Stochastic modeling of oil futures prices

Mina Hosseini

U.U.D.M. Project Report 2007:1

Examensarbete i matematik, 20 poängHandledare och examinator: Maciej Klimek

Januari 2007

Department of Mathematics

Uppsala University

Abstract In this study we consider the two and three factor modeling of oil futures prices under the risk neutral measure as well as the volatility term structure of futures returns. We analyze the two and three-factor modeling of oil futures prices developed by Schwartz (1997) and Cortazar & Schwartz (2003) and use mathematical and financial definitions to derive analytical solutions to futures contracts on the commodity.

Acknowledgement I would like to thank my supervisor Prof. Maciej Klimek, at the Department of Mathematics, for answering my questions and assisting me in the best way when writing this master’s thesis. I also wish to thank, all of my teachers at the Department of Mathematics, especially Prof. Johan Tysk for his consideration to the financial mathematics program. And

“With the special thanks to my father for his patience”

Contents 1 Introduction……………………………………………………………………….......2 2 Definitions of some mathematical terms used in this paper………………………….4 2.1 Wiener process……………………………………………………………………4 2.1.1 One-dimensional Wiener process…………………………………………...4 2.1.2 n-dimensional (independent) Wiener process...……………………………...4 2.1.3 n-dimensional (correlated) Wiener process………………………………....4 2.2 Ito’s lemma……………………………………………………………………….5 2.2.1 One-dimensional Ito’s lemma……………………………………………….5 2.2.2 Multi dimensional Ito’s lemma for independent Wiener processes………....5 2.2.3 Multi dimensional Ito’s lemma for correlated Wiener processes…………...6 2.3 Black-Scholes equation………………………………………………………......7 2.3.1 One dimensional Black-Scholes equation………………………………......7 2.3.2 Multi dimensional Black-Scholes equation……………………………........7 2.4 Feynman-Kac theorem……………………………………………………………8 2.4.1 Feynman-Kac theorem when our processes are independent.........................8 2.4.2 Feynman-Kac theorem when our processes are correlated……………….....9 2.5 Pricing equation………………………………………………………………........9 2.6 Ornstein-Uhlenbeck process (Gauss Markov process).........................................11 2.7 Stochastic integrals……………………………………………………………….…...11 2.8 Risk neutral valuation…………………………………………….……………...12 3 The two-factor modeling of oil futures prices and two alternatives for its improvement………………………………………………………………….……..13

3.1 The two-factor modeling and valuation of oil futures prices with the analytical result …………………………………………………………………………....13

3.2 A brief description on the three-factor modeling of oil futures prices when has a r stochastic process (Schwartz 1997)……………………………………………...22

3.3 The parsimonious two-factor modeling of oil futures prices (Schwartz 2003)……………………………………………………………………………..23 4 The three-factor modeling of oil futures prices……………………………………..25

4.1 The three-factor modeling and valuation of oil futures prices with the analytical result (Schwartz 2003)………………………………………………………......25 4.2 The volatility term structure of futures returns………………………………......34

5 Conclusion…………………………………………………………………………..36 References…………………………………………………………………………….....37

1

1. Introduction Since two decades ago, when the early model of oil futures prices developed by Brennan and Schwartz (1985), there has been a vast change in oil markets arising from both economical and political factors, oil markets are more volatile now compared with twenty or even ten years ago. To understand the behavior of the oil market there is a need to understand the stochastic models of oil prices. Over the last two decades different models have been proposed to justify the stochastic behavior of oil futures contracts. The latest model has been the parsimonious three-factor model by Cortazar and Schwartz (2003). The goal of this article is analysis of the parsimonious three-factor modeling of oil futures prices and the solutions to the futures contracts evaluation formula under the risk neutral measure, as well as the volatility term structure model of futures returns. Before developing a parsimonious three-factor model by Cortazar and Schwartz, some other models had been proposed; however, these models were not satisfactory enough. The first model (Brennan & Schwartz 1985) assumed a geometric Brownian motion and a constant convenience yield but very soon it was replaced by a one-factor mean-reverting model.1 After that a more realistic model which was a two-factor model with mean-reverting convenience yield was proposed by Gibson & Schwartz (1990), and some other versions of that were proposed in 1997 and 2000 by different researchers; however, these models as Schwartz and Cortazar say adopted rather slowly by practitioners. Difficulties of fitting well model to data for some days showed the need for a three-factor model, thus the parsimonious three-factor model developed by Cortazar and Schwartz in 2003. Before this model, the latest three factor model had been proposed by Schwartz (1997), in that model the third factor was the stochastic interest rate of return r. The latest three-factor model (2003) is based on the parsimonious two-factor model (Cortazar & Schwartz 2003). The best aspects of this three-factor model are: firstly, all information is from the futures prices in contrast with the three-factor model (1997), secondly, instead of using long-term (unobservable) convenience yield, long term price appreciation is used. In this study, by analyzing the two and three-factor modeling of oil futures prices and using mathematical and financial definitions, analytical solutions to evaluate futures contracts on the commodity have been derived. In this article, after considering the definitions of some financial and mathematical terms in Section 2, the first part of Section 3 focuses on the two-factor modeling of oil futures prices (Schwartz 1997) as well as the analytical solution for evaluation of futures prices. Afterwards, there is a brief description on the three-factor modeling of oil futures prices (Schwartz 1997) where a stochastic r (the stochastic rate of interest) as a third factor has been included in the model. The third part of Section 3 is an introduction to parsimonious two factor modeling (Cortazar & Schwartz 2003) which is the basis of the three-factor

1 As it has been pointed out by Schwartz (1997) the mean- reversion characteristic of the commodity spot price corresponds to the fact that when prices are high, higher cost producers of the commodity will enter the market, putting a downward pressure on prices as they had increased the supply; on the other hand, when prices are low, some of the higher cost producers will exit the market, putting upward pressure on prices as the supply had been decreased.

2

modeling of oil futures prices mentioned in Section 4, and the fifth section of this essay pays attention to the three-factor modeling of oil futures prices as well as futures prices evaluation with analytical results to the related 3-dimensional pricing equation. Moreover, the volatility term structure model of futures returns proposed by Cortazar & Schwartz (2003) will be considered in this section too.

3

2. Definitions of some mathematical terms used in this paper: 2.1 Wiener process 2.1.1 One-dimensional Wiener process A process ( )Z t is a Wiener process (or alternatively, Brownian motion) if it satisfies the following:

i. (0) 0Z =

ii. Z has continuous trajectories. iii. Z has independent increments: if 1 2 30 t t t t4≤ < ≤ < , the random variables

2( ) ( )1Z t Z t− and 4( ) ( )3Z t Z t− are independent.

iv. Z has Gaussian increments: if 1t t2< then 2 1 2( ( ) ( )) (0, )1Z t Z t N t t− −∼ , where

2 1t t− represents the standard deviation of ( 2 1( ) ( )Z t Z t− ). 2.1.2 n-dimensional (independent) Wiener process By an n-dimensional Wiener process we mean 1 2( , ,..., )nZ Z Z Z= where each KZ is a Wiener process and the components are mutually independent. [5] 2.1.3 n-dimensional (correlated) Wiener process By an n-dimensional correlated Wiener process we mean 1 2( , ,..., )nZ Z Z Z= , where each

kZ satisfies 1

d

jk kjj

Z Zδ=

=∑ , 1, 2,...,k n= ,

'jZ s are d independent standard Wiener processes and

'ij sδ are elements of a ( ) matrix n d× δ ( 1iδ = 1, 2,...,i n= ), such that Z Zδ= . We define a correlation matrix ρ of Z by: [ij i jdt E dZ dZ ]ρ = ⋅ in other words *ρ δδ= . [7]

4

2.2 Ito’s lemma 2.2.1 One-dimensional Ito’s lemma Suppose that the random process X is defined by the stochastic process

( ) ( ) ( )dX t t dt t dZµ σ= + where Z is a standard Wiener process, µ is the drift and σ is the diffusion. Let , and define1,2 ( nF +∈ × →C ) ( , ( )) ( )F t X t Y t= , then satisfies the Ito’s equation:

( )Y t

2

22

1( ) ( )2

F F F FdY t dt dZt x x x

µ σ σ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂

Notation 1: denotes the family of all functions which are continuously differentiable with respect to the first variable and twice continuously differentiable with respect to the remaining variables. It has the same meaning in the other sections too.

