20 Swinging on a Tree *

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1 20 Swinging on a Tree * Key words: interruptible contracts, energy prices, dynamic programming A swing contract grants the holder a number of transaction rights on a given asset for a fixed strike price. Each right consists of the double option to select timing and quantity to be delivered under certain limitations. Transactions are specified 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 of energy commodity markets such as oil, gas and electricity. There, swing features are usually embodied in a base-load contract providing a constant flow of the commodity for a fixed tariff. A typical scheme involves a retailer selling gas to a final consumer. The contract contains an option to interrupt delivery for a predetermined number of times. From a financial 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 is often 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 the commodity spot price and availability. The net cash flow for the retailer is given by the forward price received upon delivery minus the option premium paid to the consumer and usually settled as a discount premium over the forward price; plus any cash flow stemming from exercising the option. The purpose of this chapter is to evaluate a swing option with non-trivial constraints by means of dynamic programming. General treatments on swing contracts are discussed by Clewlow and Strickland (2000) and Eydeland and Wolyniec (2002). The seminal paper by Thompson (1995) tackles the issue of multiple exercise derivatives. Keppo (2004) proves that the optimal exercise policy in the case of no load penalty is bang-bang (i.e., an all-or-nothing clause) and derives explicit hedging with Michele Lanza and Valerio Zuccolo.

2 Swinging on a Tree strategies involving standard derivatives such as forwards and vanilla options. Other papers devoted to the analysis of swing options include Baldick, Kolos and Tompaidis (2003), Barbieri and Garman (1996, 1997), Pilipovich and Wengler (1998), Clewlow and Strickland (2000), Clewlow, Strickland and Kaminski (2001), Carmona and Dayanik (2003), and Lund and Ollmar (2003). Cartea and Williams (2007) analyze the interplay between interruptible clauses and the price of risk in the gas market. Our development moves the main problem through a simplified argument and defer a rigorous treatment of the model and its solution methodology to the next section Problem Statement The simplest swing option is defined by two input parameters: (1) a number n of transaction rights and (2) an exercise price K. The option holder, e.g., a gas retailer, manages the option exercise policy through control variables τ 1,...,τ n signalling the exercise times and q 1,...,q n indicating the transacted quantities (q i > 0for purchase and q i < 0 for selling). The control signal is defined by { 1 ift is an exercise time, i(t) = 0 otherwise. The pair (i(t), q(t)) defines the control chosen by the option holder at time t. The option fair price is computed as the maximal expected pay-off obtainable by the holder over the contract lifetime [0,T] through an admissible control policy {(i(t), q(t)), t [0,T]}. Since markets randomly evolve, 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 market instance. More precisely, we look for a number of variables whose knowledge at any time t uniquely determines the optimal control at that time. To detect the state variable underlying a swing option, we examine the spectrum of choices available at any time t. Suppose that no swing right is available anymore. Then, the only possible action is to continue the contract. The corresponding pay-off is given by the expected present value of the option price displayed one period later (i.e., at time t + 1). If instead a swinging gas load is still possible, then the holder decides to exercise the option whenever the resulting revenue is greater than the expected present value of the option price shown one period later. The revenue from exercising the option is given by three terms. First, the option pay-off (G K) q, where G is the gas spot price and q is the stricken load. Second, a 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 the same features and a number of swing rights reduced by one. Atfirstglance, thistermmayresemblethecashflowstemmingfrom q forward positions on gas. In reality, this is not the case as long as both delivery time and load are optional. In standard financial jargon, one may call this term of the swing pay-off a callable forward with flexible load. Figure 20.1 illustrates the described procedure.

