Model FST Greeks Hedging
Pricing and Hedging of Commodity Derivatives
using the Fast Fourier Transform
Vladimir Surkov
vladimir.surkov@utoronto.ca
Department of Statistical and Actuarial Sciences, University of Western Ontario
The Fields Institute, University of Toronto
December 9, 2009
1 / 33
Model FST Greeks Hedging
1
Generalized Model for Commodity Spot Prices
2
Fourier Space Time-stepping framework
3
Computing Option Greeks
4
Dynamic and Static Hedging
2 / 33
Model FST Greeks Hedging
1
Generalized Model for Commodity Spot Prices
2
Fourier Space Time-stepping framework
3
Computing Option Greeks
4
Dynamic and Static Hedging
3 / 33
Model FST Greeks Hedging
WTI Crude Oil and Henry Hub Gas
4 / 33
Model FST Greeks Hedging
Great Britain System Sell Prices
5 / 33
Model FST Greeks Hedging
Generalized Model for Commodity Spot Prices
Commodity prices
Exhibit high volatilities and spikes and prices
Tend to revert to long run equilibrium prices
6 / 33
Model FST Greeks Hedging
Generalized Model for Commodity Spot Prices
Commodity prices
Exhibit high volatilities and spikes and prices
Tend to revert to long run equilibrium prices
The Model
dX(t) = (Θ(t) − κX(t)) dt + dW(t) + dJ(t)
S(t) = S(0) exp{X(t)}
6 / 33
Model FST Greeks Hedging
Generalized Model for Commodity Spot Prices
Commodity prices
Exhibit high volatilities and spikes and prices
Tend to revert to long run equilibrium prices
The Model
dX(t) = (Θ(t) − κX(t)) dt + dW(t) + dJ(t)
S(t) = S(0) exp{X(t)}
Multi-factor, mean-reverting model with jumps
Generalizes a number of well-known models, such as Gibson and Schwartz
(1990), Clewlow and Strickland (2000), Hikspoors and Jaimungal (2007)
Mean-reversion rate is time dependent - allows to incorporate seasonality
6 / 33
Model FST Greeks Hedging
One-Factor Mean-Reverting Process with Jumps
7 / 33
Model FST Greeks Hedging
Two-Factor MR Process with Different Decay Rates
8 / 33
Model FST Greeks Hedging
One-Factor MR Processes with Co-dependent Jumps
9 / 33
Model FST Greeks Hedging
Numerical Methods for Option Pricing
Monte Carlo methods
Tree methods
Finite difference methods
Alternating Direction Implicit-FFT - Andersen and Andreasen (2000)
Implicit-Explicit (IMEX) - Cont and Tankov (2004)
IMEX Runge-Kutta - Briani, Natalini, and Russo (2004)
Fixed Point Iteration - d’Halluin, Forsyth, and Vetzal (2005)
Quadrature methods
Reiner (2001)
QUAD - Andricopoulos, Widdicks, Duck, and Newton (2003)
Q-FFT - O‘Sullivan (2005)
Transform-based methods
Carr and Madan (1999)
Raible (2000)
Lewis (2001)
Lord, Fang, Bervoets, and Oosterlee (2008)
10 / 33
Model FST Greeks Hedging
1
Generalized Model for Commodity Spot Prices
2
Fourier Space Time-stepping framework
3
Computing Option Greeks
4
Dynamic and Static Hedging
11 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Framework Overview
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
12 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Framework Overview
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
A framework for numerical pricing of financial derivatives
Fast and precise pricing of a wide range of European and path-dependent,
single- and multi-asset, vanilla and exotic derivatives
Efficient handling of path-independent and discretely monitored derivatives
Generic handling of different spot-price models and option payoffs
12 / 33
Model FST Greeks Hedging
Pricing Framework in Real Space
