# Options/1. Prof. Ian Giddy

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1 Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

2 Options/2 Options Payoff Call Option 0 Stock Price Options3 Payoffs and Profits on Options at Expiration - Calls Notation Stock Price = ST Exercise Price = X Payoff to Call Holder (ST - X) if ST >X 0 if ST < X Profit to Call Holder Payoff - Purchase Price Options6

3 Options/3 Payoff Profiles for Calls Payoff Call Holder 0 Call Writer Stock Price Options7 Payoff Profiles for Puts Payoffs Put Writer 0 Put Holder Stock Price Options8

4 Options/4 Put-Call Parity Relationship S T < X S T > X Payoff for Call Owned 0 S T - X Payoff for Put Written -( X -ST) 0 Total Payoff S T - X S T - X Options9 Payoff of Long Call & Short Put Payoff Combined = Leveraged Equity Long Call Short Put Stock Price Options10

5 Options/5 Arbitrage & Put Call Parity Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. C - P = S 0 - X / (1 + r f ) T If the prices are not equal arbitrage will be possible Cost of buying stock by borrowing \$X Cost of buying a call and selling a put Options11 Put Call Parity - Disequilibrium Example Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free rate = 10.25% Maturity =.5 yr Strike price X = 105 C - P > S 0 - X / (1 + r f ) T 17-5 > (105/1.05) 12 > 10 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative Options12

6 Options/6 Put-Call Parity Arbitrage Position Immediate Cashflow in Six Months Cashflow S T <105 S T > 105 Buy Stock -110 S T S T Borrow X/(1+r) T = Sell Call (S T -105) Buy Put S T 0 Total Option Strategies Protective Put Long Stock Long Put Covered Call Long Stock Short Call Straddle (Same Exercise Price) Long Call Long Put Options14

7 Options/7 Option Combinations Four classic ways of combining an option with a futures to create the opposite option Profit + 0 _ Profit + 0 _ LONG CALL PLUS SHORT FORWARD CREATES SYNTHETIC LONG PUT Strike LONG PUT PLUS LONG FORWARD CREATES SYNTHETIC LONG CALL Strike Forward Forward Rate Profit + 0 _ Profit + 0 _ SHORT CALL PLUS LONG FORWARD CREATES SYNTHETIC SHORT PUT Strike SHORT PUT PLUS SHORT FORWARD CREATES SYNTHETIC SHORT CALL Strike Forward Rate Forward Rate Options16 When Use Options to Take a View? View on direction View on volatility Direction: Volatility Volatility increasing Volatility falling No trend in volatility Security rising Security falling No trend Buy call Buy put Buy straddle Sell put Sell call Sell straddle Buy Sell Arbitrage forward forward Options17

8 Options/8 View on Direction, Volatility or Both? Options18 Option Valuation

9 Options/9 Options Payoff Call Option 0 Stock Price Options20 Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value Options21

10 Options/10 Option Pricing Model CALL OPTION PRICE EURODOLLAR OPTION PRICES Measured in percent per annum FUTURES PRICE Options22 Factors Influencing Option Values: Calls Factor Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend Rate Effect on value increases decreases increases increases increases decreases Options23

11 Options/11 Option Pricing Time Option Price value depends on Time Volatility Distance from the strike price Option Price = Intrinsic value + Time value Underlying Price Options24 Option Pricing Model ENTER THESE DATA: ================= -> FUTURES PRICE > STRIKE PRICE > TIME IN DAYS 300 -> INTEREST RATE 7 -> STD DEVIATION CALL PRICE IS PUT PRICE IS CALL OPTION PRICE FUTURES PRICE Options25

12 Options/12 Value of Call Option FUTURES PRICE STRIKE SHADED AREA: Probability distribution of the log of the futures price on the expiration date for values above the strike. INTRINSIC VALUE TIME VALUE EXPECTED VALUE OF PROFIT GIVEN EXERCISE Options26 Black-Scholes Option Valuation Call value = S o N(d 1 ) - Xe -rt N(d 2 ) d 1 = [ln(s o /X) + (r + σ 2 /2)T] / (σ T 1/2 ) d 2 = d 1 - (σ T 1/2 ) where S o = Current stock price X = Strike price, T = time, r = interest rate N(d) = probability that a random draw from a normal distribution will be less than d. Options27

13 Options/13 Black-Scholes Option Valuation X = Exercise price. e = , 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 σ = Standard deviation of annualized cont. compounded rate of return on the stock Options28 How a Change in the Futures Price Changes the Option s Price Profit (gain or loss) Time Value + F-K 0 K F Market price of instrument _ Time Value K F E(FJ*FJ>K) Options29

14 Options/14 Call Option Example S o = 100 X = 95 r =.10 σ =.50 T =.25 (quarter) d 1 = [ln(100/95) + (.10+(.5 2 /2))] / ( /2 ) =.43 d 2 =.43 - ((.5)(.25 1/2 ) =.18 Options30 Probabilities from Normal Dist. N (.43) =.6664 d N(d) Interpolation Options31

