Option Pricing: The Multi Period Binomial Model

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

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

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

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

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

The Multi Period Binomial Model 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 i Gurzuf, 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,… Gurzuf, Crimea, June 2001

The Multi Period Binomial Model • Choose an arbitrary vector (1, 2, …, n-1) • If A={X1= 1, X2= 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{X1= 1, X2= 2,…, Xn-1= n-1} • Probability that the stock goes up pn= P{Xn=1| X1= 1,…, Xn-1= n-1} Gurzuf, Crimea, June 2001

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

The Multi Period Binomial Model • (1, 2, …, n-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 Gurzuf, Crimea, June 2001

Limitations: Two outcomes only The same increase & decrease for all time periods The same probabilities Qualities: Simple mathematics Arbitrage pricing Easy to implement The Multi Period Binomial Model Gurzuf, Crimea, June 2001

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

S i The Binomial Process Gurzuf, Crimea, June 2001

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

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

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

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

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

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

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

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

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

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

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

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

Option Pricing: The Multi Period Binomial Model