Electricity Swing Options: Behavioral Models and ... - Semantic Scholar
Electricity Swing Options: Behavioral Models and Pricing Georg C.Pflug University of Vienna, [email protected] Nikola Broussev University of Vienna, [email protected]
ABSTRACT. Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name ”option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.
KEYWORDS. Stochastic programming; OR in Energy; Pricing; Swing Option; Behavioral Model; Equilibrium price
1
INTRODUCTION
Along with the deregulation of the energy markets came growth in the importance of power derivatives such as futures, swaps, caps, floors, collars and options. Most of these contracts are relatively easy to price, given that a spot market model is available. In contrast, there is no generally accepted way 1
to price such flexible contracts as swing options. In this paper, we present a method to determine a rational ask price for swing options, which is based on the notion of the least acceptable price for the option seller and a behavioral model for the option holder. The pricing method is based on dynamic stochastic optimization and works for large classes of contracted restrictions on the exercise pattern of the option holder, such as the global volume, the number and intervals of change times, the amounts of allowed changes, etc. The option seller is confronted with the problem that the ask price must be determined long before exercising starts. The seller must make a hereand-now decision, while the option holder can base his exercise decision on data becoming available later (wait-and-see). This fact makes the problem a stochastic asymmetric game. The option seller has to anticipate the exercise pattern of the holder (or at least some worst-case pattern), bearing in mind that this exercise pattern will depend on his asked price if the contract is realized. Thus the price must satisfy a equilibrium in the sense that both parties have no reason to change their behavior. Notice however that the behavior of the contract holder is the modeled behavior, it is the behavior which the seller has much reason to assume that the holder will follow. But even if the seller can model the holder’s behavior correctly, there is no fair ask price in the sense of the no-arbitrage theory, because there are no already priced hedging instruments for replication or superreplication. To put it differently, hedging with futures may reduce risk and reshape the profit/loss distribution, but it can never bring the risk to zero. This is why we introduce the notion of the least acceptable ask price for a contract seller: An ask price is called acceptable, if there is some hedging strategy such that for the reshaped profit/loss distribution the probability of making a loss is below some predetermined threshold, typically 10-15%. Among all acceptable ask prices, the smallest is chosen in order to win against the competitors of the option seller. The proposed approach is new. Other pricing models for energy derivatives and the difficulties coming along with these models can be found in (Eydeland and Geman, 1998), (Barbieri and Garman, 2000) and (Davison and Anderson, 2003). These models use some no-arbitrage arguments, which are questionable, because of the non-storability and incompleteness of the electricity market. Other of the proposed models are special for the commodities markets (for example the gas markets) where stronger storability assumptions are possible. Notice that in situations with early exercise possibility and incomplete markets, no closed formula exists even for the simplest type of spot-price processes. Therefore the numerical analysis has to be done by solving a stochastic dynamic optimization problem. An overview on power markets and derivatives can be found in (Pilipovic, 1997). 2
The paper is organized as follows: In section 2, the principle of the least acceptable price for electricity delivery contracts is presented. Section 3 deals with swing options and rational exercise strategies. The two-person game and its equilibrium price together with some extensions are discussed in section 4. Conclusions can be found in section 5.
2
ACCEPTABLE PRICES FOR SPOT-PRICE DEPENDENT DEMANDS
To begin with, we assume that the seller knows the demand pattern of the holder, which may be spot-price dependent. To be more precise, he knows an exercise plan of the holder (who exercises conditional on the development of the spot-market), but he does not know the future spot prices themselves. The time index will be denoted by t, where t runs between 1 and T . We will exclusively work with a discrete scenario model and hence we assume that there is a finite number of possible scenarios s = 1, . . . , S together with the scenario probabilities p = (p1 , . . . , pS ), that is we assume a finite probability space Ω = {ω1 , . . . , ωS }. The spot-price model is given by a large [S × T ] matrix (ξs,t ). The (possibly spot-price dependent) demand pattern is described by another [S × T ] matrix (ds,t ). To reduce his own risk, the seller may buy energy futures, which gives standardized deliveries in particular times. However these contracts may never completely replicate the real demand pattern and a considerable residual risk remains with the option seller.
future market
- swing option seller
- swing option holder
We assume that there are M hedging instruments (types of energy futures) available. A hedging instrument is characterized by its today’s price πm and its delivery pattern τ , where τ (m, t) is the amount delivered in time period (e.g. hour) t. Typical patterns are • the base future: τ (·, t) ≡ 1 for a year, a quarter or a month 3
1. Method for the development of GH0 Thepeak basis for the wholesale product GH0 is the quarterly hour profile H0 of VDEW. It is scaled up to a • the future: volume that has a standard size for wholesale trading; afterwards it is converted into an hourly profile. The delivery periodτ (·, is at) complete calendar = 1, for Monyear. through Fri, 8 - 20 h
otherwise τ = 0. 2. The representative VDEW load profile H0 (households) TheVattenfall H0 profile isGH0 basedprofile. on a statistical of how households power. The profile • the This isanalysis a standardized profileconsume which follows consists of nine types of days. It is divided into three seasons (summer, winter, shoulder) and three typical requirement of a German household with seasonal and daily types of weekdays (Saturday, Sunday, working day). There is a profile with auxiliary values for every variation of hour the on demand (see Vattenfall (2005)), see Figure 1. in the appendix. The quarter of an every type of day. A table with those values is available profile is then rolled out according to the calendar. A household profile is standardised to the energy of 1 MWh/a. [W]
250,0 200,0 150,0 100,0 50,0 0,0 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23:
Winter Workday Winter Saturday Winter Sunday
Summer Workday Summer Saturday Summer Sunday
shoulder Workday shoulder Saturday shoulder Sunday
Figure 1: Auxiliary values for the representative VDEW load profile H0 for a standardised annual consumption of 1 MWh/a
Figure 1: The wholesale product GH0: Auxiliary values for the representative VDEW3.load profile H0 for a standardised annual consumption of 1 MWh/a Seasons The following seasons are applicable according to the specification for the representative VDEW load
In order to avoid unboundedness of the hedging problem, we assume that profile: the spot price model is calibrated to the known future prices πm such a way, that expectation-neutralityfrom holds, i.e.until Winter Shoulder Summer Shoulder Winter
01.01. 20.03. T 14.05. S 21.03. X X 14.09. π15.05. = τ (m, t) ps ξs,t m 15.09. t=1 31.10. s=1 01.11. 31.12.
The decision to be made by the option seller consists of 8 • the number xm of units of future contract m to be bought,
• the ask price K.
4
(1)
We collect the hedge amounts in a hedge vector x = (x1 , . . . , xM ). We assume in this paper that the option seller appears on the future marker only as a buyer, i.e. the hedges xm must be nonnegative. PMThe unmatched surplus/shortage in time period t is the difference m=1 xm τ (m, t) − ds,t . This amount will be sold/bought on the spot market. Given a hedge x, the surplus/shortage value in hour t, for scenario s is "M # X ∆s,t = ξs,t xm τ (m, t) − ds,t . m=1
We remark that it is easy to incorporate transaction costs in this frame+ work. Suppose that ξs,t is the revenue of selling one unit at the spot market − and ξs,t is the cost of buying one unit on the spot market. Then "M #+ "M #− X X + − ∆s,t = ξs,t xm τ (m, t) − ds,t + ξs,t xm τ (m, t) − ds,t . m=1
m=1
However, for the sake of simplicity, we will not work with transaction costs here. Denote by Y the profit/loss variable of this contract seen from the option seller. It consists of revenue from the swing option contract minus/plus the activity on the spot-market minus the costs for the hedges. Depending on the hedging decision x and the ask price K, the profit/loss variable takes the values # "M M T T X X X X πm xm xm τ (m, t) − ds,t − ξs,t ds,t + Ys (x, K) = K t=1
m=1
t=1
m=1
with probability ps . This random variable Y (x, K) is the basis for the acceptable ask price calculation. Notice that due to the expectation-neutrality (1), T S X X ds,t [K − ξs,t ], E[Y (x, K)] = ps t=1
s=1
that is, the expected profit does not depend on the hedging decisions. For this reason, the acceptability of a hedge cannot be based on a high profit expectation but must be based on another criterion, as is discussed next.
2.1
Acceptable ask prices and optimal hedging
A revenue variable Y is called acceptable, if the probability that it falls below 0 is less than α: P{Y (x, K) < 0} ≤ α. 5
where α is a given probability, typically set by the management of the sellers’ company. Among all acceptable return variables, the one with the smallest unit price for the delivery contract is called the least acceptable contract price. It is found by the following optimization problem
Minimize K
subject to (2)
P{Y (x, K) < 0} ≤ α This problem is unfortunately non-convex. By a slight modification one may convexify it. We replace the condition P{Y < 0} ≤ α by the stronger condition AV@Rα (Y ) ≥ 0, where AV@R is the average value-at-risk Z 1 α −1 GY (u) du AV@Rα [Y ] = α 0 with GY (u) = P {Y ≤ u}. Other names for AV@R are the conditional value at risk or the expected shortfall. It is easy to see that AV@Rα (Y ) ≥ 0 implies that G−1 Y (α) ≥ 0, i.e. P {Y < 0} ≤ GY (0) ≤ α. Rewriting the above problem for the AV@R, we get the AV@R constrained contract price problem
Minimize K
subject to (3)
AV@Rα [Y (x, K)] ≥ 0 which is a linear program in the variables x and K (see Rockefellar and Uryasev (2000) and Pflug (2000)). This follows from the fact that the condition AV@Rα [Y ] ≥ 0 is equivalent to: There is an a and a nonnegative vector z = (z1 , . . . , zs )> such that 1 a − p> z ≥ 0 and Ys − a + zs ≥ 0 for all s. α For further use introduce the following quantities P ζs,m = Tt=1 τ (m, t)ξs,t the spot-value of the m-th hedge in the s-th scenario P δs = Tt=1 ds,t ξs,t the spot-value of the the demand in the s-th scenario PT ¯ ds = t=1 ds,t the total demand in the s-th scenario In this notation, the profit/loss takes the values Ys (x, K) = d¯s K +
M X m=1
6
xm [ζs,m − πm ] − δs
with probability ps . Here is the structure of the convexified least acceptable ask price program:
Minimize (in a, x, z and K) : K
subject to
P
−a + 1 S ps zs ≤ 0 (4) s=1 α
−d¯s K + a + PM xm [πm − ζs,m ] − zs ≤ −δs ; s = 1, . . . , S
m=1
x, z ≥ 0 which can be solved by any LP-solver.
3 3.1
SWING OPTIONS Definitions
The typical swing option has the following characteristics (Jaillet et al. 2000): There are predefined exercise times τi , i ∈ {1, 2, . . . n}, 1 ≤ τ1 < τ2 < τ3 < · · · < τn ≤ T at which a fixed volume of d0 units of electricity may be obtained. With one day ahead notice, the owner of the option may use his swing right to receive more (up-swing) or less (down-swing) than this volume d0 units at any of this n moments. Some contracts allow swings only at N out of the possible n time moments (N ≤ n). This is the so called swing number constraint. There is often also a freeze time constraint that forbids swings at times which have a smaller distance than a given value ∆τ . Other constraints are the local boundaries, which define how much the requested amount di at time τi may differ from the nominal volume d0 . A typical constraint is di − d0 ∈ [li1 , li2 ) ∪ (li3 , li4 ], (li1 ≤ li2 ≤ 0 ≤ li3 ≤ li4 ). Moreover, there are also global constraints, which restrict the total requested volume D within the contract period by boundaries M in and M ax. In some contracts, the global constraints are replaced by penalty functions, which define the penalty costs for violating the constraints. Examples of penalty functions are: when D < M in C1 0 when M in ≤ D ≤ M ax ϕ(D) = ξT (D − M ax) when D > M ax where ξT is the spot price at the end of the period. Hard constraints can be modeled by the following penalty costs: 7
∞ when D < M in 0 when M in ≤ D ≤ M ax ϕ(D) = ∞ when D > M ax For further use, we define the following decision and volume functions: 1 Up-swing at time τj + χj = 0 else 1 Down-swing at time τj χ− j = 0 else dj − d0 when χ+ + j = 1 dj = 0 else dj − d0 when χ− − j = 1 dj = 0 else Now we are ready to write down the constraints of the swing contract in a formal way: − χ+ j , χj ∈ {0, 1} − 0 ≤ χ+ j + χj ≤ 1 for all 1 ≤ j ≤ n τj − + − for all 1 ≤ i < j ≤ n (χ+ i + χi ) + (χj + χj ) ≤ 1 + τi + ∆τ n X − 0≤ (χ+ j + χj ) ≤ N j=1 + 4 + lj3 χ+ j ≤ dj ≤ lj χj for all 1 ≤ j ≤ n − 2 − lj1 χ− j ≤ dj ≤ lj χj for all 1 ≤ j ≤ n − Notice that the complementary slackness condition χ+ j χj = 0 is automatically fulfilled (either Up or Down swing).
3.2
The structure of the exercise patterns
Let us investigate the set of the feasible exercise patterns in a more precise manner. From Figure 2 one sees that it has a pretty complex structure. If we analyze this set and consider it to it as a subset of the vector space of all possible (feasible and not feasible) and suppose li2 = li3 = 0 for simplicity, under the assumption of absence of swing number and freeze time constraint we can show the following result. 8
6 nominal volume
exercise pattern local maximum
5
Volumen
4
3
2
1 local minimum
0
0
2
4
6
8
10 12 Period
14
16
18
20
22
Figure 2: An example of swing option feasible exercise pattern Proposition 1 The set of the feasible exercise patterns of a swing option D = D(d0 , n, Lmin , Lmax , M in, M ax, N, ∆i) is bounded and convex when N = n and ∆i = 1. Here d0 is the nominal volume, n the number of swing possibilities, Lmin := {l11 , l21 . . . ln1 } the lower local boundaries set, Lmax := {l14 , l24 . . . ln4 } the upper local boundaries set, M in, M ax global boundaries N the maximal number of swings and ∆i the freeze time. We assume that swing may only be exercised at integer times. Proof. Let observe the convex combination d00 of two exercise patterns: d00 = αd + βd0 d, d0 ∈ D(d0 , n, Lmin , Lmax , M in, M ax, n, 1) d00 = (d001 d002 . . . d00n ) = (αd1 + βd01 αd2 + βd02 . . . αdn + βd0n ) where α, β ≥ 0 and α + β = 1 For the local constraints we have: d00i − Lmin (i) = αdi + βd0i − Lmin (i) = = αdi + βd0i − (α + β)Lmin (i) = = α(di − Lmin (i)) + β(d0i − Lmin (i)) ≥ 0,
9
in an analogous manner for Lmax (i). And for the global constraints we have: n X
d00i
− M in =
i=1
n X
(αdi + βd0i ) − M in =
i=1
=
n X
(αdi + βd0i ) − (α + β)M in =
i=1 n n X X = α( di − M in) + β( d0i − M in) ≥ 0 i=1
i=1
and similarly analog for M ax. So we have shown that d00 ∈ [Lmin , Lmax ], n X d00i ∈ [M in, M ax], i=1
which means that the set is convex. 2 We remark that non-convex set of feasible exercise patterns may lead to stochastic mixed-integer problems, which are outside the scope of this paper.
3.3
An exercise model
Similar to (Haarbr¨ ucker and Kuhn(2006)) we regard the option holder as a player in a multistage stochastic game and model his behavior as the optimal strategy in this game. Assume that a spot price scenario tree is given. The option holder is fully informed about the actual node of the tree. Based on this knowledge and on his own past decisions he maximizes his expected profit till the end of the exercise period. More precisely, assume that a finite probability space (Ω, P(Ω), P), Ω = {ω1 , . . . , ωS } is given, on which a filtration is defined F = (Ft )t∈{1,...,T } , with F1 = {∅, Ω} and FT = P(Ω). The spot price process (ξt )t∈{1,...,T } (ξt (ωs ) = ξs,t ) is adapted to F. The process ξ can be represented by a tree whose terminal nodes are the scenarios {ω1 , . . . , ωS }. At each node of the tree corresponding to F, the holder chooses his exercise volume to maximize his expected profit under the constraints coming from the contract and his past decisions. Further his decision at moment t may be dependent on the information about the spot prices only up to this moment which means the exercise process (dt )t∈{1,...,T } (dt (ωs ) = ds,t ) should be a adapted regarding the filtration F process. Thus we can represent the optimal exercise process 10
as the solution of the following multistage stochastic program. dmax ∈ argmax{EP Z(d, ξ) : d ∈ D, d adapted to F} Here Z is the (random) profit if the exercise process (dt ) is chosen. Z(d, ξ) :=
T X
(ξt − K)dt
t=1
We assume that the profit per unit is the difference between spot price ξt and contract strike price K. We assume further the bid-price of the swing contract is equal to the maximal expected profit as described above. To determine the bid-price we suppose that the contract issuer has no possibility to hedge and uses only the spot market to cover the demand of the holder. A model that gives the issuer the possibility to hedge with future contracts is given in the next section. Notice the asymmetric structure of the model: The option holder may implement wait and see strategies for exercising, while the issuer in a here and now situation when he has to determine the asked price. For this reason, the filtration is irrelevant for the option seller’s optimization program (4), while it is crucial for the option holder’s decision problem (5) below. t1 , F1
t2 , F2
t3 , F3
t4 , F4 GFED 1/3 105 ddd2 @ABC ddddddd
ddd ?>=< 75 ZdZZZZZZZZZZ j4 89:; ZZZZ, 89:; ?>=< 2/3
3/10 jjjjj
j 40 jjjj jjjj ?>=< 89:; T 67 TTTT : TTTT uu ?>=< TT uu 3/7 75 ddd2 89:; u 7/10 TTTT* ddddddd 1/2 uuu d d d d d 89:; ?>=< u Z Z Z 35 ZZZZZZZZ u uu ZZZZ, 89:; ?>=< uu 4/7 u 30 u u ?>=< 89:; 45 II II ?>=< II 4/7 ddddd2 89:; 45 II ddddddd d d d II 89:; ?>=< Z Z 4 Z Z 35 I j ZZZZZZZ 1/2 III 7/10 jjjjj ZZZZ, 89:; ?>=< II jj 3/7 j 50 j j I$ j j j j ?>=< 89:; T 27 TTTT TTTT ?>=< TT 2/3 dddddd2 89:; 15 d d d 3/10 TTTT* d d d d dZdZZ 89:; ?>=< 45 ZZZZZZZZ ZZZ, @ABC GFED 1/3 100
Figure 3: The spot price scenario tree for our numerical example
11
3.4
The Linear Programming formulation
According to Proposition 1 under the assumption of no swing number constraint and no freeze time constraint the set of the feasible exercise patterns is convex. Moreover the chosen target function is linear in the time components of the exercise process, thus our multistage stochastic program can be represented as a linear program for the vector consisting of the volume decisions in each node of the tree. Enumerating the nodes of the tree in the standard way we obtain the following linear program.
P P
Maximize (in dn ): P ps P (ξ − K)d = (ξ − K)( n n n s∈S n∈N (s) n∈N s∈S(n) ps )dn
subject to:
d − d ≤ l4 ∀n ∈ N 0
n t(n)
d − d ≥ l1 ∀n ∈ N
P n 0 t(n)
n∈N (s) dn ≤ M ax ∀s ∈ S
P
n∈N (s) dn ≥ M in ∀s ∈ S
dn ≥ 0 ∀n ∈ N (5) Here we use the following terms: N S ps ξn t(n) S(n) N (s)
The The The The The The The
set of all nodes set of all scenarios probability of scenario s spot price in node n time of node n set of all scenarios to which node n belongs nodes belonging to scenario s and all time horizons
The dimension of the problem is clearly equal to the number of nodes in the tree. An example of such tree can be seen in Figure 3.
3.5
Parameter analysis
The solution of the option holder’s problem (5) depends on the value of the strike price. Thus the value of the maximum as well the dn , n ∈ N for which this maximum is attained are functions of K which we denote by E and D: X X E(K) := max{ (ξn − K)( ps )dn : dn ∈ D} (6) n∈N
D(K) := argmax {
X
s∈S(n)
(ξn − K)(
n∈N
X
s∈S(n)
12
ps )dn : dn ∈ D}
(7)
Here D is the constraint set of the problem (5). Because of the structure of the set D and the linearity of the target function in K it is obvious that: Proposition 2 The function D(K) is a piecewise constant function, mapping R to R|N | . The following result about the function E is easily shown: Proposition 3 The real function E(K) is a continuous piecewise linear decreasing convex function. Proof. We have E(K) := max{Ed (K) : d ∈ D} where Ed (K) = Σn∈N (ξn −K)(Σs∈S(n) ps )dn are linear functions of K. As it is known that the maximum of linear functions is a continuous convex function, this clearly follows for E(K) too. Moreover for the finite set of the extreme points D∗ of the feasible set we have E(K) := max{Ed (K) : d ∈ D} = max{Ed (K) : d ∈ D∗ } which means E(K) is a piecewise linear function with finite number of different linear parts. From the fact that all Ed (K) are decreasing it follows that their maximum E(K) is also decreasing. This completes the proof. 2 For further analysis of the dependence of the expected maximal achievable profit from the flexibility of the contract we take a special form of our linear 1 4 := l, M in = T d0 − L and M ax = T d0 + L. Now = −lt(n) program where lt(n) we obtain the following LP with two parameters l and L: P P P P
Maximize: s∈S ps n∈N (s) (ξn − K)dn = n∈N (ξn − K)( s∈S(n) ps )dn
subject to:
dn ≤ d0 + l ∀n ∈ N
dn ≥ d0 − l ∀n ∈ N
P
n∈N (s) dn ≤ T d0 + L ∀s ∈ S
P
n∈N (s) dn ≥ T d0 − L ∀s ∈ S In this case we use the following notation for the feasible set of patterns. D(l, L) := {d : |dn − d0 | ≤ l
∀n ∈ N , |Σn∈N (s) dn − T d0 | ≤ L ∀s ∈ S} (8)
In an analogous manner as we did for E(K) we can define the following function: X X E(l, L) := max{ (ξn − K)( ps )dn : dn ∈ D(l, L)} (9) n∈N
s∈S(n)
13
Now from the fact that D(l1 , L) ⊆ D(l2 , L) when l1 ≤ l2 and D(l, L1 ) ⊆ D(l, L2 ) when L1 ≤ L2 follows that E(l, L) is nondecreasing in both arguments. Moreover observing the dual problem and proceeding as in the last proof, we can show the following result: Proposition 4 The real function E(l, L) is a continuous piecewise linear non-decreasing concave function in both arguments.
6
5.5
5 1
Volume
4.5
2 3
4
4 5
3.5
6 7
3 8
2.5
2
1
1.5
2
2.5 Time
3
3.5
4
Figure 4: Optimal paths depending on spot price scenario
3.6
A numerical example
We present some test results from our implementation. As a simple example we used a binary spot price scenario tree with 4 stages, having the structure and the probabilities as in Figure 3. We consider a Swing Option with li2 = li3 = 0 without swing number and freeze time constraints. The local boundaries were set to the following lmin := (3 2 3 4) lmax := (6 5 6 5) and the price per unit volume of the contract was set to K = 46. The optimal exercise patterns (stochastic process) acording to our model for this tree are presented in Figure 4. 14
1000
Profit
500
0
−500
−1000 10
20
30
40
50
60 70 Strike price
80
90
100
110
Figure 5: The dependence of the expected maximal profit from the strike price One can easily recognise that the paths of the presented solution are really paths of a tree process which is adapted to the original spot price tree process. In Figure 5 there is the dependence of the expected maximal profit E(K) as a function of the strike price K. As shown in Proposition 3 it is a continuous piecewise linear decreasing convex function For this particular swing contract we used the Newton method to find the strike price where the expected maximal profit would be 0. The was the value K = 49, 3114. Further we used the same binary spot price scenario tree with 4 stages and tested the model on a swing contract with nominal volume 5 and examined how the expected profit depends on the allowed local and global variation (l and L) of the feasible exercise patterns, i.e.: |di − 5| ≤ l
|D − 20| = |
i = 1, 2, 3, 4
4 X
di − 20| ≤ L
i=1
The dependence of the bid price from both l and L for this case can be seen in Fig.6. As seen from the figure the profit is continuous piecewise linear non-decreasing concave function of both the local and global constraints. 15
600
500
Max Profit
400
300
200
100
0
−100 30
25
25
20
20
15
15 10
10 5 L
5 0
0 l
Figure 6: The expected maximal profit of the swing option as a function of the allowed local and global swing bounds (This surface was calculated by solving 11250 linear programming problems.)
4
THE EQUILIBRIUM PRICE AND EXTENSIONS
Recall that the option holder determines his exercise pattern by solving, for a fixed unit price K, problem (5), while the option seller solves (4) for a given scenario-dependent demand d. Thus one may view the swing option problem as a two player game: The option seller has to decide about the unit price K and the the hedge vector x, the option buyer decides about the demand pattern d. Let us denote the objective function to be minimized for the seller by f (K, x, d) (in our case f (K, x, d) := K if the profit variable is acceptable and ∞ otherwise), the reward for the buyer by g(K, d) (in our case g(K, d) := EP Z(d, ξ)). From the game-theoretic view, this is an asymmetric, sequential, non-zero sum stochastic game.
16
The solution of this sequential game (the seller has to decide beforehand, the buyer decides later) is the following: First, one has to determine the solution of the buyer’s problem as a function of K, i.e. D(K) = argmax d∈D g(K, d). Then the solution of the seller’s problem is (K ∗ , x∗ ) = argmin
(K,x) f (K, x, D(K)).
The pertaining final demand is d∗ = D(K ∗ ). We find the solution of this game by the following numerical method: Since the function D(K) is piecewise constant(see Proposition 2) we may determine the intervals of constancy κi , κi+1 of this function, that is D(K) = di ,
if κi ≤ K ≤ κi+1 .
(10)
The seller’s problem is solved for each interval separately: (Ki∗ , x∗i ) = argmin
(K,x) {f (K, x, d
i
) : κi ≤ K ≤ κi+1 }.
(11)
If such a problem is infeasible, we set Ki∗ = ∞. The final solution is K ∗ = min Ki∗ . i
(12)
Let us show that this solution can also be interpreted as a fixpoint: Let H(K) = argmin K minx {f (K, x, argmax d∈D g(K, d)}. We claim that K ∗ is the lowest fixpoint of H, i.e. the smallest K such that H(K) = K: If K ∗ is a fixpoint, say κi ≤ K ∗ ≤ κi+1 , then K ∗ is a solution of the i-th problem (11). Conversely, if a solution for (11) exists in [κi , κi+1 ], then this K ∗ leads to demand di and the demand di leads back to K ∗ , thus K ∗ is a fixpoint. Fixpoint iteration i.e. the algorithm K (n+1) = H(K (n) ) does however not always lead to a solution: First, the contraction property of H is not guaranteed and second, the function H may have several fixpoints and only the smallest is of interest. Thus we propose to find the solution by finding the intervals of constancy of the buyer’s problem first and then solve all seller’s problems to find the feasible ones. The numerical determination of the intervals of constancy of the buyer’s problem is the well known sensitivity problem in linear programming. As the objective function is affine-linear in K, the intervals of constancy, i.e. the intervals for K, which belong to the same dual variables, can be found 17
using the final tableau of Dantzig’s Simplex method, see e.g. Ignizio (1982), page 234. We have implemented this method for the example from section (3.6), using only base and peak futures, which satisfy the expectation neutrality condition, found the intervals of constancy and solved the overall pricing problem (12), leading to an optimal strike price of K ∗ = 57.70. Let us finally mention that there is an alternative setup for finding the optimal strike price K, which is simpler, and not based on a game model. Some option buyers do not implement profit optimization strategies for the determining the demand. In particular, if option buyers are service or production companies, their demand pattern may dependent on outside temperature, on other commodity or financial markets, on some independent factors, etc, but not depend on the strike price K. For such customers, one could estimate sets of possible demand patterns D, by extrapolating some past behavior of the option buyer. Let D be a set of admissible, possibly scenario dependent demands, i.e. D ⊆ D. Let Y (x, K, d) be the profit variable for the option seller, which depends on the hedge x, the strike price K and the demand process d. For every d ∈ D, one may define the acceptability as before, i.e. AV@R(x, K, d) ≥ 0 for all d ∈ D and form the single optimization problem
Minimize K
subject to
AV@Rα [Y (x, K, d)] ≥ 0 for all d ∈ D For a practical solution, one assumes that the demand pattern set D is the (1) (J) convex hull of finitely many patterns ds,t , . . . , ds,t . Notice that by convexity AV@Rα [Y (x, K, d(j) )] ≥ 0
for j = 1, . . . , J
implies that AV@Rα [Y (x, K, d] ≥ 0
for all d ∈ D, the convex hull of d(j) , j = 1, . . . , J.
Therefore the pricing problem leads to one linear program of the form
Minimize K
subject to (13)
AV@Rα [Y (x, K, d(j) )] ≥ 0 for all j = 1, . . . , J Notice that it is assumed in (13), that the demands do not depend on the strike price, but they may depend on the spot market. If there is reasonable ground to assume that the option holder will exhibit a strike-price dependent behavior, the just described approach is not applicable and the game model or variants of it are more appropriate. 18
5
CONCLUSIONS
In this paper we proposed a stochastic multistage game theoretic behavioral model for the holder of a swing option, with a dynamical choice of exercise patterns. Having the information about spot price development on a binary scenario tree, we suppose the possibility to change the future part of the exercise path at every moment, to maximize the expected profit. The results of the model show the bid price dependence from the degree of flexibility of this swing contract. Furthermore, we presented a pricing model of scenario dependent demands, which solves the problem of evaluating the price of a demand pattern given a number of hedging instruments. Combining the two models has allowed us to determine the equilibrium price for such contracts. Future work will concentrate on the extension of the behavioral model to holders playing different economic roles of dealer and customer. Consequently, we have to analyse the price as a function of the holder’s degree of disposition to speculate.
