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

.