Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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650 A List of Special Features, Probability Laws, and Functions
Reflection Principle.(Page 136)for y ≥ 0,x≤ y
P(W
t
≤ x,M
t
≥ y)=P(W
t
≥ 2y − x)
Joint law of W
t
and M
t
(Page 137)
P(W
t
∈ dx, M
t
∈ dy)=1
{y≥0}
1
{x≤y}
2(2y − x)
√
2πt
3
exp
−
(2y − x)
2
2t
dx dy
Tanaka’s Formulae. (Pages 214 and 215)
(W
t
− x)
+
=(W
0
− x)
+
+
t
0
1
{W
s
>x}
dW
s
+
1
2
L
x
t
|W
t
− x| = |W
0
− x| +
t
0
sgn (W
s
− x) dW
s
+ L
x
t
f(W
t
)=f(W
0
)+
t
0
(D
−
f)(W
s
) dW
s
+
1
2
R
L
a
t
f
(da)
A.1.4 Diffusions
Scale Function. (Page 271)
s(x)=
x
c
exp
−2
u
c
b(v)/σ
2
(v) dv
du
Speed Measure Density. (Page 272)
m(x)=
2
σ
2
(x)s
(x)
A.1.5 Finance
Dupire’s Formula. (Page 228)
1
2
K
2
σ
2
(T,K)=
∂
T
C(K, T)+rK∂
K
C(K, T)
∂
2
KK
C(K, T)
Let dS
t
= S
t
((r − δ)dt + σdW
t
), and let C
E
(x, K; r, δ; T − t) be the price of
a call option at time t, with strike K.
A.1 Main Formulae 651
Black and Scholes’ Formula. (Pages 97 and 160)
C
E
(x, K; r, δ; T − t)=xe
−δ(T−t)
N
d
1
xe
−δ(T−t)
Ke
−r(T −t)
,T − t
− Ke
−r(T −t)
N
d
2
xe
−δ(T−t)
Ke
−r(T −t)
,T − t
where
d
1
(y, u):=
1
√
σ
2
u
ln(y)+
1
2
√
σ
2
u
d
2
(y, u):=d
1
(y, u) −
√
σ
2
u
A.1.6 Girsanov’s Theorem
(Page 534)LetX be a local martingale with respect to P and
Q|
F
t
= L
t
P|
F
t
Then, X
t
−
t
0
d[X, L]
s
L
s
is a Q-local martingale.
If [X, L]isP-locally integrable, X
t
−
t
0
dX,L
s
L
s
−
is a Q-local martingale.
A.1.7 Hitting Times
Laplace transform of the first hitting times for diffusions. (Page 278)
E
x
e
−λT
y
=
Φ
λ↑
(x)/Φ
λ↑
(y)ifx<y
Φ
λ↓
(x)/Φ
λ↓
(y)ifx>y
A.1.8 Itˆo’s Formulae
Integration by parts formula for general semi-martingales. (Page 469)
X
t
Y
t
= X
0
Y
0
+
t
0
Y
s
−
dX
s
+
t
0
X
s
−
dY
s
+[X, Y ]
t
652 A List of Special Features, Probability Laws, and Functions
Optional Itˆo’s Formula for General Semi-martingales. (Page 528)
f(X
t
)=f(X
0
)+
t
0
f
(X
s
−
)dX
s
+
1
2
t
0
f
(X
s
)dX
c
s
+
0<s≤t
[f(X
s
) − f (X
s
−
) − f
(X
s
−
)ΔX
s
]
Itˆo–Kunita-Ventzel’s Formula. (Page 40)
dF
t
(x)=
n
j=1
f
j
t
(x)dM
j
t
Let X =(X
1
,...,X
d
) be a continuous semi-martingale. Then
F
t
(X
t
)=F
0
(X
0
)+
n
j=1
t
0
f
j
s
(X
s
)dM
j
s
+
d
i=1
t
0
∂F
s
∂x
i
(X
s
)dX
i
s
+
d
i=1
n
j=1
t
0
∂f
s
∂x
i
(X
s
)dM
j
,X
i
s
+
1
2
d
i,k=1
t
0
∂
2
F
s
∂x
i
∂x
k
dX
k
,X
i
s
.
Mixed Processes.
Let dX
t
= h
t
dt + f
t
dW
t
+ g
t
dM
t
where dM
t
= dN
t
− λ(t)dt.
