Option Pricing The Multi Period Binomial Model Henrik

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

Contents • • • European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit Gurzuf, Crimea, June 2001 2

European Call Option • • • C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e. g. Gurzuf, Crimea, June 2001 3

Geometric Brownian Motion S(y), 0 y<t, follows a geometric Brownian motion if • independent of all prices up to time y • Gurzuf, Crimea, June 2001 4

Black-Scholes Formula The price at time zero of a European call option (non-dividend-paying stock): where Gurzuf, Crimea, June 2001 5

The Multi Period Binomial Model S i=1, 2, … Note: • u and d the same for all moments i i Gurzuf, Crimea, June 2001 • d < 1+r < u, where r is the risk-free interest rate 6

The Multi Period Binomial Model • Let i=1, 2, … • Let (X 1, X 2, …, Xn) be the vector describing the outcome after n steps. • Find the set of probabilities P{X 1=x 1, X 2 =x 2, …, Xn =xn}, xi=0, 1, i=1, …, n, such that there is no arbitrage opportunity. Gurzuf, Crimea, June 2001 7

The Multi Period Binomial Model • Choose an arbitrary vector ( 1, 2, …, n-1) • If A={X 1= 1, X 2= 2, …, Xn-1= n-1} is true buy one unit of stock and sell it back at moment n • Probability that the stock is purchased qn-1=P{X 1= 1, X 2= 2, …, Xn-1= n-1} • Probability that the stock goes up pn= P{Xn=1| X 1= 1, …, Xn-1= n-1} Gurzuf, Crimea, June 2001 8

The Multi Period Binomial Model S Example: 1 2 3 n=4 i Gurzuf, Crimea, June 2001 9

The Multi Period Binomial Model • Expected gain = qn-1[pn(1+r)-1 u. Sn-1+(1 - pn) (1+r)-1 d. Sn-1 -Sn-1] r = risk-free interest rate • No arbitrage opportunity implies Gurzuf, Crimea, June 2001 10

The Multi Period Binomial Model • ( 1, 2, …, n-1) arbitrary vector • No arbitrage opportunity X 1, …, Xn independent with P{Xi=1}=p, i=1, …, n Risk-free interest rate r the same for all moments i Gurzuf, Crimea, June 2001 11

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 Gurzuf, Crimea, June 2001 12

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

The Binomial Process S i Gurzuf, Crimea, June 2001 14

GBM as a limit Let and , Y ~ Bin(n, p) Gurzuf, Crimea, June 2001 15

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

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

GBM as a limit Expected value of W Variance of W EY = np Var. Y = np(1 -p) Gurzuf, Crimea, June 2001 18

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

GBM as a limit The multi period Binomial model becomes geometric Brownian motion when n → ∞, since • are independent • Gurzuf, Crimea, June 2001 20

B-S Formula as a limit • Let , Y ~ Bin(n, p) • The value of the option after n periods = max[S(t)-K, 0] = [S(t)-K]+ where S(t)= u. Y dn-Y S(0) • No arbitrage Gurzuf, Crimea, June 2001 21

B-S formula as a limit The unique non-arbitrage option price As n → ∞ X~N(0, 1) Gurzuf, Crimea, June 2001 22

B-S formula as a limit where X~N(0, 1) and Gurzuf, Crimea, June 2001 23

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

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

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

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