1,2C

2.2.2 Multi dimensional Ito’s lemma for independent Wiener processes Consider an n-dimensional Ito process driven by a d-dimensional Wiener process

( ) ( ) ( )dX t t dt t dZµ σ= + with the following meaning:

1

( ) ( ) ( ) ( )d

i i ij jj

dX t t dt t dZ tµ σ=

= +∑ , 1, 2,...,i n=

where:

1

2

n

µµ

µ

µ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

is the drift vector , 11 1

1

d

d n

σ σσ

σ σ d

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

is the diffusion matrix and

1

2

d

ZZ

Z

Z

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

is a vector with d independent Wiener processes.

5

Let , and define 1,2 ( nF +∈ × →C ) ( ) ( , ( ))Y t F t X t= , then the following stochastic differential , called multi-dimensional Ito’s equation, is derived:

1 , 1 1

1{ }2

n n n

t i i ij ij i ii i j i

dY F F C F dt F dZµ σ= = =

= + + +∑ ∑ ∑ i

where stands for iFi

Fx∂∂

, means ijF2

i j

Fx x∂∂ ∂

and [ ]ijC C σσ ∗= = .

In other words:

1 , 1

1( , ( ))2

n n

t i i ij ii i j

dF t X t F dt F dX F dX dX= =

= + +∑ ∑ j

where we apply the following formula multiplication rules:

2( ) 0dt = . 0idt dZ = , , 2( )idZ dt= 1, 2,...,i d=

. 0i jdZ dZ = , [5, 7] i ≠ j 2.2.3 Multi dimensional Ito’s lemma for correlated Wiener processes Consider an n-dimensional correlated Ito process with correlation matrix ρ driven by a d-dimensional Wiener process, then for , the stochastic differential of the process is given by: 1,2F ∈C ( , ( ))F t X t

1 , 1

1( , ( ))2

n n

t i i ij ii i j

dF t X t F dt F dX F dX dX= =

= + +∑ ∑ j

0

where we apply the following formula multiplication rules:

2( ) 0dt = . idt dZ = 1, 2,...,i d=

.i j ijdZ dZ dtρ= and if and for any n d= iX , i i idX dt dZiµ σ= + 1, 2,...,i n= then the stochastic differential of the process is given by: ( , ( ))F t X t

1 , 1 1

1{ }2

n n n

t i i i j ij ij i ii i j i

dY F F F dt F dZµ σ σ ρ σ= = =

= + + +∑ ∑ ∑ i [7]

6

2.3 Black-Scholes equation 2.3.1 One dimensional Black-Scholes equation Suppose that our arbitrage free market contains:

• An underlying security which its price is governed by a geometric Brownian motion

( ) ( ) ( ) ( )dS t S t dt S t dZ tµ σ= + ,

process over a time interval [0 , where , ]T µ and σ are constants.

• A risk free asset with dynamics ( ) ( )dB t rB t dt= where the interest rate r is constant.

• A simple contingent claim of the form ( ( ))S tχ = Φ which can be traded on the

market with the price process ( )tΠ . Then the only pricing function of the form ( ) ( , ( ))t F t S tΠ = which is consistent with the absence of arbitrage, is when satisfies the following partial differential equation: F

22 2

2

1( , ) ( , ) ( , ) ( , ) 02

F F Ft s rS t s S t s rF t st s s

σ∂ ∂ ∂+ + −

∂ ∂ ∂=

Subject to the boundary condition:

( , ) ( )F T s s= Φ in the strip [0 . [5, 7] , ]T +×R 2.3.2 Multi dimensional Black-Scholes equation When the option depends on several underlying assets the Black-Scholes equation generalizes into:

2*

1 1 1

1 02

d d d

i i ji i ji iij

F F FrS S S rFt s s s

σσ= = =

∂ ∂ ∂⎡ ⎤+ + −⎣ ⎦∂ ∂ ∂ ∂∑ ∑∑j

=

where d is the number of underlying assets, and

( , ) ( )F T s s= Φ [5, 8]

7

2.4 Feynman-Kac theorem 2.4.1 Feynman-Kac theorem when our processes are independent Let and let . Consider the (multi-dimensional) SDE0r > 0T > 2,

( , ) ( , )s s s sdX s X dt s X dWµ σ= + )

) )

(† (Where the coefficients satisfy the assumption of the existence and uniqueness theorem) If is given and is a solution to the boundary value problem

2 2(Φ∈C 1,2 ( nF +∈ × →C

( )∗∗∗

( , ) ( )

FF rF

t

F T x x

∂+ =

∂

=Φ

⎧⎪⎪⎨⎪⎪⎩

A

in the strip [0 then for , ] nT × ( ),( , ) [ ( )]r T t

t x TF t x e E X− −= Φ ( , ) [0, ] nt x T∈ × , where the subscripts of the expected value operator indicate that the expected value is to be calculated for the solution

,t XX of († satisfying the initial condition ) tX x= .

Recall that

2121 , 1

n nF FF Cj ijx x xj i jj iµ ∂ ∂

= +∑ ∑∂ ∂= =

Aj∂

*[ ]ijC σσ= [5]

Solving ( gives us the current price, and using this current price to obtain the futures

price when time to maturity is ()

)

∗∗∗

T t− , we should multiply that by ( ( )r T te − ).

2 “Stochastic Differential Equation”

8

2.4.2 Feynman-Kac theorem when our processes are correlated We have the same result when our processes are correlated with the correlation coefficient ρ . [3] Corollary 1: Let X be a (vector valued) stochastic process with the infinitesimal operator and let

be a function satisfying the assumptions of the Feynman-Kac theorem. A

F Then:

(a) is an martingale ( , )tF t X xF ⇔ 0F Ft

∂+ =

∂A

(b) If T then [5] t> ,( , ) [ ( , )]t x TF t x E F T X= 2.5 Pricing equation Consider the following assumptions

• a k-dimensional stochastic process ( ) ( , ( )) ( , ( )) ( )dX t t X t dt t X t dW tµ δ= + where components of X are not necessarily price processes of traded assets and dW is an n-dimensional Wiener process

• A risk free asset: ( ) ( )dB t rB t dt= • There is a liquid market for all contingent claims written on X • The claims ( ( ))i

iy X T= Φ 1, 2,3,...i = are chosen and that their price processes are with ( ) ( , ( ))i it F t X tΠ = ( , ) ( )i iF T x x= Φ 1, 2,3,...i =

Then by Ito’s formula: i i i

i idF F dt F dWα σ= + where

1( ) [ ]2

i i it x xx

i i

F F tr F

F

µ δ δα

∗ ∗+ += and ( )i

xi i

FFδσ =

9

We make a technical assumption that

1

2

n

σσ

σ

σ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

is an invertible matrix at each

point ( , )t x

• Another T-claim with ( ( ))y X T= Φ ( ) ( , ( ))t F t X tΠ = and • ( , ) ( )F T x x= Φ

Then by Ito’s formula F FdF F dt F dWα σ= + where

1 [ ]2t x xx

i

F F tr F

F

µ δ δα

∗ ∗+ += and x

iFFδσ∗

=

If the market is arbitrage free then the process

1

21

n

rr

r

αα

λ σ

α

−

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥−⎣ ⎦

satisfies the

identity F Frα σ λ− = (λ is called “risk premium per unit of volatility”) Corollary 2: Under the above assumptions is the solution of F

1( ) [ ]2

( , ) ( )

t x xxF F tr F rF

F T x x

µ δλ δ δ∗ ∗+ − + =

= Φ

Corollary 3:

In the above model there exists a martingale measure Q and a n-dimensional Wiener process W with respect to Q such that

( )

,( , ) [ ( ( ))]r T t Qt xF t x e E X T− −= Φ

moreover ( )dX dt dWµ δλ σ= − + [5,7]

10

2.6 Ornstein-Uhlenbeck process (Gauss Markov process) A stochastic process { is an Ornstein-Uhlenbeck process: 0}tX t ≥ 3 or a Gauss-Markov process if it is stationary, Gaussian, Markovian 4, and continuous in probability. From general theory { necessarily satisfies the following linear stochastic differential equation:

: 0}tX t ≥

( )t tdX X dt dWtρ µ σ= − − + (See e.g. Finch (2004)) where { is Brownian motion and : 0}tW t ≥ ,µ ρ and σ are constants and in unconditional (strictly stationary) case we have:

( )tE X µ= and 2

| |( , )2

s ts tCov X X e ρσ

ρ− −= . “ [9]

2.7 Stochastic integrals To calculate a stochastic integral we can use basic properties of the (Ito’s) stochastic integrals. [5, 13]

Def: and 2[ , ] { : [ ( )]b

aL a b f E f s ds= < ∞∫ ( ) }W

tf s ∈F

If 2[ , ]f L a b∈ then:

[ ( ) ]b

uaE f u dW =∫ 0

2[ ( ) ] [ ( )]

b b

ua aVar f u dW E f u du=∫ ∫

( ( ) , ( ) ) [( ( ) ).( ( ) )] [ ( ) ( )]b b b b b

u u u ua a a a aCov f u dW g u dW E f u dW g u dW E f u g u du= =∫ ∫ ∫ ∫ ∫

where is a stochastic process and is the Wiener process. f uW

3 Mean-reverting process 4 A stochastic process { : is 0}tX t ≥

• Stationary if, for all and the random n-vectors 1 2 ... nt t t< < < 0h >1 2

{ , ,..., }nt t tX X X and

are identically distributed, that is, time shifts leave joint probabilities unchanged.

1 2{ , ,..., }

h h n ht t tX X X+ + +

• Gaussian if, for all , the n-vector 1 2 ... nt t t< < <1 2

{ , ,..., }nt t tX X X is multivariate normally

distributed. • Markovian if, for all , 1 2 ... nt t t< < <

1 2 1 1( | , ,..., ) ( | )

n n nt t t t t tP X x X X X P X x X− −n

≤ = ≤ ; that is, the future is determined only by the present and not the past. [9]

11

2.8 Risk neutral valuation Consider a market given by the following equations:

( )dB rB t dt= and

( ) ( ) ( , ( )) ( ) ( , ( )) ( )dS t S t t S t dt S t t S t dW tα σ= + ( )∗ where ( denotes the P-dynamic of the S-process, and also a contingent claim )∗

( ( ))S Tχ = Φ . Then the arbitrage free price is given by ( , ) ( , ( ))t F t S tΠ Φ = where is the solution of the Black-Scholes equation (pricing equation). We know that by applying Feynman-Kac, the solution is given by:

F

( ),( , ) [ ( ( ))]r T t

t sF t s e E X T− −= Φ where X process is defined by

( ) ( ) ( ) ( , ( )) ( )( )

dX u rX u du X u u X u dW uX t s

σ= +=

( )∗∗

(Where W is a Brownian motion.) It’s clear that the price process of S in ( )∗ is similar to the price process of X in ( )∗∗ , with a difference in their drifts. It can be shown that there exists another probability measure Q under which the S-process is described by the SDE:

( ) ( ) ( ) ( , ( )) ( )dS t rS t dt S t t S t dW tσ= + , where W is a Brownian motion with respect to Q. We call this representation of S, the Q-dynamics of S. Then we define the expectation under the martingale probability measure Q, and considering the Q-dynamics, we have the following result:

QE

( ),( , ) [ ( ( ))]r T t Q

t sF t s e E X T− −= Φ This Q-measure is called risk adjusted measure, and the last formula is called the risk neutral valuation formula. [7] We have similar assumptions and result when in general we use pricing equation.

12

3. The two-factor modeling of oil futures prices and two alternatives for its improvement After the first model (1985), which had a constant convenience yield5, several versions of the one-factor model were proposed by different experts6. In that one-factor model the logarithm of the spot price of the oil had been described by an Ornstein- Uhlenbeck process; however, this new one-factor mean reverting model wasn’t satisfactory either, because all futures returns were correlated and it was in contrast with the empirical evidence. [2, 14] To improve this one-factor model, the two-factor model with a mean reverting convenience yield was proposed. Gibson and Schwartz (1990) have pointed out that based on the theory of storage which assumes an inverse relationship between the level of inventories and the net convenience yield, the constant convenience yield assumption is refused in the case of crude oil, and the mean reverting tendency as well as the variability of its changes suggest a mean reverting stochastic process for convenience yield. They have also motivated (1989) Equation (2) by studying of the time series properties of the forward convenience yields of crude oil.7 The following model explained here is based on the two-factor model Schwartz (1997). As this 2-factor model is the basis of the other models in this essay, it would be helpful to know it in details. 3.1 The two-factor modeling and valuation of oil futures prices with the analytical result In this model we have the following assumptions: [2, 10]

• is the spot price of oil described by the stochastic process S 1( ) ( ) ( ) ( )dS t S t dt S t dz1µ δ σ= − + (1)

• δ is the instantaneous convenience yield described by the Ornstein-Uhlenbeck process:

2 2( ) ( )d t dt dzδ κ δ δ σ= − + (2) And in the above models:

1) µ is the long-term total return on oil 2) is the mean reverting coefficient κ3) δ is the long-term convenience yield 4) 1σ and 2σ are the volatilities of the spot price of oil and the convenience yield

5 Convenience yield is the benefit or premium associated with holding a commodity like oil, rather than the contract or derivative product. 6 Laughton & Jacoby (1993, 95), Ross (1997), Schwartz (1997), Schwartz & Smith (2000) 7 Gibson and Schwartz (1990) have pointed out that there are some assumptions based on which a constant convenience yield can be held, and these assumptions are rejected in the case of crude oil.

13

We also know the following assumptions:

and1( )z t12

2 ( ) (0, )z t N t∼ (3)

where 12t represents the standard deviation,

and because we have two correlated stochastic processes: 1 2dz dz dtρ⋅ = (4)

To show the cumulative convenience yield rate from the date 0 to date t , we use Bjerksund’s notation (1990):

(Later we need this definition to obtain 0

( ) ( )t

X t x dxδ≡ ∫ ( )Tt x dxδ∫ ) (5)

and considering the economy presented in 2.8, the relation between the Brownian motion with respect to the true probability measure and a modified Brownian motion with respect to the martingale probability measure is:

*1 1

1

rdz dz dtµσ−

= − (6)

*2 2dz dz dtλ= − (7)

where 1

rµσ− deducts the market price per unit of oil risk8 and λ indicates the market

price of the convenience yield risk.

The convenience yield risk arises from the fact that risks are priced relative to the market under the risk neutral measure. Thus, the growth rate attributable to a non-traded state variable must be adjusted for the market price of risk. The risk adjusted convenience yield process has the market price of risk as convenience yield risk can not be hedged. [2, 10, 11, 14]

(Based on the Girsanov theorem Processes 1z∗ and 2z∗ are Brownian motions under the

martingale probability measure.) 8 In a frictionless continuous time economy the representative agent can invest in a risky commodity, in a contingent claim and in an instantaneously risk free security, the spot price of the commodity process, S(t),

satisfies the following SDE, with initial condition 0( )S S= , 1 1( ( ))dS

t dt dzS

µ δ σ= − + , 1rµ λσ= + where λ

stands for the constant market price of risk associated with the spot price process. Thus the following is obtained:

[ ]1 11

1

( ( ))dS r

r t dt dz dtS

dz

µδ σ

σ+

∗

−= − + where 1z

∗ is a Brownian motion under the martingale probability

measure (Q).

