Dorje Brody

Pricing Storable Commodities and Associated Derivatives

Dorje C. Brody

Department of Mathematics,Imperial College London,

London SW7 2AZwww.imperial.ac.uk/people/d.brody

(London: 18 June 2010)

- 1 -

Pricing Storable Commodities and Associated Derivatives -2 - 18 June 2010

Market information about future supply, demand, and inventory

We consider the valuation of a storable commodity.

Alternatively, we may think of the valuation of a real estate.

We shall be speaking in terms of commodity prices, although the sameconstruction applies as well to real estate prices.

Let us assume that one unit of the commodity provides a “convenience benefit”equivalent to a cash flow {Xt}t≥0.

Note that we work directly with the actual flow of convenience from the storageor “possession” of the commodity, rather than the convenience yield.

The point is that the convenience yield is a secondary notion since it depends onthe price, which is what we are trying to determine.

Thus when a storable commodity is consumed, one can think of it as beingexchanged for a consumption good of identical value.

Think of the difference between a corked bottle of wine (of known quality) and

Practical Quantitative Analysis in Commodities c© DC Brody 2010


an uncorked bottle of the same wine.

The consumption good in the latter example is not storable, and must beconsumed immediately.

The value of the commodity is then given in the risk-neutral measure by:

St =1

PtE

Qt

[∫ ∞

t

PuXudu

]

, (1)

where

Pu = exp

(

−∫ u

0

rsds

)

(2)

is the discount factor.

We shall write Et[−] = EQ[−|Ft] for the expectation in the risk-neutral measureconditional on the information flow {Ft}.

For simplicity, let us now assume that the interest rate system is deterministic.

Once we work things out for deterministic rate {rt} then we can consider thegeneral situation.



Then Pt is determined by the initial term structure.

We shall assume that the market filtration is generated jointly by the followingprocesses:

(a) an information process {ξt}t≥0, given by

ξt = σt

∫ ∞

t

PuXudu +Bt, (3)

where the Q-Brownian motion {Bt} is independent of {Xt}; and

(b) the commodity convenience benefit flow process {Xt}t≥0.

Thus, at time t we have

Ft = σ(

{ξs}0≤s≤t, {Xs}0≤s≤t)

. (4)

In other words, the market information is generated jointly by the conveniencebenefit flow up to time t and the noisy information of the future benefit flow.

Modelling the convenience benefit process



As a simple model for the commodity convenience benefit, let us consider thecase where {Xt} is given by an Ornstein-Uhlenbeck (OU) process.

Then we have

dXt = κ(θ −Xt)dt + ψdβt, (5)

where {βt} is a Brownian motion that is independent of {Bt}.

Here θ is the mean reversion level, κ is the mean reversion rate, and ψ is thevolatility.

We shall be looking at the constant parameter case first, and then extend theresults into time-dependent situation.

A standard calculation making use of an integration factor shows that

Xt = e−κtX0 + θ(1 − e−κt) + ψe−κt∫ t

0

eκsdβs. (6)

Thus, starting from the initial value X0, the process tends in mean towards thelevel θ, and has the variance

Var[Xt] =ψ2

2κ

(

1 − e−2κt)

(7)



and the intertemporal covariance

Cov[Xt, XT ] =ψ2

2κe−κT

(

eκt − e−κt)

. (8)

Applications of Ornstein-Uhlenback bridges

In what follows we need some further properties of the OU process.

A short calculation establishes that for T > t we have

XT = e−κ(T−t)Xt + θ(1 − e−κ(T−t)) + ψe−κT∫ T

t

eκudβu. (9)

This expression is appropriate when we “re-initialise” the process at time t.

Furthermore, by use of the variance-covariance relations one can easily verifythat Xt is independent from XT − e−κ(T−t)Xt.

This independence relation illustrates the Markov property of the OU process.

This property corresponds to an orthogonal decomposition of the form

XT = (XT − e−κ(T−t)Xt) + e−κ(T−t)Xt (10)



for T > t.

Interestingly, there is another orthogonal decomposition as well, this time for Xt,which plays a crucial role in what follows.