19
REFERENCES -Barbieri A, Garman M.B. Understanding the Valuation of Swing Contracts, FEA Tech. Rep. , 2002 -Davison M, Anderson L. Approximate Recursive Valuation of Electricity Swing Options, 2003, (http://www.apmaths.uwo.ca/mdavison/ library/preprints/EJOR.pdf) -Eydeland A, Geman H. Some Fundamentals of Electricity Derivatives, Paris IX Dauphine and ESSEC, 1998 -Haarbr¨ ucker G, Kuhn D., Valuation of Electricity Swing Options by Multistage Stochastic Programming, Retrieved 2006-12-13, (http://www.alexandria.unisg.ch/publications/29493) -Ignizio J.,Linear Programming in single- & multiple-objective systems, PrenticeHall, 1982 -Jaillet P, Ronn E.I. Tompaidis S., Valuation of Commodity-Based Swing Options, Management Science, 50, 909–921, 2003 -Pflug G. Some remarks on the value-at-risk and the conditional value-atrisk, Probabilistic Constrained Optimization: Methodology and Applications, Kluwer Academic Publishers, Dordrecht, Uryasev S. ed.,2000 -Pilipovic D. Energy Risk, Valuing and Managing Energy Derivatives, McGrawHill, 1997 -Rosen J.B. Existence and Uniqueness of Equilibrium Points for Concave N-Person Games, Econometrica,Vol 33, No. 3, 520–534, 1965 -Rockafellar, R.T. and Uryasev, S., Optimization of Conditional Value-atRisk, Journal of Risk, vol 2,21-41, 2000 -Vattenfall Trading Services GmbH, The wholesale product GH0, 2005, (http://www.vattenfall.de/handelsprodukte)
20
ABSTRACT. Electricity swing options are supply contracts for power, which give the owner the right to change the required delivery on short time notice. It gives more flexibility than fixed base load or peak load contracts. The name ”option” is a bit misleading, since it gives the owner multiple exercise rights at many different time horizons with exercise amounts on a continuous scale. We look at the problem to determine a rational ask price for such a contract from the viewpoint of the contract seller. The pricing of these contracts differs drastically from the pricing of financial options. First, peculiar properties arise from the non-storability of the underlying (the energy) and therefore the impossibility to hedge with the underlying, hedging is only possible with some future contracts. Second, the behavior of the owner plays an important role. Based on some behavioral model for the option holder, we develop a game-theoretic model, which allows to identify the equilibrium price. Besides some theoretical results, we present some numerical results which clarify the dependence of the asked price on the amount of flexibility offered in the swing option.
KEYWORDS. Stochastic programming; OR in Energy; Pricing; Swing Option; Behavioral Model; Equilibrium price
1
INTRODUCTION
Along with the deregulation of the energy markets came growth in the importance of power derivatives such as futures, swaps, caps, floors, collars and options. Most of these contracts are relatively easy to price, given that a spot market model is available. In contrast, there is no generally accepted way 1
to price such flexible contracts as swing options. In this paper, we present a method to determine a rational ask price for swing options, which is based on the notion of the least acceptable price for the option seller and a behavioral model for the option holder. The pricing method is based on dynamic stochastic optimization and works for large classes of contracted restrictions on the exercise pattern of the option holder, such as the global volume, the number and intervals of change times, the amounts of allowed changes, etc. The option seller is confronted with the problem that the ask price must be determined long before exercising starts. The seller must make a hereand-now decision, while the option holder can base his exercise decision on data becoming available later (wait-and-see). This fact makes the problem a stochastic asymmetric game. The option seller has to anticipate the exercise pattern of the holder (or at least some worst-case pattern), bearing in mind that this exercise pattern will depend on his asked price if the contract is realized. Thus the price must satisfy a equilibrium in the sense that both parties have no reason to change their behavior. Notice however that the behavior of the contract holder is the modeled behavior, it is the behavior which the seller has much reason to assume that the holder will follow. But even if the seller can model the holder’s behavior correctly, there is no fair ask price in the sense of the no-arbitrage theory, because there are no already priced hedging instruments for replication or superreplication. To put it differently, hedging with futures may reduce risk and reshape the profit/loss distribution, but it can never bring the risk to zero. This is why we introduce the notion of the least acceptable ask price for a contract seller: An ask price is called acceptable, if there is some hedging strategy such that for the reshaped profit/loss distribution the probability of making a loss is below some predetermined threshold, typically 10-15%. Among all acceptable ask prices, the smallest is chosen in order to win against the competitors of the option seller. The proposed approach is new. Other pricing models for energy derivatives and the difficulties coming along with these models can be found in (Eydeland and Geman, 1998), (Barbieri and Garman, 2000) and (Davison and Anderson, 2003). These models use some no-arbitrage arguments, which are questionable, because of the non-storability and incompleteness of the electricity market. Other of the proposed models are special for the commodities markets (for example the gas markets) where stronger storability assumptions are possible. Notice that in situations with early exercise possibility and incomplete markets, no closed formula exists even for the simplest type of spot-price processes. Therefore the numerical analysis has to be done by solving a stochastic dynamic optimization problem. An overview on power markets and derivatives can be found in (Pilipovic, 1997). 2
The paper is organized as follows: In section 2, the principle of the least acceptable price for electricity delivery contracts is presented. Section 3 deals with swing options and rational exercise strategies. The two-person game and its equilibrium price together with some extensions are discussed in section 4. Conclusions can be found in section 5.
2
ACCEPTABLE PRICES FOR SPOT-PRICE DEPENDENT DEMANDS
To begin with, we assume that the seller knows the demand pattern of the holder, which may be spot-price dependent. To be more precise, he knows an exercise plan of the holder (who exercises conditional on the development of the spot-market), but he does not know the future spot prices themselves. The time index will be denoted by t, where t runs between 1 and T . We will exclusively work with a discrete scenario model and hence we assume that there is a finite number of possible scenarios s = 1, . . . , S together with the scenario probabilities p = (p1 , . . . , pS ), that is we assume a finite probability space Ω = {ω1 , . . . , ωS }. The spot-price model is given by a large [S × T ] matrix (ξs,t ). The (possibly spot-price dependent) demand pattern is described by another [S × T ] matrix (ds,t ). To reduce his own risk, the seller may buy energy futures, which gives standardized deliveries in particular times. However these contracts may never completely replicate the real demand pattern and a considerable residual risk remains with the option seller.
future market
- swing option seller
- swing option holder
We assume that there are M hedging instruments (types of energy futures) available. A hedging instrument is characterized by its today’s price πm and its delivery pattern τ , where τ (m, t) is the amount delivered in time period (e.g. hour) t. Typical patterns are • the base future: τ (·, t) ≡ 1 for a year, a quarter or a month 3
1. Method for the development of GH0 Thepeak basis for the wholesale product GH0 is the quarterly hour profile H0 of VDEW. It is scaled up to a • the future: volume that has a standard size for wholesale trading; afterwards it is converted into an hourly profile. The delivery periodτ (·, is at) complete calendar = 1, for Monyear. through Fri, 8 - 20 h
otherwise τ = 0. 2. The representative VDEW load profile H0 (households) TheVattenfall H0 profile isGH0 basedprofile. on a statistical of how households power. The profile • the This isanalysis a standardized profileconsume which follows consists of nine types of days. It is divided into three seasons (summer, winter, shoulder) and three typical requirement of a German household with seasonal and daily types of weekdays (Saturday, Sunday, working day). There is a profile with auxiliary values for every variation of hour the on demand (see Vattenfall (2005)), see Figure 1. in the appendix. The quarter of an every type of day. A table with those values is available profile is then rolled out according to the calendar. A household profile is standardised to the energy of 1 MWh/a. [W]
250,0 200,0 150,0 100,0 50,0 0,0 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 0: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23:
Winter Workday Winter Saturday Winter Sunday
Summer Workday Summer Saturday Summer Sunday
shoulder Workday shoulder Saturday shoulder Sunday
Figure 1: Auxiliary values for the representative VDEW load profile H0 for a standardised annual consumption of 1 MWh/a
Figure 1: The wholesale product GH0: Auxiliary values for the representative VDEW3.load profile H0 for a standardised annual consumption of 1 MWh/a Seasons The following seasons are applicable according to the specification for the representative VDEW load
In order to avoid unboundedness of the hedging problem, we assume that profile: the spot price model is calibrated to the known future prices πm such a way, that expectation-neutralityfrom holds, i.e.until Winter Shoulder Summer Shoulder Winter
01.01. 20.03. T 14.05. S 21.03. X X 14.09. π15.05. = τ (m, t) ps ξs,t m 15.09. t=1 31.10. s=1 01.11. 31.12.
The decision to be made by the option seller consists of 8 • the number xm of units of future contract m to be bought,
• the ask price K.
