March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
International Journal of Theoretical and Applied Finance
Vol. 14, No. 2 (2011) 295–312
c World Scientific Publishing Company
DOI: 10.1142/S021902491100636X
HEDGING SWING OPTIONS
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
JESÚS F. RODRÍGUEZ
Mathematics Department, Rutgers University
Hill Center — Busch Campus
Piscataway, NJ 08854-8019, USA
jesusR@math.rutgers.edu
Received 3 August 2009
Accepted 12 November 2010
We study models for electricity pricing and derivatives in the context of a deregulated
market setting. In particular we value swing options, since these are the electricity derivatives that attract the most attention from market participants. These are American style
options in that they allow for multiple exercises subject to a set of constraints on the
consumption process. Through the use of a penalty function, we generalize the problem
by allowing for the consumption restrictions to be broken. We characterize the price
function as a stochastic optimal control problem, and show that the option is exercised
in a bang-bang fashion. The value of the swing option is the solution to a backward
stochastic differential equation, and we show how European calls, along with forward
contracts, can be used to hedge them.
Keywords: Electricity derivatives; swing options; energy markets; stochastic optimal
control.
1. Introduction
Energy markets around the world have undergone rapid deregulation in the past
decade and the trend appears to be continuing. Of course, as with the deregulation
of any market, there are specific economic and policy issues that have been raised
in the electricity sector. For a treatment of these market-specific issues the reader is
directed to [3] and [17] for an introduction. We will, however, concern ourselves here
with energy trading and risk management issues. A good overview is given in [6]
and [15]. This deregulation has naturally led to increased levels of volatility in the
price of electricity, and hence a need to reduce exposure to risk for participants in
the market.
To reduce this risk, we need to know how to price derivatives in the electricity
market. Several types of financial derivatives are traded that are of interest. There
are monthly physical options, which when traded have several specifications, such
as the location, exercise month, time of day for delivery (e.g. on-peak, off-peak,
or round-the-clock), the strike price, and the amount of megawatt hours (MWh).
295
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
296
104-IJTAF
SPI-J071
J. F. Rodrı́guez
There are also daily options which can be traded each day over a given time period.
The most typical of these are the plain vanilla daily options, swing options, and
spread options. Their primary use is for managing load and managing assets. Similarly, hourly options are traded with analogous characteristics. Due to the infancy
of the field, it is a question that has not been well studied. There is a lack of historical data, and different regions have differing rates, as well as approaches, to
deregulation.
The electricity market is similar to the well-studied stock and bond markets
in some respects, but it is very different in some crucial aspects. There are issues
with cost of production such as ramp up costs, or the costs to initiate production
capabilities of a generator, and also with differing generator efficiencies. Also, seasonality plays a role in price fluctuations, as does the fact that the electricity market
is decentralized and thus participants are exposed to geographic risks.
However, the most important features for our purposes are the demand inelasticity and the non-storability of electricity. Electricity demand is highly inelastic.
Because we lack efficient ways to store electricity, when there is a spike in demand,
the power provider has to produce the electricity at that moment. But immediate high electricity production requires using less efficient power generators, thus
creating the price spikes exhibited in the market.
The price volatility leads us to search for hedging strategies to reduce risk.
However, the non-storability property of electricity makes it impossible to value
options using the standard cash and carry arguments used to price options for most
assets. Our aim in this paper is to use forwards to value options on spot electricity.
We explain in detail how swing options are used, and establish that the consumption
strategy satisfies the bang-bang property. We then give a simple approach to pricing
them, and demonstrate how they may be hedged.
2. Swing Options
It has become prevalent in practice to address the issue of non-storability of electricity using swing options. Because of the storability limitations in electricity, contracts
are needed to allow for flexibility in delivery with respect to amount and time.
Swing options address the need to hedge in markets that are subject to frequent
spikes in price and demand followed by a return to normal levels. A swing option in
the electricity market gives the owner the right to consume energy for a fixed price
during a fixed time interval. This flexibility-of-delivery option provides the owner of
the swing option with flexibility to receive differing amounts of energy depending on
the demand. However, the contract stipulates obligations that have to be met as far
as instantaneous consumption is concerned, as well as upper and lower constraints
on the total cumulative consumption over the life of the contract. The restrictions
to the electricity consumption, u(t), being used at any time t, must be satisfied,
while the cumulative constraints may be broken, in which case the holder would
have to pay some penalty.
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
297
There are two categories of swing option contracts, and they depend on the
duration of the effect that exercising the option causes. The two types, as described
in [10] are:
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
(1) Local effect: The exercise of a right modifies the delivery volume only at the
time of exercise, i.e., the delivery reverts to the level specified in the base-load
contract thereafter.
(2) Global effect: The exercise of a right modifies the delivery volume from the
exercise date onward, i.e., the delivery remains at the new level until the next
exercise, if any.
