Multi-Factor Levy Processes and Regime
Switching for Commodities:
University of Calgary
Lunch at the Lab
Sebastian Jaimungal
sebastian.jaimungal@utoronto.ca
Department of Statistics and
Mathematical Finance Program,
University of Toronto, Toronto, Canada
http://www.utstat.utoronto.ca/sjaimung
May 19, 2009
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Introduction
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Introduction
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Commodities are NOT like equities, interest rates, or
currencies!
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Commodities are real assets: produced, consumed,
transported and stored
Customers of commodity derivatives are typically
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Industrial producers / consumers
Governments
Customers tend to be very risk averse due to
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Required consumption
Legal risks
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Crude Oil Forward Price Data
Forward price curves from Oct-2006 to Oct-2008
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Crude Oil Forward Price Data
Daily deviations in Forward price curves Oct-2006, 2007, and 2008
Oct−2006
Forward Price
70
65
60
55
0
1
2
3
4
5
6
4
5
6
4
5
6
Term (years)
Oct−2007
Forward Price
85
80
75
70
0
1
2
3
Term (years)
Oct−2008
Forward Price
105
100
95
90
85
0
1
2
3
Term (years)
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Crude Oil Forward Price Data
Long end and short end of the Forward price curves from Oct-1998
to Oct-2008
150
Forward Price
Short End
Middle
Long End
100
50
0
3.6
3.65
3.7
3.75
3.8
3.85
3.9
3.95
Time
4
4
x 10
Notice periods of Backwardation and Contango.
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Crude Oil Forward Price Data
Term Structure of volatilities
0.34
0.32
Volatility
0.3
0.28
0.26
0.24
0.22
0.2
0.18
0
1
2
3
4
5
6
7
Term
Exponentially decaying volatilities
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Crude Oil Forward Price Data
Correlation
Correlation of constant maturity forward price returns
0.95
0.9
0.85
0.8
0.75
0.7
0.65
8
6
8
6
4
4
2
Term
2
0
0
Term
As expected, terms further apart less correlated
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Electricity Data
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Electricity data can have huge spikes – but mean revert
quickly
Figure: Nord Pool Data
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Electricity Data
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Electricity data can have huge spikes – but mean revert
quickly
Figure: PJM Data
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Electricity Data
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Electricity data can have huge spikes – but mean revert
quickly
Figure: EW Data
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Diffusion Models
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Schwartz’s One-Factor Spot Model
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Schwartz(1997) introduced a mean-reverting spot model:
dSt = κ(θ − ln St ) St dt + σ St dWt
Wt is a Wiener process under the real-world measure P. Wt
is a Wiener process under the real-world measure P.
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κ controls speed of mean-reversion
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θ controls target level
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σ controls volatility of paths
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Schwartz’s One-Factor Spot Model
$65 Spott Price
$60 $55 k=1
k = 1
$50 k = 10
k = 100
$45 $40 0
0.5
1
1.5
2
2.5
3
Time (years)
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Schwartz’s One-Factor Spot Model
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Choosing market price of risk as
λs = λ0 + λ1 ln St
maintains mean-reverting model class with new parameters
dSt = κ(θ − ln St ) St dt + σ St dW t
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Forward prices are then easily obtained
Ft (T ) , EQ
t [ST ]
0
0
= exp θ + (ln(St ) − θ )e
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−κ (T −t)
σ2 +
1 − e −2κ(T −t)
4κ
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Schwartz’s One-Factor Spot Model
Sample path of forward price curves:
65
Forward Price
60
55
50
45
40
35
5
4
6
3
4
2
7
5
3
1
Time (years)
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0
1
0
2
Term (years)
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Two-Factor Spot Models: Pilipovic
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Pilipovic(1997) introduced the following model to correct for
this
(1)
dSt = β(θt − St ) dt + σS St dWt
(2)
dθt = αθt dt + σθ θt dWt
(1)
with Wt
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(2)
and Wt
uncorrelated Wiener processes
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Long-run mean θt blows up
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No invariant distribution of commodity spot prices