Option price at time t is the discounted expected future payoff
V (t, S(t)) = e −r (T −t) E ϕ(S(T ))
13 / 33
Model FST Greeks Hedging
Pricing Framework in Real Space
Option price at time t is the discounted expected future payoff
V (t, S(t)) = e −r (T −t) E ϕ(S(T ))
The discount-adjusted and log-transformed price process
v (t, X(t)) , e r (T −t) V (t, S(0)e X(t) ) satisfies a PIDE
(∂t + L(t, x) − κx′ ∂x ) v (t, x) = 0 ,
v (T , x)
= ϕ(S(0) e x )
where L acts on twice-differentiable functions g (x) as follows:
Z
′
1 ′
(g (x+y)−g (x)) ν(dy)
L(t, x)g (x) = Θ(t) ∂x + 2 ∂x Σ∂x g (x) +
R
n
13 / 33
Model FST Greeks Hedging
Fourier Transform
A function in the space domain g (x) can be transformed to a function in the
frequency domain ĝ (ω), where ω is given in radians per second, and
vice-versa using the continuous Fourier transform
Z ∞
′
F [g ](ω) ,
g (x)e −i ω x dx
−∞
Z ∞
′
1
−1
F [ĝ ](x) ,
ĝ (ω)e i ω x dω
2π −∞
14 / 33
Model FST Greeks Hedging
Fourier Transform
A function in the space domain g (x) can be transformed to a function in the
frequency domain ĝ (ω), where ω is given in radians per second, and
vice-versa using the continuous Fourier transform
Z ∞
′
F [g ](ω) ,
g (x)e −i ω x dx
−∞
Z ∞
′
1
−1
F [ĝ ](x) ,
ĝ (ω)e i ω x dω
2π −∞
Continuous Fourier transform is a linear operator that maps spatial
derivatives ∂x into multiplications in the frequency domain
F [∂xn g ](ω) = iω F ∂xn−1 g (ω) = · · · = (iω)n F [g ](ω)
14 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space
Applying the Fourier transform to the pricing PDE we obtain a PDE in
frequency space
( ∂t + L̂(t, ω) + κ + κω∂ω v̂ (t, ω) = 0 ,
v̂ (T , ω)
= Φ̂(T , ω)
15 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space
Applying the Fourier transform to the pricing PDE we obtain a PDE in
frequency space
( ∂t + L̂(t, ω) + κ + κω∂ω v̂ (t, ω) = 0 ,
v̂ (T , ω)
= Φ̂(T , ω)
The Fourier transform of the operator L(t, x) can be computed analytically
Z ′
L̂(t, ω) = iωΘ(t) − 12 ω′ Σω +
e i ω z − 1 ν(dz)
15 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space
Applying the Fourier transform to the pricing PDE we obtain a PDE in
frequency space
( ∂t + L̂(t, ω) + κ + κω∂ω v̂ (t, ω) = 0 ,
v̂ (T , ω)
= Φ̂(T , ω)
The Fourier transform of the operator L(t, x) can be computed analytically
Z ′
L̂(t, ω) = iωΘ(t) − 12 ω′ Σω +
e i ω z − 1 ν(dz)
Introduce a new coordinate system via frequency scaling
′
ṽ (t, ω) = v̂ (t, e κ (t−t⋆ ) ω)
15 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space
Applying the Fourier transform to the pricing PDE we obtain a PDE in
frequency space
( ∂t + L̂(t, ω) + κ + κω∂ω v̂ (t, ω) = 0 ,
v̂ (T , ω)
= Φ̂(T , ω)
The Fourier transform of the operator L(t, x) can be computed analytically
Z ′
L̂(t, ω) = iωΘ(t) − 12 ω′ Σω +
e i ω z − 1 ν(dz)
Introduce a new coordinate system via frequency scaling
′
ṽ (t, ω) = v̂ (t, e κ (t−t⋆ ) ω)
The PDE reduces to an ODE in time parameterized by ω
∂t + L̃(t, ω) + κ ṽ (t, ω) = 0 ,
ṽ (T , ω)
= Φ̃(T , ω)
15 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space (cont.)
Given the value of ṽ (t, ω) at time t2 ≤ T , the constant coefficient ODE is
easily solved to find the value at time t1 < t2 :
ṽ (t1 , ω) = ṽ (t2 , ω) · e Ψ̃κ (t1 ,ω;t2 ) ,
where the frequency space propagator is
Z t2
Ψ̃κ (t1 , ω; t2 ) =
L̃(s, ω) ds + Tr κ (t2 − t1 )
t1
16 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space (cont.)