The former has been studied by many authors. In this scenario, the option
is treated as an American type claim with multiple exercise possibilities. In [18],
Thompson studied this and developed a numerical pricing algorithm using a lattice
based technique. Jaillet et al. present a numerical scheme in [10] for the valuation
of swing options by using a multiple layer extension of the usual binomial/trinomial
method to the so-called forrest-tree approach and achieve numerical success for the
one factor model. In [1] Barrera-Esteve et al. extend the work of [12] to numerically
price the swing option using Monte-Carlo simulations. Also, [4] and [5] apply Snell
envelope theory to study the exercise regions, and are able to prove existence of
multiple exercise policies. Local effects are also studied in [7] by characterizing the
value of the option as an impulse control problem, and approximating numerically
by solving a system of quasi-variational inequalities.
We however will focus on the second category of contracts, which is more prevalent in practice. In [11], Keppo studies this, and shows how to price the swing option
with a basket of forwards and calls. We attain similar results, but find a solution
which is easier for implementation.
The pricing of the swing option is a stochastic control problem, and some preliminaries will first need to be introduced. In the following, (Ω, F , F, P ) is a filtered
probability space with filtration F = (Ft )t≥0 which satisfies the usual hypotheses
(see e.g. [16]). Let W = (Wt1 , Wt2 )t≥0 be a standard 2-dimensional Brownian motion
defined on (Ω, F , F, P ).
Definition 2.1. Given a subset U of R, we denote by U0 the set of all progressively
measurable processes u = {ut , t ≥ 0} valued in U . The elements of U0 are called
control processes.
Let
β : (t, x, u) ∈ R+ × Rn × U → β(t, x, u) ∈ R
and
σ : (t, x, u) ∈ R+ × Rn × U → σ(t, x, u) ∈ Rn×2
be two given functions satisfying the uniform Lipschitz condition
|β(t, x, u) − β(t, y, u)| + |σ(t, x, u) − σ(t, y, u)| ≤ C1 |x − y|,
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
298
104-IJTAF
SPI-J071
J. F. Rodrı́guez
for some constant C1 that does not depend on (t, x, y, u). For each control process
u ∈ U, we consider the state stochastic differential equation:
dXt = β(t, Xt , ut )dt + σ(t, Xt , ut )dWt .
(2.1)
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
Definition 2.2. If equation (2.1) has a unique solution X for given initial data,
then the process X is called the controlled process.
Definition 2.3. Let T > 0 be some given time horizon. We shall denote by U the
subset of all control processes u ∈ U0 which satisfy the additional requirement:
T
2
(|β(t, x, ut )| + |σ(t, x, ut )| )dt < ∞ for x ∈ Rn .
(2.2)
E
0
Any such u is an admissible control process.
The condition given in Definiton 2.3 guarantees the existence of a controlled
process for each given initial condition and control under the uniform Lipschitz
condition on the coefficients β and σ. This a consequence of a general existence
theorem (see e.g. [16]).
Let
f, k : [0, T ] × Rn × U → R
and g : Rn → R
be given functions. We assume that k − ∞ < ∞ (i.e. max{−k, 0} is uniformly
bounded), and f and g satisfy the quadratic growth condition:
|f (t, x, u)| + |g(x)| ≤ C2 (1 + |x|2 )
for some constant C2 that does not depend on (t, u).
Definition 2.4. We define the cost function J on [0, T ] × Rn × U by:
T
J(t, x, u) ≡ Et,x
D(t, s)f (s, Xs , us )ds + D(t, T )g(XT )
t
where
D(t, s) ≡ e−
Rs
t
k(z,Xz ,u(z))dz
(2.3)
and Et,x is the expectation conditioned on Xt = x where X is the solution to the
above SDE with initial condition Xt = x and control u.
We now formulate the swing option mathematically. If t is the time when
the contract is written, the swing option is defined by the set of parameters
{ulow (·), uup (·), elow , eup , T0 , T1 , K}. The option takes effect during the time interval
[T0 , T1 ] where t ≤ T0 < T1 . We will say that
u(·) : [T0 , T1 ] → R+
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
299
is the swing option’s consumption process, and it is stochastic. Also,
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
ulow (·) : [T0 , T1 ] → R+ and uup (·) : [T0 , T1 ] → R+ ,
with ulow (t) ≤ uup (t) ∀ t ∈ [T0 , T1 ], are deterministic right continuous functions
that are the lower and upper boundary functions for consumption. We say that
ulow and uup are the instantaneous consumption constraints. In addition, elow and
eup are given constants and the lower and upper boundaries respectively for the
total purchased energy. We say that elow and eup are the cumulative consumption
constraints. As usual, K is the price paid for electricity when the option is exercised,
or the strike
t price. If we let the total cumulative consumption process be given by
C u (t) = T0 u(s)ds, then the set of admissible consumption strategies for the owner
are the admissible control processes u which satisfy
elow ≤ C u (T1 ) ≤ eup ,
ulow (t) ≤ u(t) ≤ uup (t),
and
∀ t ∈ [T0 , T1 ] ∀ ω ∈ Ω
(2.4)
(2.5)
where it is assumed that uup (t) − ulow (t) is constant. The above is the standard
framework for the swing option considered in the literature.
