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Class Business

Class Business. Upcoming Groupwork Course Evaluations. Stock Price Tree. Option Price Tree. Binomial Option Pricing: Call Option on Dell. Find value of a corresponding call option with X=65:. 66. 1. H. 60. ?. L. 54. 0. Binomial Option Pricing: Call Option on Dell.

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Class Business

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  1. Class Business • Upcoming Groupwork • Course Evaluations

  2. Stock Price Tree Option Price Tree Binomial Option Pricing:Call Option on Dell • Find value of a corresponding call option with X=65: 66 1 H 60 ? L 54 0

  3. Binomial Option Pricing:Call Option on Dell • Do we know how to price the replicating portfolio? Yes: • We know the price of the stock is $60 • 1/12 shares of the stock will cost$5 • When we short $4.49 of the bond • we get $4.48 • Total cost of replicating portfolio is • 5.00- 4.48 = 0.52 • This is the price of the option. Done.

  4. Stock Price Tree Option Price Tree Binomial Option Pricing:Put Option On Dell • Find value of a corresponding put option with X=65: 66 0 60 ? 54 11

  5. Binomial Option Pricing:Put Option on Dell • We want to find D and B such that • D66 and D54 are the payoffs from holding D shares of the stock • B(1.01)1/2 is the payoff from holding B shares of the bond • Mathematically possible • Two equations and two unknowns

  6. Binomial Option Pricing:Put Option on Dell • Shortcut to finding D: • Subscripts: • H – the state in which the stock price is high • L – the state in which the stock price is low

  7. Binomial Option Pricing:Put Option on Dell • Once we know D, it is easy to find B • So if we • short 11/12 shares of stock • buy $60.20 of the bond • Then we have a portfolio that replicates the option

  8. Binomial Option Pricing:Put Option on Dell • Do we know how to price the replicating portfolio? Yes: • The price of the stock is $60 • When we short 11/12 shares of the stock we will get$55.00 • To buy $60.20 of the bond • This will cost $60.20 • Total cost of replicating portfolio is • 60.20 - 55.00 = 5.20 • This is the price of the option. Done.

  9. Insights on Option Pricing • The value of a derivative • Does not depend on the investor’s risk-preferences. • Does not depend on the investor’s assessments of the probability of low and high returns. • To value any derivative, just find a replicating portfolio. • The procedures outlined above apply to any derivative with any payoff function

  10. Binomial Trees in Practice S0u4 S0u3 S0u2 S0u2 S0u S0u S0 S0 S0 S0d S0d S0d2 S0d 2 S0d3 S0d4

  11. Black-Scholes Option Valuation Co= Soe-dTN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2) d2 = d1 - (s T1/2) where Co = Current call option value. So= Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.

  12. Black-Scholes More definitions Co= Soe-dTN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2) d2 = d1 - (s T1/2) X = Exercise (strike) price. d = Annual dividend yield of underlying stock e = 2.71828, the base of the natural log. r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option. T = time to maturity of the option in years. ln = Natural log function s = Standard deviation of annualized cont. compounded rate of return on the stock

  13. Call Option ExampleDell Stock So = 100 X = 95 r = .10 T = .25 (quarter) s = .50 d = 0 d1 = [ln(100/95)+(.10-0+(.52/2))*.25]/(.5*.251/2) = .43 d2 = .43 - ((.5)( .251/2) = .18

  14. Probabilities from Normal Dist.Get from Table 15.2 (page 544-5) N (.43) = .6664 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700 N (.18) = .5714

  15. Call Option Value Co= Soe-dTN(d1) - Xe-rTN(d2) Co = 100*(.6664) – 95*(e- .10 X .25)*.5714 Co = 13.70

  16. Put Option Value: Black-Scholes P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)] Price a put option with X=95 and T = .25 Using the example data P = $95*(e(-.10X.25))*(1-.5714) - $100*(1-.6664) P = $6.35

  17. Implied Volatility • Suppose the actual price of the Dell call that expires in 6 months is currently $15.00 • But Black-Scholes says it should be $13.70 • What is going on? • Is there an arbitrage opportunity? • To compute the Black-Scholes values we assumed that the volatility of the stock over the life of the option is constant at 50%. • Is this a good assumption? Maybe not.

  18. Implied Volatility Two ways to use Black-Scholes model: Inputs: S, X, T, s, h Output = price MODEL Inputs: S, X, T, h, option price Output = s MODEL

  19. Implied Volatility • Which volatility is consistent with the call price of Dell? • The Black-Scholes formula gives: • A call price of $13.70 if the volatility is 50%, • A put price of $6.35 if the volatility is 50%. • This volatility is called “implied volatility.”

  20. Implied Volatility of S&P 500 1990-2003 Source: CBOE

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