Given the value of ṽ (t, ω) at time t2 ≤ T , the constant coefficient ODE is
easily solved to find the value at time t1 < t2 :
ṽ (t1 , ω) = ṽ (t2 , ω) · e Ψ̃κ (t1 ,ω;t2 ) ,
where the frequency space propagator is
Z t2
Ψ̃κ (t1 , ω; t2 ) =
L̃(s, ω) ds + Tr κ (t2 − t1 )
t1
The solution in terms of original coordinates (with t⋆ = t1 ) is given by
′
v̂ (t1 , ω) = v̂ (t2 , e κ (t2 −t1 ) ω) · e Ψ̂κ (t1 ,ω;t2 )
and the frequency space propagator is
Z t2
′
Ψ̂κ (t1 , ω; t2 ) =
L̂(s, e κ (s−t1 ) ω) ds + Tr κ (t2 − t1 )
t1
16 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space (cont.)
The scaled option prices in frequency space can be obtained from the scaled
option prices in real space
′
F [g ](t, e κ (t2 −t1 ) ω) = F [ğ ](t, ω) · e −Tr κ (t2 −t1 ) ,
′
where ğ (t, x) , g (t, x e −κ (t2 −t1 ) )
17 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space (cont.)
The scaled option prices in frequency space can be obtained from the scaled
option prices in real space
′
F [g ](t, e κ (t2 −t1 ) ω) = F [ğ ](t, ω) · e −Tr κ (t2 −t1 ) ,
′
where ğ (t, x) , g (t, x e −κ (t2 −t1 ) )
The final solution becomes
h
i
v (t1 , x) = F −1 F [v̆ ](t2 , ω) · e Ψ̂(t1 ,ω;t2 ) (x)
17 / 33
Model FST Greeks Hedging
Pricing Framework in Fourier Space (cont.)
The scaled option prices in frequency space can be obtained from the scaled
option prices in real space
′
F [g ](t, e κ (t2 −t1 ) ω) = F [ğ ](t, ω) · e −Tr κ (t2 −t1 ) ,
′
where ğ (t, x) , g (t, x e −κ (t2 −t1 ) )
The final solution becomes
h
i
v (t1 , x) = F −1 F [v̆ ](t2 , ω) · e Ψ̂(t1 ,ω;t2 ) (x)
mrFST Method for Propagating Option Prices
h
i
vm−1 = FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm )
17 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Numerical Method
European options
h
i
v0 = FFT−1 FFT [v̆1 ] · e Ψ̂(t,ω;T )
18 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Numerical Method
European options
h
i
v0 = FFT−1 FFT [v̆1 ] · e Ψ̂(t,ω;T )
Bermudan/American options
i
h
⋆
vm−1
= FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm ) ,
⋆
vm−1 = max vm−1
, vM ,
⋆
where vm−1
represents the holding value of the option
18 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Numerical Method
European options
h
i
v0 = FFT−1 FFT [v̆1 ] · e Ψ̂(t,ω;T )
Bermudan/American options
i
h
⋆
vm−1
= FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm ) ,
⋆
vm−1 = max vm−1
, vM ,
⋆
where vm−1
represents the holding value of the option
Barrier options
h
i
vm−1 = FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm ) · 1{x<B} + R · 1{x≥B}
18 / 33
Model FST Greeks Hedging
Fourier Space Time-stepping Numerical Method
European options
h
i
v0 = FFT−1 FFT [v̆1 ] · e Ψ̂(t,ω;T )
Bermudan/American options
i
h
⋆
vm−1
= FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm ) ,
⋆
vm−1 = max vm−1
, vM ,
⋆
where vm−1
represents the holding value of the option
Barrier options
h
i
vm−1 = FFT−1 FFT [v̆m ] · e Ψ̂(tm−1 ,ω;tm ) · 1{x<B} + R · 1{x≥B}
Exotic options, such as swings, can also be handled
18 / 33
Model FST Greeks Hedging
Discrete Barrier Option Results
N
2048
4096
8192
16384
32768
M
252
252
252
252
252
Value
2.75818698
2.77495289
2.77164607
2.77315499
2.77395701
Change
Convergence
0.0167659
0.0033068
0.0015089
0.0008020
2.3420
1.1319
0.9118
Time (sec.)
0.048
0.099
0.210
0.523
0.974
Option: Down-and-out barrier put S = 100, K = 105, T = 1, B = 90, R = 3
with daily monitoring
Model: Merton jump-diffusion with mean reversion
σ = 0.2, λ = 1.0, µ
e = −0.1, σ
e = 0.25, θ = 90.0, κ = 0.75, r = 0.05
Monte Carlo: 2.77533300 – 95% CI width of 0.00323116 @ 114 sec.