We will in fact consider a more general version of this in which condition in (2.4)
is not an absolute constraint. We study the more general version because, although
the swing option contract has a cumulative consumption constraint, in practice it
is often possible to not meet that obligation. In this case the owner of the option
has to pay a penalty fee for the over- or under-consumption of power over the life
of the contract. The payoff to the owner of the swing option on the time interval
[T0 , T1 ] with strike price K is given by
T1
W (ω; T1 , K, u) =
u(t)(St − K)dt + P(C u (T1 ))
T0
where u is the consumption strategy used, and P is our penalty function which
is the dollar amount the owner of the option has to pay for either exceeding the
upper cumulative constraint or for not meeting the lower cumulative constraint.
The penalty function can take different forms. We will take P to be
P(x) = −B(elow − x) · 1{x<elow } − A(x − eup ) · 1{x>eup } ,
(2.6)
where A and B are fixed non-random constants.
Remark 2.1. Sometimes the penalty can depend on the spot price of electricity
at expiry, in which case A and B in (2.6) are replaced by the random variable ST1 ,
where ST1 denotes the spot price of electricity at time T1 . Or, in some cases, the
penalty is just a constant, and it does not matter by how much the owner fails to
meet the requirement.
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
300
104-IJTAF
SPI-J071
J. F. Rodrı́guez
Suppose our model for the spot price S is given by
dSt = m(St )dt + v(St )dWt1 .
Now for x ∈ R2 , let
m(x1 )
β(t, x, u) =
,
u
v(x1 ) 0
and σ(t, x, u) =
.
0
0
Let
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
Uu =
max uup (s)
s∈[T0 ,T1 ]
and Ul =
min
s∈[T0 ,T1 ]
ulow (s).
Then we have that our consumption processes are admissible control processes
belonging to the compact set U ≡ [Ul , Uu ], and since β and σ satisfy the desired
integrability conditions, there exists a controlled process X given by
dXt = β(t, Xt , u(t))dt + σ(t, Xt , u(t))dWt
for each initial condition and control. Furthermore, we know that Xt = (St ,
C u (t))T where St is the spot price of electricity at time t, and C u is the cumulative consumption process.
Now if we take P as defined in (2.6), then for any t, if we let f and g be given by
f (t, x, u) = u(x1 − K),
and g(x) = P(x2 ),
(2.7)
then we can see that f, g satisfy the quadratic growth condition, and hence the cost
function
T1
u
J(u) = Et,x
D(t, s)u(s)(Ss − K)ds + D(t, T1 )P(C (T1 )) ,
T0 ∨t
= Et,x
T1
T0 ∨t
D(t, s)f (s, Xs , u)ds + D(t, T1 )g(XT1 )
Rs
where D(t, s) = e− t r(z)dz , is well-defined.
The writer of the swing option, in order to hedge her position, must assume
the buyer will try to maximize J(u) over all possible strategies u. So our goal is to
determine this value. Intuitively, it seems that the optimal consumption strategy
is to consume as much as possible when one exercises the swing option. This is
known as acting in a bang-bang fashion. In [10], it is discussed in the setting of
swing options whose exercises have local effects. It is shown that if it is optimal to
exercise at all, without other constraints, then it is optimal to exercise as much as
possible. To prevent this from occurring, the notion of refraction time is introduced.
In our setting we do not need a refraction time because of the absolute instantaneous
constraints, but we show the option is exercised in a bang-bang fashion nonetheless.
We say that u∗ is optimal if
J(u∗ ) = sup J(u).
u∈U
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
301
One can show the existence of an optimal control, u∗ , using general results
from stochastic control (see e.g. [9]). We show here that if the owner decides to
exercise the swing option, the exercise will be for the maximum allowable amount.
Otherwise, the owner of the option consumes ulow as required by the instantaneous
consumption constraints.
Theorem 2.1. Under the constraints above, the optimal consumption process u∗
can only take two values, and in fact
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
u∗ (t) ∈ {ulow (t), uup (t)}
∀ t ∈ [T0 , T1 ].
Proof. Let ū ∈ U be some admissible control process. Let τ1 and τ2 be two stopping
times such that T0 ≤ τ1 < τ2 ≤ T1 a.s. Now, for ∈ [0, 1], define
u (t) = u∗ (t) + · (ū(t) − u∗ (t))1(τ1 ,τ2 ) (t).
(2.8)
For any ∈ [0, 1], u is clearly an admissible control process, and so we have
J(u ) ≤ J(u∗ )
∀ ∈ [0, 1].
t
Since C u (t) = T0 u(s)ds for t ∈ [T0 , T1 ] is our total cumulative consumption up to
time t using strategy u, then evaluating a Gâteaux-like derivative, we see that
J(u ) − J(u∗ )
↓0
T1
1
= lim
E
e−r(s−t) u (s)(Ss − K)ds + e−r(T1 −t) P(C u (T1 ))
↓0 T0
0 ≥ lim
T1
−E
= lim
↓0
1
E
∗
e−r(s−t) u∗ (s)(Ss − K)ds + e−r(T1 −t) P(C u (T1 ))
T0
τ2
e−r(s−t) (ū − u∗ )(s)(Ss − K)ds
τ1
∗
+e−r(T1 −t) (P(C u (T1 )) − P(C u (T1 )))
τ2
=E
e−r(s−t) (ū − u∗ )(s)(Ss − K)ds
τ1
+E e
−r(T1 −t)
∗
P(C u (T1 )) − P(C u (T1 ))
lim
↓0
.
