Financial Math
1998
9
14
1. 1.1 Markov Property 1.2 Wiener Process 1.3 2. 2.1 2.2 2.3 2.4 2.5 2.6 Taylor Expansion 2.7 3. Stochastic 3.1 3.2 SDE(Stochastic Differential Equation) 4. Stochastic 4.1 Stochastic integration 4.2 Ito Integral 4.3 Ito Integral 4.4 5. Ito’s Lemma 5.1 Stochastic 5.1.1 5.1.2 5.1.3 First Order Term Second Order Term Cross Product Terms “ ” – Ito Integral Riemann (Ordinary Differential Equation) (Chain rule)
5.2 Ito’s Lemma 6. 6.1 6.1.1 6.1.2 Closed-Form Solution Numerical Solution 2
. stochastic process Stochastic process . Stochastic process . stochastic calculus . stochastic calculus . (continuous time) (discrete time)
3
1.
1.1 Markov Property ( Markov ) .
. , . 1. 2 Wiener Process Wiener Process Wiener process PROPERTY 1 Markov stochastic process .
Markov
.
z
∆ z = ε ∆t
ε ~ N (0,1)
PROPERTY 2
∆t
∆z
.
∆z ~ N 0, ∆t
.
∆t → 0 ,
(
)
z
Markov
∆z
.
dz = ε dt x dx = a dt + b dz
4
dz
Wiener process
.
a
(drift rate) b (dt)
(diffusion rate)
.
x
dx = a dt + b ε dt
Wiener process dx = a( x, t )dt + b( x, t )dz
Ito process
.
,
1.3
x
.
, 14% 14% . (drift) . , , 10,000 100,000
S dS = µSdt
. . . Ito process
µ
.
.
σS
.
dS = µSdt + σSdz dS = µ dt + σ dz S process geometric Brownian motion .
5
1 30% dS = 0.15 dt + 0.30 dz S 1,000 . Wiener process
∆S = 1000(0.00288 + 0.0416ε )
, .
15%
1 ,1 0.0192 dz
,
1 .
2.88
41.6
6
2.
. stochastic process , stochastic . Stochastic chain rule chain rule
ITO’S LEMMA F (S t , t )
, . . stochastic Ito’s Lemma .
t Ito process .
stochastic process
St
. St
dSt = a(St , t ) dt + σ (St , t ) dWt dFt dFt = 1 ∂2F 2 ∂F ∂F dS t + dt + σ t dt ∂t 2 ∂S t2 ∂S t
.
⎡ ∂F ∂F ∂F 1 ∂ 2 F 2 ⎤ dFt = ⎢ at + σ t ⎥ dt + σ t dWt + 2 ∂S t ∂t 2 ∂S t ⎦ ⎣ ∂S t F (S t , t )
Ito process
.
2.1 f (x )
f x = lim
∆ →0
f ( x + ∆) − f ( x ) ∆ f (x )
x .
9
14
1. 1.1 Markov Property 1.2 Wiener Process 1.3 2. 2.1 2.2 2.3 2.4 2.5 2.6 Taylor Expansion 2.7 3. Stochastic 3.1 3.2 SDE(Stochastic Differential Equation) 4. Stochastic 4.1 Stochastic integration 4.2 Ito Integral 4.3 Ito Integral 4.4 5. Ito’s Lemma 5.1 Stochastic 5.1.1 5.1.2 5.1.3 First Order Term Second Order Term Cross Product Terms “ ” – Ito Integral Riemann (Ordinary Differential Equation) (Chain rule)
5.2 Ito’s Lemma 6. 6.1 6.1.1 6.1.2 Closed-Form Solution Numerical Solution 2
. stochastic process Stochastic process . Stochastic process . stochastic calculus . stochastic calculus . (continuous time) (discrete time)
3
1.
1.1 Markov Property ( Markov ) .
. , . 1. 2 Wiener Process Wiener Process Wiener process PROPERTY 1 Markov stochastic process .
Markov
.
z
∆ z = ε ∆t
ε ~ N (0,1)
PROPERTY 2
∆t
∆z
.
∆z ~ N 0, ∆t
.
∆t → 0 ,
(
)
z
Markov
∆z
.
dz = ε dt x dx = a dt + b dz
4
dz
Wiener process
.
a
(drift rate) b (dt)
(diffusion rate)
.
x
dx = a dt + b ε dt
Wiener process dx = a( x, t )dt + b( x, t )dz
Ito process
.
,
1.3
x
.
, 14% 14% . (drift) . , , 10,000 100,000
S dS = µSdt
. . . Ito process
µ
.
.
σS
.
dS = µSdt + σSdz dS = µ dt + σ dz S process geometric Brownian motion .
5
1 30% dS = 0.15 dt + 0.30 dz S 1,000 . Wiener process
∆S = 1000(0.00288 + 0.0416ε )
, .
15%
1 ,1 0.0192 dz
,
1 .
2.88
41.6
6
2.
. stochastic process , stochastic . Stochastic chain rule chain rule
ITO’S LEMMA F (S t , t )
, . . stochastic Ito’s Lemma .
t Ito process .
stochastic process
St
. St
dSt = a(St , t ) dt + σ (St , t ) dWt dFt dFt = 1 ∂2F 2 ∂F ∂F dS t + dt + σ t dt ∂t 2 ∂S t2 ∂S t
.
⎡ ∂F ∂F ∂F 1 ∂ 2 F 2 ⎤ dFt = ⎢ at + σ t ⎥ dt + σ t dWt + 2 ∂S t ∂t 2 ∂S t ⎦ ⎣ ∂S t F (S t , t )
Ito process
.
2.1 f (x )
f x = lim
∆ →0
f ( x + ∆) − f ( x ) ∆ f (x )
x .