19 / 33
Model FST Greeks Hedging
1
Generalized Model for Commodity Spot Prices
2
Fourier Space Time-stepping framework
3
Computing Option Greeks
4
Dynamic and Static Hedging
20 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
The discrete method for computing Deltas is then given by
∆k,m−1 = FFT−1 [iωk · v̂m−1 ]
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
The discrete method for computing Deltas is then given by
∆k,m−1 = FFT−1 [iωk · v̂m−1 ]
Higher order derivatives computed in similar manner
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
The discrete method for computing Deltas is then given by
∆k,m−1 = FFT−1 [iωk · v̂m−1 ]
Higher order derivatives computed in similar manner
Theta – ∂t v (t, x)
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
The discrete method for computing Deltas is then given by
∆k,m−1 = FFT−1 [iωk · v̂m−1 ]
Higher order derivatives computed in similar manner
Theta – ∂t v (t, x)
Obtained directly from the pricing ODE
∂t ṽ (t, ω) = − L̃(t, ω) + κ ṽ (t, ω)
21 / 33
Model FST Greeks Hedging
Computation of Greeks - State Variables
Delta – ∂Sk v (t, x) = ∂xk v (t, x)/Sk (t)
Differentiation in real space computed via scaling in Fourier space
∂xk v (t, x) = F −1 [iωk · v̂ (t, ω)](x) .
The discrete method for computing Deltas is then given by
∆k,m−1 = FFT−1 [iωk · v̂m−1 ]
Higher order derivatives computed in similar manner
Theta – ∂t v (t, x)
Obtained directly from the pricing ODE
∂t ṽ (t, ω) = − L̃(t, ω) + κ ṽ (t, ω)
The discrete method for computing Theta is then given by
h
i
Θm−1 = FFT−1 − L̂(t, ω) + κ · v̂m−1
21 / 33
Model FST Greeks Hedging
Computation of Greeks - Model Parameters
In Fourier space, the sensitivity satisfies an ODE with source term
n
o
∂⋆ ∂t + L̃κ ṽ (t, ω) = ∂t + L̃κ ∂⋆ ṽ (t, ω) + ∂⋆ L̃κ · ṽ (t, ω) = 0
22 / 33
Model FST Greeks Hedging
Computation of Greeks - Model Parameters
In Fourier space, the sensitivity satisfies an ODE with source term
n
o
∂⋆ ∂t + L̃κ ṽ (t, ω) = ∂t + L̃κ ∂⋆ ṽ (t, ω) + ∂⋆ L̃κ · ṽ (t, ω) = 0
The ODE can be solved explicitly
h
i
′
∂⋆ v (t, x) = F −1 ∂⋆ Ψ̂κ (t, e κ (T −t) ω; T ) · v̂ (t, ω) (x)
22 / 33
Model FST Greeks Hedging
Computation of Greeks - Model Parameters
In Fourier space, the sensitivity satisfies an ODE with source term
n
o
∂⋆ ∂t + L̃κ ṽ (t, ω) = ∂t + L̃κ ∂⋆ ṽ (t, ω) + ∂⋆ L̃κ · ṽ (t, ω) = 0
The ODE can be solved explicitly
h
i
′
∂⋆ v (t, x) = F −1 ∂⋆ Ψ̂κ (t, e κ (T −t) ω; T ) · v̂ (t, ω) (x)
The discrete method for computing the sensitivity is then given by
h
i
′
∇⋆, m−1 = FFT−1 ∂⋆ Ψ̂κ (tm−1 , e κ ∆tm ω; tm ) · v̂m−1
22 / 33
Model FST Greeks Hedging
Computation of Greeks - Model Parameters
In Fourier space, the sensitivity satisfies an ODE with source term
n
o
∂⋆ ∂t + L̃κ ṽ (t, ω) = ∂t + L̃κ ∂⋆ ṽ (t, ω) + ∂⋆ L̃κ · ṽ (t, ω) = 0
The ODE can be solved explicitly
h
i
′
∂⋆ v (t, x) = F −1 ∂⋆ Ψ̂κ (t, e κ (T −t) ω; T ) · v̂ (t, ω) (x)
The discrete method for computing the sensitivity is then given by
h
i
′
∇⋆, m−1 = FFT−1 ∂⋆ Ψ̂κ (tm−1 , e κ ∆tm ω; tm ) · v̂m−1