(2.9)
To analyze this we first need to evaluate
∗
1
lim [P(C u (T1 )) − P(C u (T1 ))].
↓0 (2.10)
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
302
104-IJTAF
SPI-J071
J. F. Rodrı́guez
∗
We first notice that if C u (T1 ) ∈ (0, elow ) or (elow , eup ) or (eup , ∞), then there
exists ˆ > 0 such that ∀ < ˆ, C u (T1 ) belongs to the same open interval. This is
because
T1
T1
τ2
∗
u (s)ds −
u (s)ds =
(ū − u∗ )(s)ds
T0
T0
τ1
≤ · d · (T1 − T0 ),
(2.11)
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
where d = supt∈[T0 ,T1 ] (uup (t) − ulow (t)). Therefore, for small enough, C u (T1 ) is
∗
arbitrarily close to C u (T1 ), and hence for all such , both consumption strategies
at time T1 will lie in the same penalty region. So using this fact, there are three
cases we will investigate:
∗
(1) Suppose C u (T1 ) > eup . Then there exists ˆ s.t ∀ < ˆ, C u (T1 ) > eup , and for
all such ,
T1
∗
P(C u (T1 )) − P(C u (T1 )) = −A
u (s)ds − eup
T0
−
= −A
T1
∗
u (s)ds − eup
T0
τ2
· (ū(s) − u∗ (s))ds.
τ1
∗
(2) If elow < C u (T1 ) < eup , then again for small enough, neither consumption
strategy will result in the owner of the option being penalized, and
∗
P(C u (T1 )) − P(C u (T1 )) = 0 − 0 = 0.
∗
(3) Finally, if C u (T1 ) < elow then again for small enough C u (T1 ) < elow , and so
as in the first case
τ2
∗
P(C u (T1 )) − P(C u (T1 )) = B
· (ū(s) − u∗ (s))ds.
τ1
u∗
(4) Suppose C (T1 ) = elow . Then can be chosen small enough so that C u (T1 ) <
eup . Then, for sufficiently small ,
∗
P(C u (T1 )) − P(C u (T1 )) = P(C u (T1 ))
= −B(elow − C u (T1 )) · 1{C u (T1 )<elow }
T1
τ2
∗
∗
= −B elow −
u (s)ds −
(ū − u )ds · 1{C u (T1 )<elow }
= B ·
T0
τ2
τ1
= B·1
τ1
(ū − u∗ )ds · 1{C u (T1 )<elow }
{C u (T1 )<elow }
·
T1
T0
(u (s) − u∗ (s))ds.
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
303
∗
(5) Similar to Case 4, we can show that if C u (T1 ) = eup , then
τ2
∗
P(C u (T1 )) − P(C u (T1 )) = −A ·
(ū − u∗ )ds · 1{C u (T1 )>eup }
τ1
= −A · 1{C u (T1 )>eup } ·
T1
(u (s) − u∗ (s))ds.
T0
Now let
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
G(x) ≡ B · 1{x<elow } − A · 1{x>eup } .
So combining the results from the five cases above we evaluate (2.10) by applying
the dominated convergence theorem
∗
1
E lim [P(C u (T1 )) − P(C u (T1 ))]
↓0 ∗
1
= lim E{P(C u (T1 )) − P(C u (T1 ))}
↓0 T1
1
u
∗
(u (s) − u (s))ds .
= lim E G(C (T1 ))
↓0 T0
Therefore applying Fubini’s Theorem and the tower property for conditional expectation, we then have
∗
1
E lim [P(C u (T1 )) − P(C u (T1 ))]
↓0 1 T1
= lim
E{G(C u (T1 ))(u (s) − u∗ (s))}ds
↓0 T
0
T1
1
= lim
E{E(G(C u (T1 ))(u (s) − u∗ (s))|Fs )}ds
↓0 T
0
1 T1 ∗
u
(u (s) − u (s))E(G(C (T1 ))|Fs )ds
= E lim
↓0 T0
1 τ2
= E lim
(ū(s) − u∗ (s))E(G(C u (T1 ))|Fs )ds
↓0 τ
1
τ2
∗
=E
(ū(s) − u∗ (s))E(G(C u (T1 ))|Fs )ds ,
(2.12)
τ1
where the second to last equality comes from substituting the value of u given by
(2.8), and the last equality follows from (2.11). Hence, substituting (2.12) back into
(2.9) gives
τ2
∗
(ū − u∗ )(s)[e−r(s−t) (Ss − K) + e−r(T1 −t) E{G(C u (T1 ))|Fs }]ds ≤ 0.
E
τ1
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
304
104-IJTAF
SPI-J071
J. F. Rodrı́guez
Now for fixed t, let us define the process H = (Hs )s≤t to be
∗
Hs = e−r(s−t) (Ss − K) + E{e−r(T1 −t) G(C u (T1 ))|Fs }.
