Option Pricing with Greek Calculation
Andreas Griewank
Institute for Applied Mathematics, Humboldt Universität zu Berlin, Germany
Matheon, DFG - Research Center, Berlin, Germany
griewank@math.hu-berlin.de
November 14, 2007
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
November 14, 2007
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References
F.Y. Kuo and I. Sloan (2005):
Lifting the Curse of Dimensionality
M. Giles and P. Glassermann (2007):
Smoking Adjoints for Monte Carlo Greeks
A. Griewank and A. Walther (2000, 2008):
Evaluating Derivatives,
Principles and Techniques of Algorithmic Differentiation
A. Griewank (2008):( in preparation)
Evaluating and Bounding Cross-derivatives
Andreas Griewank (HUB, Matheon)
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Contents
Quadratures by Quasi-Monte Carlo (QMC)
Cross-Derivative Evaluation by AD
Cross-Derivative Bounding by AD
Application to Asia Options of 80 stocks
(Tentative) Conclusions
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Option Pricing with Greek Calculation
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High Dimensional Quadrature by QCD
Z
I(f ) =
f (x)dx
with n ≥ 50
[0,1]n
≈
N
N
k ·z +∆
1 X
f (xk ) with xk =
QX (f ) ≡
N k=1
N
k=1
for some generating vector z ∈ Z.
Doable if f is in some sense largely separable, i.e., small cross terms
X
f (x) =
f J (xJ ) for xJ = (xj )j∈J .
J⊂{1,...,n}
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
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High Dimensional Quadrature by QCD
This NOVA decomposition is unique if
Z1
f J (xJ )dxj = 0 for all j ∈ J.
0
We call f partially separable if equivalently
f {1,2,...,n} (x) = 0
⇐⇒
f{1,2,,...,n} (x) = 0,
where
fJ (x) ≡
∂ |J| f
(x) denotes cross − derivative
(∂xj )j∈J
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Option Pricing with Greek Calculation
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High Dimensional Quadrature by QCD
Multiplicative bounds g = (γ1 , . . . , γn ) ∈ IR n s.t.
Y
|fj (x)| ≤
γj for x ∈ Ω
j∈J
yield unanchored Sobolev space F w.r.t
hf , hiF
≡
X
J⊂{1,...,n}
Andreas Griewank (HUB, Matheon)
1
Q
γj
j∈J
Z
fJ (x) · gJ (x) dx
[0,1]n
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Reproducing Kernel by Grace Waba
With
B2 (z) = x 2 − x +
1
2
the kernel
k(x, y ) =
n Y
1 + γj
j=1
1
1 1
B2 (|xj − yj |) + (xj − )(yj − )
2
2
2
yields
Z
hk(·, y ), f (·)i =
k(x, y )f (x)dx = f (y )
[0,1]n
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Option Pricing with Greek Calculation
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High Dimensional Quadrature by QCD
Quadrature Error Bound for X = (xk )Nk=1...
eN (X ) ≡ max [QX (f ) − I(f )]2
f ∈F
=
N
N
1 XX
k(xk , x` )
N 2 k=1 `=1
Average error w.r.t. X ∈ [0, 1]d·N yields
ēN =
≤
Andreas Griewank (HUB, Matheon)
! 21
γk 1+
−1
6
k=1
!
N
X
1
1
√ exp
γk
N
12 k=1
√1
N
n Y
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(In)Tractability Results
γj = 1
=⇒
eN2 (X )
γ̄ ≡
1
≥
N
∞
X
for j = 1, . . . , N
1/p
γj
13
12
n
−1
<∞
for any X
with p ∈ {1, 2}
j=1
=⇒
(∗)
ēN ≤
Andreas Griewank (HUB, Matheon)


√1 exp(γ̄/12)
N
if p = 1

Cδ
N 1−δ
if p = 2
for any c > 0
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High Dimensional Quadrature by QCD
Suitable choice: Shifted Lattice Rule
n z
o
xk = k + ∆
with ∆ ∈ [0, 1]n
N
Averaging
w.r.t ∆ yields
ēN2 (z1 , . . . , zn )
n
n 1 XY
kzj
= −1 +
(1 + γj B2
N k=1 j=1
N
allows recursive = greedy construction from z1 = 1 for j = 2, . . . , n
zj = argmin ēN (z2 , . . . , zj−1 , z)
z
result achieves (∗).
Andreas Griewank (HUB, Matheon)
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Cross-Derivative Evaluation by AD
Number cross-derivatives versus all derivatives of order ≤ n
n
2 2n
n
≈
4n
p
πn/2
Cost of propagation through multiplication operation
n
3 Andreas Griewank (HUB, Matheon)
3n
n
≈
6.75n
p
4πn/3
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Workhorse: Incremental Convolution
for(int j=0;j< h;j++) v[j] = 0; // with h = pow(2,n)
void inconv(int h,double* u,double* w,double* v)
if(h==1)
v[0]+=u[0]*w[0]; // -= u[0]*w[0]; for deconv
else
h = h/2
inconv(h,u,w,v);
inconv(h,u+h,w,v+h);
inconv(h,u,w+h,v+h);
Significant speed up by unrolling lowest 3 levels of recursion!!