This decomposition is given by

Xt =

(

Xt −eκt − e−κt

eκT − e−κTXT

)

+eκt − e−κt

eκT − e−κTXT . (11)

The process {btT}0≤t≤T defined for fixed T by

btT = Xt −eκt − e−κt

eκT − e−κTXT (12)

is an Ornstein-Uhlenbeck bridge (OU bridge).

An alternative way of expressing the OU bridge is to write

btT = Xt −sinh(κt)

sinh(κT )XT . (13)

Clearly we have b0T = X0 and bTT = 0.



The OU bridge is a Gaussian process with mean

E[btT ] =sinh(κ(T − t))

sinh(κT )X0 +

(

1 − sinh(κt) + sinh(κ(T − t))

sinh(κT )

)

θ (14)

and variance

var[btT ] =ψ2

2κ

(

cosh(κT ) − cosh(κ(T − 2t))

sinh(κT )

)

. (15)

The mean and variance of the OU bridge are plotted in Figure 1.

Valuation formula for the commodity price

Armed with these facts, now we shall show that

E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

{ξs}0≤s≤t, {Xs}0≤s≤t

]

= E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ξt, Xt

]

. (16)

This will simplify the calculations that follow later.

First we note that

Ft = σ(

{ξs}0≤s≤t, {Xs}0≤s≤t)

= σ(

{ηs}0≤s≤t, {Xs}0≤s≤t)

, (17)



0 50 100 150 200 250 300 350-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time

OU Bridge

data1

data2

Mean

Variance

Figure 1: Mean (red) and variance (blue) of the OU bridge. The parameters are set as κ = 0.15, T = 1, X0 = 0.6,

θ = 1.4, and ψ = 0.5.



where

ηt = σt

∫ ∞

0

PuXudu +Bt. (18)

This follows from the fact that

ξt = ηt − σt

∫ t

0

PuXudu. (19)

We then observe further that

Ft = σ

(

ηt,{ηtt− ηss

}

0<s≤t, {Xs}0≤s≤t

)

. (20)

We note that {ηt} has the Markov property, since

Q [ηT < x|{ηs}0≤s≤t] = Q

[

ηT < x

∣

∣

∣

∣

ηt,{ηtt− ηss

}

0<s≤t

]

= Q [ηT < x|ηt] . (21)

This follows from the fact that ηt and ηT are independent from Gt, where

Gt = σ

(

{ηtt− ηss

}

0≤s≤t

)

, (22)

which follows in turn from properties of the standard Brownian motion.



This can be seen as follows.

We note first thatηtt− ηss

=Bt

t− Bs

s. (23)

Now it is a property of Brownian motion that for any times t, s, s1 satisfyingt > s > s1 > 0 the random variables Bt and Bs/s− Bs1/s1 are independent.

More generally, if s > s1 > s2 > s3 > 0, we find that Bs/s−Bs1/s1 andBs2/s2 −Bs3/s3 are independent.

It follows that ηt and ηT are independent from Gt.

We remark, furthermore, that Xs is independent of Gt.



Thus, we conclude that the price is given by

PtSt = E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

Ft

]

= E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ηt,Gt, {Xs}0≤s≤t

]

= E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ηt, {Xs}0≤s≤t

]

= E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ξt, {Xs}0≤s≤t

]

. (24)

But note on the other hand that

σ(

ξt, {Xs}0≤s≤t)

= σ (ξt, Xt, {bst}0≤s≤t) , (25)

and that the OU bridge {bst}0≤s≤t is independent of {Xu}u≥t.

Thus {bst} is independent of ξt and∫∞t PuXudu.

We deduce therefore that

St =1

PtE

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ξt, Xt

]

. (26)



Commodity price process

We can use the orthogonal decomposition (10) to isolate the dependence of thecommodity price on the current level of the benefit rate Xt.

Remarkably, this turns out to be linear in our model.