Higher order derivatives computed in similar manner
22 / 33
Model FST Greeks Hedging
Greeks Computation Errors
N=4096
N=8192
N=16384
N=32768
70
N=4096
N=8192
N=16384
N=32768
85
100
115
Stock Price (S)
Theta Error
N=4096
N=8192
N=16384
N=32768
70
85
100
115
Stock Price (S)
N=4096
N=8192
N=16384
N=32768
70
130
85
100
115
Stock Price (S)
Vega Error
130
−5
10
N=4096
N=8192
N=16384
N=32768
130
−5
10
−5
10
130
Absolute Error
100
115
Stock Price (S)
Gamma Error
Absolute Error
85
Absolute Error
−5
10
70
Absolute Error
Delta Error
70
Absolute Error
Absolute Error
Price Error
85
100
115
Stock Price (S)
Rho Error
130
−5
10
N=4096
N=8192
N=16384
N=32768
70
85
100
115
Stock Price (S)
130
23 / 33
Model FST Greeks Hedging
1
Generalized Model for Commodity Spot Prices
2
Fourier Space Time-stepping framework
3
Computing Option Greeks
4
Dynamic and Static Hedging
24 / 33
Model FST Greeks Hedging
Dynamic Hedging
Hedging portfolio for the option V consists of B units of cash, e units of the
~
underlying asset S and N hedging instruments ~I with weights φ
~ · ~I (t, S(t)) + eS(t) + B − V (t, S(t))
Π=φ
25 / 33
Model FST Greeks Hedging
Dynamic Hedging
Hedging portfolio for the option V consists of B units of cash, e units of the
~
underlying asset S and N hedging instruments ~I with weights φ
~ · ~I (t, S(t)) + eS(t) + B − V (t, S(t))
Π=φ
The portfolio’s value remains unchanged under small movements in stock
~ · ∂S~I (t, S(t)) + e − ∂S V (t, S(t)) = 0
∂S Π = φ
25 / 33
Model FST Greeks Hedging
Dynamic Hedging
Hedging portfolio for the option V consists of B units of cash, e units of the
~
underlying asset S and N hedging instruments ~I with weights φ
~ · ~I (t, S(t)) + eS(t) + B − V (t, S(t))
Π=φ
The portfolio’s value remains unchanged under small movements in stock
~ · ∂S~I (t, S(t)) + e − ∂S V (t, S(t)) = 0
∂S Π = φ
Can also hedge against small movements in interest-rates, volatility, etc.
~ · ∂⋆~I (t, S(t)) − ∂⋆ V (t, S(t)) = 0
∂⋆ Π = φ
25 / 33
Model FST Greeks Hedging
Dynamic Hedging
Hedging portfolio for the option V consists of B units of cash, e units of the
~
underlying asset S and N hedging instruments ~I with weights φ
~ · ~I (t, S(t)) + eS(t) + B − V (t, S(t))
Π=φ
The portfolio’s value remains unchanged under small movements in stock
~ · ∂S~I (t, S(t)) + e − ∂S V (t, S(t)) = 0
∂S Π = φ
Can also hedge against small movements in interest-rates, volatility, etc.
~ · ∂⋆~I (t, S(t)) − ∂⋆ V (t, S(t)) = 0
∂⋆ Π = φ
What about large movements?
25 / 33
Model FST Greeks Hedging
Static Hedging - Minimization of Portfolio Variance
Minimize portfolio price variance under expected asset price movement
h
i2
~n · ∆~In + en ∆Sn − ∆Vn + (1 − ξ)Υn .
arg min ξ Etn φ
~n
en ,φ
where Υn is the transaction cost to rebalance the portfolio:
Υn =
N h
i2 h
i2
X
~k,n − φ
~k,n−1 )~Ik,n + β(en − en−1 )Sn ,
α
~ k (φ
k=1
26 / 33
Model FST Greeks Hedging
Static Hedging - Minimization of Portfolio Variance
Minimize portfolio price variance under expected asset price movement
h
i2
~n · ∆~In + en ∆Sn − ∆Vn + (1 − ξ)Υn .
