The binomial model for option pricing презентация

Gurzuf, Crimea, June 2001 Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit

Слайд 1Gurzuf, Crimea, June 2001
Option Pricing: The Multi Period Binomial Model
Henrik Jönsson
Mälardalen University
Sweden


Слайд 2Gurzuf, Crimea, June 2001
Contents
European Call Option
Geometric Brownian Motion
Black-Scholes Formula
Multi period

Binomial Model
GBM as a limit
Black-Scholes Formula as a limit

Слайд 3Gurzuf, Crimea, June 2001
European Call Option
C - Option Price
K - Strike

price
T - Expiration day
Exercise only at T
Payoff function, e.g.

Слайд 4Gurzuf, Crimea, June 2001
Geometric Brownian Motion
S(y), 0≤y

geometric Brownian motion if

independent of all prices up to time y



Слайд 5Gurzuf, Crimea, June 2001
Black-Scholes Formula
The price at time zero of a

European call
option (non-dividend-paying stock):


where



Слайд 6Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
i
S
i=1,2,…
Note:
u and

d the same for all moments i
d < 1+r < u, where r is the risk-free interest rate

Слайд 7Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
Let

Let (X1, X2,…,

Xn) be the vector describing the outcome after n steps.
Find the set of probabilities P{X1=x1, X2 =x2,…, Xn =xn}, xi=0,1, i=1,…,n, such that there is no arbitrage opportunity.

i=1,2,…


Слайд 8Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
Choose an arbitrary vector

(α1, α2, …, αn-1)
If A={X1= α1, X2= α2, …, Xn-1= αn-1} is true buy one unit of stock and sell it back at moment n
Probability that the stock is purchased qn-1=P{X1= α1, X2= α2, …, Xn-1= αn-1}
Probability that the stock goes up pn= P{Xn=1| X1= α1, …, Xn-1= αn-1}

Слайд 9Gurzuf, Crimea, June 2001
The Multi Period Binomial Model


Слайд 10Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
Expected gain =



No

arbitrage opportunity implies

qn-1[pn(1+r)-1uSn-1+(1- pn) (1+r)-1dSn-1-Sn-1]


r = risk-free interest rate


Слайд 11Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
(α1, α2, …, αn-1)

arbitrary vector
No arbitrage opportunity


X1,…, Xn independent with P{Xi=1}=p, i=1,…,n

Risk-free interest rate r the same for all moments i


Слайд 12Gurzuf, Crimea, June 2001
The Multi Period Binomial Model
Limitations:
Two outcomes only
The

same increase & decrease for all time periods
The same probabilities

Qualities:
Simple mathematics
Arbitrage pricing
Easy to implement


Слайд 13Gurzuf, Crimea, June 2001
Geometric Brownian Motion as a Limit
The Binomial process:









Слайд 14Gurzuf, Crimea, June 2001
The Binomial Process


Слайд 15Gurzuf, Crimea, June 2001
GBM as a limit
Let


and

, Y ~ Bin(n,p)


Слайд 16Gurzuf, Crimea, June 2001
GBM as a Limit
The stock price after n

periods




where

Слайд 17Gurzuf, Crimea, June 2001
GBM as a Limit
Taylor expansion



gives


Слайд 18Gurzuf, Crimea, June 2001
GBM as a limit
Expected value of W
Variance of

W

EY = np
VarY = np(1-p)




Слайд 19Gurzuf, Crimea, June 2001
GBM as a limit
By Central Limit Theorem




Слайд 20Gurzuf, Crimea, June 2001
GBM as a limit
The multi period Binomial model

becomes geometric Brownian motion when n → ∞, since

are independent



Слайд 21Gurzuf, Crimea, June 2001
B-S Formula as a limit
Let

, Y ~ Bin(n,p)

The value of the option after n periods =


where S(t)= uY dn-Y S(0)


max[S(t)-K,0] = [S(t)-K]+

No arbitrage ⇒


Слайд 22Gurzuf, Crimea, June 2001
B-S formula as a limit
The unique non-arbitrage option

price



As n → ∞

X~N(0,1)


Слайд 23Gurzuf, Crimea, June 2001
B-S formula as a limit





where X~N(0,1) and


Слайд 24Gurzuf, Crimea, June 2001
B-S formula as a limit


Слайд 25Gurzuf, Crimea, June 2001
B-S formula as a limit
Φ(·) is the N(0,1)

distribution function

Слайд 26Gurzuf, Crimea, June 2001
B-S formula as a limit


Слайд 27Gurzuf, Crimea, June 2001
B-S formula as a limit
where


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