Specifically, we have the following decomposition into orthogonal components:∫ ∞

t

PuXudu =

∫ ∞

t

Pu

(

Xu − e−κ(u−t)Xt

)

du

+

(∫ ∞

t

Pue−κ(u−t)du

)

Xt. (27)

Thus, we deduce that

PtSt =

(∫ ∞

t

Pue−κ(u−t)du

)

Xt + E

[

At

∣

∣

∣σtAt +Bt

]

, (28)

where

At =

∫ ∞

t

Pu

(


)

du, (29)

and Bt is the value of the Brownian motion at time t.



But now the problem is essentially solved, since the remaining expectation is ofthe form

E [A|A + B] , (30)

where A and B are independent Gaussian random variables each with a knownmean and variance.

That is to say, we have:

A =

∫ ∞

t

Pu

(


)

du (31)

and

B =Bt

σt. (32)

Writing

A = x(A + B) + (1 − x)A− xB, (33)

we observe in particular that A +B and (1 − x)A− xB are orthogonal andhence independent if we set

x =Var[A]

Var[A] + Var[B]. (34)



This then enables us to work out the expectation to determine the value of thecommodity.

We proceed as follows.

First we note from (9) that

Xu − e−κ(u−t)Xt = θ(1 − e−κ(u−t)) + ψe−κu∫ u

t

eκsdβs. (35)

Therefore, we have

A =

∫ ∞

t

Pu

(


)

du

= θ

∫ ∞

t

Pudu− θeκt∫ ∞

t

Pue−κudu

+ψ

∫ ∞

u=t

Pue−κu∫ u

s=t

eκsdβsdu. (36)

It follows that

E[A] = θ(

pt − eκtqt)

, (37)



where

pt =

∫ ∞

t

Pudu (38)

and

qt =

∫ ∞

t

Pue−κudu. (39)

By interchanging the order of integration we can write∫ ∞

u=t

Pue−κu(∫ u

s=t

eκsdβs

)

du =

∫ ∞

s=t

eκs(∫ ∞

u=s

Pue−κudu

)

dβs, (40)

and therefore we have

A− E[A] = ψ

∫ ∞

s=t

eκs(∫ ∞

u=s

Pue−κudu

)

dβs

= ψ

∫ ∞

t

eκsqsdβs. (41)

Thus, by the Wiener-Ito isometry, we obtain

Var[A] = ψ2

∫ ∞

t

e2κsq2sds. (42)



We also have

Var[B] =1

σ2t. (43)

The commodity price can then be worked out as follows.

We have

PtSt = E

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ξt, Xt

]

= eκtqtXt + x(A +B) + (1 − x)E[A] − xE[B]. (44)

Note that E[B] = 0, and that A + B is given by

A + B =1

σtξt − eκtqtXt, (45)

and that E[A] is given by

E[A] = θ(

pt − eκtqt)

. (46)

Gathering terms, we therefore obtain

PtSt = (1 − xt)[

θpt + eκtqt(Xt − θ)]

+ xtξtσt. (47)



The weighted factor xt is given by

xt =σ2ψ2t

∫∞t e2κsq2

sds

1 + σ2ψ2t∫∞t e2κsq2

sds. (48)

Thus we see that for large ψ and/or large σ the value of x tends to unity.

On the other hand, for small ψ and/or small σ the value of x tends to zero.

Hence, if the market information has a low noise content (high σ), then themarket information is what mainly determines the price of the commodity.

On the other hand, if the volatility of the benefit is high, then marketparticipants also rely heavily of “the latest information” in their determination ofprices.

The other term in the expression for St is essentially an annuitised valuation of aconstant benefit rate set at the mean reversion level, together with a correctionterm to adjust for the present level of the benefit rate.

This term dominates in situations when the market information is of low quality.



It also dominates in situations when the benefit volatility is low.

In other words, in the absence of information our judgements are formed on thebasis of a kind of average of the status quo and the long term average.

But we also rely on the status quo in situations where there is little uncertainty.

It should be evident that there is an interesting and rather complex set ofrelations at work here.

The calculation above has been carried out in a deterministic interest ratesetting.

We can however pursue a similar analysis in a random interest rate environment.

