Energy Swing Options with Load Penalty

Transcription

1 Energy Swing Options with Load Penalty Andrea Roncoroni ESSEC Business School Valerio Zuccolo Politecnico di Milano Corresponding author, Finance Department, ESSEC Business School, Av.B.Hirsch BP 105, Cergy-Pontoise, France. Tel.: ; Fax.: ; Acknowledgements: Zuccolo acknowledges nancial support from the Doctoral Program at ESSEC Business School. Key words: Swing Options, Energy Markets, Dynamic Programming. JEL Classi cation: C31, E43, G11. 1

2 Energy Swing Options with Load Penalty

3 1. Introduction A swing contract grants the holder a number of transaction rights on a given asset for a xed strike price. Each right consists in the double option to select timing and quantity to be delivered under certain limitations. Transactions are speci ed by the contract structure and usually involve a supplementary right to choose between purchase and selling. Swing contracts are very popular in markets where delivery is linked to consumption or usage over time. This is the case for energy commodity markets such as oil, gas and electricity. There, swing-like features are usually embodied into a base-load contract providing a constant ow of the commodity for a xed price. A typical scheme involves a retailer selling gas to a nal consumer under the option to interrupt delivery for a determined number of times. From a nancial viewpoint, the retailer is short one strip of forward contracts, one contract per delivery day, and long one swing option allowing for adjustments in the gas delivery according to contingent market conditions. The joint position described is referred to as a callable forward contract. Beyond side commitments existing between the two counterparts, the swing option exercise policy depends on standing market conditions such as commodity price and availability. The net cash ow for the retailer is given by the forward price received upon delivery, minus the option premium paid to the consumer (usually liquidated in the form of a discount over the forward price), plus any cash ow stemming from the option exercise. The purpose of this case is to evaluate a swing option with non-trivial constraints by means of dynamic programming. Chapter XXX provides a detailed account on this methodology. General treatments on swing contracts are Clewlow and Strickland (2000), Eydeland and Krzysztof (2002) and, to a broader perspective, Geman (2005). The seminal paper by Thompson (1995) tackles the issue of multiple exercise

4 derivatives. Keppo (2004) proves that the optimal exercise policy in the case of no load penalty is "bang-bang" and derives explicit hedging strategies involving standard derivative (i.e., forward and vanilla options). Other papers devoted to the analysis of swing options include Baldick et al. (2003), Barbieri and Garman (1996, 1997), Pilipovich and Wengler (1998), Clewlow et al. (2000, 2001), Carmona and Dayanik (2003), and Lund and Ollmar (2003). Our development moves from the tree-based approach developed in Lari Lavassani et al. (2000) and Jaillet et al. (2003). We now introduce the main problem through a simpli ed argument and defer rigorous treatment of the model and its solution methodology to the next section. The simplest swing option is de ned by two input parameters: 1) a number n of transaction rights and 2) an exercise price K. The option holder, i.e., the gas retailer in our example above, manages his option through control variables 1 ; :::; n signalling the exercise times and q 1 ; :::; q n indicating the transacted quantities (q i > 0 for purchase and q i < 0 for selling). The control signal is de ned by 8 < 1 if t is an exercise time i (t) =. : 0 otherwise The pair (i (t) ; q (t)) determines the control chosen by the option holder at time t. The option fair price is de ned as the maximal expected payo obtainable by the holder over the contract lifetime [0; T ] through a admissible control policy f(i (t) ; q (t)) ; t 2 [0; T ]g. In a market evolving over time with randomness, the optimal control cannot be fully determined at the contract inception. However, it is possible to specify a rule selecting the optimal control corresponding to any realization in the market. More precisely, we look for a number of variables whose knowledge at any time t univocally determines the optimal control at that time. To detect the state variable underlying a swing option, we examine the spectrum of choice available at any time t. Suppose that no swing right is available anymore. Then, the only possible action is to continue

