1. 1 OPTIONS Call Option Put Option Option premium Exercise (striking) price Expiration date In, out-of, at-the-money options American vs European Options

2. 2 Option Valuation Valuation of a call option at Expiration = max{P-X, 0} P VcVc X Valuation of a put option at expiration: max{X - P, 0} P VpVp X

3. 3 Option Valuation (Cont’d) Binominal Call Pricing (one period) 7040% P 0 = 50 45-10% 70 - 50 =20 V 0 = ? 0 70 - 45 255 Hedge Ratio = = = 20 - 0 20 4 HR: number of calls sold for each stock bought Buy 1 shr of stock, sell 1.25 calls If P 1 =$45, portfolio value = $45 If P 1 =$70, portfolio value = 70 - 20(1.25)=45 Return = 45/(50-1.25V c )-1 = 0.10 V c = $7.27

4. 4 Option Valuation (Cont’d Binominal Call Pricing (two periods) P 2 =98.00 V 2 =48.00 P 1 =70.00 V 1 =24.55 P 2 =63.00 V 2 =13.00 P 0 =50.00 V 0 =11.60P 2 =63.00 V 2 =13.00 P 1 =45.00 V 1 =4.73 P 2 =40.50 V 2 =0

5. 5 Option Valuation (Cont’d At T=1, If P 1 = $70.00 HR = (98.00 - 63.00)/(48.00 - 13.00) = 1 Buy 1 stock, sell 1 call If P 2 = 98.00 Port. Value = 98 - 48 = 50 P 2 = 63.00 Port. Value = 63 - 13 = 50 1+Return = 50/(70 - V 1 ) = 1.1 V1 = $24.55 At T=1, If P 1 = $40.50 HR = (63.00 - 40.50)/(13.00) = 1.73 Buy 1 stock, sell 1.73 call If P 2 = 63.00 Port. Value = 63 - 1.73x13 = 40.50 P 2 = 40.50 Port. Value = 40.50 - 0 = 40.50 1+Return = 40.50/(40.50 - 1.73V 1 ) = 1.1 V1 = $4.73

6. 6 Option Valuation (Cont’d At T=0 HR = (70.00 - 45.00) / (24.55 - 4.73)= 1.26 Buy 1 stock, sell 1.26 call If P 1 = 70.00 Port. value = 70 - 1.26x24.55 =39.07 P 1 = 45.00 Port. Value = 45 - 1.26x4.73 = 39.07 Return = 39.07 / (50 - 1.26V 0 ) = 1.1 V 0 = $11.60

7. 7 Black and Scholes OPM d 1 and d 2 are deviations from the expected value of a unit normal distribution. N(d) is the probability of getting a value below d.

8. 8 Black and Scholes Eg. P 0 = $50.00X = $50.00 R f =10%  =0.60 d 1 ={ ln(50/50) + [0.10+ (1/2)0.60 2 ]1} / 0.60 = 0.28 / 0.60 = 0.4667 d2 = 0.4667 - 0.60 = -0.1333 N(0.4667) = 0.6796N(-0.1333) = 0.4470 V c = 50 (0.6796) - 50 e -0.10 (0.4470) = $13.76

9. 9 Put-Call Parity Buy a share at P, sell a call, buy a put at the same exercise price (X) as call. Value of Portfolio if P X Stock P P call 0X-P putX-P 0 Portfolio X X Therefore the value of the portfolio today must be equal to the PV of X: P + V p -V C = X/(1 +R f ) or V p = V c + X/(1 +R f ) - P

10. 10 Option Investment Strategies Writing covered calls - buy stock, write cals Synthetic long: Buy call, sell put

11. 11 Option Investment Strategies Straddle: simultaneously buying puts and calls with the same X and t on the same underlying asset Long Straddle Short Straddle

12. 12 Option’s Delta, Gamma, and Theta Delta: Rate of change in position value in response to a change in the value of the underlying asset. Gamma: Rate of change in delta in response to change in the value of the underlying asset. Theta: Change in position value as time to expiration gets closer (other things being the same) delta zero; gamma +

13. 13 Portfolio Insurance Investing in a portfolio of stocks and a put option on the portfolio simultaneously. The problem is when you cannot find a put option on your portfolio.

14. 14 Portfolio Insurance Cont’d Alternatively one can combine stock portfolio with the risk free asset to have the same portfolio insurance, using OPM: N(d 1 ) = slope of the call option value. It gives the fall in position value for a decline of $1 in stock value. For portfolio insurance, invest 1 -N(d 1 ) in t-bills, and N(d 1 ) in the risky portfolio. Potential problem