Constant interest rate case

When the short rate is constant, we can make further simplification for thecommodity price valuation.



In this case we have

Pu = e−ru, (49)

from which it follows that

pt =1

re−rt, qt =

1

r + κe−(r+κ)t, (50)

and

xt =σ2ψ2t

2r(r + κ)2e2rt + σ2ψ2t. (51)

A short calculation then shows that

St = (1 − xt)

[(

1

r− 1

r + κ

)

θ +1

r + κXt

]

+ xt ert ξtσt. (52)



0 50 100 150 200 250 300 35020

30

40

50

60

70

80

90

100

110

120psi = 0.4, kappa = 0.05, sigma = 0.05

time

Bre

nt C

rude P

rice

Sim

Sim

Sim

Sim

Sim

Market

Figure 2: Sample paths for the price process (colour) vs the market price for crude oil (black).



Pricing commodity derivatives

Let us now consider the problem of pricing a commodity derivative.

Specifically, we consider the valuation of a call option:

C0 = e−rT E[

(ST −K)+]

. (53)

Recall that the commodity price process {St} in the OU model is a linearfunction of the convenience yield

Xt = e−κtX0 + θ(1 − e−κt) + ψe−κt∫ t

0

eκsdβs, (54)

and also in the information process

ξt = σt

∫ ∞

t

e−ruXudu +Bt. (55)

Thus we have three Gaussian processes {Xt}, {∫∞t e−ruXudu}, and {Bt} at

hand, where {Xt} and {Bt} are independent.

Since the sum of Gaussian processes is also Gaussian, the evaluation of the callprice reduces to the determination of the mean mT and the variance γ2

T of ST .



A short calculation shows that

mT = (1 − xT )

[

θ

r+X0 − θ

r + κe−κT

]

(56)

and that

γ2T =

ψ2

2κ(r + κ)2(

1 − e−2κT)

+ x2T

[

ψ2

2r(r + κ)2+

e2rT

σ2T

]

. (57)

These can be obtained from the orthogonal decomposition (27):∫ ∞

t

PuXudu = At +

(∫ ∞

t

Pue−κ(u−t)du

)

Xt

= At +1

r + κe−rtXt, (58)

where {At} is defined in (29).

Therefore, the commodity price (52) can be expressed in the form

St = (1 − xt)

(

1

r− 1

r + κ

)

θ +1

r + κXt + xt e

rtAt + xt ertBt

σt. (59)

However, the Gaussian processes {Xt}, {At}, and {Bt} are independent.



It follows that the variance of St is determined by the variance of the threeGaussian variables Xt in (7), At in (42), and Bt in (43).

Putting these together we obtain (57).

Returning to the call price valuation, we thus have

C0 = e−rT1√

2πγT

∫ ∞

K

(z −K) exp

(

−(z −mT )2

2γ2T

)

dz. (60)

If we write

N(x) =1√2π

∫ x

−∞exp(

−12z

2)

dz (61)

for the cumulative normal density function, then we obtain

C0 = e−rT[

γT√2π

exp

(

−(mT −K)2

2γ2T

)

+ (mT −K)N

(

mT −K

γT

)]

(62)

for the price of a commodity option.



Figure 3: Option price surface as functions of the initial asset price and option maturity.



6 8 10 12 140

1

2

3

4

Figure 4: The call option prices as functions of the initial asset price in the OU model. The parameters are set asκ = 0.15, σ = 0.25, X0 = 0.6, ψ = 0.15, r = 0.05, and K = 10. The three maturities are T = 0.5 (blue), T = 1.0

(green), and T = 3.0 (brown).



Extended mean-reversion models for the convenience dividend

We now consider the time-inhomogeneous Ornstein-Uhlenbeck process as asimple model for the commodity convenience benefit.

In this case we have

dXt = κt(θt −Xt)dt + ψtdβt, (63)

where {βt} is again a Brownian motion that is independent of {Bt}.