Optional Itˆo’s Formula for Mixed Processes. (Page 553)
F (t, X
t
)=F (0,X
0
)+
t
0
∂
s
F (s, X
s
) ds +
t
0
∂
x
F (s, X
s
−
)dX
s
+
1
2
t
0
∂
xx
F (s, X
s
)f
2
s
ds
+
s≤t
[F (s, X
s
) − F (s, X
s
−
) − ∂
x
F (s, X
s
−
)ΔX
s
]
A.1 Main Formulae 653
Predictable Itˆo’s Formula for Mixed Processes. (Page 612)
F (t, X
t
)=F (0,X
0
)+
t
0
∂
x
F (s, X
s
)f
s
dW
s
+
t
0
[F (s, X
s
−
+ g
s
) − F (s, X
s
−
)] dM
s
+
t
0
∂
t
F (s, X
s
)+h
s
∂
x
F (s, X
s
)+
1
2
f
2
s
∂
xx
F (s, X
s
)
+λ(s)[F (s, X
s
+ g
s
) − F (s, X
s
) − ∂
x
F (s, X
s
)g
s
]
ds
Optional ItˆoformulaforL´evy Processes. (Page 612)
f(t, X
t
)=f(0,X
0
)+
t
0
∂
t
f(s, X
s
)ds
+
σ
2
2
t
0
∂
xx
f(s, X
s
)ds +
t
0
∂
x
f(s, X
s
−
)dX
s
+
s≤t
(f(s, X
s
−
+ ΔX
s
) − f (s, X
s
−
) − (ΔX
s
) ∂
x
f(s, X
s
−
))
Predictable ItˆoFormulaforL´evy Processes.(Page 612)
df (t, X
t
)=∂
x
f(t, X
t
)σdW
t
+
R
(f(t, X
t
−
+ x) − f (t, X
t
−
))
N(dt, dx)
+
1
2
σ
2
∂
xx
f(t, X
t
)+m∂
x
f(t, X
t
)+∂
t
f(t, X
t
)
+
R
f(t, X
t
−
+ x) − f (t, X
t
−
) − x∂
x
f(t, X
t
−
)1
|x|≤1
ν(dx)
dt
A.1.9 L´evy Processes
L´evy-Khintchine Representation. (Page 593)
μ(u) = exp
iu m −
1
2
u Au +
R
d
(e
iu x
− 1 − iu x1
|x|≤1
)ν(dx)
654 A List of Special Features, Probability Laws, and Functions
Laplace and Characteristic Exponents. (Page 607)
Φ(u)=−iu m +
1
2
u Au −
(e
iu x
− 1 − iu x1
|x|≤1
)ν(dx)
Ψ(λ)=λ m +
1
2
λ σ
2
λ +
(e
λ x
− 1 − λ x1
|x|≤1
)ν(dx)
Subordination. (Page 636)
The characteristics of
X
t
= X
Z
t
are
a = βa
X
+
ν
Z
(ds)1
{|x|≤1}
x P(X
s
∈ dx)
A = βA
X
ν(dx)=βν
X
(dx)+
∞
0
ν
Z
(ds)P(X
s
∈ dx)
A.1.10 Semi-martingales
Tanaka-Meyer Formula. (Page 224)
|X
t
− x| = |X
0
− x| +
t
0
sgn (X
s
− x) dX
s
+ L
x
t
(X)
(X
t
− x)
+
=(X
0
− x)
+
+
t
0
1
{X
s
>x}
dX
s
+
1
2
L
x
t
(X)
A.3 Some Main Models 655
A.2 Processes
Process Law Equation Page
Drifted BM W
(μ)
drift μ dX
t
= μdt + dW
t
32
Squared Bessel Q
δ
δ dρ
t
= δdt +2
√
ρ
t
dW
t
334
BESQ
δ
Bessel dimension 334
BES
δ
P
δ
δ>1 dR
t
= dW
t
+
δ − 1
2
1
R
t
dt
CIR
a
Q
b,σ
dr
t
=(a − br
t
)dt 357
+σ
√
r
t
dW
t
CEV dS
t
= S
t
(μdt + σS
β
t
dW
t
) 365
A.3 Some Main Models
Model Parameters Dynamics Page
Black and 94
Scholes r, σ dS
t
= S
t
(rdt + σdW
t
)
Garman and r, δ, σ 129
Kohlhagen dS
t
= S
t
((r − δ)dt + σdW
t
)
Hull and White k,θ, σ dr
t
= k(t)(θ(t) − r
t
)dt 120
(interest rate) Borel funct. +σ(t)dW
t
Vasicek k, θ, σ dr
t
= k(θ − r
t
)dt + σdW
t
120
CIR or square root k, θ, σ dr
t
= k(θ − r
t
) dt + σ
√
r
t
dW
t
357
CEV μ, σ, β dS
t
= S
t
(μdt + σS
β
t
dW
t
) 365