14

Substituting (6) and (7) into (1) and (2) we obtain the following models:

*1( ) ( ) ( ) ( )dS t r S t dt S t dzδ σ= − + 1 (8)

*

2 2 2( ) [ ( ) ]d t dt dzδ κ δ δ σ λ σ= − − + (9)

* *1 2dz dz dtρ=

As it has been pointed out by Gibson and Schwartz (1990) we assume that the price of the oil contingent claim is a twice continuously differentiable function of and S δ , By defining:

( ) ( , , )Y t F t s δ= (10) and applying Ito’s lemma for correlated stochastic processes to the stochastic processes, (8) and (9), the following result is obtained:

2{ ( ) [ ( ) ]t SdY F r SF Fδδ κ δ δ σ λ= + − + − − 2 2 21 2 1 2

1 1 }2 2SS SS F F SF dtδδ δσ σ σ σ ρ+ + +

*1 1 2 2S

*F S dz F dzδσ σ+ + (11) which shows instantaneous price change. Thus the current price of a futures contract, ( , , )F t s δ , on one barrel of crude oil [6] satisfies the following partial differential equation which is a 2-dimensional pricing equation:

2 2 22 1 2 1 2

1 1( ) [ ( ) ]2 2t S SS SF r SF F S F F SF rδ δδδ κ δ δ σ λ σ σ σ σ ρ+ − + − − + + + = Fδ

( , , ) ( )F T s S Tδ = (12)

To solve the above equation, we apply corollary (3):

( ), ,( , , ) [ ( ( ))]r T t Q

t sF t s e E S Tδδ − −= Φ (13) Where

( ( )) ( )S T S TΦ = (14) So we need to obtain and substitute it in (13). ( )S T

15

How to evaluate : ( )S T We know that:

1( ) ( ) ( ) ( )dS u S u du S u dz1µ δ σ= − + (Here u represents time.) ( )S t s=

This equation can be solved explicitly. By Ito’s formula

22

2

1[ ( , )] ( )2

g g gd g u S du dS dSu S S∂ ∂ ∂

= + +∂ ∂ ∂

Assume [12]. We know that , ( , ) logg u S S= 2

1dz du= 2 0du = and 1. 0dz du =

22

1 1 1(log ) 0 ( )( )2

d S du dS dSS S

−= + +

21 1 1 12

1 1 1( ( ) ) ( ( ) )2

Sdu Sdz u Sdu Sdu Sdz u SduS S

µ σ δ µ σ δ= + − − + −

2 21 1 1 1 1 1

1 1( ) ( ( ) )2 2

du dz u dt du u du dzµ σ δ σ µ δ σ σ= + − − = − − +

So we obtain:

21 1

1(log ) ( ( ) )2

d S u du dµ δ σ σ= − − + 1z

log ( ) logS t s= By applying integrals from to T we will have: u t=

21 1

( ) 1log ( )( ) ( )( ) 2

T T

t t

S T T t s ds dzS t

µ σ δ σ= − − − +∫ ∫ 1

And at last the following result is obtained:

21 1 1

1( ) ( ).exp{( )( ) ( ) }2

T T

t tS T S t T t dz s dsµ σ σ δ= − − + −∫ ∫ (15)

A B

in (6) we had *1 1

1

rdz dz dtµσ−

= − , so:

16

= *1 1 ( )

T T

t tdz r dtσ µ− −∫ ∫

= (16) *1 1 ( )(

T

tdz r T tσ µ− − −∫

A

)

To obtain a more explicit expression for we have several steps being stated by Bjerksund (1991): [10]

B

Based on (5) we have (17) ( ) ( ) ( )T

ts ds X T X tδ ≡ −∫

We also know ( ) ( ) ( )T

td s T tδ δ δ= −∫ (18)

And from (2) we have:

2 2( ) ( ( ))T T T

t t td s s ds dzδ κ δ δ σ= − +∫ ∫ ∫

2 2( ) ( )T T

t tT t s ds dzκδ κ δ σ= − − +∫ ∫

(17) and (18) ⇒ 2( ) ( ) ( ) ( ( ) ( ))T

tT t T t X T X t dδ δ κδ κ σ− = − − − + 2z∫ (19)

and in (19) ( )Tδ can be modified too. From 2 2( ) ( )d t dt dzδ κ δ δ σ= − + we have:

( ) ( )2( ) ( ) (1 ) ( )

TT t T t T s

tT e t e e e dz sκ κ κ κδ δ δ σ− − − − −= + − + ∫ 2 (20)

(See [7], section on linear SDE.) Substituting (20) into (19) we will have:

( )2 22 2

1( ) ( ) ( ) (1 )( ( ) )T TT s T t

t tX T X t T t dz e e dz e tκ κ κσ σδ δ δ

κ κ κ− − −− = − + − + − −∫ ∫ (21)

17

And in (21) we replace by its equal term as we had in (7), 2dz *2 2dz dz dtλ= − :

( )2 22 2

1( ( ) (1 )( ( ) ))T TT s T t

t tT t dz e e dz e tκ κ κσ σδ δ δ

κ κ κ− − −= − − + − + − −∫ ∫ (22)

B

⇒

* *2 2 22 2( ) ( )

T T TT s

t t tT t dz dt e e dzκ κσ σ σδ λ

κ κ κ−= − − − − − +∫ ∫ ∫ 2 ( )

TT s

te eκ κ dtσ λ

κ−+ −∫ B

( )1 (1 )( ( ) )T te tκ δ δκ

− − −− −

thus:

= * *2 2 2 22 2 2( ) ( ) (1

T TT s T t

t tT t dz T t e e dz eκ κ κ ( ) )σ λσ σ λσδ

κ κ κ κ− −− − − + − + − −∫ ∫ − B

( )1 (1 )( ( ))T te κ δ δκ

− − − t+ − (23)

After substituting and into (15) and rearranging that the following is resulted: A B

2 *2 21 1 1

1( ) ( ).exp{( )( )2

T T

t tS T S t r T t dz dz*

2λσ σσ δ σκ κ

= − + − + − + −∫ ∫

* ( )2 22

1 (1 )( ( ))}TT s T t

te e dz e tκ κ κσ λσ δ δ

κ κ κ− − −+ + (24) − − + −∫

18

Recall: The purpose of obtaining was solving Equation (12), so inserting (24) into (13) results:

( )S T

( ) 2 * *2 21 1 1

1( , , ) [ .exp{( )( )2

T Tr T t Q

t t 2F t s e E s r T t dz dzλσ σδ σ δ σκ κ

− −= ⋅ − + − + − + −∫ ∫

* ( )2 22

1 (1 )( ( ))}]TT s T t

te e dz e tκ κ κσ λσ δ δ

κ κ κ− − −+ + − − + −∫

⇒

2 *2 21 1 1

1( , , ) [ .exp{( )( )2

T TQ

t t

*2F t s E s T t dz dzλσ σδ σ δ σ

κ κ= − − + − + −∫ ∫

* ( )2 22

1 (1 )( ( ))}]TT s T t

te e dz e tκ κ κσ λσ δ δ

κ κ κ− − −+ + − − + −∫

(25) [QE≡ exp{ }]s D⋅ To take this expectation we need the expectation and variance of D

(As an example 21 ( )( ( ) ( )) 2( . ) .

T tW T W tE A e A eαα −− = where W is a Wiener process, and A and α

are constants.)