Defining the integral

ft =

∫ t

0

κsds, (64)

we find, by use of the standard method involving an integration factor, that thesolution to (63) is given by

Xt = e−ft(

X0 +

∫ t

0

efsκsθsds +

∫ t

0

efsσsdβs

)

. (65)

Orthogonal decompositions: time-inhomogeneous OU bridge



In what follows we need to use some further properties of the OU process.

A short calculation establishes that for T > t we have

XT = e−∫ Tt κsds

(

Xt + e−∫ t0 κsds

∫ T

t

e∫ u0 κsdsκuθudu

+e−∫ t0 κsds

∫ T

t

e∫ u0 κsdsψudβu

)

. (66)

This expression is appropriate when we “re-initialise” the process at time t.

Furthermore, by use of the variance-covariance relations one can easily verify

that Xt is independent from XT − e−∫ Tt κsdsXt.

Similarly to the time-homogeneous case, this independence relation illustratesthe Markov property of the time-inhomogeneous OU process.

This property corresponds to an orthogonal decomposition of the form

XT =(

XT − e−∫ Tt κsdsXt

)

+ e−∫ Tt κsdsXt (67)

for T > t.



As before, there is another orthogonal decomposition, this time for Xt, whichplays a crucial role in the time-inhomogeneous setup.

This decomposition is given by

Xt =

(

Xt −e−ft

∫ t

0 e2fsψ2sds

e−fT∫ T

0 e2fsψ2sds

XT

)

+e−ft

∫ t

0 e2fsψ2sds

e−fT∫ T

0 e2fsψ2sds

XT . (68)

The process {btT}0≤t≤T defined for fixed T by

btT = Xt −e−ft

∫ t

0 e2fsψ2sds

e−fT∫ T

0 e2fsψ2sds

XT (69)

is the time inhomogeneous Ornstein-Uhlenbeck bridge.

Clearly we have b0T = X0 and bTT = 0.

Valuation of the commodity price

The arguments presented in the foregoing material carry through in the case ofan extended OU model.



We are required therefore to calculate

St =1

PtE

[∫ ∞

t

PuXudu

∣

∣

∣

∣

ξt, Xt

]

. (70)

We can use the orthogonal decomposition (67) to isolate the dependence of thecommodity price on the current level of the benefit rate Xt.

As before, this turns out to be linear in the benefit rate:∫ ∞

t

PuXudu =

∫ ∞

t

Pu

(

Xu − e−(fu−ft)Xt

)

du

+

(∫ ∞

t

Pue−(fu−ft)du

)

Xt. (71)

Thus, we deduce that

PtSt =

(∫ ∞

t

Pue−(fu−ft)du

)

Xt + E

[

At

∣

∣

∣σtAt +Bt

]

, (72)

where

At =

∫ ∞

t

Pu

(

Xu − e−(fu−ft)Xt

)

du, (73)

and Bt is the value of the Brownian motion at time t.



Valuation of general assets: real estate

We close by remarking that in the case of an equity-type asset a model based ona geometric Brownian motion might be feasible for the cash flow.

Consider a simple example in which the dividend process satisfies the stochasticequation

dXt = µXtdt + νXtdβt, (74)

where µ and ν > 0 are constants, and {βt} is a standard Q-Brownian motion.

Assuming for simplicity that the short rate {rt} is also a constant given by r, wehave, for the cumulative dividend, the expression

Z∞ = X0

∫ ∞

0

eνβs−(r+12ν

2−µ)sds. (75)

We assume, further, that

r +1

2ν2 − µ > 0. (76)

Then a standard result on geometric Brownian motion shows that Z∞ is



inverse-gamma distributed with density

g(z) =

(

2X0

ν2

)αz−1−αe−ν

2/(2X0z)

Γ(α), (77)

where α = 1 + 2ν−2(r − µ).

The price process of the asset is then obtained by defining the informationprocess {ηt} of (18) according to

ηt = σtZ∞ + Bt. (78)

Specifically, the problem reduces to evaluating

St =1

PtE [Z∞| ηt, Xt] −

ZtPt, (79)

where

Zt = X0

∫ t

0

eνβs−(r+12ν

2−µ)sds. (80)


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