3 Gas spot price G(t) Residual rights N(t) Exercise signal i(t) Exercise load q(t) 20.1 Problem Statement 459 Option pay-off +(G K) q(t) Penalty P(q) Future present value e r E(V (t + 1)) State Control Pay-off Fig Exercise decision scheme. Notice 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. Swing option valuation can be carried out through Dynamic Programming. The idea is to write a recursive algorithm for computing the swing option value as a function of its possible value one period later. For the sake of clarity, we consider a contract with swing rights to call for any quantity q under a penalty P(q) affecting the option pay-off. The backward procedure starts at the contract maturity T. For all possible values of G and N,thevalueV(T,G,N)is computed as the corresponding option pay-off: { 0 ifn = 0, V(T,G,n)= max q {(G K) + q P(q)} if n>0, where (x) + := max{0,x}. The general term in the backward recursion reads as pay-off from continuation {}}{ V(t 1,G,n) = e r ( ) E t 1 V(t,,n) if n = 0, { V(t 1,G,n) = max (G K)+ q P(q)+ e r ( ) E t 1 V(t,,n 1), q }{{} pay-off from exercise e r ( )} E t 1 V(t,,n) if n>0, }{{} pay-off from continuation where r denotes the continuously compounded one-period risk-free rate of interest and E t 1 indicates the conditional expectation given the information available at time t 1. The swing option 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 beyond the simplified setting illustrated above. Exercise dates may be required to differ by more than a fixed refraction period ρ, i.e., τ i+1 τ i ρ; delivery may be delayed; constraints may be strictly binding (e.g., q i (t) = q) or floating (a(t) q i (t) b(t)); the strike price may differ according to whether the holder decides to buy or sell gas upon exercise; its value can also be contingent upon other specified market variables.

4 Swinging on a Tree An important feature of real-world swing contracts is the penalty function P(q), which may be either local or global. A local penalty affects the revenue from exercising the option depending on the exercised quantity q, whereas a global penalty applies to the overall exercised quantity Q = 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) offer a deeper analysis of the optimal exercise policies for swing options under both local and global penalties Model and Solution Methodology We consider a swing option with lifetime [0,T] providing the holder with 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 evolving over a discrete set of evenly spaced times. We assume these times constitute a refinement of the option lifetime, namely T/Δ is an integer for a time lag Δ. We allow for cash rolling-over at a rate r. Table 20.1 reports basic notation for the proposed model. The time t spot price of gas is denoted by G(t). This variable evolves over time according to a stochastic process (G(t), t T). The process is assumed to follow a trinomial discretization of a geometric Brownian motion. The exact specification is detailed in the implementation section below. We use symbols u and d to denote the standing number of upswing and downswing rights, respectively. Table 20.2 indicates the state variables of the swing valuation problem. At the outset, u 0 (resp. d 0 ) upswing (resp. downswing) rights are made available to the option holder. Each exercise consists of the delivery of a constant load q for a fixed price K. A final penalty P is applied to the gas load accrued over the lifetime of the contract, that is: Table Input parameters and basic definitions Symbol Quantity [0,T] Time horizon Δ Time period length n =[T/Δ] Periods T ={kδ, k = 0,...,n} Horizon refinement G(t), t T Spot price process of gas G 0 Initial spot price of gas G t Image space of G(t), t T r One-year risk-free rate of interest Symbol t G u d Table State variables Quantity Time point Spot gas price Standing number of upswing options Standing number of downswing options

5 20.3 Implementation and Algorithm 461 Symbol T O T n O = #T O u 0 N d 0 N q K Q(u, d) P(Q) Table Contract features Quantity Set of optional dates Number of optional dates Initial number of upswing options Initial number of downswing options Fixed callable/puttable load Strike price Overall load Global penalty function Q(u, d) = q(u 0 u) q(d 0 d). 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 times available at the outset. Table 20.3 shows the contractual features defining the swing option under investigation. The problem can be cast as a maximization of the present value of the option future cash flow over a set of admissible exercise policies. The value function for this problem is denoted by V G. Notice that if the swing load q is fixed, then each upswing right corresponds to an American call option and is therefore exercised at the last available time Implementation and Algorithm Gas Price Tree The benchmark model for spot price dynamics in continuous time is the geometric Brownian motion (GBM) G(t) = exp ((r σ 2 ) ) t + σw(t), (20.1) 2 where constants r and σ represent the instantaneous annualized risk-free rate of interest and price volatility, respectively. In energy markets, the use of this model is rather questionable. The main reason is that energy demand is driven by periodical trends and prices tend to revert back to a periodical mean level. Moreover, some markets display peculiar features such as stochastic volatility and spikes (see, e.g., Roncoroni (2002)). However, the popularity gained in the last thirty years by the GBM, mainly due to the Black and Scholes model for option pricing, suggests considering this specification as a theoretical reference. Also, this model is widely employed by energy traders for the purpose of obtaining benchmark values for exotic derivatives. Accordingly, we develop our implementation under this specification for the gas price process and leave the final user the option of selecting alternative dynamics. For simplicity, we set an initial price G 0 = 1. The time horizon is refined into a set of 2n periods {kδ, k = 1,...,2n}, where δ = Δ/2 (i.e., half the time period