µ = [ (26) ]QE D2

σ = [ (27) ]QVar D

At first we obtainµ :

µ = 2 21

1( )( )2

T tλσσ δκ

− − + − ( ) 21 (1 )( ( ))T te tκ λσ δ δκ κ

− −+ − − + − (28)

( , *1 1[ ] 0

TQ

tE dzσ =∫ *2

2[ ] 0TQ

tE dzσ

κ=∫ , and *2

2[TQ T S

tE e e dzκ κ ] 0σ

κ− =∫ arising from the

definition (2.1.1, (iv)) and also see (2.7))

Secondly we obtain 2

σ : We know

2 2[ ] ( [ ])Q Q 2E D E Dσ = − (29)

19

⇒ 2 * *2 2

1 1 2 2[( ) ]T T TQ T

t t tE dz dz e e dzκ κ * 2sσ σσ σ

κ κ−= − +∫ ∫ ∫

* 2 * 2 * 22 21 1 , , 2 , , 2[( ) ] [( ) ] [( ) ]

T T TQ Q Q TS t S tt t t

E dz E dz E e e dzκ κδ δ

sσ σσκ κ

−= + − +∫ ∫ ∫

* * *2 21 1 2 1 1 22 [( )( )] 2 [( )( )]

T T T TQ Q

t t t t

*T sE dz dz E dz e e dzκ κσ σσ σκ κ

−+ − +∫ ∫ ∫ ∫

*2 222 [( )( )

T TQ T

t t

*2 ]sE dz e e dzκ κσ σ

κ κ−+ − ∫ ∫ (30)

So our task is to take several expectations that we have in (30) 9 To calculate the following integrals, we apply methods presented in (2.7).

* 2 21 1 1

2* 22 22 2

* 2 2 2 ( )2 22

* *2 1 21 1 2

* * (2 1 21 1 2 2

[( ) ] ( )

[( ) ] ( )

1[( ) ] ( ) ( )(1 )2

2 [( )( )] 2 ( )

2 [( )( )] 2 (1

TQ

t

TQ

t

TQ T s T t

t

T TQ

t t

T TQ T s T t

t t

E dz T t

E dz T t

E e e dz e

E dz dz T t

E dz e e dz e

κ κ κ

κ κ κ

σ σ

σ σκ κ

σ σκ κ κ

σ ρσ σσκ κ

σ ρσ σσκ κ

− − −

− −

= −

− = −

= −

− = − −

= −

∫

∫

∫

∫ ∫

∫ ∫ − )

* * 22 2 22 2

)

12 [( )( )] 2( ) ( )(1T TQ T s

t tE dz e e dz eκ κ κσ σ σ

κ κ κ κ− −

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪

− = −⎪⎩ ∫ ∫ ( ) )T t−−

(31)

⇒

2σ =

22 2 1 21 2( 2 )( )T tσ ρσ σσ

κ κ+ − −

2( )1 2 2

2 32( )(1 )T te κρσ σ σκ κ

− −+ − − 2 22 1( ) ( )(12

T te κ ( ) )σκ κ

− −+ −

(32) Therefore (28) and (32) result in (33):

9 see (2.7)

20

[QE exp{ }]s D =21exp{ }

2s µ σ+

2

2 2 1 22

1exp{( )( )2

s Tλσ σ ρσ σδκ κ κ

= − + + − t−

2

( )2 1 2 22

1 ( ( ) )(1 T tt e κλσ ρσ σ σδ δκ κ κ κ

− −+ − + − + − − )

2 2 (2 1( ) ( )(1 )}4

T te κ )σκ κ

− −+ − (33)

which is the current price of our futures contract. To obtain the futures price of a futures contract, ( , , )fF t s δ , on one barrel of crude oil, we multiply the obtained “current futures price” by ( )r T te − , where T t− is time to maturity:

( )( , , ) ( , ,f r T tF T t s e F t s )δ δ−− = ⋅ ⇒

22 2 1 2

2

1( , , ) exp{( )( )2

fF T t s s r T tλσ σ ρσ σδ δκ κ κ

− = − + + − −

2

( )2 1 2 22

1 ( ( ) )(1 T tt e κλσ ρσ σ σδ δκ κ κ κ

− −+ − + − + − − )

2 2 (2 1( ) ( )(1 )}4

T te κ )σκ κ

− −+ −

which is the price of a futures contract on one barrel of crude oil. The above model was more realistic compared with the existed one-factor models; however, it had some specifications which showed the necessity to include the third factor in the model:

1. At first the futures price was equal to the forward price in this model because of the constant interest rate of return . r

2. Secondly, the implemented model wasn’t well-fitted for some days. (Schwartz 97).

To improve this model, the 3-factor model (Schwartz 1997) which has , explained by a stochastic process instead of a constant , was proposed. The following is a short description on this model.

rr

21

3.2 A brief description on the three-factor modeling of oil futures prices when rate of return r has a stochastic process: (Schwartz 1997) This 3-factor model is exactly based on the 2-factor model presented at the first part of Section 3, meaning that

*1( ) ( ) ( ) ( )dS t r S t dt S t dzδ σ= − + 1

*

2 2 2( ) [ ( ) ]t dt dzδ κ δ δ σ λ σ= − − +d

which are risk adjusted processes. In addition, it is assumed that, risk free interest rate in the first equation above follows an Ornstein-Uhlenbeck stochastic process, (Vasicek 1977)10: *

3 3( ) [ ( ) ]dr t a r r dt dzσ λ σ′= − − + 3 And also we have the following assumptions as our processes are correlated:

* *1 2 12* *1 3 13* *2 3 23

dz dz dt

dz dz dt

dz dz dt

ρ

ρ

ρ

=

=

=

Then the futures prices of the futures contracts on oil are obtained by solving the following multi-dimensional pricing equation:

2 3 1 2 12 2 3 23( ) ( ( ) ) [ ( ) ]T t S r S rF r SF F a r r F S F Fδ δ δδ κ δ δ σ λ σ λ σ σ ρ σ σ ρ− ′− + − + − − + − − + +

2 2 2 21 3 13 1 2 3

1 1 1 02 2 2Sr SS rrS F S F F Fδδσ σ ρ σ σ σ+ + + + =

Subject to the boundary condition:

( 0, , , ) (F T t s r S T )δ− = = (Here shows futures price) F In the above model, the futures price and the forward price are not equal as interest rate is a stochastic process. However, this model still isn’t satisfactory enough as futures prices are just used to estimate the parameters of the two-factor model, and the risk free interest

10 “The feature of mean reversion is considered to be quite important by many researchers and practitioners since it is felt that interest rates have a natural home (of about 6%) and that if rates differ widely from this home value, there is a strong tendency to move back to it. “[1]

22

rate is obtained from the bond data which is clear in case of the model implementation. Moreover, the risk free interest rate obtained from the bond data affects on the estimation of futures prices. To improve this 3-factor model (1997), Cortazar and Schwartz developed a 3-factor model in 2003. This new model was based on the parsimonious 2-factor model explained in the next part. We will pay more attention to the 3-factor model 2003 in Section 4. 3.3 The parsimonious two-factor modeling of oil futures prices (Schwartz 2003) Cortazar and Schwartz (2003) developed this parsimonious 2-factor model, by introducing two new variables. In fact they have modified their two-factor model presented in 3.1. This parsimonious model has some advantages compared with the last 2-factor model. The advantages that Cortazar and Schwartz have pointed out are:

Firstly, the simplicity of this model because of fewer parameters whereas it has the same explanatory power.

Secondly, there is no need to estimate the risk free interest rate from bond data.

Thirdly, the model is more intuitive for practitioners as returns are defined in

terms of the long-term price appreciation instead of the long-term convenience yield.