Hull and White μ, a, λ dS
t
= S
t
(μdt + σ
t
dW
t
) 393
(stochastic vol.) dσ
t
= σ
t
(adt + λdB
t
)
656 A List of Special Features, Probability Laws, and Functions
A.4 Some Important Probability Distributions
A.4.1 Laws with Density
Law Density Characteristic function
Φ(u)=E(e
iuX
)
or Mellin function
M(λ)=E(X
λ
)
Exponential λe
−λx
1
{x>0}
Φ(u)=
λ
λ − iu
Gaussian
1
σ
√
2π
exp
−
(x − m)
2
2σ
2
Φ(u)=exp(ium −
1
2
σ
2
u
2
)
Cauchy
1
π
a
a
2
+ x
2
Φ(u)=exp(−a|u|)
Arcsine
1
π
1
s(1 − s)
1
0≤s≤1
M(λ)=
B(λ +1/2, 1/2)
B(1/2, 1/2)
Beta (a,b)
t
a−1
(1 − t)
b−1
B(a, b)
1
0≤t≤1
M(λ)=
B(a + λ, b)
B(a, b)
Gamma(α)
t
α−1
e
−t
Γ (α)
1
0≤t
M(λ)=
Γ (α + λ)
Γ (α)
A.4.2 Some Algebraic Properties for Special r.v.’s
• If C follows a Cauchy law with parameter a = 1, the random variable
1
1+C
2
follows the Arcsine law.
• If G and
G are two independent standard Gaussian r.v’s., then
G
G
follows
a Cauchy law with parameter a =1.
A.4 Some Important Probability Distributions 657
• “Beta-Gamma” algebra: If γ
a
is a gamma r.v. with parameter a,then
γ
a
law
= β
a,b
γ
a+b
where β
a,b
is a beta(a, b) r.v. independent of γ
a+b
.
A particular case of the beta-gamma algebra is
G
2
law
=2γ
1/2
law
=2β
1/2,1/2
e
where G is a gaussian variable, e
law
= γ
1
is a standard exponential variable,
independent of the arc-sine variable β
1/2,1/2
.
See Chaumont and Yor [161] and Dufresne [278] for other algebraic properties.
A.4.3 Poisson Law
P(X = n) e
−λ
λ
n
n!
Characteristic function exp
λ(e
iu
− 1)
Poisson law, λ>0
Mean λ
Variance λ
658 A List of Special Features, Probability Laws, and Functions
A.4.4 Gamma and Inverse Gaussian Law
Density
ν
a
Γ (a)
x
a−1
e
−xν
1
{x>0}
Characteristic function (1 − iu/ν)
−a
Γ (a, ν) Mean a/ν
a, ν > 0 Variance a/ν
2
L´evy density ax
−1
e
−νx
1
x>0
Density
a
√
2π
e
aν
x
−3/2
×exp
−
1
2
(a
2
x
−1
+ ν
2
x)
1
{x>0}
Characteristic function exp
−a(
√
ν
2
− 2iu − ν)
IG(a, ν) Mean a/ν
Variance a/ν
3
L´evy density
a
√
2πx
3
exp
−
1
2
ν
2
x
1
{x>0}
If X follows a Γ (a, ν)law,thencX follows a Γ (a, ν/c)law.
A.4 Some Important Probability Distributions 659
A.4.5 Generalized Inverse Gaussian and Normal Inverse Gaussian
GIG(θ, a, ν),a>0,ν >0
Density
(ν/a)
θ
2K
θ
(aν)
x
θ−1
exp
−
1
2
(a
2
x
−1
+ ν
2
x)
1
{x>0}
Characteristic f.
1
K
θ
(aν)
1 − 2iu/ν
2
θ/2
K
θ
(aν
1 − 2iuν
−2
)
Mean aK
θ+1
(aν)/(νK
θ
(aν))
Variance a
2
ν
2
K
−2
θ
(aν)
K
θ+2
(aν)K
θ
(aν)+K
2
θ+1
(aν)
L´evy density x
−1
e
−ν
2
x/2
a
2
∞
0
e
−xy
g(y)dy +max(0,θ)
g(y)=
π
2
a
2
y
J
2
|θ|
(a
√
2y)+N
2
|θ|
(a
√
2y))
−1