4 Why can’t we apply Black–Scholes theory to energy derivatives?
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Part Five advanced topics
45%
40%
35%
30%
Volatility
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25%
20%
15%
10%
5%
0%
Jul-98
Feb-99
Aug-99
Mar-00
Oct-00
Apr-01
Figure 72.4 Forward volatilities of at-the-money Brent crude oil options.
72.5 THE CONVENIENCE YIELD
The convenience yield is to energy what dividend yield is to stocks. Speciﬁcally, it measures
the net beneﬁt less cost of holding the energy. Users of energy are willing to pay a premium
for the ability to get the energy they require, when they require it. On the other hand, there
is the cost of energy storage to take into account. The beneﬁts minus the costs become the
convenience yield which can be quantiﬁed by examining the future prices of energy. If the
convenience yield were constant then we would expect the usual relationship between forward
and spot prices,
F = Se(r−q)(T −t) ,
where q is the convenience yield.
Because the convenience yield plays such an important role in shaping the forward price
curve, it is usual to use a much more sophisticated model for it than a simple constant.
´ TWO-FACTOR
72.6 THE PILOPOVIC
MODEL
The Pilopovi´c two-factor model takes the form
dS = α(L − S) dt + σ S dX1
and
dL = µL dt + ξ L dX2
energy derivatives Chapter 72
where dX1 and dX2 are uncorrelated. S is the spot price and L is a long-term price or an
‘equilibrium’ price. In this model the spot price reverts to L, while L grows lognormally. The
speed of mean reversion is much faster than the long-term growth so that
α
µ.
We want to relate the spot and equilibrium price processes, together with the convenience
yield and any market prices of risk, to the shape of the forward price curve. To do this we must
ﬁrst write down the equation governed by the forward price F (S, L, t).
This equation is
∂F
∂F
∂F
∂ 2F
∂ 2F
+ 12 σ 2 S 2 2 + 12 ξ 2 L2 2 + (r − q)S
+ (µ − λξ )L
= 0.
∂t
∂S
∂L
∂S
∂L
(72.1)
This has the ﬁnal condition
F (S, L, T ) = S.
Note that there is no −rF term because the forward price is paid at expiry, not at the initiation
of the contract. λ is the market price of risk for the equilibrium price. Also note that I have
assumed risk-neutrality in that the coefﬁcient of ∂F /∂S is the risk-adjusted drift of the spot
price.1
We need to model the convenience yield, perhaps as a function of S, L and t. As we often
do, let’s choose a functional form so that we get a nice convenient explicit solution for F .
Working backwards, let’s suppose
F (S, L, t) = a(t)S + b(t)L.
Plugging this into (72.1), and assuming that λ is a constant, we get
q = q0 (t) +
q1 (t)L
S
,
with
q0 (t) =
a˙
+r
a
and
q1 (t) =
b˙ + (µ − λξ )b
.
a
Here the dots over a and b mean differentiation with respect to time.
Final conditions are
a(T ) = 1 and b(T ) = 0.
There is plenty of freedom (i.e. time dependence) here to ﬁt the forward rate curve and also
the volatility structure of the curve.
1 Since electricity is so difﬁcult to store this is a bad assumption, but one that most people probably make. Combine
the impossibility of hedging with the rapid price ﬂuctuations and you’ll see that it might be better to use a mean-variance
model, such as those discussed in Chapter 59. As I said, this chapter is only the briefest of intros to a complex subject.
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The governing equation for the value of non-path-dependent energy derivatives, under the
Pilopovi´c two-factor model, is
∂V
∂ 2V
∂ 2V
∂V
∂V
+ 12 σ 2 S 2 2 + 12 ξ 2 L2 2 + ((r − q0 )S − q1 L)
+ (µ − λξ )L
− rV = 0. (72.2)
∂t
∂S
∂L
∂S
∂L
72.6.1
Fitting
From the above we get
T
log(a(t; T )) = −r(T − t) +
t
q0 (τ ) dτ
and
T
b(t; T ) = e(µ−λξ )(T −t)
T
exp −(µ − λξ )(T − s) − r(T − s) +
t
s
q0 (τ ) dτ ds.
