occur to the most eæcient plant but to the plant with heat rate close to the market implied heat rate 5. To compare the theoretical valuation of generation capacity with the market valuation, I plot a capacity value curve in Figure è3è using the NYMEX electricity futures prices at Palo Verde on è5è97 and the parameters obtained by ætting Model a to the NYMEX electricity futures prices at Pale Verde and the natural gas futures prices at Henry Hub. Figure è3è also plots the capacity value curve obtained by using the discounted cash æow èdfè method for comparison purpose 6.At the heat rate level of 95, the capacity value of a natural gas power plant is around $2èkW under Model a. The discounted cash æow method predicts a value of $29èkW. To put things in perspective, I take a look at the four gas-æred power plants which Southern alifornia Edison recently sold to Houston Industries. Unfortunately, not all of the individual plant dollar investments have been made public yet. As a proxy I use the total investment made by Houston Industries è$237 million 34