1. Gurzuf, Crimea, June 20011 Option Pricing: The Multi Period Binomial Model Henrik Jönsson Mälardalen University Sweden

2. Gurzuf, Crimea, June 20012 Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit

3. Gurzuf, Crimea, June 20013 European Call Option C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e.g.

4. Gurzuf, Crimea, June 20014 Geometric Brownian Motion S(y), 0  y<t, follows a geometric Brownian motion if independent of all prices up to time y

5. Gurzuf, Crimea, June 20015 Black-Scholes Formula The price at time zero of a European call option (non-dividend-paying stock): where

6. Gurzuf, Crimea, June 20016 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

7. Gurzuf, Crimea, June 20017 The Multi Period Binomial Model Let Let (X 1, X 2,…, X n ) be the vector describing the outcome after n steps. Find the set of probabilities P{X 1 =x 1, X 2 =x 2,…, X n =x n }, x i =0,1, i=1,…,n, such that there is no arbitrage opportunity. i=1,2,…

8. Gurzuf, Crimea, June 20018 The Multi Period Binomial Model Choose an arbitrary vector (  1,  2, …,  n-1 ) If A={X 1 =  1, X 2 =  2, …, X n-1 =  n-1 } is true buy one unit of stock and sell it back at moment n Probability that the stock is purchased q n-1 =P{X 1 =  1, X 2 =  2, …, X n-1 =  n-1 } Probability that the stock goes up p n = P{X n =1| X 1 =  1, …, X n-1 =  n-1 }

9. Gurzuf, Crimea, June 20019 The Multi Period Binomial Model Example: i S 123n=4

10. Gurzuf, Crimea, June 200110 The Multi Period Binomial Model Expected gain = No arbitrage opportunity implies q n-1 [p n (1+r) -1 uS n-1 +(1- p n ) (1+r) -1 dS n-1 -S n-1 ] r = risk-free interest rate

11. Gurzuf, Crimea, June 200111 The Multi Period Binomial Model (  1,  2, …,  n-1 ) arbitrary vector No arbitrage opportunity  X 1,…, X n independent with P{X i =1}=p, i=1,…,n Risk-free interest rate r the same for all moments i

12. Gurzuf, Crimea, June 200112 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

13. Gurzuf, Crimea, June 200113 Geometric Brownian Motion as a Limit The Binomial process:

14. Gurzuf, Crimea, June 200114 S i The Binomial Process

15. Gurzuf, Crimea, June 200115 GBM as a limit Let and, Y ~ Bin(n,p)

16. Gurzuf, Crimea, June 200116 GBM as a Limit The stock price after n periods where

17. Gurzuf, Crimea, June 200117 GBM as a Limit Taylor expansion gives

18. Gurzuf, Crimea, June 200118 GBM as a limit Expected value of WVariance of W EY = np VarY = np(1-p)

19. Gurzuf, Crimea, June 200119 GBM as a limit By Central Limit Theorem

20. Gurzuf, Crimea, June 200120 GBM as a limit The multi period Binomial model becomes geometric Brownian motion when n → ∞, since are independent

21. Gurzuf, Crimea, June 200121 B-S Formula as a limit Let, Y ~ Bin(n,p) The value of the option after n periods = where S(t)= u Y d n-Y S(0) max[S(t)-K,0] = [S(t)-K] + No arbitrage 

22. Gurzuf, Crimea, June 200122 B-S formula as a limit The unique non-arbitrage option price As n → ∞ X~N(0,1)

23. Gurzuf, Crimea, June 200123 B-S formula as a limit where X~N(0,1) and

24. Gurzuf, Crimea, June 200124 B-S formula as a limit

25. Gurzuf, Crimea, June 200125 B-S formula as a limit  (·) is the N(0,1) distribution function

26. Gurzuf, Crimea, June 200126 B-S formula as a limit

27. Gurzuf, Crimea, June 200127 B-S formula as a limit where