introduction to the Brownian motion derive the continuous model of option pricing giving the definition and relevant properties Brownian motion derive stochastic calculus based on the Brownian motion including the Ito integral & Ito formula.
∆
∆
4. S (tk + δ ) − S (tk ) & S (tk ) are independent
∆ ∆ ∆ ^ where tk = k ∆,1 ≤ k ≤ N − 1; δ > 0;0 ≤ t2 , t1^ ≤ T .
Central Limit Theorem
For any random sequence
Application of Central Limit Them.
Consider limit S (t ) / t as ∆→ 0.
S (t ) 1 t − tk ∆ tk +1 − t ∆ = ∆ S k +1 + ∆ S k t t tk +1 − t k +1 1 t − tk k = ∑ Ri + ∆ ∑ Ri t ∆ i =1 i =1 t − tk = ∆ tk tk k tk − t tk +1 k +1 ∑ Ri + ∑ Ri → X t i =1 ∆ t (k + 1) i =1
S k∆ (t ), t = tk , ∆ S (t ) = t − t ∆ t − t ∆ k S k +1 + k +1 S k , tk ≤ t ≤ tk +1. ∆ ∆
S ∆ (t ) is called the path of the random walk.
Distribution of the Path
1 2 i.e. ln Sk = ln S + ln Bk = rtk + σ ∑ R − σ tk 2 i =1
* k ∆ i
k
Geometric Brownian Motion cont.Let ⇒ ln S (t ) = (r − σ / 2)t + σ W (t )
2
∆t → 0
i.e.
( S (t ) = S0 e
Let T=1,N=4,∆=1/4,
S = 0, S1∆ = 1/ 4 R1 = {−1/ 2,1/ 2} ,
∆ S 2 = 1/ 4( R1 + R2 ) = {−1, 0,1} , ∆ 0
S3∆ = 1/ 4( R1 + R2 + R3 ) = {−3 / 2, −1/ 2, 0,1/ 2,3 / 2} ,
= qu (eσ
∆t
− 1) 2 + qd (1 − e −σ
∆t 2
)
Proof of the Lemma cont.
by the assumption of the lemma,
eσ
∆t
− 1 = σ ∆t + o( ∆t )
∆t
1 − e −σ
= σ ∆t + o( ∆t )
qu , qd = 1/ 2 + O( ∆t ) qu + qd = 1, qu − qd = O( ∆t )
With the random variable, define a Ri∆ = ∆ Ri and a random variable random sequence ∆ S k , (k = 0,1,L) :
S0∆ = 0, S = ∑ R = ∑ ∆ Ri , k = 1, 2,L
∆ k i =1 ∆ i i =1 k k
∆
∆
Definition of Winner Process (Brownian Motion)
1) Continuity of path: W(0)=0,W(t) is a continuous function of t. 2) Normal increments: For any t>0,W(t)~ N(0,t), and for 0 < s < t, W(t)-W(s) is normally distributed with mean 0 and variance t-s, i.e., W (t ) − W ( s ) ~ N (0, t − s ) 3) Independence of increments: for any choice of ti in [0,T] with 0 < t1 < t2 < L < tn , the increments
Random Walk
Consider a time period [0,T], which can be divided into N equal intervals. Let ∆=T\ N, t_n=n∆ ,(n=0,1,\cdots,N), then 0 = t0 < t1 < L t N = T . ∆ A random walk S (t ) is defined in [0,T]:
All of the description and discussion emphasize clarity rather than mathematical rigor.
Coin-tossing Problem
Define a random variable
1, ω = head Ri (ω ) = , (i = 1, 2,K) −1, ω = tail
k
{Ri } defined above, when k → ∞ 1 ∑R → X, k
i =1 i
where the random variable X~ N(0,1), i.e. the random variable X obeys the standard normal distribution: E(X)=0,Var(X)=1.
W (tn ) − W (tn −1 ), W (tn −1 ) − W (tn − 2 ),LW (t2 ) − W (t1 ),
are independent.
Continuous Models of Asset Price Movement
St* Introduce the discounted value
of an underlying asset as follows:
St* = St / Bt , r * = 0, ρ * = 1 in time interval [t,t+∆t], the BTM can be written as St*u / ρ *
St
St*d / ρ
Lemma
If ud=1, σis the volatility, letting u / ρ = eσ ∆t , d / ρ = e −σ ∆t then under the martingale measure Motion & Itô Formula
Stochastic Process
The price movement of an underlying asset is a stochastic process. The French mathematician Louis Bachelier was the first one to describe the stock share price movement as a Brownian motion in his 1900 doctoral thesis.
r −σ 2 / 2 t +σ W ( t )
)
Geometric Brownian Motion cont.-This means the underlying asset price movement as a continuous stochastic process, its logarithmic function is described by the Brownian motion. The underlying asset price S(t) is said to fit geometric Brownian motion. This means: Corresponding to the discrete BTM of the underlying asset price in a riskneutral world (i.e. under the martingale measure), its continuous model obeys the geometric Brownian motion .
* S0 = 1, B0 = 1 By definition
therefore after partitioning [0,T], at each instant t = tk ,
k St*+∆t 1 2 * ∆ ln S k = ∑ ln * = σ ∑ Ri − σ tk St 2 i =1 i =1 k
St +∆t − St Bt +∆t − Bt E = ρ −1 = St Bt
Q
thus by straightforward computation,
St*+∆t − St* Q Q St +∆t Bt / Bt +∆t − St E =E = 1−1 = 0 * St St