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4 Why can’t we apply Black–Scholes theory to energy derivatives?

# 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) =

+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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Part Five advanced topics

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

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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Part Five advanced topics

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 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.

• 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

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Part Five advanced topics

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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