1. 1 16-Option Valuation

2. 2 Pricing Options Simple example of no arbitrage pricing: Stock with known price: S 0 =$3 Consider a derivative contract on S: Payoff is 2*S T, where S T is the value of the stock at time T Assume stock pays no dividends What is price of derivative contract? Answer is simple to find since payoff is a linear function of the payoff on A. Can “replicate” derivative by buying 2 shares of A

3. 3 Binomial Option Pricing: Call Option on Dell The current price is S 0 = $60. After six months, the stock price will either grow to $66 or fall to $54. Pick what ever probabilities you want. The annual risk-free interest rate is 1%. Assume yield curve is flat What is the value of a call option with a strike price of $65 that expires after 6 months?

4. 4 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:

5. 5 Binomial Option Pricing: Call Option on Dell Claim: we can use the stock along with a risk- free bond to replicate the option Replicating portfolio: Position of  shares of the stock If  is positive, that means you “own” the stock If  is negative, that means you are “short” the stock Position of $B in bonds (B=present value) If B is positive, that means you “own” the bond If B is negative, that means you are “short” the bond

6. 6 Binomial Option Pricing: Call Option on Dell Strategy: If we know that holding  shares of stock and $B in bonds will replicate the payoffs of the option, then we know the cost of the option is  S 0 + B Example: Suppose the stock is currently $60, and we find that holding 1 share of stock and shorting $55 in bonds will give us the exact same payoffs as the option (in either state). Then we know the price of the option is ________. 60-55 = 5

7. 7 Binomial Option Pricing: Call 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 of the bond Mathematically possible Two equations and two unknowns

8. 8 Binomial Option Pricing: Call 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

9. 9 Binomial Option Pricing: Call Option on Dell Once we know , it is easy to find B So if we buy 1/12 shares of stock Short $4.48 of the bond Then we have a portfolio that replicates the option

10. 10 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.

11. 11 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:

12. 12 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

13. 13 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

14. 14 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

15. 15 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.

16. 16 Binomial Option Pricing: Call Option on Dell You can assume  can be a fraction – that is you can buy a fraction of a share of stock. This assumption does not make the answers “unrealistic” Suppose you could replicate the payoff of one option by buying 4.25 shares of the stock and shorting $25 in bonds. Then 17 shares of stock and shorting $100 in bonds would replicate the payoff of four options. The price of four options would be _______________________________________________ Or rather, the price of one option would be _______________________________________________ the price of 17 shares of stock minus $100. the price of 4.25 shares of stock less $25.

17. 17 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

18. 18 Multi-Period Binomial Model A shortcoming of the binomial model is that future stock prices can only take two possible values and that stock prices change only once during the period. We can generalize our binomial model by cutting time into smaller pieces and modeling what prices can do over those sub-periods.

19. 19 Two-period Binomial Model for Stock Prices 70 56 84 70 77 63 Time=0 Time=3 mo Time=6 mo

20. 20 Two-period Binomial Model for Stock Prices Price a call option with strike of 80 Suppose the stock price in 3 months is $77. Use method above to price call option at this point  = B= Price of call is $77 $84 $70 0.2857 -19.95 [Solve 0.2857(70) + B(1.01) 1/4 = 0] for B 0.2857(77) – 19.95= 2.05 Stock Call 4 0

21. 21 Two-period Binomial Model for Stock Prices Suppose the stock price in 3 months is $63. What is the value of the call option struck at 80? $63 $70 $56

22. 22 Two-period Binomial Model for Stock Prices In 3 months the call value will be either 2.05 (if the stock price is at $77) 0 (if the stock price is at $63) If you buy the call option now and were to “sell it” in three months, what would be payoff? 2.05 (if the stock price is at $77) 0 (if the stock price is at $63)

23. 23 Two-period Binomial Model for Stock Prices The payoff trees over the next three months: Use method above to price call option at this point  = B= Price of call is Stock Option 70 77 63 2.05 0.00 ? 2.05/14= 0.1464 -9.20 [Solve 0.1464(63) + B(1.01) 1/4 = 0] for B (0.1464)70-9.20 =1.048

24. 24 Two-period Binomial Model for Stock Prices To model stock prices, pick up and down movements to match expected return (e.g. from the CAPM) estimated volatility Involves solving two equations and two unknowns. Reference: Hull “Options, Futures, and Other Derivatives” page 213-214

25. 25 What Did Black-Scholes Do? Showed how to replicate an option assuming the stock price has a “log-normal” distribution, not just two possible outcomes. Actually, the simple binomial approach was developed after Black and Scholes solved the more complex problem. If we cut time into infinitely small pieces, the binomial model converges to the Black-Scholes solution.

26. 26 Delta Hedging Delta Hedging: the practice of replicating an option by using just the stock and the bond. When would you do it? When the option doesn’t exist You want to “tailor” the risk of an option position. How do you do it? Figure out the delta (position in stock) and B (the position in bonds). Delta and B will change as other factors change, such as the stock price and time to maturity. Requires heavy portfolio rebalancing.

27. 27 Black-Scholes Formula The Black-Scholes Formula for European options on stocks paying no dividends is: where S = current stock price X = strike price h = “continuously compounded” risk-free rate T = time until option expires = standard deviation of stock return (not price)

28. 28 Black-Scholes Formula is a “continuous time” discounting factor In discrete time the discounting factor would be Example: suppose the 1-year risk-free rate is 10% and the continuously compounded rate is 9.53% What is the PV of $100 received 1-year from now? 100/1.10 =90.91 100e -.0953 =90.91

29. 29 Black-Scholes Formula N(d) is the cumulative distribution function for a standard normal random variable - use Excel normsdist function (not normdist).

30. 30 Black-Scholes Formula The price of a European put is given by: