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 . fx

, .

x x

f (x )

(differential)

7

(continuous)

(smooth)

. .

2.2

(chain rule)

. . . x xt = g (t ) yt = f ( g (t )) (composite function) . . t t ,

dy df ( g (t )) dg (t ) = dt dg (t ) dt

. .

x
. . . stochastic version . stochastic calculus “ ” , Ito’s Lemma

2.3

(Integral)

. countable . , .

Σ

.

uncountably infinite, 8

. f (t )

.

Riemann .

(

[0, T ]

) .

.

∫ f (s )ds
T 0

[0, T ]
0 = t0 < t1 < L < tk < Ltn = T ( Riemann

n

.

)

.

∑ f⎜ ⎝ i =1

n

⎛ t i + t i −1 ⎞ ⎟(t i − t i −1 ) 2 ⎠

(t i − t i −1 ) → 0

.

∑ f⎜ ⎝ i =1

n

T ⎛ t i + t i −1 ⎞ ⎟(t i − t i −1 ) → ∫0 f (s )ds 2 ⎠

.

2.4

(Partial Derivatives)

, . . Ct = F (St , t )

.

Ct

St

t

St
.
Fs = ∂F (St ,t ) ∂St

Ct

9

.
∂F (St ,t ) ∂t

Ft =

.

, . . , stochastic .

,

stochastic calculus

2 .

F (St , t ) = .3St + t 2 St
( stochastic , .

t St
) St .

Fs = 0.3
)

(

delta hedging

2.5

(Total Differentials)

F (S t , t ) t
⎡ ∂F (St , t ) ⎤ ⎡ ∂F (St , t ) ⎤ dF = ⎢ ⎥ dSt + ⎢ ⎥ dt ⎣ ∂t ⎦ ⎣ ∂St ⎦ = Fs dSt + Ft dt

,

St
.

10

3

T

, .

rt

F (rt , t ) = e − rt (T − t )100

dF (rt , t ) = −(T − t ) e − rt (T −t )100 drt + rt e − rt (T −t )100 dt

[

]

[

]

. Ito’s Lemma .

2.6 Taylor Expansion f (x ) x∈R

x
.

x0

Taylor series expansion

f ( x ) = f ( x0 ) + f x ( x0 )( x − x0 ) + =∑
1 i i f ( x0 )( x − x0 ) i = 0 i!
∞

1 1 2 3 f xx ( x0 )( x − x0 ) + f xxx ( x − x0 ) + L 2 3!

x0
2

x
3

,

x − x0

0

,

x1 − x0 > x1 − x0 > x1 − x0 > L
, Taylor series expansion . First-order approximation: f ( x ) ≅ f ( x0 ) + f x ( x0 )( x − x0 ) .

11

(x − x0 ) ≅ dx f ( x ) − f ( x0 ) = df ( x ) df ( x ) = f x ( x )dx x0
Second-order approximation: .

f ( x ) ≅ f ( x0 ) + f x ( x0 )( x − x0 ) +

1 2 f xx ( x0 )( x − x0 ) 2

4

T

100

.

Bt = 100 e − r (T − t ) r0 second-order approximation Taylor series expansion .

1 ⎡ 2 2⎤ Bt ≅ 100e − r (T −t ) ⎢1 − (T − t )(r − r0 ) + (T − t ) (r − r0 ) ⎥ 2 ⎣ ⎦ dBt 1 2 2 ≅ −(T − t )(r − r0 ) + (T − t ) (r − r0 ) Bt 2 duration . , convexity

(

)

2.7

(ordinary differential equation)

dBt = − rt Bt dt Bt . Bt = e ∫0
− ru du t B0 t

. Bt dBt . . Bt zero-coupon t Bt

12

. . , , . , . stochastic differential equation(SDE) SDE . stochastic .

13

3. Stochastic
3.1

Random process x .

f (x )

x0

Taylor expansion

f ( x ) = f ( x0 ) + f x ( x0 )( x − x0 ) + R( x, x0 )

1 1 2 3 f xx ( x0 )( x − x0 ) + f xxx ( x − x0 ) + R( x, x0 ) 2 3! . .

Taylor series expansion

∆x = x − x0 f ( x0 + ∆x ) − f ( x0 ) ≅ f x (∆x ) +
∆x

1 1 2 3 f xx (∆x ) + f xxx (∆x ) 2 3! . x random

random process x deterministic . random

(∆x )2 first-order approximation , ∆x ) “ ” random . 0 , E [∆x ] > 0
2

.

x

(random

(∆x )2
.

.

random process

Taylor series expansion

3.2 SDE

[0, T ]
0 = t0 < t1 < L < tk < Ltn = T k

h

n

.

14

t k − t k −1 = h

t k = kh .

[0, T ]
∆S k = S (kh ) − S ((k − 1)h ) k ∆Wk = [S k − S k −1 ] − E k −1 [S k − S k −1 ] E k −1 [⋅] k −1

S k = S (kh )

.

∆Wk

.

I k −1

(conditional expectation). ∆Wk ∆Wk measurable with respect to I k W0 = 0 . . E k −1 [∆Wk ] = 0 Wk = ∆W1 + L + ∆Wk , Wk martinglale

Ek −1Wk = Ek −1 [∆W1 + L + ∆Wk ]

Ek −1Wk = [∆W1 + L + ∆Wk ] = Wk −1 ∆Wk h .

E [∆Wk ] = σ k2 h
2

, ∆Wk “ ” .

. ,

15

. ∆Wk . E k −1 [S k − S k −1 ] E k −1 [S k − S k −1 ] = A(I k −1 , h ) A(I k −1 , h ) h=0 .