Clearly then H is F-adapted. Also, for any ū ∈ U, and any stopping times τ1 , τ2
with τ1 < τ2 a.s. we have
τ2
E
(ū(s) − u∗ (s))Hs ds ≤ 0.
(2.13)
τ1
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
Therefore in particular, (2.13) holds for
τ1 ≡ inf{s > T0 : Hs > 0},
and τ2 ≡ inf{s > τ1 : Hs < 0}.
For an arbitrary admissible process uc ∈ U, we let Γ = (Γs )s≤t be given by
Γs = (uc (s) − u∗ (s))Hs
(2.14)
ū1 = u∗ + (uc − u∗ )1{Γs >0} 1{s∈(τ1 ,τ2 )} .
(2.15)
and define
Since (2.13) holds for any ū ∈ U, then in particular it holds for ū1 given above.
Substituting (2.15) into (2.13) yields
τ2
(uc − u∗ )1{Γs >0} 1{s∈(τ1 ,τ2 )} Hs ds ≤ 0.
E
Therefore E{
that
τ2
τ1
τ1
Γs 1{Γs >0} ds} ≤ 0, and so
τ2
τ1
Γs 1{Γs >0} ds ≤ 0 a.s. Hence we have
Γs ≤ 0 ∀ s ∈ (τ1 , τ2 ) a.s.
This holds for any uc ∈ U, and so in particular it holds for

uup (s) if Hs ≥ 0
uc (s) =
ulow (s) if Hs < 0.
But since on (τ1 , τ2 ), Hs > 0 and Γs ≤ 0, it follows from the definition of Γ that
uc (s) − u∗ (s) ≤ 0. Therefore because of the way we chose uc , we then have that
uup − u∗ ≤ 0 on (τ1 , τ2 ). But since u∗ ∈ U we also know that u∗ (s) ≤ uup (s) ∀ s ∈
[T0 , T1 ] and hence
u∗ (s) = uup (s)
on (τ1 , τ2 ).
Now for each k ∈ N we let
τ2k+1 ≡ inf{s > τ2k : Hs > 0} and τ2k+2 ≡ inf{s > τ2k+1 : Hs < 0}.
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
305
We can apply the same argument as above to show that u∗ (s) = uup (s) on the
interval (τ2k+1 , τ2k+2 ) for each k ∈ N, and therefore we get that
u∗ (s) = uup (s)
on
∞
(τ2k+1 , τ2k+2 ) = {Hs > 0}.
k=0
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
So whenever Hs is positive, the optimal strategy is to purchase as much as possible.
It remains to show what happens when H is not positive. Again let uc ∈ U
be arbitrary and let Γs be given by (2.14), but now we consider the admissible
consumption process
ū2 = u∗ + (uc − u∗ )1{Γs >0} 1{s∈[τ2 ,τ3 ]} .
Then, substituting this value of ū2 back into (2.13) and using the same argument
as above gives that
Γs ≤ 0
∀ s ∈ [τ2 , τ3 ] a.s.
Again, this holds in for any arbitrary choice of uc ∈ U, and so in particular it holds
if we choose uc such that uc (s) = ulow (s) on {Hs ≤ 0}. Therefore we have that
on [τ2 , τ3 ], ulow − u∗ ≥ 0 and since u∗ ∈ U we then have u∗ = ulow on the closed
interval [τ2 , τ3 ]. Reasoning inductively we see indeed that u∗ = ulow on {Hs ≤ 0}.
Hence, using this and the above, it follows that the optimal consumption strategy
is given by
u∗ (s) = uup (s)1{Hs >0} + ulow (s)1{Hs ≤0} .
So when it is optimal to consume, the owner takes as much as possible, uup ,
and otherwise she would only purchase the required amount, ulow . Suppose we
write u = ulow + uS where uS denotes the swing part of our consumption process.
By doing this, the consumption process is decomposed into the required purchase
amount, ulow , and the additional amount being purchased at any given time, uS .
From the above result we have that for any t ∈ [T0 , T1 ], then either
uS (t) = uup (t) − ulow (t),
or uS (t) = 0.
Remark 2.2. In fact, this gives us an exercise strategy. When the discounted profit
of exercising an option with strike price K is greater than the expected value of the
derivative of the penalty function we exercise as much as possible.
We have established the bang-bang property of the swing option’s consumption.
While this is stated as true in the literature for discrete time swing options, as well
as swing options with local effect, we have yet to see a rigorous treatment of the
bang-bang property of the continuous time swing option with global effect.
Now we will show that the swing option can be priced and hedged using forward
contracts and plain vanilla options. The goal of the option’s owner is to maximize
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
306
104-IJTAF
SPI-J071
J. F. Rodrı́guez
the total payout of the swing option,
T1
u(t)(St − K)dt + P(C u (T1 )),
T0
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
by controlling the electricity consumption by the rules indicated earlier.
For each control u, let ξ u = P(C u (T1 )) be the penalty incurred at expiration
time T1 by using consumption strategy u, and let Ytu correspond to the objective
function at time t, i.e.