5 Gas Spot Price G (t) Residual Rights N (t)! Exercise Signal i (t) Exercise Load q (t)! Option Pay-o + (G K) q (t) Penalty P (q) Future Present Value + f E (V (t + 1)) Situation Control Revenue Table 1: Exercise decision scheme the contract. The corresponding pay-o is given by the expected present value of the option price displayed one period later (i.e., at time t + 1). If instead gas "swinging" is still possible, then the holder decides to exercise the option whenever the resulting revenue is greater than the expected present value of option price shown one period later. The revenue for exercising the option is given by three terms. First, the option pay-o (G K)q, where G is the spot gas price and q is the striken load. Second, an eventual penalty P (q) on the exercised quantity must be computed and subtracted to the previous term. Third, the present value V (t + 1) of the residual contract, namely a swing option with same features and one swing right less. At a rst glance, this term may resemble the cash ow stemming from q forward positions on gas. In actuality, this is not the case as both delivery time and load are optional. In standard nancial gerg, one may call this term of the swing pay-o a "callable foward with exible load". Table 1 illustrates the described procedure. This scheme shows that the time t optimal choice depends on two state variables, the gas price G and the number N of standing swing rights. Correspondingly, the swing option value is a function V = V (t; G (t) ; N (t)) of time, gas price and the number of residual swing rights. The swing option evaluation can be carried out by dynamic programming. The

6 idea is to write a recursive algorithm computing the swing option value at one time as a function of its possible value one period later. For sake of clarity, let us consider a contract with swing rights to call for any quantity q under a penalty P (q) a ecting the option pay-o. The backward procedure starts at the contract maturity T. For possible values of G and N, the value V (T; G; N) is computed as the corresponding option pay-o : 8 < 0 if n = 0 V (T; G; n) = : max q (G K)+ q P (q) if n > 0. The general term in the backward recursion reads as pay-o from continuation z } { V (t 1; G; n) = e r E t 1 (V (t; ; n)) if n = 0, 8 >< V (t 1; G; n) = max q >: (G K) + q P (q) + e r E t {z 1 (V (t; ; n 1)) } pay-o from exercise9 >= ; e r E t 1 (V (t; ; n)) {z } >; if n > 0. pay-o from continuation where r denotes the continuously compounded risk-free rate of interest and E t 1 indicated the conditional expectation give the information available at time t 1. The time 0 price of a swing option obtains through recursively calling the backward program until time 0 is reached. The resulting value V (0; G 0 ; n) depends on the number n of swing rights and the standing market situation as represented by the current gas price G 0. In actuality, swing contracts include additional constraints to the simpli ed setting above illustrated. refraction period, i.e., i+1 Exercise dates may be required to di er by more than a xed i ; delivery may be delayed by default; quantities may be required to undergo sti, e.g., q i (t) = q, or oating, e.g., a (t) q i (t) b (t), limitations; the strike price may di er according to whether buying or selling gas

7 upon exercise; its value can also be contingent to other speci ed market variables. One of the most important features of any swing contract is the penalty function P (q). This penalty may be local or global. A local penalty a ects the revenue from exercising the option depending on the exercised quantity q, whereas a global penalty applies to the overall exercised quantity Q = P i q i at the end of the contract. We provide a framework for evaluating swing options to call or put gas under global penalties. Roncoroni and Zuccolo (2004) o er a deeper analysis of the optimal exercise policies for swing options under both local and global penalties. 2. Model We consider a swing option with lifetime [0; T ] providing a number u 0 of upswing rights (i.e., call options) and a number d 0 of downswing rights (i.e., put options) on a gas load. The gas spot price is described by a stochastic process over a discrete set of evenly spaced times. We assume these times constitute a re nement of the option lifetime, namely T= is an integer for a time lag. As for the process, we consider a trinomial discretization of a geometric Brownian motion. The exact process speci cation is detailed in the implementation section below. We suppose the market allows for cash rolling over at a constant continuously compounded risk-free rate r. Table 2 summarizes parameters de nition and basic notation. The time t spot price of gas is denoted by G (t). This variable evolves through time according to a stochastic process (G (t) ; t 2 T). We use symbols u and d to denote the standing number of upswing and, respectively, downswing options. Table 3 reports the state variables of the swing valuation problem. At the outset, u 0 (resp. d 0 ) upswing (resp. downswing) rights are available. Each exercise consists in the delivery of a constant load q for a xed price K. A nal

8 Symbol Quantity [0; T ] time horizon time period length n = [T=] periods T = fk; k = 0; :::; ng horizon re nement G (t) ; t 2 T spot price process of gas G 0 G t r initial spot price of gas image space of G (t), t 2 T one-period risk-free rate of interest Table 2: Input parameters and basic de nitions Symbol t G u d Quantity time point spot gas price standing number of upswing options standing number of downswing options Table 3: State variables