Two state variables; y andν , are defined as follows: y δ δ= − (34)

ν µ δ= − (35)

where y is the demeaned convenience yield, and ν is the expected long-term spot price return on oil11, then by substituting (34) and (35) in (1) and (2), we obtain12:

11 We know that: change in price(or price appreciation) = (long term total return) − (long term convenience yield) (⇒ ν = µ − δ ) 12 Even a mathematical approach gives us these 2 new definitions: We knew that 2 2( ) ( ( ) )d t t dt dzδ κ δ δ σ= − − + , and also it’s clear that ( )d dδ δ− = δ where δ is a

constant, so 2 2( ( ) ) ( ( ) )d t t dt dzδ δ κ δ δ σ− = − − + and by defining ( ) ( )y t tδ δ= − we get the eq. (37); ; based on which is an Ornstein-Uhlenbeck process with long-term value zero. 2 2dy ydt dzκ σ=− + yAnd also we had 1( ) ( ( )) ( ) ( )dS t t S t dt S t dz1µ δ σ= − + which is equal to

1( ) ( ( )) ( ) ( )dS t t S t dt S t dzµ δ δ δ σ= − + − + 1 , and by defining ν µ δ= − , gives us the eq. (36)

23

1( )dS y Sdt Sdz1ν σ= − + (36) (37) 2 2dy ydt dzκ σ= − + But now and both are treated such that they are non-traded state variables, hence for the purpose of risk neutral valuation, in (36) and (37)

S y

1λ and 2λ are introduced as risk premiums: (38) *

1 1 1 1( )dS y Sdt Sdzν σ λ σ= − − +

(39) *2 2 2 2( )dy y dt dzκ σ λ σ= − − +

and: * *

1 2dz dz dtρ= (40) As Cortazar and Schwartz (2003) have indicated, this parsimonious two-factor model is the basis of the three-factor model, the fact which leads them to develop this three factor model is that, where their two-factor model is used to show the behavior of the market, the obtained curve is not fitted well for some days. In contrast the new three-factor model presents a better fit for those days, without worsening the last fitted line.

24

4. The three-factor modeling of oil futures prices 4.1 The three-factor modeling and valuation of oil futures prices with the analytical result (Schwartz 2003) In this model we have the following assumptions:

• S is the spot price of oil depends on the demeaned convenience yield ( ) and the long term total return (

yν ), both of which are stochastic processes.

1( )dS y Sdt Sdz1ν σ= − + (41) • y is the demeaned convenience yield, described by an Ornstein-Uhlenbeck

process, meaning that it is a mean revering model: . (42) 2 2dy ydt dzκ σ= − +

• ν is the long-term spot price return which is an Ornstein-Uhlenbeck process, as

Cortazar and Schwartz have mentioned (2003): 3 3( )d a dt dzν ν ν σ= − + . (43)

And in the above models:

1) has the same meaning as in the two-factor model in Section 3. κ2) is the mean reverting coefficient for the third risk factor. a3) ν is the expectation of the long-term spot price return. 4) 1σ , 2σ and 3σ are the volatilities of the spot price of oil, the demeaned

convenience yield and the long term spot price return.

and because we have three correlated stochastic processes:

(44) 1 2 12

1 3 13

2 3 23

dz dz dtdz dz dtdz dz dt

ρρρ

=⎧⎪ =⎨⎪ =⎩

25

The SDE (41), states in particular that the drift of the spot price process yν − (the instantaneous expected change in price) involves two stochastic processes. Cortazar and Schwartz (2003), defined as the deviation of spot price returns (or change in price) from its long-term value.

y

To show the cumulative demeaned convenience yield rate and the expected cumulative long-term spot price return from the date 0 to date t, we define:

0( ) ( )

tX t y x dx≡ ∫ (45)

0( ) ( )

tL t x dxν≡ ∫ . (46)

We also define the following equivalency to show the relation between the Brownian motion , , with respect to the true probability measure and martingale probability 1z 2z 3zmeasure:

*i i idz dz dtλ= − for (47) 1, 2,3i =

where iλ are prices of risk.

1 1 1 1( )dS y Sdt Sdzν σ λ σ ∗= − − + (48)

2 2 2 2( )dy y dt dzκ σ λ σ ∗= − − + (49)

3 3 3 3[ ( ) ]d a dt dzν ν ν σ λ σ ∗= − − + (50)

The same as what we have done in two-factor model, we define:

( ) ( , , , )Y t F t s δ ν= (51)

26

and we apply Ito’s lemma to three correlated stochastic processes to obtain the following result:

1 1 2 2 3 3{ ( ) ( ) [ ( ) ]t S ydY F y SF y F a Fνν σ λ κ σ λ ν ν σ λ= + − − + − − + − − 1 2 12 2 3 23 1 3 13Sy y SS F F S Fν νσ σ ρ σ σ ρ σ σ ρ+ + +

2 2 2 21 2 3 1 1 2 2 3

1 1 1 }2 2 2SS yy S yS F F F dt S F dz F dz F dzνν νσ σ σ σ σ σ 3

∗ ∗+ + + + + + ∗ (52)

which shows instantaneous price change. Thus the current price of a futures contract, ( , , , )F t s y ν , on one barrel of crude oil [6] satisfies the following partial differential equation which is a 3-dimensional pricing equation:

1 1 2 2 3 3 1 2 12( ) ( ) [ ( ) ]t S y SyF y SF y F a F Sν Fν σ λ κ σ λ ν ν σ λ σ σ ρ+ − − + − − + − − +

2 2 2 21 2 3 2 3 23 1 3 13

1 1 12 2 2SS yy y SS F F F F S F rFνν ν νσ σ σ σ σ ρ σ σ ρ+ + + + + = (53)

Subject to the boundary condition:

( , , , ) ( )F T s y S Tν =

To solve the above equation, we use corollary (3):

( ), , ,( , , , ) [ ( ( ))]r T t Q

t s yF t s y e E S Tνν − −= ⋅ Φ (54) where

( ( )) ( )S T S TΦ = So we should obtain and substitute it in (54). ( )S T

27

How to evaluate for the three factor model: ( )S T The estimation procedure is similar to our estimation in 2-factor model. We know that:

1( ) ( )( )

dS u y Sdu SdzS t s

ν σ= − +⎧⎨ =⎩

1 (55)

and also

22

1 1 1(log ) 0 ( )( )2

d S du dS dSS S

= + + − (56)

By substituting dS into (56), we get the following equation:

21 1

1(log ) ( )2

d S y du dν σ σ= − − + 1z (57)

where *

1 1 1dz dz duλ= − ⇒

2 *1 1 1 1 1

1( ) ( ) exp{ ( ) ( ) ( ) ( )2

T T T

t t tS T S t s ds y s ds T t dz T tν σ σ σ= − − − + −∫ ∫ ∫ }λ − (58)

A B To find a more convenient expression for we can do the following: A

Based on (46), we have ∫ (59) ( ) ( ) ( )T

ts ds L T L tν ≡ −

We also know that ( ) ( ) ( )T

td s T tν ν ν= −∫ (60)

and from (43) and dz *

3 3 3 we will have: dz dtλ= −

*3 3 3 3( ) ( )

T T T

t t td a T t a ds dz T tν ν ν σ σ λ= − − + − −∫ ∫ ∫

Thus the following equation is obtained:

28

*3 3 3 3( ) ( ) ( ) ( ( ) ( )) ( )

T

tT t a T t a L T L t dz T tν ν ν σ σ λ− = − − − + − −∫ (61)

Now in above equation we can replace ( )Tν by its equivalent expression obtained by integrating Eq. (43). As a result we get: (See [7], section on linear SDE.)

( ) ( ) * ( )3 33 3( ) (1 ) ( ) (1 )

Ta T t a T t aT as a T t

tT e e t e dz e e

aσ λν ν ν σ− − − − − − −= − + + − −∫ (62)

Substituting (62) into (61), gives us

( ) ( ) *3 3

1( ) ( ) (1 ) (1 )Ta T t a T t aT as

tL T L t e e e dz e

a a aν ν σ− − − − −− = − − + − − ∫

( ) *3 3 3 3 332 (1 ) ( ) ( )

Ta T t

te T t dz T

a at

aσ λ σν− −− + − + −∫

σ λ− (63)

To find a more explicit form of we can proceed as follows: B

Based on (45), we have ∫ (64) ( ) ( ) ( )T

ty s ds X T X t≡ −

We also know that . (65) ( ) ( ) ( )T

tdy s y T y t= −∫

From (42) and *

2 2 2dz dz dtλ= − we obtain:

∫ ∫ *2 2 2 2 ( )

T T T

t t tdy yds dz T tκ σ σ λ= − + − −∫

Therefore, the following equation is obtained:

*2 2 2 2( ) ( ) ( ( ) ( )) ( )

T

ty T y t X T X t dz T tκ σ σ λ− = − − + − −∫ (66)

Now in the above equation we replace by its equivalent obtained by solving Eq. (42).