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Recursive Multiplications
void mult(int h, double* u, double* w, double* v)
{
v[0]=u[0]*w[0];
for(int j=1;j<h;j++) v[j] = 0; //Initialize to zero
for(int i=1;i<h;i*=2) // perform two convolutions
{ inconv(i,u+i,w,v+i)
inconv(i,u,w+i,v+i) }
}
The validity of this form can be derived from the Leibniz formula
X
X
vi∪{k} (x) =
uj∪{k} (x) w{i−j} (x) +
uj (x) w{i∪{k}−j} (x)
j⊂i
Andreas Griewank (HUB, Matheon)
j⊂i
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Division
ui∪{k} (x)
=
X
vj∪{k} (x) w{i−j} (x) +
j⊂i
=
X
uj (x) w{i∪{k}−j} (x)
j⊂i
w∅ vi∪{k} (x) +
X
vj∪{k} (x) w{i−j} (x) +
i6=j⊂i
X
wj (x) w{i∪{k}
j⊂i
void divide(int h, double* u, double* w, double* v)
int i,j; double w0; w0=w[0]
for(j=0;j<h;j++) {v[j]=u[j]/=w0; w[j]/=w0 }
w[0]=0
for(i=1;i<h;i*=2) // perform two convolutions
deconv(i,w+i,v,v+i)
deconv(i,w,v+i,v+i)
w[0] = w0
for(j=1;j<h;j++) { u[j]*=w0; w[i]*=w0; }
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
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Exponential
vi∪{k} =
X
v{i−j} (x) vj∪{k} (x)
j⊂i
void exponent(int h, double* u, double* v)
v[0]=exp(u[0])
for(int j=1;j<h;j++) v[j] = 0
for(int i=1;i<h;i*=2) // perform one convolution
conv(i,v,u+i,v+i)
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
November 14, 2007
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Cross-Derivative Bounding by AD
Cheaper Propagation of Bounds
Task
for all v = vi = vi (x) find
g v ≡ (γ1v , . . . , γnv )
v
|vJ (x)| ≤ |J| δ|J|
and δ0v , δ1v , . . . , δnv s.t.
Y
γjv for x ∈ Ω.
j∈J
w.l.o.g δ1V = 1 =⇒ g v ≡ Lipschitz vector.
Initialization (v = xj )
δ0v ≡ max{|xj |},
x∈Ω
Andreas Griewank (HUB, Matheon)
δ1v = 1 = γjv
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Cross-Derivative Bounding by AD
Addition/Subtraction (v = u ± w )
δ0v = δ0u + δ0w ,
gv = gu + gw
1
1
1
|vJ (x)| ≤
|uJ (x)| +
|wJ (x)|
|J|
|J|
|J|
Y
Y
u
w
≤ δ|J|
γju + δ|J|
γjw
≤
j∈J
j∈J
w
u
{δ|J| + δ|J| } max{γjn , γjn }
|
Andreas Griewank (HUB, Matheon)
for J 6= ∅
{z
v
δ|J|
}
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Cross-Derivative Bounding by AD
Multiplication (v = u ∗ w )
δuv = δ0u · δ0w , g v = δ0w g u + δ0u g w
X
Y
v
|vJ | ≤
(Leibniz)
uJ 0 · vJ\J 0 ≤ δ|J|
γjv
J 0 ⊂J
where
δjv =
j∈J
1
w
(δ0u )j−k }
max {δku (δ0u )k · δj−k
(δ0u δ0w )j 1≤k≤j
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
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Cross-Derivative Bounding by AD
Exponential
v = exp(u) =⇒ v{j} = v · u{j}
δ0v = exp(δ0u ), g v = δ0v · g u
j−1
X
u
v
S(j, k)
δkv δj−k
δj =
k=0
where S(j, k) is Stirling Number.
Total Complexity
OPS{δ0f , . . . , δnf , g f }
≤ c n2
OPS(f )
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Reverse Mode Gradient Calculation
Complexity:
OPS(∇f (x))
≤ 0 30
OPS(f (x))
but
for any n
MEM(∇f (x))
≤ ` ' OPS(f (x))
MEM(f (x))
Andreas Griewank (HUB, Matheon)
Option Pricing with Greek Calculation
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Asian Option Application
Parameters:
80
=
number of securities = stocks
r
=
0.05 interest rate
T
=
250 = 1/∆t = number of periods
K
=
100 strike price
Si
=
K for i ≤ 80 initial stock price
σ ∈
IR 80
= volatility vector
Σ
=
ΣT ∈ IR 80×80 = Correlation Matrix
Precalculation
AT A = diag (σ) Σ diag (σ), AT ≡ [a2 , . . . , a80 ]
Z
= [z1 , . . . , zT ] with zt ∈ Normal −1 (Rand[0, 1]) ∈ IR 80 )
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Asian Option Application
Path Evaluation
R=0
for 1 = 1 . . . T
for i = 1 . . . 80
Si
= Si exp
h
√
√ i
r − σi2 /2 ∆t ai zt
R = R + Si
R = max{R/(T · 80) − K , 0}
Single Sample
f = R(K , r , S, σ, Σ, Z )
∇(K ,r ,S,σ,Σ) f ∈ IR 3402
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Experimental Results
ADOL-C
Runtime (∇f )
'
Runtime (f )
TADBAD
(Giles)
Runtime (∇f )
' 15 for LIBOR option
Runtime (f )
val., hc-b,
0.55, 0.70,
Andreas Griewank (HUB, Matheon)
2.5
4
with − O0
with − O4
hc f, FADBA, AD
1.54, 37.15, 3.06
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Conclusions, tentative comme tojours
Lattice rules improvable using problem (class) specific bounds?
Cross-derivative propagation efficient, regular memory access!
Bound propagation very similar to univariate Taylor scheme!
Reverse mode differentiation of (Q)MC is cheap
and yields good results if pay-off is Lipschitz
For digital options some ’smoothing’ seems unavoidable’
Possibly mathematics of finance is the Killer Application?
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