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Does not lead to closed form option prices
Xu(2004) generalized this model by incorporating seasonality,
making σ time dependent and θt an OU process
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Two-Factor Spot Models: BGL & HJ
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Barlow, Gusev, and Lai(2004) and Hikspoors & J.(2007)
introduced a more tractable generalization as follows
St
= exp{gt + Xt }
(1)
dXt = α (Yt − Xt ) dt + σ dWt
(2)
dYt = β (φ − Yt ) dt + η dWt
(1)
(2)
Wt and Wt are correlated Wiener processes, gt
incorporates seasonality, and σ can easily be made
deterministic
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Two-Factor Spot Models: BGL & HJ
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Sample paths in the two factor model:
$100 Spot Price
$90 $80 alpha = 0.25
alpha 0.25
$70
$70 alpha = 0.5
$60 alpha = 1
Long Run
$50 $40 0
1
2
3
4
5
Term
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Two-Factor Spot Models: BGL & HJ
Sample path of forward price curves in the two-factor BGL & HJ
model
70
Forward Price
65
60
55
50
45
5
4
7
6
3
5
4
2
3
1
Time (years)
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2
0
1
0
Term (years)
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Jump Models
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One-Factor Jump-Diffusion Model
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Clewlow & Strickland (2001) and Cartea & Figueroa
(2005) simple extension of Gaussian OU process:
St
= exp{Xt }
dXt = κ(θ − Xt− ) dt + σ dWt + dJt
where Jt is a compound Poisson process
Jt =
Nt
X
ji
n=1
with Nt a Poisson process with activity rate λ and {j1 , j2 , . . . }
i.i.d. random variables.
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One-Factor Jump-Diffusion Model
120
κ = 5 ; σ = 70%
κ = 20 ; σ = 95%
κ = 100 ; σ = 300%
100
Spot Price
80
60
40
20
0
0
50
100
150
200
250
300
350
400
450
500
Time
Notice that as reversion rate increases, need to increase volatility
to compensate – otherwise diffusion will be washed out
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Two-factor Model
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Hikspoors & J. (2007) propose instead
St
= exp{Xt + Yt }
dXt = κ(θ − Xt ) dt + σ dWt
dYt = −α Yt− dt + dJt
where Jt is a pure jump process – such as compound Poisson
Jt =
Nt
X
ji
n=1
with Nt a Poisson process with activity rate λ and {j1 , j2 , . . . }
i.i.d. random variables.
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Two-factor Model
160
κ=1
κ = 10
κ = 100
140
Spot Price
120
100
80
60
40
20
0
0
50
100
150
200
250
300
350
400
450
500
Time
Notice that diffusion and jumps are uncoupled
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Two-factor Model
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Can incorporate frequent small jumps via Lévy processes
St
= exp{Xt + Yt }
dXt = κ(θ − Xt− ) dt + σ dWt + dQt
dYt = −α Yt− dt + dJt
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Qt – small, frequent, slowly mean-reverting jumps
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Carr, Madan, Chang (1998): Variance Gamma (VG) model
ν(dy ) =
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Carr, Geman, Madan, Yor (2002): CGMY model
ν(dy ) =
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1 α y −β|y |
e
dy
ν|y |
C −G |y |
−M y
dy
e
1
+
e
1
y
<0
y
>0
|y |1+Y
Jt – large, infrequent, fast mean-reverting jumps
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Multi-Factor Cointegration Model
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A Multi-Factor Cointegration Model
J. & Surkov (2008) introduce the affine multi-factor framework
for commodities [à la Duffie, Pan and Singleton (2000) ]
dYt = −κYt− dt + dJt ,
Xt = θ + BYt ,
St (i) = exp{Xt (i) }
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n log spot-prices Xt are modeled as a linear transformation of
a set of d−fundamental market factors Yt
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θ a d-dimensional vector of long-run means
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κ a d × d matrix with positive eigenvalues representing the
mixing of the market factors
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B a d × n matrix representing the linear transformation of the
market factors into the observed log-prices
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Jt a d-dimensional Lévy process with Lévy triple (γ, C, ν)
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A Multi-Factor Model
JS Model contains several known models:
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One factor mean-reverting model with jumps (Clewlow and
Strickland 2000):
θ = θ, κ = κ, C = σ 2 , B = 1, and ν(dZ) = λF (dz)
dXt = κ(θ − Xt ) dt + σ dWt + dJt
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A Multi-Factor Model
JS Model contains several known models:
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Mean-reverting jump-diffusion model (Hikspoors and
Jaimungal 2007) with different decay rates for the jumps
and diffusion.
