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Option Pricing

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

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  1. Option Pricing BA 543 Aoyang Long

  2. Binomial pricing model • Black—Scholes model Agenda

  3. t0 t1 60% • Interest rate =8% • Price0 = (60%*$ 80+40%*$ 55)/(1+8%) = $ 64.81 40% Binomial Option Pricing Model

  4. t0 t1 60% • Interest rate =8% • Exercise price= $70 • Value of call = (60%*$ 10) / (1+8%) = $ 5.56 40% Binomial Option Pricing Model

  5. 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

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

  7. Lognormal Distribution

  8. Lognormal Distribution

  9. 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

  10. 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

  11. 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

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

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