( )y T

As a consequence we get:

29

( ) * ( )2 22 2( ) ( ) (1 )

TT t T s T t

ty T e y t e dz e eκ κ κ κσ λσ

κ− − − − −= + − −∫ (67)

Substituting (67) into (66), gives us

( ) *2 2

1( ) ( ) (1 )TT t kT s

t

yX T X t e e dz eκ κσκ κ

− − −− = − − ∫

( ) *2 2 2 2 222 (1 ) ( )

TT t

te dzκ T tσ λ σ σ λ

κ κ κ− −− + −∫ − (68)

After replacing and in (58) and rearranging the terms, the following result is obtained:

A B

( ) ( ) * ( )3 33 3 2

1( ) ( ) exp{ (1 ) (1 ) (1 )Ta T t a T t aT as a T t

tS T S t e e e dz e e

a a a aσ λν ν σ− − − − − − −= − − + − − + −∫

* ( )3 3 3 23 2( ) ( ) (1 )

T TT t T s

t t

yT t dz T t e e e dza a

κ κ *κσ σ λ σνκ κ

− − −+ − + − − − − +∫ ∫

( ) * 22 2 2 2 22 12

1(1 ) ( ) ( )2

TT t

te dz T tκ T tλ σ σ λ σ σ

κ κ κ− − − + − −∫− − −

}−∫

+ − (69) *1 1 1 1( )

T

tdz T tσ σ λ

Recall: The purpose of obtaining the above form of was to solve Eq. (53). Substituting (69) into (54) results in the following:

( )S T

( ) ( ) ( ) *

3 31( , , , ) [ exp{ (1 ) (1 )

Tr T t Q a T t a T t aT as

tF t s y e E s e e e dz e

a a aν νν σ− − − − − − −= ⋅ − − + − − ∫

( ) * ( )3 3 3 3 332 (1 ) ( ) ( ) (1 )

Ta T t T t

t

ye T t dz T t ea a a

κσ λ σ σ λνκ

− − − −+ − + − + − − − −∫ * ( ) * 22 2 2 2 2 2

2 22

1(1 ) ( ) ( )2

T TT s T t

t te e dz e dz T t Tκ κ κ

1 tσ λ σ σ λ σ σκ κ κ κ

− − −+ − − − + − − −∫ ∫]−∫ + − *

1 1 1 1( )}T

tdz T tσ σ λ

(70) ⇒

30

( ) ( ) *

3 31( , , , ) [ exp{ (1 ) (1 )

TQ a T t a T t aT

t

asF t s y E s e e e dz ea a aν νν σ− − − − −= − − + − − ∫

( ) * ( )3 3 3 3 332 (1 ) ( ) ( ) (1 )

Ta T t T t

t

ye T t dz T t ea a a

κσ λ σ σ λνκ

− − − −+ − + − + − − − −∫

* ( ) * 22 2 2 2 2 22 22

1(1 ) ( ) ( )2

T TT s T t

t te e dz e dz T t Tκ κ κ

1 tσ λ σ σ λ σ σκ κ κ κ

− − −+ − − − + − −∫ ∫ −

]

*1 1 1 1( ) ( )}

T

tdz T t r T tσ σ λ+ − − − −∫

[ exp{ }]QE s M≡ To calculate the above expectation we need to calculate the expectation and variance of M . µ = (71) [QE M ]

2σ = [ (72) ]QVar M

At first we obtainµ :

µ = ( ) ( ) ( )3 3 3 32(1 ) (1 ) (1 ) ( ) ( )a T t a T t a T te e e T t

a a a aT tσ λ σ λν ν ν− − − − − −− − + − + − + − − −

( ) ( ) 22 2 2 21 1 12

1(1 ) (1 ) ( ) ( ) (2

T t T ty e e T t T t T tκ κ )λ σ λ σ σ σ λκ κ κ

− − − −− − − − + − − − − −

(73) (r T t− − )

Secondly we obtain 2

σ : We know

2 2[ ] ( [ ])Q QE M E Mσ = − 2

⇒

2 * *3 2

3 3 3 21[( ) ( ) ( )

T T TQ aT as T s

t t t

*E e dz e dz e e dza a

κ κσ σσ σκ

− −= − + +∫ ∫ ∫

*22 1 1( ) (

T T

t tdz dz* 2)]σ σ

κ+ − +∫ ∫ (74)

31

So now our task is to take several expectations that we have in (74):

* 2 2 2 ( )3 3 32

1 1 1[( ) ] (1 )2

TQ aT as a T

tE e dz e e

a a aσ σ− −− = −∫ t−

2

* 23 33 2[( ) ] ( )

TQ

tE dz T

a aσ σ

= −∫ t

* 2 2 2 ( )2 22

1[( ) ] ( ) (1 )2

TQ T s T

tE e e dz eκ κ κ tσ σ

κ κ κ− −= ⋅ −∫ −

2

* 22 22 2[( ) ] ( )

TQ

tE dz Tσ σ

κ κ− =∫ t−

t

* 2 2

1 1 1[( )] ( )TQ

tE dz Tσ σ= −∫

2

* *3 33 3 3 3

21[2( )( )] (1 )T TQ aT as a T

t tE e dz e dz e

a a aσ σσ− −− = −∫ ∫ ( )t−−

* * 2 3 232

3 3 221 1[2( )( )] (1 )

( )T TQ aT as T s a T

t tE e dz e e e dz e

a aκ κ κ( )( )t

aσ σ ρσσ

κ κ κ− − −− = − ⋅

+∫ ∫ + −−

* * 2 3 232

3 3 2 2

21[2( )( )] (1 )T TQ aT as a T

t tE e dz e dz e

a a( )tσ σ ρσσ

κ κ− −− − = −∫ ∫ −

* * 1 3 13

3 3 1 1 2

21[2( )( )] (1 )T TQ aT as a T

t tE e dz e dz e

a a( )tσ σ ρσ σ− −− = −∫ ∫ −−

* *3 2 323 2 2

2[2( )( )] (1 )T TQ T s

t tE dz e e dz e

a aκ κ κ ( )23 T tσ σ σ ρσ

κ κ− −= −∫ ∫ −

* *3 223 2

2[2( )( )] ( )T TQ

t t3 23E dz dz T t

a aσ σ σ ρσ

κ κ− = −∫ ∫ −

* *3 13 1 1

2[2( )( )] ( )T TQ

t t3 13E dz dz T t

a aσ σ σ ρσ = −∫ ∫

2

* *2 2 22 2 3

2[2( )( )] (1 )T TQ T s T

t tE e e dz dz eκ κ κσ σ σ

κ κ κ− −− = − −∫ ∫ ( )t−

32

* * (2 1 22 1 1 2

2[2( )( )] (1 )T TQ T s T

t tE e e dz dz eκ κ κ )12 tσ σ σ ρσ

κ κ− −= −∫ ∫ −

* *2 12 1 1

2[2( )( )] ( )T TQ

t t2 12E dz dz T tσ σ σ ρσ

κ κ− = −∫ ∫ −

(75) ⇒

2 22 2 2 ( ) 2 2 ( )3 2 232 2

1 1 1(1 ) ( ) ( ) (1 ) ( )2 2

a T t T te T t ea a a

κσ σ σσ σκ κ κ

− − − −= − + − + ⋅ − + 2 T t−

2

2 ( )3 2 3 231 3

2 2 1( ) (1 ) (1( )

a T t a T tT t e ea a a

κσ σ σ ρσκ κ

− − − + −+ − − − − ⋅ −+

( )( ) )

( ) ( ) ( )2 3 23 1 3 13 2 3 232 2 2

2 2 2(1 ) (1 ) (1 )a T t a T t T te ea a a

κeσ σ ρ σ σ ρ σ σ ρκ κ

− − − − − −+ − − − + −

2

( )2 3 23 1 3 13 23

2 2 2( ) ( ) (1 T tT t T t ea a

κσ σ ρ σ σ ρ σκ κ

− −− − + − − − )

