1. Black-Scholes Pricing & Related Models

2. Option Valuation  Black and Scholes  Call Pricing  Put-Call Parity  Variations

3. Option Pricing: Calls  Black-Scholes Model: C = Call S = Stock Price N = Cumulative Normal Distrib. Operator X = Exercise Price e = 2.71..... r = risk-free rate T = time to expiry = Volatility

4. Call Option Pricing Example  IBM is trading for $75. Historically, the volatility is 20% (  A call is available with an exercise of $70, an expiry of 6 months, and the risk free rate is 4%. ln(75/70) + (.04 + (.2) 2 /2)(6/12) d 1 = -------------------------------------------- =.70, N(d 1 ) =.7580.2 * (6/12) 1/2 d 2 =.70 - [.2 * (6/12) 1/2 ] =.56, N(d 2 ) =.7123 C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98 Intrinsic Value = $5, Time Value = $2.98

5. Put Option Pricing  Put priced through Put-Call Parity: Put Price = Call Price + X e -rT - S From Last Example of IBM Call: Put = $7.98 + 70 e -.04(6/12) - 75 =$1.59 Intrinsic Value = $0, Time Value = $1.59 )()( 12 dSNdNXeP rT ---= - (or :)

6. Black-Scholes Variants  Options on Stocks with Dividends  Futures Options (Option that delivers a maturing futures)  Black’s Call Model (Black (1976))  Put/Call Parity  Options on Foreign Currency  In text (Pg. 375-376, but not req’d)  Delivers spot exchange, not forward!

7. The Stock Pays no Dividends During the Option’s Life  If you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium  Robert Merton developed a simple extension to the BSOPM to account for the payment of dividends

8. The Stock Pays Dividends During the Option’s Life (cont’d) Adjust the BSOPM by following (  =continuous dividend yield): Tdd T T  R X S d dNXedSNeC RT TT                       * 1 * 2 2 * 1 * 2 * 1 * and 2 ln where )()(

9. Futures Option Pricing Model  Black’s futures option pricing model for European call options:

10. Futures Option Pricing Model (cont’d)  Black’s futures option pricing model for European put options:  Alternatively, value the put option using put/call parity:

11. Assumptions of the Black- Scholes Model  European exercise style  Markets are efficient  No transaction costs  The stock pays no dividends during the option’s life (Merton model)  Interest rates and volatility remain constant, but are unknown

12. Interest Rates Remain Constant  There is no real “riskfree” interest rate  Often use the closest T-bill rate to expiry

13. Calculating Volatility Estimates  from Historical Data: S, R, T that just was, and  as standard deviation of historical returns from some arbitrary past period  from Actual Data: S, R, T that just was, and  implied from pricing of nearest “at-the-money” option (termed “implied volatility).

14. Intro to Implied Volatility  Instead of solving for the call premium, assume the market- determined call premium is correct  Then solve for the volatility that makes the equation hold  This value is called the implied volatility

15. Calculating Implied Volatility  Setup spreadsheet for pricing “at-the- money” call option.  Input actual price.  Run SOLVER to equate actual and calculated price by varying .

16. Volatility Smiles  Volatility smiles are in contradiction to the BSOPM, which assumes constant volatility across all strike prices  When you plot implied volatility against striking prices, the resulting graph often looks like a smile

17. Volatility Smiles (cont’d)

18. Problems Using the Black- Scholes Model  Does not work well with options that are deep-in- the-money or substantially out-of-the-money  Produces biased values for very low or very high volatility stocks  Increases as the time until expiration increases  May yield unreasonable values when an option has only a few days of life remaining