The binomial model for option pricing

Сентябрь 14, 2022

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The binomial model for option pricing

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2. Gurzuf, Crimea, June 2001 Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial
3. Gurzuf, Crimea, June 2001 European Call Option C - Option Price K - Strike price T
4. Gurzuf, Crimea, June 2001 Geometric Brownian Motion S(y), 0≤y independent of all prices up to time
5. Gurzuf, Crimea, June 2001 Black-Scholes Formula The price at time zero of a European call option
6. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model i S i=1,2,… Note: u and d
7. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model Let Let (X1, X2,…, Xn) be the
8. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model Choose an arbitrary vector (α1, α2, …,
9. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model
10. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model Expected gain = No arbitrage opportunity implies
11. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model (α1, α2, …, αn-1) arbitrary vector No
12. Gurzuf, Crimea, June 2001 The Multi Period Binomial Model Limitations: Two outcomes only The same increase
13. Gurzuf, Crimea, June 2001 Geometric Brownian Motion as a Limit The Binomial process:
14. Gurzuf, Crimea, June 2001 The Binomial Process
15. Gurzuf, Crimea, June 2001 GBM as a limit Let and , Y ~ Bin(n,p)
16. Gurzuf, Crimea, June 2001 GBM as a Limit The stock price after n periods where
17. Gurzuf, Crimea, June 2001 GBM as a Limit Taylor expansion gives
18. Gurzuf, Crimea, June 2001 GBM as a limit Expected value of W Variance of W EY
19. Gurzuf, Crimea, June 2001 GBM as a limit By Central Limit Theorem
20. Gurzuf, Crimea, June 2001 GBM as a limit The multi period Binomial model becomes geometric Brownian
21. Gurzuf, Crimea, June 2001 B-S Formula as a limit Let , Y ~ Bin(n,p) The value
22. Gurzuf, Crimea, June 2001 B-S formula as a limit The unique non-arbitrage option price As n
23. Gurzuf, Crimea, June 2001 B-S formula as a limit where X~N(0,1) and
24. Gurzuf, Crimea, June 2001 B-S formula as a limit
25. Gurzuf, Crimea, June 2001 B-S formula as a limit Φ(·) is the N(0,1) distribution function
26. Gurzuf, Crimea, June 2001 B-S formula as a limit
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Contents
European Call Option
Geometric Brownian Motion
Black-Scholes Formula
Multi period

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

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European Call Option
C - Option Price
K - Strike

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

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Geometric Brownian Motion
S(y), 0≤y

Brownian motion if
independent of all prices up to time y

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Black-Scholes Formula
The price at time zero of a

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

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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

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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,…

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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}

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The Multi Period Binomial Model

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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

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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

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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

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Geometric Brownian Motion as a Limit
The Binomial process:

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The Binomial Process

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GBM as a limit
Let
and , Y ~

Bin(n,p)

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GBM as a Limit
The stock price after n

periods
where

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GBM as a Limit
Taylor expansion
gives

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GBM as a limit
Expected value of W
Variance of

W

EY = np
VarY = np(1-p)

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GBM as a limit
By Central Limit Theorem

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GBM as a limit
The multi period Binomial model

becomes geometric Brownian motion when n → ∞, since
are independent

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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 ⇒

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B-S formula as a limit
The unique non-arbitrage option

price
As n → ∞

X~N(0,1)

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B-S formula as a limit
where X~N(0,1) and

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B-S formula as a limit

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B-S formula as a limit
Φ(·) is the N(0,1)

distribution function

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B-S formula as a limit