IMPA Commodities Course : Forward Price · PDF fileBlack’s Forward Price Models Multifactor Forward Price Models HJM Forward Price Models IMPA Commodities Course : Forward Price

Black’s Forward Price ModelsMultifactor Forward Price Models

HJM Forward Price Models

IMPA Commodities Course :Forward Price Models

Sebastian [email protected]

Department of Statistics andMathematical Finance Program,

University of Toronto, Toronto, Canadahttp://www.utstat.utoronto.ca/sjaimung

February 19, 2008

Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models



Table of contents

1 Black’s Forward Price Models

2 Multifactor Forward Price ModelsBasic ModelPrincipal Component AnalysisFunctional Principal Component Analysis

3 HJM Forward Price ModelsInitial FormulationsString / Market Models




Basic Forward Price Models

Basic Black(1976) Forward price model assumes

dFt(T )

Ft(T )= σ dWt

where Wt is a Q-Wiener process

Each forward price evolves like a GBM on its own

Option prices are trivial. Here is a call price

Ct = e−r(T−t) FT (U) Φ(d+)− K Φ(d−)

d± =ln(FT (U)/K )± 1

2σ2(T − t)√

σ2(T − t)




Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis

Multifactor Forward Price Models

Multifactor forward price model assume instead

dFt(T )

Ft(T )=

K∑k=1

σ(k)t (T ) dW

(k)t

where W(k)t are correlated Q-Wiener process with

d [W(k)t ,W

(l)t ] = ρkl dt.

The covariance structure is estimated from principlecomponent analysis (PCA) of forward price curves

Volatility functions often assumed to be deterministic

A further simplifying assumption is often made:

σ(k)t (T ) = σt σ

(k)(T − t)






A principal component analysis on forward curves is used todetermine main factors

The forward prices at a constant set of terms τ1, τ2, . . . , τnare interpolated from given data

Ft1(τ1) Ft1(τ2) Ft1(τ3) . . . Ft1(τn)Ft2(τ1) Ft2(τ2) Ft2(τ3) . . . Ft2(τn)...

......

......

FtN (τ1) FtN (τ2) FtN (τ3) . . . FtN (τn)






The daily returns are then computed

Rti (τk) =Fti+1(τk)− Fti (τk)

Fti (τk)

The resulting time series used to estimate the covariancematrix Σ

µk =1

N

N∑i=1

Rti (τk)

Σjk =1

N

N∑i=1

(Rti (τj)− µj) (Rti (τk)− µk)






Given the estimated covariance matrix, obtain eigenvectors(v1, v2, . . . , vn) and eigenvalues (λ1, λ2, . . . , λn):

Σ = V ΛV T

The eigenvectors corresponding to the first n-largesteigenvalues are called the principal components

For crude oil and heating oil, first three components accountfor 99% of variability:

PC-1 accounts for parallel shiftsPC-2 accounts for tiltingPC-3 accounts for bending

For electricity more than 10 PCs are required to account for99% of variability






Crude Oil first 3 Principal Components

0 1 2 3 4 5 6 7−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Term

PC

Wei

gh

t

PC 1 −− λ =92.6934PC 2 −− λ =6.4458PC 3 −− λ =0.49371






Heating Oil first 3 Principal Components

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Term

PC

Wei

gh

t

PC 1 −− λ =96.1985PC 2 −− λ =2.9411PC 3 −− λ =0.46599






Simulation of crude oil forward curves using 3 principal components






Simulation of crude oil forward curves using 3 principal components

0 1 2 3 4 5 6 760

65

70

75

80

85

90

95

100

105

110

Term (years)

Fo

rwar

d P

rice





Functional PCA

Functional Principal Component Analysis(FPCA)developed by Ramsay & Silverman (book in 2005) views thedata as sequence of random functions

Allows domain specific knowledge to augment PCs

Produces smooth PCs

Allows interpolation and extrapolation between observationpoints

Easily handles non-equal spaced and non-equal number ofdata per curve





Functional PCA

Jaimungal & Ng (2007) introduced a consistent FPCAapproach appropriate for commodity time-series:

Project data onto a functional basisFit a Vector Autoregressive (VAR) process to time-seriescoefficientsDetrend the coefficientsRemove predictable part of VAR process to extract truestochastic degrees of freedomProject covariance matrix onto a distortion metric associatedwith basis functionsSolve a modified eigen-problem for PCs of basis coefficientsWeight basis functions according to eigen-vectors





Consistent Functional PCA

Project data points onto basis functions φ1(τ), . . . , φK (τ)for each trading date t1, . . . , tM

Ftm(tm + τi ) =K∑

k=1

βm,k φk(τi )

βm,i = arg minβm,i

Nm∑i=1

∥∥∥∥∥F ∗tm(tm + τi )−

K∑k=1

βm,k φk(τi )

∥∥∥∥∥2

to produce a time-series of fitting coefficient estimates βm,i






10/11/02 02/23/042

4

6

β1(t)

10/11/02 02/23/04−5

0

5

β2(t)

10/11/02 02/23/04−2

0

2

β3(t)

10/11/02 02/23/04−1

0

1

β4(t)

10/11/02 02/23/04−0.1

0

0.1

β5(t)

(a) βm,i

0 2 4 618

19

20

21Jan 4, 2002

τ

Pric

es [$

]

0 2 4 625

30

35Jan 5, 2004

τ

Pric

es [$

]

0 2 4 652

54

56

58Jun 6, 2005

τ

Pric

es [$

]

0 2 4 660

65

70Dec 14, 2006

τ

Pric

es [$

]

07/03/02 01/19/03 08/07/03 02/23/04 09/10/04 03/29/05 10/15/05 05/03/06 11/19/060

0.5

1

1.5

Rel

. RM

S E

rr [%

]

(b) Data Fit






Estimate vector auto-regressive (VAR) on time-series ofprojection coefficients

βm = m + d t + Aβm−1 + εm

m is a constant mean vectord is a linear trend vector (any detrending is allowed)A is a K × K cross-interaction matrixεm are iid N (0,Ω).

Extract “true” stochastic degrees of freedom, i.e. theresiduals

εm = βm −(

m + d t + Aβm−1

)Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models





Let E = (ε1ε2 . . . εN)T denote the matrix form of theresiduals

Define a variance-covariance function

v(τ1, τ2) ,1

N(Eφ(τ1))T Eφ(τ2)

The eigen-function problem is now

< v , ξ > (τ) = λ ξ(τ)

where the inner product < f , g > (τ) ,∫ τmax

τminf (τ, s) g(s) ds






To solve the eigen-function problem, expand ξ onto the basisfunctions

ξ(τ) = zφ(τ)

Then the eigen-problem becomes

1

N(Eφ(τ))T EWz = λφT (τ)z

here Wij ,< φi , φj >

Take inner product with φ to find that z satisfy theeigen-problem

1

N

(WET E

)(Wz) = λ (Wz)






Comparison of extracted principal components Crude Oil data

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

1.5

2

Time to Maturity [Years]

Top 3 EigenFunctions

PC1 : Score 92.5%PC2 : Score 6.27%PC3 : Score 0.76%

(c) Consistent FPCA

0 1 2 3 4 5 6 7−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Term

PC

Wei

gh

t

PC 1 −− λ =92.6934PC 2 −− λ =6.4458PC 3 −− λ =0.49371

(d) Standard PCA






Principal component perturbations on a given curve

2 4 655

60

65

70Std. Dev.

τ2 4 6

55

60

65

70PC1

τ

2 4 655

60

65

70PC2

τ2 4 6

55

60

65

70PC3

τ






Simulation forward curves using 3 principal components







0 1 2 3 4 5 6 75

10

15

20

25

30

35

40

45

Term (years)

Fo

rwar

d P

rice







0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−9

−8

−7

−6

−5

−4

−3

−2

−1

0

1

γ1

γ2

γ3




Initial FormulationsString / Market Models


Heath-Jarrow-Morton (HJM) inspired forward price models

Ft(T ) = const.× exp

∫ T

tyt(s) ds

dyt(s) = µt(s) dt + σt(s) dWt

This is an infinite system of SDEs (one for every maturity).

The processes yt(s) are called forward cost of carry– analogs of instantaneous forward rates of interest

To avoid arbitrage, the drift and volatility must satisfy theHJM drift restrictions

µt(T ) = −σt(T )

∫ T

tσt(s) ds






Forward prices then evolve as

dFt(T )

Ft(T )= σF

t (T ) dWt , σFt (T ) =

∫ T

tσt(s) ds

Entire forward price curve is matched exactly

With deterministic volatilities, the forward prices are GBMs

Ft(T ) = Ft0(T ) exp

−1

2

∫ t

t0

(σFu (T ))2 du +

∫ t

t0

σFu (T ) dWt

Can match implied volatility term structure with constantvolatilities

Vol smiles will require state dependent vol or stochastic vol






Spot price models can be recast into forward price modelsfairly easily

The Schwarz (1997) stochastic convenience modelcorresponds to setting

σF (1)t (T ) = σ1 − ρσ2

1− e−κ(T−t)

κ

σF (2)t (T ) = −σ2

√1− ρ2

1− e−κ(T−t)

κ






Volatility components in the Schwartz model

0 1 2 3 4 5 6 7

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Term

Vol

Fun

ctio

n

σ1 ; ρ = 0

σ2 ; ρ = 0

σ1 ; ρ = −0.3

σ2 ; ρ = −0.3

σ1 ; ρ = +0.3

σ2 ; ρ = +0.3

Notice that both components are affected by correlation






Spot price models can be recast into forward price modelsfairly easily

The HJ (2007) two-factor spot model corresponds to setting

σF (1)t (T ) = ηγ

(e−βτ − e−ατ

)σ

F (2)t (T ) =

[σ2e−2βτ + η2γ2

(e−βτ − e−ατ

)2

+ρησγ(e−βτ − e−ατ

)e−βτ

]1/2γ =

β

α− β






Volatility components in the HJ model

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Term

Vol

Fun

ctio

n

σ1

σ2 ; ρ = 0

σ2 ; ρ = −0.5

σ2 ; ρ = 0.5

Notice that only the tilting component is affected by correlation





String / Market Models

A continuum of maturity dates do not exist

Model instead a discrete set of forward prices directly – notthrough forward cost of carry

dFt(Ti )

Ft(Ti )= σF

t (Ti ) dWt

Exactly fits market forward prices

Exactly fits a given term structure of at-the-money impliedvolatilities

Easy to compute call / put options on forward contracts






Another way to account for the discrete nature of availablematurity dates...

Between pairs of maturity dates (Ti ,Ti+1), define a discreteforward cost of carry

y(i)t ,

1

Ti − Ti−1

(Ft(Ti )

Ft(Ti−1)− 1

)Then forward prices can be recovered as

Ft(Tn) = St

n∏k=1

(1 + (Tk − Tk−1)y(k)t )

where T0 = t and recall that Ft(t) = St .






Each y(i)t is martingale under a measure induced by the

Radon-Nikodym derivative process(dQ(i)

dQ

)t

, η(i)t =

Ft(Ti−1)

F0(Ti−1)

Then assuming a diffusive model, can write

dy(i)t

y(i)t

= σ(i)t dW

(i)t

where W(i)t are Q(i)-Wiener processes






Further assuming σ(i)t are deterministic (or even just constant)

provides Magrabe like formula for nearby calendar spreadoptions

Vt = EQt [(FT (Ti )− FT (Ti−1))+]

=EQ(i)

t

[(FT (Ti )− FT (Ti−1))+

(dQ

dQ(i)

)T

]EQ(i)

t

[(dQ

dQ(i)

)T

]= Ft(Ti−1)EQ(i)

t

[(Ft(Ti )

Ft(Ti−1)− 1

)+

]


Documents

IMPA Commodities Course : Forward Price · PDF fileBlack’s Forward Price Models Multifactor Forward Price Models HJM Forward Price Models IMPA Commodities Course : Forward Price