T1
u
u
Yt = E
D(t, s)f (s, Xs , u(s))ds + D(t, T1 )ξ |Ft .
t∨T0
So if we let Ȳt be the value function at time t, we have
Ȳt = sup Ytu ,
u
∗
∗
and we say that a control u is optimal if Ȳt = Ytu . We will give the value process
in terms of a backward stochastic differential equation (BSDE). For an overview
see [14].
Theorem 2.2. The value of the swing option at time t, Ȳt , is the solution to a
BSDE, and in fact when r = 0, ∃ a constant χ such that
T1
T1
Ȳt = χ +
ulow (s)(F (t, s) − K)ds +
(uup (s) − ulow (s))C(t, s, K)ds
t
t
where F denotes the forward price process, and C(t, s, K) is the value of a call option
with expiry date s.
Proof. Let us take the discount process to be
D(t, s) = e−
Rs
t
r(u)du
,
where r(t) ≥ 0 ∀ t, and f, g are given by (2.7). We know
T1
u
u
D(t, s)f (s, Xs , u(s))ds + D(t, T1 )ξ |Ft
Yt = E
t∨T0
is a solution to
dYtu = −[f (t, Xt , u(t)) − r(t)Ytu ]dt + Ztu dWt
YTu1 = ξ u
for some process Z (see e.g. [8]). Let
αut = −[f (t, Xt , u(t)) − r(t)Ytu ].
Then, if ∃ û such that
α̂t ≡ αût = sup αut
u
(2.16)
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
307
and
ξˆ ≡ ξ û = sup ξ u ,
(2.17)
u
then Ȳ satisfies
dȲt = −α̂t dt + Zt dWt
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
ȲT1 = ξ̂.
Applying the dynamic programming principle above, we have that for any h < T1 −t,
t+h
Ȳt = sup E
u∈U
D(t, s)f (s, Xs , u(s))ds + D(t, t + h)Ȳt+h |Ft
,
t
and so for any particular control u
t+h
0≥E
D(t, s)f (s, Xs , u(s))ds + D(t, t + h)Ȳt+h − Ȳt |Ft
.
(2.18)
t
Using Itô’s formula,
D(t, t + h)Ȳt+h − Ȳt =
t+h
t
t+h
t+h
Ȳs dD(t, s) +
D(t, s)dȲs +
t
d[D, Ȳ ]s
t
where [·, ·] denotes the bracket process (see [16]). Assume Ȳ satisfies a stochastic
differential equation of the form
dȲt = −αt dt + Zt dWt
ȲT1 = ξ,
for some α and Z satisfying the appropriate hypothesis. We then get
t+h
t+h
−D(t, s)αs ds +
D(t, s)Zs dWs
D(t, t + h)Ȳt+h − Ȳt =
t
−
t
t+h
Ȳs D(t, s)r(s)ds.
(2.19)
t
So substituting (2.19) back into (2.18) gives
t+h
0≥E
D(t, s)[f (s, Xs , u(s)) − αs − r(s)Ȳs ]ds|Ft
.
t
This is true for all sufficiently small h, and therefore
t+h
1
D(t, s)[f (s, Xs , u(s)) − αs − r(s)Ȳs ]ds|Ft ≤ 0,
lim E
h↓0 h
t
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
308
104-IJTAF
SPI-J071
J. F. Rodrı́guez
so we see that indeed
f (t, Xt , u(t)) − αt − r(t)Ȳt ≤ 0.
In fact, looking back at the beginning of the proof, we see that it is precisely when
u = u∗ that we get
f (t, Xt , u(t)) − αt − r(t)Ȳt = 0.
Therefore since r(t) ≥ 0 and Ȳt = supu Ytu we know that
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
|f (t, Xt , u(t)) − r(t)Ytu | ≤ |f (t, Xt , u(t)) − r(t)Ȳt |.
(2.20)
Also, it is clear that
sup{f (t, Xt , u(t)) − r(t)Ytu } ≥ sup f (t, Xt , u(t)) − r(t) sup Ytu
u
u
u
= sup{f (t, Xt , u(t)) − r(t)Ȳt }.
(2.21)
u
Hence, combining (2.20) and (2.21) we get
α̂(t, Yt , Zt ) = sup{f (t, Xt , u(t)) − r(t)Ytu }
u
∗
= f (t, Xt , u∗ (t)) − r(t)Ytu ,
and thus (2.16) is satisfied. It is easy to see that (2.17) also holds for some û, and
therefore (Ȳ , Z) is the solution to the BSDE
dȲt = −α̂(t, Yt , Zt )dt + Zt dWt ;
ȲT1 = ξ̂
where
α̂(t, Ȳt , Zt ) = sup{f (t, Xt , u(t)) − r(t)Ȳt : u ∈ U}
and
ξ̂ = sup{ξ u : u ∈ U}.
This shows the first part of Theorem 2.2.
This solution can be approximated using numerical techniques, by a scheme
for a discretization and simulation of the decoupled forward-backward stochastic
differential equation given above, developed in [13], or [2]. Specifically, we now
consider when r = 0. We want to show how a swing option can be hedged using a
portfolio of call options and forward contracts.
