1. Class Business Upcoming Groupwork Course Evaluations

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

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

5. Binomial Option Pricing: Put Option on Dell We want to find  and B such that  and  54 are the payoffs from holding  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  : 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 , 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 S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u 2S0u 2 S0d 2S0d 2 S0u 2S0u 2 S0u 3S0u 3 S0u 4S0u 4 S 0 d 2 S0uS0u S0dS0d S0d 4S0d 4 S0d 3S0d 3

11. Black-Scholes Option Valuation C o = S o e -  T N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r –  +  2 /2)T] / (  T 1/2 ) d 2 = d 1 - (  T 1/2 ) where C o = Current call option value. S o = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.

12. Black-Scholes More definitions C o = S o e -  T N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r –  +  2 /2)T] / (  T 1/2 ) d 2 = d 1 - (  T 1/2 ) X = Exercise (strike) price.  = 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  Standard deviation of annualized cont. compounded rate of return on the stock

13. Call Option Example Dell Stock S o = 100X = 95 r =.10T =.25 (quarter)  =.50  = 0 d 1 = [ln(100/95)+(.10-0+(  5 2 /2))*.25]/(  5*.25 1/2 ) =.43 d 2 =.43 - ((  5 .25 1/2 ) =.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 C o = S o e -  T N(d 1 ) - Xe -rT N(d 2 ) C o = 100*(.6664) – 95*(e -.10 X.25 )*.5714 C o = 13.70

16. Put Option Value: Black- Scholes P=Xe -rT [1-N(d 2 )] - S 0 e -  T [1-N(d 1 )] 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, , h MODEL Output = price Inputs: S, X, T, h, option price MODEL Output = 

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