6 Swinging on a Tree selected for the option lifetime discretization). On this doubly refined time horizon, we establish a binomial random walk { G B G(kδ) I with probability p, ((k + 1)δ) = G(kδ) 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 figures are selected so that the two processes G and G B match in their conditional mean and variance over each time step. To this aim, we set and exp(rδ)g(t) = E(G(t + δ) G(t)) = E ( G B (t + δ) G B (t) ) = p I G(t) + (1 p) D G(t) e (2t+σ 2 )δt X 2 i exp( 2t + σ 2) δ G(t) = Var(G(t + δ) G(t)) = Var ( G B (t + δ) 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 has the same effect on the price quotation as a downward move followed by an upward move). Solving for I,D and p gives I = B + B 2 1, D = B + B 2 + 1, p = exp(rδ) D I D, where 2B = exp( rδ) + exp[(r + σ 2 )δ]. 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 Δ = 2δ: the resulting random walk reads as G T (kδ) I 2 with probability p 2, G T ((k + 1)Δ) = G T (kδ) with probability 2p(1 p), G T (kδ) D 2 with probability (1 p) 2, where I,D and p are defined as above. Notice that the intermediate point is constant since I D = 1.

7 20.3 Implementation and Algorithm Backward Recursion We now provide an algorithm for computing the option value at contract inception. In a general swing contract, the option holder has the right to choose delivery time and quantity. To avoid cumbersome algorithms, we simplify the context and suppose that upon exercise a fixed 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 specified as contractual clauses. This method considerably simplifies the valuation procedure in the case examined herein. Notice that under different contract specifications, e.g., a penalty function affecting each single exercise, the load constraint can be relaxed without incurring into the above mentioned complications. The solution algorithm is a Dynamic Programming backward recursion. Time T Value Function The holder has the choice to maximize the profit from exercising either upswing or downswing rights provided that both of them are still available. An indicator function signals the availability of the corresponding right. In particular, if all option rights have been already exercised, the contract is worthless. The final value reads as V G (T,G,u,d)= max{q(g K) + 1 u 0,q(K G) + 1 d 0 } P (Q(u, d)), for all G G T,u u 0,d d 0.HereG T denotes the set of all possible values taken by the gas price at time T (see Table 20.1) and 1 u 0 denotes the indicator function of the set {u 0}, i.e., 1 u 0 = 1ifu 0 and 0 otherwise. Time t Value Function If t T is not an optional time, i.e., t / T O, or neither upswing nor downswing rights are available anymore, i.e., u = d = 0, then the swing is worth the conditional expected value of its discounted price one time step forward: π N (t,g,u,d)= e rδ E t [ VG ( t + Δ, G(t + Δ),u,d )]. Notice that expectation has been taken conditional to the information available at time t. If any swing right is available at time t (i.e., t T O and u 0ord 0), the option holder compares the value π N from continuing the contract (i.e., no exercise) to the pay-off from exercising a swing right. An upswing right exercise leads to an immediate cash flow q(g K) + plus the expected value of its discounted price one time step later, namely the time t + Δ value of a swing contract with one upswing right less: π U (t,g,u,d)= q(g K) + + e rδ E t [ VG ( t + Δ, G(t + Δ), u 1,d )]. A similar argument leads to the following value of a downswing exercise:

8 Swinging on a Tree π D (t,g,u,d)= q(k G) + + e rδ E t [ VG ( t + Δ, G(t + Δ),u,d 1 )]. Combining these expressions, we obtain the following recursive relation for the time t value function: V G (t,g,u,d) = max { π U (t,g,u,d)1 u 0,t T O,π D (t,g,u,d)1 d 0,t T O,π N (t,g,u,d) }, where t T \{T }, G 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 Code Our code splits into three main parts. A first part computes the tree knots. A second part evaluates penalty. A final part calculates the option value. The penalty is computed as a fixed fare π for each unit of the net recourse to swing load (i.e., modulus of the number of exercised upswing minus the number of exercised downswing) over a threshold τ representing the minimal number of net exercises. For instance, if this level equals 10 and the number of stricken upswing (resp., downswing) rights is 20 (resp. 5), then the penalty paid-off at the contract expiration is P = 3max{ , 0} =15. Function Swing(n,T,G 0,K,r,σ,n O,ρ,q,τ,π) implements the valuation procedure. Here, n is the number of dates in the time horizon, T is the contract maturity, G 0 is the spot price of gas at the outset, K is the option strike price, r is the one-period interest rate (i.e., it equals rδ in the previous section), σ is the annualized instantaneous volatility of gas price, n O is the number of optional dates (which we assume to be uniformly spread over the contract lifetime), ρ = u 0 = d 0 is the number of upswing and downswing rights (which, for simplicity, are assumed to be equal), q is the deliverable quantity upon each exercise, τ is the penalty threshold, π is the unit penalty Results and Comments We evaluate the swing option price across alternative parameter scenarios. In all cases, the option maturity is set to 1 year, the time step is approximately equal to one week, i.e., 1/49, the exercise quantity is 5 MMBtu (Million British termal units), current spot price of gas is $3/MMBtu, strike is set to $2.90/MMBtu, and interest rate is 4% per annum. The spot price volatility is estimated at 60%. At contract maturity, a penalty π is charged for each delivered gas unit Q(u, d) exceeding a threshold τ. We perform a comparative study of swing option prices across different values for the number n O of available exercise times, the number ρ of swing rights, the unit penalty π, and the penalty threshold τ.

9 Table Option prices at varying unit penalties 20.4 Results and Comments 465 Dates n O Rights ρ Threshold τ Penalty π Value Table Option prices at varying penalty thresholds (high unit penalty) Dates n O Rights ρ Threshold τ Penalty π Value The first experiment considers 25 exercise dates, namely one exercise opportunity each two consecutive working days, a penalty threshold τ = 10, four swing rights, each one for delivering 5 MMBtu. We examine the way the swing option fair price changes across varying overload penalties. The case π = 0 corresponds to an absence of penalty. Table 20.4 reports option values for the examined instances. Notice that swing prices sharply decrease for low level unit penalties, whereas they show steady behavior for high-level unit penalties. In particular, the price drop is rather significant moving from no penalty to a unit penalty. Another way to affect the option price through penalties is to modify the penalty threshold as defined in Sec The lower this figure, the stronger the effect of a penalty on the option value. As Table 20.5 illustrates, if the threshold is set to 5, any exercise beyond the first one is subject to a penalty, whereas the level 20 makes the penalty totally ineffective 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 penalty level. Table 20.6 shows that a sufficiently low unit penalty makes the option value quite insensitive to threshold resetting. The next experiment shows the price 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. We assume that these dates are evenly spread over the contract lifetime. Table 20.7 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 an exercise is allowed (i.e., n O ρ). Our last experiment involves price sensitivity to a varying number of swing rights. Each right affects the option value by modifying the quantity of gas that can be delivered and the effectiveness of a penalty constraint. Table 20.8 reports figures for all cases. We observe an interesting behavior whenever the number of rights deviates from a relatively low figure. For instance, if rights go from 2 to 4, the price increases

10 Swinging on a Tree Table Option prices at varying penalty thresholds (low unit penalty) Dates n O Rights ρ Threshold τ Penalty π Value Table Option prices at a varying number of optional dates Dates n O Rights ρ Threshold τ Penalty π Value Table Option prices at a varying swing rights Dates n O Rights ρ Threshold τ Penalty π Value by about 55%, whereas moving swing rights from 6 to 12, the option value only increases by about 19%. This is a typical scarcity item due to 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 first few exercise times. These distributions are not available in closedform. 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 each of the exercise times, starting with the first one, moving to the second one, and so on. We consider a swing option with 2 downswing rights and 2 upswing rights. The resulting histograms are displayed in Fig 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 noted in Section 20.2, each upswing right corresponds to an American call option and is therefore exercised at the last available time.

11 Fig Sample jump times distributions Results and Comments 467