If we know the forward curve, and S and L, at time t ∗ , F (S ∗ , L∗ , t ∗ ; T ), then
F (S ∗ , L∗ , t ∗ ; T ) = a(t ∗ ; T )S ∗ + b(t ∗ ; T )L∗
is one equation for q0 (t) and q1 (t). (Note that the qs aren’t allowed to be functions of T .)
Another equation could come from the volatility structure:
Volatility(S ∗ , L∗ , t ∗ ; T ) =
a(t ∗ ; T )2 σ 2 S ∗2 + b(t ∗ ; T )2 ξ 2 L∗2 .
As already mentioned, seasonality plays an important role in the modeling of energy prices.
So we might want to add an oscillatory (Fourier series) term or two to the basic model for S.
72.7 ENERGY DERIVATIVES
The types of derivatives seen in the energy markets are not that different from the many types
we have seen already in other markets.
72.7.1
One-day Options
One-day options on electricity are very popular in the US. Because of the possibility of enormous price ﬂuctuations these contracts are very hard to price. This effect is slightly mitigated
by a timewise averaging that is sometimes part of these contracts.
72.7.2
Asian Options
Asian options are very popular in the energy markets. Two forms of averaging are seen, averaging over several realized spot prices that have settled, and averaging over unsettled forward
prices.
The former are easier to value, in the path-dependent framework we’ve already seen.
The latter require a very accurate forward curve model. If we think of forward contracts as
derivatives then the latter type of contract is a second-order contract. The contract is, however,
not path-dependent.
energy derivatives Chapter 72
72.7.3
Caps and Floors
Caps and ﬂoors exist in the energy markets. These restrict the price that must be paid for
delivery of the energy. There is often the added, path-dependent, complication that some form
of timewise averaging takes place.
72.7.4
Cheapest to Deliver
This contract allows the delivering party to deliver the same energy source to one of two
delivery points. Presumably he will choose to deliver the cheaper. The correlation between the
prices of the underlying at the two delivery points is a crucial parameter in the valuation of
this contract. It is a multi-asset contract.
72.7.5
Basis Spreads
A basis spread is a contract on the difference or spread between two very similar, but different
underlyings, two different types of oil for example. Again, this is a multi-asset contract and the
correlation between the two assets is of paramount importance.
Figure 72.5 Time series of electricity spark spreads. Source: Bloomberg L.P.
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72.7.6
Swing Options
Swing options allow the energy user to vary his energy delivery to vary between set limits.
Perhaps also the amount is only allowed to vary a set number of times. Perhaps the user can
take any ten days’ of electricity during the summer, with the choice of precise days left up to
the option holder. These contracts are complex, having characteristics similar to the passport
option contracts seen in Chapter 27 and requiring the same mathematics of stochastic control.
72.7.7
Spread Options
Spread options are options on the spread between the price of the fuel used to produce the
electricity and the price of that electricity. They are used to hedge the electricity production
costs. Figure 72.5 shows a time series of electricity spark spreads, the relative efﬁciency of
buying gas for making electricity versus buying the electricity itself.
72.8 SUMMARY
Energy derivatives are a fascinating subject, and still relatively new. Modeling the underlying
is tricky because of the enormous spikes in prices. Modeling derivatives is difﬁcult because
hedging is often impossible. I think that there’s a long way to go before the models become
satisfactory.
FURTHER READING
• Gabillon (1995) was the ﬁrst to present a two-factor model applied to the oil market.
• Pilopovi´c (1998) is the best, by virtue of being the only affordable, book on the energy
market and derivatives.
• See Ahn et al. (2002) for an analysis of the costs of storage of energy.
CHAPTER 73
real options
In this Chapter. . .
•
•
the application of derivatives theory outside of ﬁnance
optimal investment
73.1
INTRODUCTION
We’ve seen the word ‘option’ used to describe a ﬁnancial contract containing some element of
choice. And that choice is made complicated, and interesting, by the randomness underpinning
the ﬁnancial markets. But high ﬁnance is not the only area in which randomness and choice
play important roles, every decision in life could be interpreted as trying to make the best
choice, given an unknown future.
In this chapter we will see how many of the ideas of derivatives theory can be applied to
other walks of life, and we’ll also see a few more ideas as well. This is the subject of Real
options.