4
(1)
We collect the hedge amounts in a hedge vector x = (x1 , . . . , xM ). We assume in this paper that the option seller appears on the future marker only as a buyer, i.e. the hedges xm must be nonnegative. PMThe unmatched surplus/shortage in time period t is the difference m=1 xm τ (m, t) − ds,t . This amount will be sold/bought on the spot market. Given a hedge x, the surplus/shortage value in hour t, for scenario s is "M # X ∆s,t = ξs,t xm τ (m, t) − ds,t . m=1
We remark that it is easy to incorporate transaction costs in this frame+ work. Suppose that ξs,t is the revenue of selling one unit at the spot market − and ξs,t is the cost of buying one unit on the spot market. Then "M #+ "M #− X X + − ∆s,t = ξs,t xm τ (m, t) − ds,t + ξs,t xm τ (m, t) − ds,t . m=1
m=1
However, for the sake of simplicity, we will not work with transaction costs here. Denote by Y the profit/loss variable of this contract seen from the option seller. It consists of revenue from the swing option contract minus/plus the activity on the spot-market minus the costs for the hedges. Depending on the hedging decision x and the ask price K, the profit/loss variable takes the values # "M M T T X X X X πm xm xm τ (m, t) − ds,t − ξs,t ds,t + Ys (x, K) = K t=1
m=1
t=1
m=1
with probability ps . This random variable Y (x, K) is the basis for the acceptable ask price calculation. Notice that due to the expectation-neutrality (1), T S X X ds,t [K − ξs,t ], E[Y (x, K)] = ps t=1
s=1
that is, the expected profit does not depend on the hedging decisions. For this reason, the acceptability of a hedge cannot be based on a high profit expectation but must be based on another criterion, as is discussed next.
2.1
Acceptable ask prices and optimal hedging
A revenue variable Y is called acceptable, if the probability that it falls below 0 is less than α: P{Y (x, K) < 0} ≤ α. 5
where α is a given probability, typically set by the management of the sellers’ company. Among all acceptable return variables, the one with the smallest unit price for the delivery contract is called the least acceptable contract price. It is found by the following optimization problem
Minimize K
subject to (2)
P{Y (x, K) < 0} ≤ α This problem is unfortunately non-convex. By a slight modification one may convexify it. We replace the condition P{Y < 0} ≤ α by the stronger condition AV@Rα (Y ) ≥ 0, where AV@R is the average value-at-risk Z 1 α −1 GY (u) du AV@Rα [Y ] = α 0 with GY (u) = P {Y ≤ u}. Other names for AV@R are the conditional value at risk or the expected shortfall. It is easy to see that AV@Rα (Y ) ≥ 0 implies that G−1 Y (α) ≥ 0, i.e. P {Y < 0} ≤ GY (0) ≤ α. Rewriting the above problem for the AV@R, we get the AV@R constrained contract price problem
Minimize K
subject to (3)
AV@Rα [Y (x, K)] ≥ 0 which is a linear program in the variables x and K (see Rockefellar and Uryasev (2000) and Pflug (2000)). This follows from the fact that the condition AV@Rα [Y ] ≥ 0 is equivalent to: There is an a and a nonnegative vector z = (z1 , . . . , zs )> such that 1 a − p> z ≥ 0 and Ys − a + zs ≥ 0 for all s. α For further use introduce the following quantities P ζs,m = Tt=1 τ (m, t)ξs,t the spot-value of the m-th hedge in the s-th scenario P δs = Tt=1 ds,t ξs,t the spot-value of the the demand in the s-th scenario PT ¯ ds = t=1 ds,t the total demand in the s-th scenario In this notation, the profit/loss takes the values Ys (x, K) = d¯s K +
M X m=1
6
xm [ζs,m − πm ] − δs
with probability ps . Here is the structure of the convexified least acceptable ask price program:
Minimize (in a, x, z and K) : K
subject to
P
−a + 1 S ps zs ≤ 0 (4) s=1 α
−d¯s K + a + PM xm [πm − ζs,m ] − zs ≤ −δs ; s = 1, . . . , S
m=1
x, z ≥ 0 which can be solved by any LP-solver.
3 3.1
SWING OPTIONS Definitions
The typical swing option has the following characteristics (Jaillet et al. 2000): There are predefined exercise times τi , i ∈ {1, 2, . . . n}, 1 ≤ τ1 < τ2 < τ3 < · · · < τn ≤ T at which a fixed volume of d0 units of electricity may be obtained. With one day ahead notice, the owner of the option may use his swing right to receive more (up-swing) or less (down-swing) than this volume d0 units at any of this n moments. Some contracts allow swings only at N out of the possible n time moments (N ≤ n). This is the so called swing number constraint. There is often also a freeze time constraint that forbids swings at times which have a smaller distance than a given value ∆τ . Other constraints are the local boundaries, which define how much the requested amount di at time τi may differ from the nominal volume d0 . A typical constraint is di − d0 ∈ [li1 , li2 ) ∪ (li3 , li4 ], (li1 ≤ li2 ≤ 0 ≤ li3 ≤ li4 ). Moreover, there are also global constraints, which restrict the total requested volume D within the contract period by boundaries M in and M ax. In some contracts, the global constraints are replaced by penalty functions, which define the penalty costs for violating the constraints. Examples of penalty functions are: when D < M in C1 0 when M in ≤ D ≤ M ax ϕ(D) = ξT (D − M ax) when D > M ax where ξT is the spot price at the end of the period. Hard constraints can be modeled by the following penalty costs: 7
∞ when D < M in 0 when M in ≤ D ≤ M ax ϕ(D) = ∞ when D > M ax For further use, we define the following decision and volume functions: 1 Up-swing at time τj + χj = 0 else 1 Down-swing at time τj χ− j = 0 else dj − d0 when χ+ + j = 1 dj = 0 else dj − d0 when χ− − j = 1 dj = 0 else Now we are ready to write down the constraints of the swing contract in a formal way: − χ+ j , χj ∈ {0, 1} − 0 ≤ χ+ j + χj ≤ 1 for all 1 ≤ j ≤ n τj − + − for all 1 ≤ i < j ≤ n (χ+ i + χi ) + (χj + χj ) ≤ 1 + τi + ∆τ n X − 0≤ (χ+ j + χj ) ≤ N j=1 + 4 + lj3 χ+ j ≤ dj ≤ lj χj for all 1 ≤ j ≤ n − 2 − lj1 χ− j ≤ dj ≤ lj χj for all 1 ≤ j ≤ n − Notice that the complementary slackness condition χ+ j χj = 0 is automatically fulfilled (either Up or Down swing).
3.2
The structure of the exercise patterns
Let us investigate the set of the feasible exercise patterns in a more precise manner. From Figure 2 one sees that it has a pretty complex structure. If we analyze this set and consider it to it as a subset of the vector space of all possible (feasible and not feasible) and suppose li2 = li3 = 0 for simplicity, under the assumption of absence of swing number and freeze time constraint we can show the following result. 8
6 nominal volume
exercise pattern local maximum
5
Volumen
4
3
2
1 local minimum
0
0
2
4
6
8
10 12 Period
14
16
18
20
22
Figure 2: An example of swing option feasible exercise pattern Proposition 1 The set of the feasible exercise patterns of a swing option D = D(d0 , n, Lmin , Lmax , M in, M ax, N, ∆i) is bounded and convex when N = n and ∆i = 1. Here d0 is the nominal volume, n the number of swing possibilities, Lmin := {l11 , l21 . . . ln1 } the lower local boundaries set, Lmax := {l14 , l24 . . . ln4 } the upper local boundaries set, M in, M ax global boundaries N the maximal number of swings and ∆i the freeze time. We assume that swing may only be exercised at integer times. Proof. Let observe the convex combination d00 of two exercise patterns: d00 = αd + βd0 d, d0 ∈ D(d0 , n, Lmin , Lmax , M in, M ax, n, 1) d00 = (d001 d002 . . . d00n ) = (αd1 + βd01 αd2 + βd02 . . . αdn + βd0n ) where α, β ≥ 0 and α + β = 1 For the local constraints we have: d00i − Lmin (i) = αdi + βd0i − Lmin (i) = = αdi + βd0i − (α + β)Lmin (i) = = α(di − Lmin (i)) + β(d0i − Lmin (i)) ≥ 0,
9
in an analogous manner for Lmax (i). And for the global constraints we have: n X
d00i
− M in =
i=1
n X
(αdi + βd0i ) − M in =
i=1
=
n X
(αdi + βd0i ) − (α + β)M in =
i=1 n n X X = α( di − M in) + β( d0i − M in) ≥ 0 i=1
i=1
and similarly analog for M ax. So we have shown that d00 ∈ [Lmin , Lmax ], n X d00i ∈ [M in, M ax], i=1
which means that the set is convex. 2 We remark that non-convex set of feasible exercise patterns may lead to stochastic mixed-integer problems, which are outside the scope of this paper.