9 Symbol T O T n O = #T O u 0 2 N d 0 2 N q K Q (u; d) P (Q) Quantity set of optional dates number of optional dates initial number of upswing options initial number of downswing options xed callable/puttable load strike price overall load global penalty function Table 4: Contract features penalty P is applied on the gas load accrued over the contract lifetime Q (u; d) = jq (u 0 u) q (d 0 d)j. We allow for the exercise times to be constrained within a subset T O of the time horizon T and denote by n O the number of optional times available at the outset. Table 4 shows the contractual features de ning the swing option under investigation. The problem can be cast as a maximization of the present value of the option future cash ows over the set of admissible exercise policies. The value function for this global penalty problem is denoted by V G. Notice that if the swing load q is xed, then each upswing right corresponds to an American call option and is therefore exercised at the last available time.

10 3. Valuation 3.1. Gas Price Tree The benchmark model for spot price dynamics in continuous time is the geometric Brownian motion (GBM) G (t) = exp (r 2 )t + W (t), (1) 2 where constants r and represent the instantaneous annualized risk-free rate of interest and price volatility, respectively. In energy markets, the use of this model is questionable. The main reason is that energy demand is driven by periodical trends and prices tend to revert to periodical mean level. Moreover, some markets display over peculiar features such as stochastic volatility and spikes (see, e.g., Geman and Roncoroni (2004)). However, the popularity gained in the last thirty years by the GBM, mainly due to the Black and Scholes model for option pricing, suggests to consider this speci cation as a theoretical reference. Also, this model is widely employed even in energy markets for the purpose of obtaining benchmark values for exotic derivatives. We accordingly decide to develop our implementation in this framework and leave the nal user the option to select alternative gas price models. For simplicity, we set an initial price G 0 = 1. The time horizon is re ned into a set of 2n periods fk; k = 0; :::; 2ng, where = =2 (i.e., half the time period selected for the option lifetime discretization). On this "doubly re ned" time horizon, we build a binomial random walk 8 < G B G (k) I with probability p, ((k + 1) ) = : G (k) D with probability 1 p. Constants I and D represent the one-period percentage increase and decrease in the standing price and must satisfy the inequality 0 < D < 1 < I. These gures are

11 selected so that the two processes G and G B match in their conditional mean and variance over each time step. For, we set exp (r) G (t) = E (G (t + )j G (t)) = E G B (t + ) G B (t) = p I G (t) + (1 p) D G (t), and e (2t+2 )t X 2 i exp 2t + 2 G (t) = V ar (G (t + )j G (t)) = V ar G B (t + ) G B (t) = (pi 2 + (1 p)d 2 ) G (t) 2 (pi G (t) + (1 p)d G (t)) 2, plus the symmetry condition D = 1 I ensuring that the resulting tree does recombine (i.e., an upward move followed by a downward move have the same e ect on the price quotation as a downward move followed an upward move). Solving for I; D and p gives I = B + p B 2 1, D = B + p B 2 + 1, p = where 2B = exp ( r) + exp r + 2. exp (r) D I D, From this binomial tree we build a trinomial tree with the desired time step t by merging all consecutive pairs of time periods into a single period = 2: the

12 resulting random walk reads as 8 G >< T (k) I 2 with probability p 2, G T ((k + 1) ) = G T (k) with probability p (1 p), >: G T (k) D 2 with probability (1 p) 2. where I; D and p are de ned as above. Notice that the intermediate point is constant since I D = Backward Recursion The main goal of this section is to provide an algorithm for computing the option value at the outset. In a general swing contract, the option holder has the double right to choose delivery time and quantity. To avoid cumbersome algorithms, we simplify the context and suppose that upon exercise a xed load q is delivered. Consequently, the option price is a function V G (0; G 0 ; u 0 ; d 0 ), where G 0 is the gas price prevailing in the market at time 0 and u 0 (resp. d 0 ) is the number of upswing (resp. downswing) rights speci ed as contractual clauses. This method simpli es considerably the valuation procedure in the case examined hereby. Under di erent contract speci cations, e.g., a penalty function a ecting each single exercise 1, this constraint can be relaxed without incurring into the above complications (see Roncoroni and Zuccolo (2004)). We perform the solution algorithm as a dynamic programming backward recursion. Time T value function The holder choice is to maximize the pro t from exercising either the upswing or the downswing right, if still available. The indicator function signal the availability of the corresponding right. In particular, if all option rights have been already exercised, the contract is worthless. 1 This is known as a local penalty function.