73.2
FINANCIAL OPTIONS AND REAL OPTIONS
The key points that relate ﬁnancial option theory and Real option theory are as follows:
• Randomness concerning the future introduces the idea of examining probability distributions
for outcomes;
• Decisions should be made optimally, there is some question over the timing;
• Decisions may be partially or wholly irreversible.
In this chapter we will slowly build up the concepts and math used in Real option theory.
In particular, we look at project valuation, optimal entry into and exit from a business, and
optimally and sequentially investing. Most of the math is very similar to what we’ve seen
already.
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73.3 AN INTRODUCTORY EXAMPLE: ABANDONMENT
OF A MACHINE
You own a factory that produces goods with a proﬁtability of P which is realized continuously.
So that P dt is the proﬁt made between times t and t + dt. The factory has a natural life span
of T .
If P is a constant then the present value of all the future proﬁts V (t) satisﬁes
dV
− rV + P = 0
dt
(73.1)
with
V (T ) = 0.
Notice that I have discounted at the risk-free rate; we may be discounting at a different rate
later on.
The solution of this is just
V =
P
1 − e−r(T −t) .
r
As long as P > 0 then V is always positive and there would be no reason to close down the
factory.
If P is a deterministic function of time then we can still solve (73.1). If P goes negative for
sufﬁciently long then it might be worthwhile closing the factory down. Since this is a completely
deterministic problem you just wait until V = 0. You could even open up the factory later on
if conditions improve. Let’s not worry about this until later in the chapter.
More realistically, suppose that the proﬁt ﬂuctuates due to general market conditions such as
supply and demand, and so we’ll assume that it follows the random walk
dP = a dt + b dX
with a and b constant, for the sake of argument. In a mature market you may expect a < 0
reﬂecting the general decline in proﬁt as the product becomes outmoded. It is certainly possible
that the proﬁt could become negative, again a likely outcome in practice.
If there is some randomness in the dynamics of P , because b is non-zero then we must solve
the diffusion problem
∂V
∂V
∂ 2V
+ 12 b2 2 + a
− rV + P = 0
∂t
∂P
∂P
(73.2)
and now V is a function of P and t. By solving this problem, subject to conditions below,
we will be ﬁnding out information about the present value of expected quantities. Observe
that (73.2) is similar to the equation for the transition probability density function; the only
differences are in the discounting to get the present value and the cashﬂow term, the last on
the left.
The ﬁnal condition is
V (P , T ) = 0.
real options Chapter 73
60
50
No closure
Optimal closure
V
40
30
20
10
−10
−5
P
0
−10
0
5
10
−20
−30
−40
Figure 73.1 The value of a perpetual factory, with and without closure.
The solution of this problem tells us the present value of the expected total proﬁt provided
that the factory stays open until t = T .
Now we are at the point where we can ask about how it is best to run the factory: Should
we actually keep it open even if it starts to lose money?
The optimal time at which to close down the factory, if we want to maximize the present value
of accumulated expected proﬁt, comes from solving the free boundary problem, Equation (73.2),
subject to the ﬁnal condition and
V (P , t) ≥ 0
with continuity of ∂V /∂P . This is the same in principle as the American option problem.
In Figure 73.1 are shown the theoretical values for the present value of the accumulated
proﬁts in two cases, with and without the option to close the factory down. Obviously the
former is greater and the difference between the two is the value of the option of closure. In
the example, T is taken to be inﬁnity so that the factory has no natural lifespan.
The important point to note about the above analysis is that by postponing the closure of the
factory until the optimal time we add signiﬁcant value to the factory. If we had had to decide
at the start whether to close the factory immediately, or continue forever, then we would not
be fully exploiting the opportunity offered by waiting for new information (the realization of
P ) before making the big decision.
This model is also applicable to other interesting situations, such as modeling the ‘rewards’
from marriage. Some change of notation would be needed, for instance P would have to mean
something other than proﬁtability. T could be the expected lifespan of a man after marriage.
The question is then when to separate.
73.4
OPTIMAL INVESTMENT: SIMPLE EXAMPLE #2
Here’s another example, equally straightforward. When is the optimal time to invest a given
ﬁxed amount in return for a product whose value evolves randomly? This could be the purchase
of our factory.
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