3.3
An exercise model
Similar to (Haarbr¨ ucker and Kuhn(2006)) we regard the option holder as a player in a multistage stochastic game and model his behavior as the optimal strategy in this game. Assume that a spot price scenario tree is given. The option holder is fully informed about the actual node of the tree. Based on this knowledge and on his own past decisions he maximizes his expected profit till the end of the exercise period. More precisely, assume that a finite probability space (Ω, P(Ω), P), Ω = {ω1 , . . . , ωS } is given, on which a filtration is defined F = (Ft )t∈{1,...,T } , with F1 = {∅, Ω} and FT = P(Ω). The spot price process (ξt )t∈{1,...,T } (ξt (ωs ) = ξs,t ) is adapted to F. The process ξ can be represented by a tree whose terminal nodes are the scenarios {ω1 , . . . , ωS }. At each node of the tree corresponding to F, the holder chooses his exercise volume to maximize his expected profit under the constraints coming from the contract and his past decisions. Further his decision at moment t may be dependent on the information about the spot prices only up to this moment which means the exercise process (dt )t∈{1,...,T } (dt (ωs ) = ds,t ) should be a adapted regarding the filtration F process. Thus we can represent the optimal exercise process 10
as the solution of the following multistage stochastic program. dmax ∈ argmax{EP Z(d, ξ) : d ∈ D, d adapted to F} Here Z is the (random) profit if the exercise process (dt ) is chosen. Z(d, ξ) :=
T X
(ξt − K)dt
t=1
We assume that the profit per unit is the difference between spot price ξt and contract strike price K. We assume further the bid-price of the swing contract is equal to the maximal expected profit as described above. To determine the bid-price we suppose that the contract issuer has no possibility to hedge and uses only the spot market to cover the demand of the holder. A model that gives the issuer the possibility to hedge with future contracts is given in the next section. Notice the asymmetric structure of the model: The option holder may implement wait and see strategies for exercising, while the issuer in a here and now situation when he has to determine the asked price. For this reason, the filtration is irrelevant for the option seller’s optimization program (4), while it is crucial for the option holder’s decision problem (5) below. t1 , F1
t2 , F2
t3 , F3
t4 , F4 GFED 1/3 105 ddd2 @ABC ddddddd
ddd ?>=< 75 ZdZZZZZZZZZZ j4 89:; ZZZZ, 89:; ?>=< 2/3
3/10 jjjjj
j 40 jjjj jjjj ?>=< 89:; T 67 TTTT : TTTT uu ?>=< TT uu 3/7 75 ddd2 89:; u 7/10 TTTT* ddddddd 1/2 uuu d d d d d 89:; ?>=< u Z Z Z 35 ZZZZZZZZ u uu ZZZZ, 89:; ?>=< uu 4/7 u 30 u u ?>=< 89:; 45 II II ?>=< II 4/7 ddddd2 89:; 45 II ddddddd d d d II 89:; ?>=< Z Z 4 Z Z 35 I j ZZZZZZZ 1/2 III 7/10 jjjjj ZZZZ, 89:; ?>=< II jj 3/7 j 50 j j I$ j j j j ?>=< 89:; T 27 TTTT TTTT ?>=< TT 2/3 dddddd2 89:; 15 d d d 3/10 TTTT* d d d d dZdZZ 89:; ?>=< 45 ZZZZZZZZ ZZZ, @ABC GFED 1/3 100
Figure 3: The spot price scenario tree for our numerical example
11
3.4
The Linear Programming formulation
According to Proposition 1 under the assumption of no swing number constraint and no freeze time constraint the set of the feasible exercise patterns is convex. Moreover the chosen target function is linear in the time components of the exercise process, thus our multistage stochastic program can be represented as a linear program for the vector consisting of the volume decisions in each node of the tree. Enumerating the nodes of the tree in the standard way we obtain the following linear program.
P P
Maximize (in dn ): P ps P (ξ − K)d = (ξ − K)( n n n s∈S n∈N (s) n∈N s∈S(n) ps )dn
subject to:
d − d ≤ l4 ∀n ∈ N 0
n t(n)
d − d ≥ l1 ∀n ∈ N
P n 0 t(n)
n∈N (s) dn ≤ M ax ∀s ∈ S
P
n∈N (s) dn ≥ M in ∀s ∈ S
dn ≥ 0 ∀n ∈ N (5) Here we use the following terms: N S ps ξn t(n) S(n) N (s)
The The The The The The The
set of all nodes set of all scenarios probability of scenario s spot price in node n time of node n set of all scenarios to which node n belongs nodes belonging to scenario s and all time horizons
The dimension of the problem is clearly equal to the number of nodes in the tree. An example of such tree can be seen in Figure 3.
3.5
Parameter analysis
The solution of the option holder’s problem (5) depends on the value of the strike price. Thus the value of the maximum as well the dn , n ∈ N for which this maximum is attained are functions of K which we denote by E and D: X X E(K) := max{ (ξn − K)( ps )dn : dn ∈ D} (6) n∈N
D(K) := argmax {
X
s∈S(n)
(ξn − K)(
n∈N
X
s∈S(n)
12
ps )dn : dn ∈ D}
(7)
Here D is the constraint set of the problem (5). Because of the structure of the set D and the linearity of the target function in K it is obvious that: Proposition 2 The function D(K) is a piecewise constant function, mapping R to R|N | . The following result about the function E is easily shown: Proposition 3 The real function E(K) is a continuous piecewise linear decreasing convex function. Proof. We have E(K) := max{Ed (K) : d ∈ D} where Ed (K) = Σn∈N (ξn −K)(Σs∈S(n) ps )dn are linear functions of K. As it is known that the maximum of linear functions is a continuous convex function, this clearly follows for E(K) too. Moreover for the finite set of the extreme points D∗ of the feasible set we have E(K) := max{Ed (K) : d ∈ D} = max{Ed (K) : d ∈ D∗ } which means E(K) is a piecewise linear function with finite number of different linear parts. From the fact that all Ed (K) are decreasing it follows that their maximum E(K) is also decreasing. This completes the proof. 2 For further analysis of the dependence of the expected maximal achievable profit from the flexibility of the contract we take a special form of our linear 1 4 := l, M in = T d0 − L and M ax = T d0 + L. Now = −lt(n) program where lt(n) we obtain the following LP with two parameters l and L: P P P P
Maximize: s∈S ps n∈N (s) (ξn − K)dn = n∈N (ξn − K)( s∈S(n) ps )dn
subject to:
dn ≤ d0 + l ∀n ∈ N
dn ≥ d0 − l ∀n ∈ N
P
n∈N (s) dn ≤ T d0 + L ∀s ∈ S
P
n∈N (s) dn ≥ T d0 − L ∀s ∈ S In this case we use the following notation for the feasible set of patterns. D(l, L) := {d : |dn − d0 | ≤ l
∀n ∈ N , |Σn∈N (s) dn − T d0 | ≤ L ∀s ∈ S} (8)
In an analogous manner as we did for E(K) we can define the following function: X X E(l, L) := max{ (ξn − K)( ps )dn : dn ∈ D(l, L)} (9) n∈N
s∈S(n)
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Now from the fact that D(l1 , L) ⊆ D(l2 , L) when l1 ≤ l2 and D(l, L1 ) ⊆ D(l, L2 ) when L1 ≤ L2 follows that E(l, L) is nondecreasing in both arguments. Moreover observing the dual problem and proceeding as in the last proof, we can show the following result: Proposition 4 The real function E(l, L) is a continuous piecewise linear non-decreasing concave function in both arguments.
