The binomial model for option pricing презентация

Option Pricing: The Multi Period Binomial Model Henrik Jönsson Mälardalen University Sweden

Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit

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

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

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

The Multi Period Binomial Model

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.

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}

The Multi Period Binomial Model

The Multi Period Binomial Model Expected gain = No arbitrage opportunity implies

The Multi Period Binomial Model (1, 2, …, n-1) arbitrary vector No arbitrage opportunity 

The Multi Period Binomial Model Limitations: Two outcomes only The same increase & decrease for all time periods The same probabilities

Geometric Brownian Motion as a Limit The Binomial process:

GBM as a limit Let and , Y ~ Bin(n,p)

GBM as a Limit The stock price after n periods where

GBM as a Limit Taylor expansion gives

GBM as a limit Expected value of W

GBM as a limit By Central Limit Theorem

GBM as a limit The multi period Binomial model becomes geometric Brownian motion when n → ∞, since are independent

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)

B-S formula as a limit The unique non-arbitrage option price As n → ∞

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

B-S formula as a limit

B-S formula as a limit

B-S formula as a limit

B-S formula as a limit