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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
Pricing Storable Commodities and Associated Derivatives -3 - 18 June 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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -4 - 18 June 2010
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
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -5 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -6 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -7 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -8 - 18 June 2010
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)
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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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -10 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -11 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -12 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -13 - 18 June 2010
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
(
Xu − e−κ(u−t)Xt
)
du, (29)
and Bt is the value of the Brownian motion at time t.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -14 - 18 June 2010
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
(
Xu − e−κ(u−t)Xt
)
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -15 - 18 June 2010
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
(
Xu − e−κ(u−t)Xt
)
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -16 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -17 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -18 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -19 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -20 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -21 - 18 June 2010
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).
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -22 - 18 June 2010
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 .
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -23 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -24 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -25 - 18 June 2010
Figure 3: Option price surface as functions of the initial asset price and option maturity.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -26 - 18 June 2010
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).
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -27 - 18 June 2010
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
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -28 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -29 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -30 - 18 June 2010
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.
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -31 - 18 June 2010
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
Practical Quantitative Analysis in Commodities c© DC Brody 2010
Pricing Storable Commodities and Associated Derivatives -32 - 18 June 2010
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)
Practical Quantitative Analysis in Commodities c© DC Brody 2010
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