arg min ξ Etn φ
~n
en ,φ
where Υn is the transaction cost to rebalance the portfolio:
Υn =
N h
i2 h
i2
X
~k,n − φ
~k,n−1 )~Ik,n + β(en − en−1 )Sn ,
α
~ k (φ
k=1
Since the objective function is quadratic, the optimality requires
h
i
∂F
~ · ∆~I + e∆S − ∆V 2∆Ik + (1 − ξ) ∂φ Υn = 0
= ξ Etn φ
k,n
∂φk,n
h
i
∂F
~ · ∆~I + e∆S − ∆V 2∆S + (1 − ξ) ∂e Υn = 0
= ξ Etn φ
n
∂en
26 / 33
Model FST Greeks Hedging
Static Hedging - Minimization of Portfolio Variance
27 / 33
Model FST Greeks Hedging
Static Hedging - Minimize Price and Greeks Variance
Minimize portfolio price and Greeks variance under expected asset price
movement
h
i2
X
~n · ∆(D~In ) + en ∆(DSn ) − ∆(DVn ) + (1 − ξ)Υn
arg min ξ
wD Etn φ
~n
en ,φ
D
28 / 33
Model FST Greeks Hedging
Static Hedging - Minimize Price and Greeks Variance
Minimize portfolio price and Greeks variance under expected asset price
movement
h
i2
X
~n · ∆(D~In ) + en ∆(DSn ) − ∆(DVn ) + (1 − ξ)Υn
arg min ξ
wD Etn φ
~n
en ,φ
D
Since the objective function is quadratic, the optimality requires
h
X
i
∂F
~ · ∆(D~I ) + e∆(DS)−∆(DV ) 2∆(DIk )
=ξ
wD Etn φ
∂φk,n
D
∂F
=ξ
∂en
X
D
wD Etn
h
+(1−ξ) ∂φk,n Υn = 0
i
~ · ∆(D~I ) + e∆(DS)−∆(DV ) 2∆(DS)
φ
+(1−ξ) ∂en Υn = 0
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Model FST Greeks Hedging
Static Hedging - Minimize Price and Greeks Variance
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Model FST Greeks Hedging
Loss Distribution and VaR - Constant Volatility
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Model FST Greeks Hedging
Loss Distribution and VaR - Dynamic Volatility
31 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
32 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
Independent-increment, mean-reverting and interest-rate Lévy models are
handled generically
32 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
Independent-increment, mean-reverting and interest-rate Lévy models are
handled generically
Option values are obtained for a range of spot prices - readily price
path-dependent options
32 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
Independent-increment, mean-reverting and interest-rate Lévy models are
handled generically
Option values are obtained for a range of spot prices - readily price
path-dependent options
Two FFTs per time-step are required; no time-stepping for European options
or between monitoring dates of discretely monitored options
32 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
Independent-increment, mean-reverting and interest-rate Lévy models are
handled generically
Option values are obtained for a range of spot prices - readily price
path-dependent options
Two FFTs per time-step are required; no time-stepping for European options
or between monitoring dates of discretely monitored options
One extra FFT required to compute each Greek
32 / 33
Model FST Greeks Hedging
FST Framework Summary
The Approach
Consider the PIDE for the option price
Transform the PIDE into ODE in Fourier space
Solve the resulting ODE analytically
Utilize FFT to efficiently switch between real and Fourier spaces
Independent-increment, mean-reverting and interest-rate Lévy models are
handled generically
Option values are obtained for a range of spot prices - readily price
path-dependent options
Two FFTs per time-step are required; no time-stepping for European options
or between monitoring dates of discretely monitored options
One extra FFT required to compute each Greek
Second order convergence in space and second order convergence in time for
American options with penalty method
32 / 33
Model FST Greeks Hedging
Thank You!
Jackson, K. R., S. Jaimungal, and V. Surkov (2008).
Fourier space time-stepping for option pricing with Lévy models.
Journal of Computational Finance 12 (2), 1–28.
Jaimungal, S. and V. Surkov (2008).
A general Lévy-based framework for energy price modeling and derivative
valuation via FFT.
Working paper, available at http://ssrn.com/abstract=1302887.
Surkov, V. (2009).
Efficient static hedging under generalized Lévy models.
Working paper.
More at http://ssrn.com/author=879101
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