Again, we know the value of the option is given by the value function
T1
T
α̂(s, Ȳs , Zs )ds −
Zs dWs
Ȳt = ξ +
t∨T0
t∨T0
u
where ξ and α̂ are defined above. Since ξ is defined to be the penalty the holder of
the option incurs if she uses consumption strategy u, then we must know the value
of C u (t) in order to determine ξ u .
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
309
Let us first introduce some notation. Since
for all t ∈ [T0 , T1 ],
ulow (t) < u(t) < uup (t),
then for all t ∈ [T0 , T1 ] our total consumption process, C u (t) =
the closed interval
t
t
ulow (s)ds,
uup (s)ds .
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
T0
t
T0
u(s)ds, lies in
T0
But there are three subintervals we will be particularly interested in. One subinterval
is defined by the states in which the holder of the option has consumed in such a way
as to be certain of not meeting one of the pre-determined cumulative consumption
constraints.
The first time the upper cumulative consumption constraint cannot be met is
denoted by
t
T1
uup (s)ds > eup −
ulow (s)ds .
TG3 = inf t > T0 :
T0
t
Then ∀ t ∈ (TG3 , T1 ] we define the open interval
T1
3
ulow (s)ds,
Gt ≡ eup −
t
uup (s)ds .
T0
t
as the set of possible values u(t) can take for t > TG3 if TG3 < ∞. Similarly, letting
T1
t
uup (s)ds >
ulow (s)ds ,
TG1 = inf t > T0 : elow −
T0
t
be the first time the lower cumulative consumption constraint cannot be met, we
define the interval
T1
t
1
Gt ≡
ulow (s)ds, elow −
uup (s)ds
T0
t
T1
for t ∈ (TG1 , T1 ]. Note that if T0 uup (s)ds ≤ eup , then G3t = ∅ ∀ t ∈ [T0 , T1 ], and
T
also we have that if T01 ulow (s)ds ≥ elow , then G1t = ∅ ∀ t ∈ [T0 , T1 ].
It will also be useful to consider the times when the holder of the option can act
without regard to cumulative constraints which must be satisfied. For this scenario,
it is convenient to define
T1
T1
2
ulow (s)ds, eup −
uup (s)ds ,
Gt = elow −
t
for t ∈ (TG2 , T1 ], where
TG2 = inf
t
t > T0 : eup −
t
T1
uup (s)ds > elow −
T1
ulow (s)ds .
t
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
310
104-IJTAF
SPI-J071
J. F. Rodrı́guez
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
Some properties of these intervals include:
(1) If C(t̂) ∈ G2t̂ for some t̂, then C(t) ∈ G2t ∀ t ∈ [t̂, T1 ]. If the total cumulative
consumption process ever enters G2t , we know that the owner of the option will
never incur a penalty. If the minimum instantaneous consumption is taken for
the remainder of the life of the contract, the lower cumulative constraint will
still be met. Likewise, even if the maximum allowable consumption is taken for
the remainder of the life of the contract, the upper cumulative constraint will
not be violated. Therefore, in this case, the swing option should be exercised
whenever a profit can be made.
(2) If C(t̂) ∈ G1t̂ (respectively G3t̂ ) for some t̂, then C(t) ∈ G1t (respectively G3t̂ )
∀ t ∈ [t̂, T1 ]. This means that if the owner of the swing option consumes in such
a way so that the cumulative consumption, at time t̂, is in one of these two
regions, then it will be impossible to satisfy both cumulative constraints, and
one must be violated.
T
T
(3) If T01 ulow (s)ds = elow and T01 uup (s)ds = eup , then C(t) ∈ G2t ∀ t ∈ [T0 , T1 ],
and our consumption choice will only be determined by whether or not a profit
can be made at the current time.
Proof of Theorem 2.2 (cont.).
Now we evaluate ξ and α̂
(1) If C(t) ∈ G1t , then we will definitely incur a penalty for not meeting the lower
cumulative constraint, and the smallest penalty would occur if we buy the
maximum allowable amount for the remainder of the life of the option, i.e.
t
T1
u
∗
ξ1 ≡ sup{ξ : u ∈ U} = −B elow −
u (s)ds −
uup (s)ds .
0
t
(2) Similarly if C(t) ∈ G3t , then the holder will definitely incur a penalty for exceeding the upper cumulative constraint, eup , and the smallest penalty will be
incurred if the holder buys as little as possible for the remainder of the life
of the swing option. Therefore
T1
t
u
∗
u (s)ds +
ulow (s)ds − eup .
ξ3 ≡ sup{ξ : u ∈ U} = −A
0
G1t ∪ G3t ,
(3) Now, if C(t) ∈
a penalty, and thus
t
then we can still consume in such a way as to not incur
ξ2 ≡ sup{ξ u : u ∈ U} = 0.
Therefore, we can write ξ in terms of C(t), and we have
ξ = ξ1 · 1{C(t)∈G1t } + ξ3 · 1{C(t)∈G3t } .
