Upload
vophuc
View
235
Download
2
Embed Size (px)
Citation preview
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
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM 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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price 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)
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
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)
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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)
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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)
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
Simulation of crude oil forward curves using 3 principal components
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Multifactor Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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)
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional 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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
Simulation forward curves using 3 principal components
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Basic ModelPrincipal Component AnalysisFunctional Principal Component Analysis
Consistent Functional PCA
Simulation forward curves using 3 principal components
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price 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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
HJM Forward Price Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
String / Market Models
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 .
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
String / Market Models
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
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models
Black’s Forward Price ModelsMultifactor Forward Price Models
HJM Forward Price Models
Initial FormulationsString / Market Models
String / Market Models
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
)+
]
Sebastian Jaimungal [email protected] IMPA Commodities Course : Forward Price Models