Option Pricing BA 543 Aoyang Long

Agenda Binomial pricing model Black—Scholes model

Binomial Option Pricing Model Interest rate =8% Price 0 = (60%*$ 80+40%*$ 55)/(1+8%) = $ ? $80$55 60% 40% t0t0 t1t1

Binomial Option Pricing Model 60% 40% t0t0 t1t1 Stock Price = $ 80 Stock Price = $ 55 Call option payoff $ 10$ 0$ 0 Interest rate =8% Exercise price= $70 Value of call = (60%*$ 10) / (1+8%) = $ 5.56

Multiple Periods t0t0 t1t1 t2t2 t3t3 t4t Price 0 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% 60% 40% How many path for a stock price of $80?

Pascal’s Triangle Each number in the triangle is the sum of the two directly above it.

Lognormal Distribution

Black—Scholes Model The trick is to set up an option equivalent by combing common stock investment and borrowing. The net cost of buying the option equivalent must equal the value of the option. -- Black and Scholes Assumptions  European call option only  Underlying assets does not pay dividends until expiration date  Both the interest rate and the variance of the return on the stock are constant  Stock prices are continuous ( no sudden jump)

Black—Scholes Model d1=log[P/PV(X)] /σ√t+σ√t2 d2=d1-σ√t N(d) = cumulative normal probability function X = exercise price t = number of periods to exercise date S = current stock price σ= standard deviation per period of (continuously compounded) rate of return on stock

Black—Scholes Model Example S = 55 X = 55 r = 4% per year t = 0.5 year = days σ = 40.69% Black-Scholes Calculator

Summary Binomial pricing model:  discrete model  both European and American call  slow Black—Scholes model:  continuous model  European call  quick