Furthermore, from Theorem 2.1 we know that our admissible consumption processes
can only take two values, namely uup (t) and ulow (t) for each t ∈ [T0 , T1 ]. So for any
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
104-IJTAF
SPI-J071
Hedging Swing Options
311
(t, x) ∈ [0, T ] × R2 ,
α̂(t, x) = sup{f (t, x, u) : u ∈ U}
= uup · (x1 − K) · 1{x1 ≥K} + ulow · (x1 − K) · 1{x1 <K} ,
and therefore
α̂(t, Xt ) = uup (t) · (St − K) · 1{St −K≥0} + ulow (t) · (St − K) · 1{St −K<0}
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
= ulow (t) · (St − K) + [uup (t) − ulow (t)](St − K)+ .
A càdlàg version of α̂(t, Xt ) exists, and that is what we consider. We know that our
value process can be written
T1
T1
Ȳt = ξ +
α̂s ds −
Zs dWs ,
t
t
where (Ȳ , Z) is the unique solution to the BSDE, with ξ and α̂ shown above. Thus,
since Ȳ is an adapted process, by taking the conditional expectation we get
Ȳt = E(Ȳt |Ft )
= E(ξ|Ft ) + E
T1
α̂s ds|Ft
t
= ξ1 · 1{C u (t)∈G1t } + ξ3 · 1{C u (t)∈G3t } + E
T1
+E
T1
t
ulow (s)(Ss − K)ds|Ft
+
(uup (s) − ulow (s))(Ss − K) ds|Ft
t
T1
=χ+
t
T1
ulow (s)(F (t, s) − K)ds +
(uup (s) − ulow (s))C(t, s, K)ds,
t
where χ = ξ1 · 1{C u (t)∈G1t } + ξ3 · 1{C u (t)∈G3t } , and F (t, s) is the forward price at
time t with maturity s, and C(t, s, K) is the value at time t of a call option with
maturity s and strike price K.
Remark 2.3. This theorem implies that swing options are equivalent to a portfolio of regular electricity derivatives. In particular, it can be hedged with forward
contracts and plain vanilla European call options.
Note that our swing option value given by Theorem 2.2 is similar to the one found
in [11], except our solution does not require the solving complicated optimization
problems at each time t.
3. Conclusions
The newly deregulated electricity market presents a new set of problems in mathematical finance. Swing options are the most prevalent financial instrument in the
March 31, 2011 13:57 WSPC/S0219-0249
S021902491100636X
312
104-IJTAF
SPI-J071
J. F. Rodrı́guez
electricity market, for hedging exposure to the extreme price fluctuations that have
been observed. We approach the issue as an optimal control problem, and use techniques from the theory of backward stochastic differential equations to show how
one can hedge these swing options using forwards and plain vanilla call options in
an implementable fashion.
Int. J. Theor. Appl. Finan. 2011.14:295-312. Downloaded from www.worldscientific.com
by FLINDERS UNIVERSITY LIBRARY on 01/24/15. For personal use only.
References
[1] C. Barrera-Esteve, F. Bergeret, C. Dossal, E. Gobet, A. Meziou, R. Munos and
D. Reboul-Salze, Numerical methods for the pricing of swing options: A stochastic control approach, Methodology And Computing in Applied Probability 8 (2006)
517–540.
[2] B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation
of backward stochastic differential equations, Stoch. Proc. and Their Applications
111 (2004) 175–206.
[3] T. Brennan, K. Palmer and S. Martinez, Alternating Currents: Electricity Markets
and Public Policy (RFF Press, Washington DC, 2002).
[4] R. Carmona and S. Dayanik, Optimal multiple-stopping of linear diffusions and swing
options, Mathematics of Operations Research 33(2) (2008) 446–460.
[5] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options,
Mathematical Finance 18(2) (2008) 239–268.
[6] L. Clewlow and C. Strickland, Energy Derivatives: Pricing and Risk Management
(Lacima, London, 2000).
[7] M. Dahlgren, A continuous time model to price commodity based swing options,
Review of Derivatives Research 8(1) (2005) 27–47.
[8] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations
in finance, Mathematical Finance 7 (1997) 1–71.
[9] W. Flemming and Rishel, Deterministic and Stochastic Control (Berlin and New
York: Springer-Verlag, 1975).
[10] P. Jaillet, E. Ronn and S. Tompaidis, Valuation of commodity based swing options,
Management Science 50(7) (2004) 909–921.
[11] J. Keppo, Pricing of electricity swing options, Journal of Derivatives 11 (2004) 26–43.
[12] F. Longstaff and E. Schwartz, Valuing American options by simulation: A simple
least squares approach, The Review of Financial Studies 14(1) (2001) 113–147.
[13] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential
equations explicitly — A four step scheme, Probab. Theory/Relat Fields 98 (1994)
339–359.
[14] P. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990) 55–61.
[15] D. Pilipović, Energy Risk: Valuing and Managing Energy Derivatives (McGraw-Hill,
1998).
[16] P. Protter, Stochastic Integration and Differential Equations, A New Approach
(Springer-Verlag, Berlin, 1990).
[17] S. Stoft, Power System Economics: Designing Markets for Electricity (IEEE Press,
John Wiley & Sons, Philadelphia, 2002).
[18] A. Thompson, 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 (1995) 271–293.