( )1 2 12 1 2 122

2 2(1 ) ( )T te Tκ tσ σ ρ σ σ ρκ κ

− −+ − − −

(76) Therefore (73) and (76) imply the following formula:

[QE exp{ }]s M =21exp{ }

2s µ σ+

=2 2

3 3 3 1 3 13 2 3 232 2 2 1 2 121 1 2 2exp{( )( )

2 2s T t

a a a aσ λ σ σ σ ρ σ σ ρσ λ σ σ σ ρν σ λ

κ κ κ κ− − + − + + + − − − r

2 2

( ) 2 ( )3 3 3 2 3 23 1 3 13 32 3 2 2(1 )( ) (1 )( )

4a T t a T te e

a a a a a a a3

σ λ σ σ σ ρ σ σ ρ σν νκ

− − − −+ − − + + − + − + −

2 2

( ) 2 ( )2 3 232 2 2 1 2 12 22 2 3 2(1 )( ) (1 )( )

4T t T tye e

aκ κσ σ ρ

3

σ λ σ σ σ ρκ κ κ κ κ κ

− − − −+ − − − + − + + −σ

( )( ) 2 3 23(1 )( )}( )

a T tea a

κ σ σ ρκ κ

− + −− −+

, (77)

which is the current price of our futures contract on one barrel of crude oil. Thus the futures price of a futures contract, ( , , , )fF T t s y ν− , on one barrel of crude oil is:

33

( )( , , , ) ( , , ,f r T tF T t s y e F t s y )ν ν−− = ⋅

⇒

( , , ,fF T t s y )ν−

=2 2

3 3 3 1 3 13 2 3 232 2 2 1 2 121 1 2 2exp{( )( )

2 2s T t

a a a aσ λ σ σ σ ρ σ σ ρσ λ σ σν σ λ

κ κ κ− − + − + + + − −

σ ρκ

2 2

( ) 2 ( )3 3 3 2 3 23 1 3 13 32 3 2 2(1 )( ) (1 )( )

4a T t a T te e

a a a a a a a3

σ λ σ σ σ ρ σ σ ρ σν νκ

− − − −+ − − + + − + − + −

2 2

( ) 2 ( )2 3 232 2 2 1 2 12 22 2 3 2(1 )( ) (1 )( )

4T t T tye e

aκ κσ σ ρ

3

σ λ σ σ σ ρκ κ κ κ κ κ

− − − −+ − − − + − + + −σ

( )( ) 2 3 23(1 )( )}( )

a T tea a

κ σ σ ρκ κ

− + −− −+

(78)

which is the price of a futures contract on one barrel of crude oil when time to maturity is

. T t− 4.2 The volatility term structure of futures returns Assuming T tτ = − , we apply Ito’s lemma to (78) to determine the volatility of futures returns and we obtain:

21( , , , ) ( )2

f f f f fS y SSdF s y F dS F dy F d F dt F dSν ττ ν ν= + + − + f (79)

we also know that:

1 1

2 2

3 3

(( ) )

1( ) (

1( ) ( ( )

0

ff

S

f fy

af f

fSS

FF dS y Sdt SdzS

eF dy F ydt dz

eF d F a dt dza

F

κτ

τ

ν

ν σ

κ σκ

ν ν ν

−

−

⎧⋅ = ⋅ − +⎪

⎪−⎪

⋅ = ⋅ − ⋅ − +⎪⎨⎪ −⎪ ⋅ = ⋅ ⋅ − +⎪⎪ =⎩

)

)σ

(80)

⇒

34

1 1 2 21( , , , ) (( ) ) ( ) ( )

ff fF edF s y y Sdt Sdz F ydt dz

S

κτ

τ ν ν σ κ σκ

−−= ⋅ − + + ⋅ − ⋅ − +

3 31( ) ( ( ) ) ( )

af feF a dt dz F u t dt

a

τ

ν ν σ−−

+ ⋅ ⋅ − + − ⋅ ⋅ (81)

from where we have:

2 2 2( ) ( ) ( )f

f f

fF

dF dFE EF F

σ τ = − f (82)

whereas we know 2 ( )f

f

dFEF

= 0 and

2 2 2 2 2 2

1 1 2 2 3 3 1 1 2 21 1 1( ) ( ) ( ) ( ) ( ) ( ) 2( )( )( )

f a

f

dF e e edz dz dz dz dzF a

κτ τ κτ

σ σ σ σκ κ

− − −− − −= + − + − σ

2 2 3 3 1 1 3 31 1 12( )( )( )( ) 2( )( )( )

a ae e edz dz dz dza a

κτ τ τ

σ σ σ σκ

− − −− − −− +

⇒

2 2 2 2 2 21 2 3

1 1 1( ) [( ) ( ) ( ) ( ) ( ) 2( )f a

f

dF e e eE E dt dt dt dtF a

κτ τ κτ

σ σ σ σκ κ

− − −− − −= + − + − 1 2 12σ ρ

2 3 23 1 3 131 1 12( )( ) 2( ) ]

a ae e edt dta a

κτ τ τ

σ σ ρ σ σ ρκ

− − −− − −− +

So from the last equality and (82) we get the following result:

2 22 2 2 2

1 2 32 2

(1 ) (1 ) 1( ) 2( )f

a

F

e e ea

κτ τ κτ

1 2 12σ τ σ σ σ σ σ ρκ κ

− − −− − −= + + −

2 3 23 1 3 131 1 12( )( ) 2( )

a ae e ea a

τ κτ τ

σ σ ρ σ σ ρκ

− − −− − −− + (83)

and when τ →∞ ,

222 2 3 2 3 232 1 2 12

1 2 2lim 2 2 2fF a aτ

1 3 13

aσ σ σ ρ σ σ ρσ σ σ ρσ σ

κ κ κ→∞= + + − − +

2321( )

aσσσ

κ= − + (84)

which shows the volatility of futures returns converges to a positive constant as τ goes to infinity.

35

5. Conclusion In this article, we considered the parsimonious two and three-factor modeling of oil futures prices when the drift of the stochastic spot price process contains two stochastic Ornstein-Uhlenbeck processes. We also derived analytical solutions to the futures contracts evaluation formula under the risk neutral measure and analyzed the volatility term structure model of futures returns given by Cortazar and Schwartz (2003). In addition, some brief descriptions on the other models, which help to understand the three-factor modeling of crude oil futures prices, have been presented in this study too.

36

References [1] David G. Luenberger, Investment Science, Oxford University Press, 1998 [2] Cortazar, G. and Schwartz, E., 2003, Implementing a stochastic model for oil futures prices, Elsevier. [3] Rapuch, G. and Roncalli, T., 2001, Some remarks on two-asset options pricing stochastic dependence of asset prices. [4] http://www.investorguide.com, October, 2006. [5] Klimek, M., Depatment of Mathematics, Uppsala University, Lecture Notes, 2006. [6] Gibson, R., Schwartz, E., 1990, Stochastic Convenience Yield and the Pricing of Oil Contingent Claims. [7] Tomas Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 2004 [8] Linde, G., 2005, High-order adaptive space-discretizations for the Black-Scholes equation. [9] Finch, S., 2004, Ornstein-Uhlenbeck Process, Preprint. [10] Bjerksund, P., 1991, Contingent Claims Evaluation when the Convenience Yield is Stochastic: Analytical Results. [11] http://en.wikipedia.org/wiki/Risk-neutral_measure, September, 2006. [12] Tysk, J., Depatment of Mathematics, Uppsala University, Lecture Notes, 2006. [13] http://lipas.uwasa.fi/~sjp/Teaching/Mfd/Lectures/mfdc4d.pdf, October, 2006. [14] Schwartz, E., 1997, The Stochastic Behavior of Commodity Prices: Implication for valuation and Hedging.

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