2 θ
α 0
σ 0
θ=
κ=
C=
B= 1 1
0
0 β
0 0
and ν(dZ1 × dZ2 ) = λ δZ1 dF (Z2 ).
dYt1 = α(θ − Yt1 ) dt + σ dWt
dYt2 = −βYt2 dt + dJt
Xt = Yt1 + Yt2
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A Multi-Factor Model
JS Model contains several known models:
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Two factor mean-reverting model (Barlow, Gusev, and Lai
2004) with log-prices mean-revert to a stochastic long-run
mean, which itself mean-reverts to a fixed level:
2
θ
α −α
σ
ρση
θ=
κ=
C=
B = 1 0 ν(dZ) = 0
2
θ
0 β
ρση η
dXt = α(Yt − Xt )dt + σ dWtX
dYt = β(θ − Yt )dt + η dWtY
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A Multi-Factor Model
JS model also contains many new models:
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Jump-diffusion model with correlated diffusions, idiosyncratic
and codependent jumps (via a copula).
2
θ
α γ
σ
ρση
1 0
θ=
κ=
C=
B=
φ
δ β
ρση η 2
0 1
and ν(dZ1 × dZ2 ) = dC (F1 (Z1 ), F2 (Z2 )) with C (u, v ) a
copula and Fi (z) two marginal distribution functions.
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A Multi-Factor Model
Allows for independent & codependent jumps...
80
Commodity 1
Commodity 2
75
70
65
Price
60
55
50
45
40
35
30
0
1
2
3
4
5
Time
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A Multi-Factor Model
Allows for cointegration...
90
Raw
Refined 1
Refined 2
80
70
Price
60
50
40
30
20
10
0
0.5
1
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1.5
Time
2
2.5
3
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A Multi-Factor Model
Allows for cointegration... with jumps too...
70
Raw
Refined 1
Refined 2
60
Price
50
40
30
20
10
0
0.5
1
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1.5
Time
2
2.5
3
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A Multi-Factor Model: Induced Forward Prices
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Forward prices are martingales under the risk-neutral measure
(
(∂t + L)F (t, Y) = 0 ,
F (T , Y) = exp(θ + BY) ,
with
Lf = (κ(θ − Y))0 ∂Y f + 21 ∂Y0 C∂Y f
Z
+
f (Y + Z) − f (Y) − 1{|Z|<1} Z0 ∂Y f ν(dZ) .
Rd /{0}
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A Multi-Factor Model: Induced Forward Prices
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Using the Affine ansatz
Fit = exp αit (T ) + β it (T )Yt
one finds
0
αit (T ) = θi + Ψ(−i e −κ (T −t) bi , T − t) ,
0
β it (T ) = e −κ (T −t) bi ,
where, (bi )j = Bij .