Taylor series expansion

.

A(I k −1 , h ) = A(I k −1 ,0) + a(I k −1 )h + R(I k −1 , h ) h=0 A(I k −1 ,0) = 0 .

h

random

I E k −1 [S k − S k −1 ] ≅ a(I k −1 , kh )h first-order Taylor series approximation stochastic difference equation S kh − S( k −1)h = a(I k −1 , kh )h + σ k Wkh − W(k −1)h . .

[

] h→0 stochastic differential equation . dS (t ) = a(I , t )dt + σ t dW (t )
E [∆Wk ] = σ k2 h
2

∆Wk2 ≅ h

.

W

dW (t )

. W(k −1)h + h − W(k −1)h h

lim h →0

→∞

16

h→0 f (h ) =

.

1 h1 2 = h h dW (t )

Ito Integral

.

17

4. Stochastic

– Ito Integral
. dX t Xt

0

.
= Xt

∫ dX
0

t

u

dX t . stochastic . . t +h

(integral equation)

∫ dX t u

= X t +h − X t

h Ito integral integral t
Wt = ∫ dWu
0 t

dX t random . Wiener process Wt

. stochatic

.

dW

.

(unbounded)

.

4.1 Stochastic integration

Riemann

stochastic difference equation S k − S k −1 = a(S k −1 , k )h + σ (S k −1 , k )∆Wk Riemann .

.

18

∑ [S k − S k −1 ] = ∑ [a(S k −1 , k )h] + ∑ σ (S k −1 , k )[∆Wk ] k =1 k =1 k =1

n −1

n −1

n −1

.

∫

T

0

n ⎧n ⎫ dS u = lim ⎨∑ [a(S k −1 , k )h] + ∑ σ (S k −1 , k )[∆Wk ]⎬, n →∞ k =1 ⎩ k =1 ⎭

T = nh

k −1

random .

(h

).

∫

T

0

a(S u , u )du = lim ∑ [a(S k −1 , k )h] n →∞ k =1

n

Wk − Wk −1

random (convergence)

. .

4.2 Ito Integral

Ito integral 1. σ (S t , t ) 2. σ (S t , t ) “non-explosive” .

. (non-anticipative)
T 2 E ⎡ ∫ σ (S t , t ) dt ⎤ < ∞ . ⎥ ⎢0 ⎦ ⎣

Ito integral .

∫ σ (S , t )dW
T 0 t

t

n→∞

∑ σ (S k =1

n

k −1

, k )[Wk − Wk −1 ] → ∫ σ (S t , t )dWt
T 0

mean square convergence

.

19

Mean Square Convergence X 0 , X 1 ,K, X n ,K random
X
n→∞

. sequence .
Xn

mean square converge
2

lim E [ X n − X ] = 0
Xn − X = εn

(variance)

0

.

∫ σ (S , t )dW
T 0 t

t

Ito integral
2

.

T ⎤ ⎡ n lim E ⎢∑ σ (S k −1 , k )[Wk − Wk −1 ] − ∫ σ (S u , u )dWu ⎥ = 0 0 n→∞ ⎦ ⎣ k =1

5 xt Wiener process .

∫

T

0

xt dt

.

Riemann

V n = ∑ x t i x t i +1 − x t i i =0

n −1

[

] xti x t i +1

non-anticipating

.
Vn Vn = Vn

. 1 ⎡ 2 n −1 2 ⎤ xT − ∑ ∆xti +1 ⎥ 2⎢ i =0 ⎣ ⎦ mean square limit
Z

∑ ∆x i =0

n −1

2 t i +1

mean square limit .

.

⎤ ⎡ n−1 lim E ⎢∑ ∆xt2+1 − Z ⎥ = 0 i n→∞ ⎦ ⎣ i =0

2

Z

.

20

n −1 ⎡ n −1 ⎤ n −1 E ⎢∑ ∆xt2i +1 ⎥ = ∑ E ∆xt2i +1 = ∑ (t i +1 − t i ) = T i =0 ⎣ i =0 ⎦ i =0

[

]

⎤ ⎡ n −1 lim E ⎢∑ ∆xt2i +1 − T ⎥ = 0 n →0 ⎦ ⎣ i =0

2

.1
Vn =

1 ⎡ 2 n −1 2 ⎤ xT − ∑ ∆xti +1 ⎥ 2⎢ i =0 ⎣ ⎦ 1 2 xT − T 2

.

lim E [Vn ] =
2 n→∞

[

]
.

Ito integral

∫

T

0

xt dt =

1 2 xT − T 2

[

]
.

⎤ ⎡ n −1 2 lim E ⎢∑ ∆xti +1 − T ⎥ = 0 n →0 ⎦ ⎣ i =0

2

xt

Wiener process
2

Ito integral

∫ (dx )
T 0 t

Ito integral
T ⎡ n −1 2⎤ lim E ⎢∑ ∆xt2i +1 − ∫ (dxt ) ⎥ = 0 o n →0 ⎦ ⎣ i =0 2

.

T = ∫ dt
0

T

∫ (dx )
T 0 t

2

= ∫ dt
0

T

.
Wt

Wiener process

(dW )2 = dt

.

1

An Introduction to the Mathematics of Financial Derivatives, Salih N. Neftchi pp182-183

21

4.3 Ito integral

∫

t +∆

t

σ u dWu

Ito integral (“news” “noise” ) . .

∆

.

t Et ⎡∫ ⎢t ⎣ t +∆

σ u dWu ⎤ = 0
⎥ ⎦

∫ ∫σ
0 t u

t +∆

t

σ u dWu
Martingale

Martingale difference . 0