6
5.5
5 1
Volume
4.5
2 3
4
4 5
3.5
6 7
3 8
2.5
2
1
1.5
2
2.5 Time
3
3.5
4
Figure 4: Optimal paths depending on spot price scenario
3.6
A numerical example
We present some test results from our implementation. As a simple example we used a binary spot price scenario tree with 4 stages, having the structure and the probabilities as in Figure 3. We consider a Swing Option with li2 = li3 = 0 without swing number and freeze time constraints. The local boundaries were set to the following lmin := (3 2 3 4) lmax := (6 5 6 5) and the price per unit volume of the contract was set to K = 46. The optimal exercise patterns (stochastic process) acording to our model for this tree are presented in Figure 4. 14
1000
Profit
500
0
−500
−1000 10
20
30
40
50
60 70 Strike price
80
90
100
110
Figure 5: The dependence of the expected maximal profit from the strike price One can easily recognise that the paths of the presented solution are really paths of a tree process which is adapted to the original spot price tree process. In Figure 5 there is the dependence of the expected maximal profit E(K) as a function of the strike price K. As shown in Proposition 3 it is a continuous piecewise linear decreasing convex function For this particular swing contract we used the Newton method to find the strike price where the expected maximal profit would be 0. The was the value K = 49, 3114. Further we used the same binary spot price scenario tree with 4 stages and tested the model on a swing contract with nominal volume 5 and examined how the expected profit depends on the allowed local and global variation (l and L) of the feasible exercise patterns, i.e.: |di − 5| ≤ l
|D − 20| = |
i = 1, 2, 3, 4
4 X
di − 20| ≤ L
i=1
The dependence of the bid price from both l and L for this case can be seen in Fig.6. As seen from the figure the profit is continuous piecewise linear non-decreasing concave function of both the local and global constraints. 15
600
500
Max Profit
400
300
200
100
0
−100 30
25
25
20
20
15
15 10
10 5 L
5 0
0 l
Figure 6: The expected maximal profit of the swing option as a function of the allowed local and global swing bounds (This surface was calculated by solving 11250 linear programming problems.)
4
THE EQUILIBRIUM PRICE AND EXTENSIONS
Recall that the option holder determines his exercise pattern by solving, for a fixed unit price K, problem (5), while the option seller solves (4) for a given scenario-dependent demand d. Thus one may view the swing option problem as a two player game: The option seller has to decide about the unit price K and the the hedge vector x, the option buyer decides about the demand pattern d. Let us denote the objective function to be minimized for the seller by f (K, x, d) (in our case f (K, x, d) := K if the profit variable is acceptable and ∞ otherwise), the reward for the buyer by g(K, d) (in our case g(K, d) := EP Z(d, ξ)). From the game-theoretic view, this is an asymmetric, sequential, non-zero sum stochastic game.
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The solution of this sequential game (the seller has to decide beforehand, the buyer decides later) is the following: First, one has to determine the solution of the buyer’s problem as a function of K, i.e. D(K) = argmax d∈D g(K, d). Then the solution of the seller’s problem is (K ∗ , x∗ ) = argmin
(K,x) f (K, x, D(K)).
The pertaining final demand is d∗ = D(K ∗ ). We find the solution of this game by the following numerical method: Since the function D(K) is piecewise constant(see Proposition 2) we may determine the intervals of constancy κi , κi+1 of this function, that is D(K) = di ,
if κi ≤ K ≤ κi+1 .
(10)
The seller’s problem is solved for each interval separately: (Ki∗ , x∗i ) = argmin
(K,x) {f (K, x, d
i
) : κi ≤ K ≤ κi+1 }.
(11)
If such a problem is infeasible, we set Ki∗ = ∞. The final solution is K ∗ = min Ki∗ . i
(12)
Let us show that this solution can also be interpreted as a fixpoint: Let H(K) = argmin K minx {f (K, x, argmax d∈D g(K, d)}. We claim that K ∗ is the lowest fixpoint of H, i.e. the smallest K such that H(K) = K: If K ∗ is a fixpoint, say κi ≤ K ∗ ≤ κi+1 , then K ∗ is a solution of the i-th problem (11). Conversely, if a solution for (11) exists in [κi , κi+1 ], then this K ∗ leads to demand di and the demand di leads back to K ∗ , thus K ∗ is a fixpoint. Fixpoint iteration i.e. the algorithm K (n+1) = H(K (n) ) does however not always lead to a solution: First, the contraction property of H is not guaranteed and second, the function H may have several fixpoints and only the smallest is of interest. Thus we propose to find the solution by finding the intervals of constancy of the buyer’s problem first and then solve all seller’s problems to find the feasible ones. The numerical determination of the intervals of constancy of the buyer’s problem is the well known sensitivity problem in linear programming. As the objective function is affine-linear in K, the intervals of constancy, i.e. the intervals for K, which belong to the same dual variables, can be found 17
using the final tableau of Dantzig’s Simplex method, see e.g. Ignizio (1982), page 234. We have implemented this method for the example from section (3.6), using only base and peak futures, which satisfy the expectation neutrality condition, found the intervals of constancy and solved the overall pricing problem (12), leading to an optimal strike price of K ∗ = 57.70. Let us finally mention that there is an alternative setup for finding the optimal strike price K, which is simpler, and not based on a game model. Some option buyers do not implement profit optimization strategies for the determining the demand. In particular, if option buyers are service or production companies, their demand pattern may dependent on outside temperature, on other commodity or financial markets, on some independent factors, etc, but not depend on the strike price K. For such customers, one could estimate sets of possible demand patterns D, by extrapolating some past behavior of the option buyer. Let D be a set of admissible, possibly scenario dependent demands, i.e. D ⊆ D. Let Y (x, K, d) be the profit variable for the option seller, which depends on the hedge x, the strike price K and the demand process d. For every d ∈ D, one may define the acceptability as before, i.e. AV@R(x, K, d) ≥ 0 for all d ∈ D and form the single optimization problem
Minimize K
subject to
AV@Rα [Y (x, K, d)] ≥ 0 for all d ∈ D For a practical solution, one assumes that the demand pattern set D is the (1) (J) convex hull of finitely many patterns ds,t , . . . , ds,t . Notice that by convexity AV@Rα [Y (x, K, d(j) )] ≥ 0
for j = 1, . . . , J
implies that AV@Rα [Y (x, K, d] ≥ 0
for all d ∈ D, the convex hull of d(j) , j = 1, . . . , J.
Therefore the pricing problem leads to one linear program of the form
Minimize K
subject to (13)
AV@Rα [Y (x, K, d(j) )] ≥ 0 for all j = 1, . . . , J Notice that it is assumed in (13), that the demands do not depend on the strike price, but they may depend on the spot market. If there is reasonable ground to assume that the option holder will exhibit a strike-price dependent behavior, the just described approach is not applicable and the game model or variants of it are more appropriate. 18
5
CONCLUSIONS
In this paper we proposed a stochastic multistage game theoretic behavioral model for the holder of a swing option, with a dynamical choice of exercise patterns. Having the information about spot price development on a binary scenario tree, we suppose the possibility to change the future part of the exercise path at every moment, to maximize the expected profit. The results of the model show the bid price dependence from the degree of flexibility of this swing contract. Furthermore, we presented a pricing model of scenario dependent demands, which solves the problem of evaluating the price of a demand pattern given a number of hedging instruments. Combining the two models has allowed us to determine the equilibrium price for such contracts. Future work will concentrate on the extension of the behavioral model to holders playing different economic roles of dealer and customer. Consequently, we have to analyse the price as a function of the holder’s degree of disposition to speculate.
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