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Mean-Reverting Fourier Space Time-Stepping
(mrFST)
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Fourier Space Time-Stepping
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Option payoff is given by ϕ(S)
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Recall, mean-reverting jump-diffusion process for spot price
St = e Xt ,
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dXt = κ(θ−Xt )dt +σdWt +dJt ,
X0 = ln S0
Discounted option value v satisfies the PIDE
∂t v + Lv = 0
v (T , x) = ϕ(e x )
where L is the infinitesimal generator:
1
Lf (x) = κ(θ − x)∂x f (x) + σ 2 ∂xx f (x)
2
Z
+ (f (x + y ) − f (x)) ν(dy )
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Fourier Space Time-Stepping
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The conditional characteristic function under Q is
ht,x (ω, T ) , EQ [e iωXT |Ft ]
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Assume the affine form:
T (ω)+ΦT (ω)x
t
ht,x (ω, T ) = e Ψt
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T
ΨT
t (ω) and Φt (ω) then satisfy a system of Riccati ODEs:
(
R ΦT y
1 2
T 2
T
t
− 1)ν(dy ) = 0
Ψ̇T
t + κθΦt + 2 σ (Φt ) + (e
T
T
Φ̇t − κΦt
= 0,
T
subject to ΨT
T (ω) = 0 and ΦT (ω) = iω
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Fourier Space Time-Stepping
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Can solve for ΦT
t (ω) analytically:
−κ(T −t)
ΦT
t (ω) = iωe
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ΨT
t (ω) can then be solved:
ω2σ2
−κ(T−t)
(1−e −2κ(T−t) )
ΨT
(ω)
=
iωθ(1−e
)
−
t
4κ
Z T
+
ψ̃(ωe −k(T−s) )ds
t
where ψ̃ is the characteristic function of the jump distribution
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Last term can be computed analytically for double-exponential
distribution (Kou model) and numerically using quadratures
for others
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Fourier Space Time-Stepping
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Expand the payoff (assume paid at t + ∆t) in a Fourier basis:
Z ∞
1
e iωx F[vt+∆t ](ω) dω
ϕ(x) = vt+∆t (x) =
2π −∞
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Assuming no decisions (such as barrier breach or optimal
exercise) are made during the interval (t, t + ∆t]:
Z ∞
1
vt (x) =
ht,x (ω, t + ∆t) F[vt+∆t ](ω) dω
2π −∞
The above satisfies the PIDE and the boundary condition
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Apply Fourier transform to vt (x):
Z ∞Z ∞
t+∆t
t+∆t
dx
0
0
F[vt ](ω) =
e Ψt (ω )+Φt (ω )x F[vt+∆t ](ω 0 )dω 0 e −iωx
2π
−∞ −∞
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Fourier Space Time-Stepping
The price Vt (Yt ) of a European option written on the vector of
(n)
(1)
(1)
price processes {St = e Xt , . . . , Stn = e Xt } with payoff function
ϕ(ST ) = ϕ(θ
e + B YT ) = φ(YT ) is
h
i
0
Vt (Yt ) = F −1 F[φ(YT )](e κ (T −t) ω) e Ψ(ω,T −t)+(T −t)Tr κ (Yt ) ,
where,
Z
s
0
ψ(e κ u ω) du ,
and
0
Z
0
e iω y − 1 − i 1{|y|<1} ω 0 y ν(dy) ,
ψ(ω) = − 12 ω 0 Cω +
Ψ(ω, s) =
Rd /{0}
F[.] and F −1 [.] denote the Fourier and inverse Fourier transforms
respectively, and 0 denotes transpose.
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Fourier Space Time-Stepping
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Mean-Reverting FST Method
h
i
κ0 ∆t )
0
v m−1 (Y) = FFT−1 e Ψ(ωe
· FFT[v m (Y0 e −κ ∆t )]
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Without mean reversion, mrFST reduces to the standard FST
method of Jackson, Jaimungal and Surkov (2008)
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For Bermudan style claims:
h
i
o
n
em ](ω) e Ψ(ω,∆tm ) (Y) , VM (Y) .
Vm−1 (Y) = max e −r ∆tm F−1 F[V
A penalty method can also be used.
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For Barrier style claims:
h
i
em ](ω) e Ψ(ω,∆tm ) (Y) · I{θ+BY<B}
Vm−1 (Y) =e −r ∆tm F−1 F[V
+ R · I{θ+BY≥B} .
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Numerical Examples
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Fourier Space Time-Stepping
Comparison of Tree, LSMC, and mrFST for mean-reverting
diffusion: American Put Option
50
LSMC
mrFST
Tree
49.5
49
Asset Price
48.5
48
47.5
47
46.5
46
45.5
45
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
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Fourier Space Time-Stepping
Comparison of LSMC and mrFST for mean-reverting diffusion with
Kou jumps: American Put Option
50
LSMC
mrFST
49
Asset Price
48
47
46
45
44
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
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Fourier Space Time-Stepping
Sensitivity of boundary on asymmetry using mrFST mean-reverting
diffusion with Kou jumps: American Put Option
50
p=0.5
p=0.4
p=0.3
p=0.2
49
Asset Price
48
47
46
45
44
43
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
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Swing Options
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Common in electricity and natural gas markets
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Provides constrained flexibility with respect to volume and
timing of energy delivery
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Two components: a pure forward agreement and a swing
option
An Example
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I
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The holder agrees to buy 100MWh at a cost of $45/MWh over
a period of 1 month.
