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Option Pricing. BA 543 Aoyang Long. Binomial pricing model Black—Scholes model. Agenda. t 0. t 1. 60%. Interest rate =8% Price 0 = ( 60% *$ 8 0 +40% *$ 55)/ (1 +8%) = $ 64.81. 4 0%. Binomial Option Pricing Model. t 0. t 1. 60%. Interest rate =8% Exercise price = $70
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Option Pricing BA 543 Aoyang Long
Binomial pricing model • Black—Scholes model Agenda
t0 t1 60% • Interest rate =8% • Price0 = (60%*$ 80+40%*$ 55)/(1+8%) = $ 64.81 40% Binomial Option Pricing Model
t0 t1 60% • Interest rate =8% • Exercise price= $70 • Value of call = (60%*$ 10) / (1+8%) = $ 5.56 40% Binomial Option Pricing Model
t0 t1 t2 t3 t4 90 60% 40% 60% 80 40% 60% 60% 60% 40% 40% 60% 70 How many path for a stock price of $80? • Price0 40% 60% 60% 40% 40% 60% Multiple Periods 60 40% 40% 60% 40% 50
Each number in the triangle is the sum of the two directly above it. Pascal’s Triangle
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 = 182.5 days • σ = 40.69% • Black-Scholes Calculator Black—Scholes Model
Binomial pricing model: • discrete model • both European and American call • slow • Black—Scholes model: • continuous model • European call • quick Summary