OPTIONS Call Option Put Option premium Exercise striking

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

Option Valuation • Valuation of a call option at Expiration = max{P-X, 0} Vc P X Valuation of a put option at expiration: max{X - P, 0} Vp P X 2

Option Valuation (Cont’d) Binominal Call Pricing (one period) 70 40% P 0 = 50 45 -10% 70 - 50 =20 V 0 = ? 0 70 - 45 25 5 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. 25 Vc)-1 = 0. 10 Vc = $7. 27 3

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. 60 P 2=63. 00 V 2=13. 00 P 1=45. 00 V 1=4. 73 P 2=40. 50 V 2=0 4

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 V 1 = $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. 73 x 13 = 40. 50 P 2 = 40. 50 Port. Value = 40. 50 - 0 = 40. 50 1+Return = 40. 50/(40. 50 - 1. 73 V 1) = 1. 1 V 1 = $4. 73 5

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. 26 x 24. 55 =39. 07 P 1 = 45. 00 Port. Value = 45 - 1. 26 x 4. 73 = 39. 07 Return = 39. 07 / (50 - 1. 26 V 0) = 1. 1 V 0 = $11. 60 6

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. 7

Black and Scholes Eg. P 0= $50. 00 X = $50. 00 Rf =10% =0. 60 d 1 ={ ln(50/50) + [0. 10+ (1/2)0. 602 ]1} / 0. 60 = 0. 28 / 0. 60 = 0. 4667 d 2 = 0. 4667 - 0. 60 = -0. 1333 N(0. 4667) = 0. 6796 N(-0. 1333) = 0. 4470 Vc = 50 (0. 6796) - 50 e-0. 10 (0. 4470) = $13. 76 8

Put-Call Parity Buy a share at P, sell a call, buy a put at the same exercise price (X) as call. Stock call put Portfolio Value of Portfolio if P<X P>X P P 0 X-P 0 X X Therefore the value of the portfolio today must be equal to the PV of X: P + Vp -VC = X/(1 +Rf) or Vp = Vc + X/(1 +Rf) - P 9

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

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

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 + 12

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. 13

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 14