At the start of each hour, the holder has the right to increase
power consumption to 110MW for that hour (swing up) or
decrease to 90MW (swing down) at the same cost.
The total number of swings is limited to 50.
The swing component is the right to change consumption at
holder’s choosing.
For overview of swing option valuation see Ware (2007)
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Swing Options
At each swing opportunity, a choice to exercise q swing “options”
for immediate cashflow Υ must be made:
Dynamic Programming Equation
n
o
vtm (x, Q) = max Υtm (x, q) + e −r ∆t E[vtm+1 (x, Q + q)]
q
where the expectation is readily computed using the mrFST
method
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The available choices are to do nothing q = 0, swing up q > 0
or swing down q < 0
The amount
of swings mayPbe bounded |Qtm | ≤ Q where
P
Qtm = m
q
|qtj |
j=1 tj or Qtm =
The cashflow function Υt (x, q) may include a penalty term to
enforce additional limits on Q or may be as simple as
Υt (x, q) = q(e x − K )
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Swing Options
230.0
Swing Value (V)
210.0
=0.05
=0.25
=0.5
=1.25
190.0
170.0
150.0
130.0
110.0
40.0
55.0
70.0
85.0
100.0
115.0
130.0
Spot Price (S)
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Option: Swing S = 100, K = 100, T = 2, −3 ≤ Q ≤ 5
Model: Kou jump-diffusion with mean reversion σ =
75%, λ = 0.5, p = 0.45, η+ = 0.25, η− = 0.2, θ = 80, r = 0.05
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Real Option – Regime Switching Volume
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Consider the option to invest with limited lifetime
Volume is not known, but you have estimates which are
continually updated according to a Markov chain Zt
a exp(‐b t)
V1
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V2
V3
V4
V5
Project Value Pt is then modeled as
Pt = VZt exp{Xt }
dXt = κ(θ − Xt ) dt + σdWt
I
Investment amount is I .
Option to invest is Bermudan style:
ft = sup E e −r τ VZτ e Xτ − I
τ ∈T
(c) S.Jaimungal (2009)
Ft
+
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Real Option – Regime Switching Volume
I
Between exercise dates, can show that the price f (i) (t, x)
contingent on Zt = i satisfies system of coupled PDEs:
(
P
(j) = 0
(∂t + L − r )f (i) + ht m
j=1 Aij f
f (i) (tn−1 , x)
= max(f (i) (tn+ , x) , ϕ(i) (x))
where
ϕ(i) (x) = (Vi e x − I )+
I
An mrFST regime switching algorithm can be worked out:
i
h
n
f · (tn−1 , x) = max e −r ∆tn F−1 Hn F[e
fn ](ω) e Ψ(ω,∆tm ) (x)
, (Vi e x − I )+
where
Hn = Ue Λ
(c) S.Jaimungal (2009)
R tn
tn −1
hs ds
U −1
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Real Option – Regime Switching Volume
Comparison of constant versus Markov switching Trigger curves
(Value / Investment)
vol=0.6
vol=0.8
vol=1
vol=1.2
vol=1.4
2.6
Value / Investment
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0
2
4
6
8
10
Time
(c) S.Jaimungal (2009)
sebastian.jaimungal@utoronto.ca
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Real Option – Regime Switching Volume
Comparison of constant versus Markov switching Trigger curves
(Value / Investment) – SLOWER “learning”
vol=0.6
vol=0.8
vol=1
vol=1.2
vol=1.4
2.6
Value / Investment
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0
2
4
6
8
10
Time
(c) S.Jaimungal (2009)
sebastian.jaimungal@utoronto.ca
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THE END!!
THANK YOU FOR YOUR ATTENTION!!
ANY QUESTIONS?
sebastian.jaimungal@utoronto.ca
http://www.utstat.utoronto.ca/sjaimung
(c) S.Jaimungal (2009)
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