13 V G (T; G; u; d) = max q (G K) + 1 u6=0 ); q (K G) + 1 d6=0 P (Q (u; d)) for all G 2 G T ; u u 0 ; d d 0. Here G T denotes the set of all possible values taken by the gas price at time t (see Table 2) and the 1 u6=0 denotes the indicator function of the set fu 6= 0g, i.e., 1 u6=0 = 1 if u 6= 0 and 0 otherwise. Time t value function If t 2 T is not an optional time, i.e., t =2 T O, or neither upswing nor downswing rights are available anymore, i.e., u = d = 0, the swing is worth the conditional expected value of its discounted price one time step later: N (t; G; u; d) = e r E t [V G (t + ; G (t + ) ; u; d)]. Notice that expectation has been taken conditional to the information available at time t. If any swing right is available, i.e., t 2 T O and u 6= 0 or d 6= 0, the option holder compares the value N from continuing the contract with no exercise to the pay-o resulting from the swing rights. The upswing right exercise leads to an immediate cash ow q (G K) + plus the expected value of its discounted price one time step later, namely the time t + value of a swing contract with one upswing right less: U (t; G; u; d) = q (G K) + + e r E t [V G (t + ; G (t + ) ; u 1; d)]. A similar argument leads to the following value of a downswing exercise: D (t; G; u; d) = q (K G) + + e r E t [V G (t + ; G (t + ) ; u; d 1)]. Wrapping these expressions altogether, we obtain the following recursive relation for the time t value function:

14 V G (t; G; u; d) = maxf U (t; G; u; d) 1 u6=0;t2t O; D (t; G; u; d) 1 d6=0;t2t O; N (t; G; u; d)g, where t 2 Tn ft g, G 2 G t ; u u 0 ; d d 0 and the indicator functions signal the availability of the corresponding swing rights. Notice that no penalty directly applies to any swing right exercise. 4. Simulation Experiments We evaluate the swing option price over di erent parameter scenarios. In all cases, option maturity is set to 1 year, the time step is approximately one week, i.e., 1=49, the exercise quantity is 5 MMBtu (Million British Termal Units), current spot price of gas is $ 3, strike is set to $ 2:90, and interest rate is 4% per annum. The spot price volatility in the gas price dynamics (1) is estimated at 60%: At contract maturity, a penalty is charged for each delivered net gas unit (i.e., Q (u; d)) exceeding a threshold. We perform a comparative study of swing option prices across di erent values for the number n O of available exercise times, the number of swing rights, unit penalty, and penalty threshold expressed in MMBtu. The rst experiment considers 25 exercise dates, namely one exercise opportunity each two consecutive working days, a penalty threshold = 10, 4 swing rights, each one for delivery of 5 MMBtu. We examine how the swing option fair price modi es across varying unit overload penalties, namely 1; 3; 5; and 7 $/MMBtu. The case = 0 corresponds to absence of any penalty. Table 5 reports option values for the examined instances.

15 Dates n O Rights Threshold Penalty Value : : : : :8667 Table 5: Option Prices at Varying Unit Penalties Notice that swing prices sharply decrease for low unit penalties, whereas they show steady behavior for high levels of unit penalty. In particular, the price drop is quite substantial between the cases of no penalty and unitary penalty. Another way to a ect the option price through penalties is to modify the penalty threshold as de ned in section??. The lower this gure, the stronger the e ect of penalty on the option value. As Table 6 illustrates, if the threshold is set to 5, any exercise beyond the rst one is subject to penalty, whereas the level 20 make the penalty totally ine ective for the examined case (i.e., 4 rights, each one delivering 5 MMBtu). Naturally, the impact of moving the penalty threshold depends on the unit Dates n O Rights Threshold Penalty Value : : : :9024 Table 6: Option Prices at Varying Penalty Thresholds (High Unit Penalty) penalty level. Table 7 shows how a su ciently low unit penalty makes the option value quite insensitive to threshold resetting. The following experiment shows price

16 Dates n O Rights Threshold Penalty Value :05 26: :05 26: :05 26: :05 26:9024 Table 7: Option Prices at Varying Penalty Thresholds (Low Unit Penalty) Dates n O Rights Threshold Penalty Value : : : :6317 Table 8: Option Prices at a Varying Number of Optional Dates sensitivity to a varying number of optional dates. Recall that an optional date is a point in time where a swing option can be exercised, if any available. We suppose that these dates are evenly spanned over the contract lifetime. Table 8 shows option prices corresponding to 7; 13; 25; and 49 optional dates. In general, the swing value is slightly sensitive to these dates as long as they do not forbid for any exercise, i.e., n O. Our last experiment involves price sensitivity to varying swing rights. Each right a ects the option value by modifying the quantity of gas that can be delivered and the e ectiveness of the penalty constraint. Table 9 indicates gures for the examined cases. If 2 rights are granted by the contract and the delivered quantity is 5 MMBtu, a penalty function with threshold 10 is not e ective. A higher number of rights makes this penalty e ective. Still we remark an interesting behavior when rights increase starting from a relatively low number. For instance, if rights go from 2 to 4, the price

17 Dates n O Rights Threshold Penalty Value : : : : :6269 Table 9: Option Prices at a Varying Swing Rights increase by about 55%, whereas moving swing rights from 6 to 12, the option value increases only by about 19%. This is a typical scarcity item due the relatively higher value of the former available swing rights compared to the latter ones. It is interesting to explore the optimal exercise strategy by showing the distribution of the rst few exercise times. These distributions are not available in closed-form. However, we may provide approximate versions by computing the exercise times over a large number of simulated paths of the underlying process. The procedure works as follows. First, the optimal exercise policy is computed: this is a rule applying to any point in the state variable domain. Next, several trajectories are sampled by simulation. For each simulation, the corresponding upswing and downswing times are calculated and stored. Finally, relative frequency functions are computed for all exercise times. We consider a swing option with 2 downswing rights and 2 upswing rights. The resulting histograms are displayed in Figures 1-4. Insert Figure 1 about here Insert Figure 2 about here Insert Figure 3 about here Insert Figure 4 about here

18 These graphs display a clear picture of the time distribution of optimal exercises provided that the random evolution of the market is correctly described by the given random walk. As noticed in section 2, each upswing right corresponds to an American call option and is therefore exercised at the last available time.

19 References [1] Baldick, R., Kolos, S., Tompaidis, S., Valuation and Optimal Interrruption for Interruptible Electricity Contracts, Working Paper, University of Texas. [2] Barbieri, A., Garman, M.B., Putting a Price on Swings. Energy and Power Risk Management, 1(6). [3] Barbieri, A., Garman, M.B., Ups and Downs of Swings, Energy and Power Risk Management 2(1). [4] Carmona, R., Dayanik, S., Optimal Multiple Stopping of Linear Di usions and Swing Options, Working Paper, Princeton University. [5] Clewlow, L., Strickland, C., Kaminski, V., Risk Analysis of Swing Contracts, Energy and Power Risk Management. [6] Clewlow, L., Strickland, C., 2000, Energy Derivatives: Pricing and Risk Management, Lacima Group Publications. [7] Eydeland, A., Wolyniec, K.,, Energy and Power Risk Management: New Developments in Modeling, Pricing and Hedging, Wiley, Chicago. [8] Geman, H., 2005 (forthcoming). Commodities and Commodity Derivatives: Agricultural, Metals and Energy, Wiley Finance. [9] Jaillet, P., Ronn, E., Tompaidis, S., Valuation of Commodity Based Swing Options. Management Science 50(7), [10] Keppo, J., Pricing Electricity Swing Options. Journal of Derivatives, 11,

20 [11] Lari Lavassani, A., Simchi, M., Ware, A., A Discrete Valuation of Swing Options. Canadian Applied Mathematics 9, [12] Lund, A., Ollmar, F., Analyzing Flexible Load Contracts. Working Paper. [13] Pilipovich, D., Wengler, J., Getting into the Swing. Energy and Power Risk Management 2(10). [14] Roncoroni, A., Zuccolo, V., The Optimal Exercise Policy of Volumetric Swing Options with Penalty Constraints, Working Paper, ESSEC. [15] Thompson, A.C., Valuation of Path-Dependent Contingent Claims with Multiple Exercise Decisions over Time: the Case of Take-or-Pay. Journal of Financial and Quantitative Analysis 30(2),