15 Options/15 Probabilities from Normal Dist. N (.18) =.5714 d N(d) Options32 Call Option Value C o = S o N(d 1 ) - Xe -rt N(d 2 ) C o = 100 X e -.10 X.25 X.5714 C o = Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Options33

16 Options/16 Put Option Valuation: Using Put-Call Parity P = C + PV (X) - S o = C + Xe -rt - S o Using the example data C = X = 95 S = 100 r =.10 T =.25 P = e -.10 X P = 8.35 Options34 Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d 1 ) Put = N (d 1 ) - 1 Option Elasticity Percentage change in the option s value given a 1% change in the value of the underlying stock Options35

17 Options/17 Portfolio Insurance - Protecting Against Declines in Stock Value Buying Puts - results in downside protection with unlimited upside potential Limitations Tracking errors if indexes are used for the puts Maturity of puts may be too short Hedge ratios or deltas change as stock values change Options36 Trading Options: Risk Management Concepts Hedger s view: An option is an insurance premium--you can gain, but you cannot lose more than the premium you pay. Trader s view: An option is just another security--i can buy it or sell it, take a long or short position in puts or calls. Its price fluctuates with market variables. For any position, the gain or loss is the change in price of the position. Options37

18 Options/18 Trading Options: Delta Hedging Delta We ve written an option Options38 Delta Hedging Option Value HEDGE RATIO =80% HEDGE RATIO =50% HEDGE RATIO =15% Forward or Futures Price Options39

19 Options/19 Trading Options: Delta Hedging with Futures Delta or hedge ratio is the change in the option price for a given change in the price of the underlying. Delta is also the probability of exercise. Delta is also the hedge ratio that tells one how many futures you need to hedge a given options position. Options40 Trading Options: Delta Hedging Hedged with 40% short futures Delta We ve written a put option Options41

20 Options/20 Options Trading: Delta Hedging and the Gamma Delta is not constant: it s low for out of the money options, high for in the money options. Change in delta makes it tough to know exactly how many futures to use. Change in delta is the gamma. Positive gamma is nice. Writing options produces negative gamma, which is nasty. Options42 Goal: Understand Options Sensitivity An option trader has a portfolio of options with different deltas, gammas, etc. The goal is to discover the sensitivities of the portfolio to changes in rates, time, volatility, etc, and to neutralize them. Greek Measures Delta Sensitivity of portfolio value to change in price of the underlying asset Γ Gamma Sensitivity of delta to change in price of underlying asset θ Theta Sensitivity of portfolio value to change in time Λ Lambda (Vega) Sensitivity of portfolio value to change in volatility Ρ Rho Sensitivity of portfolio to change in interest rate Options43

21 Options/21 Options: Summary Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options44 Appendix: Option Pricing: Trees The Idea: A risk-free portfolio must earn the risk-free rate of return. Options45

22 Options/22 Binomial Tree Consider a call option on a Treasury Bill The strike price is set at \$21. The underlying bill could go to \$22 or \$18 in 1 month. UP Bill=\$22 Bill=\$20 Call=\$1 Call=? Bill=\$18 DOWN Call=\$0 A risk-free portfolio must earn the risk-free rate of return. Options46 Binomial Tree Consider a call option on a Treasury Bill The strike price is set at \$21. The underlying bill could go to \$22 or \$18 in 1 month. UP Bill=\$20 Call=? DOWN Bill=\$22 Call=\$1 Bill=\$18 Call=\$0 PORTFOLIO: Long bills Short 1 Call Portfolio value = 22 1 Portfolio value = 18 What s the delta that will make the portfolio risk free? A risk-free portfolio must earn the risk-free rate of return. Options47

23 Options/23 Binomial Tree Consider a call option on a Treasury Bill The strike price is set at \$21. The underlying bill could go to \$22 or \$18 in 1 month. UP Bill=\$20 Call=? DOWN Bill=\$22 Call=\$1 Bill=\$18 Call=\$0 PORTFOLIO: Long bills Short 1 Call Portfolio value = 22 1 Portfolio value = 18 If =0.25, Portfolio = \$4.5 \$4.5 A risk-free portfolio must earn the risk-free rate of return. Discounting the certain outcome of \$4.5 by the risk-free rate (10%) and subtracting cost of bills leads to the call price: PV(4.5)-5, ie \$0.91. Options48 Plain Vanilla PLAIN OPTION PLAIN OPTION Option call Option call Maturity 3 <5 Maturity 3 <5 Strike Strike Type Type European European Expired Expired On the bond 39.33% On the bond 39.33% Maturity 5 <6 Maturity <6 5 Coup. Rate Coup. Rate 1% 1% % % Expired Expired % 26.36% 23.78% 26.36% % 22.89% 12.52% 22.89% Option 0.09 Expired Option 0.09 Expired Bond Bond Fwrd rate 2.00% 15.94% 17.67% Fwrd rate 2.00% 15.94% 17.67% % 15.35% 8.39% 15.35% 0.39 Expired 0.39 Expired % 11.85% 10.69% 11.85% % 10.29% Expired Expired % 7.94% Options49

24 Options/24 Barrier Options This Down-and-In option is an example of pathdependent options, where how you get there matters. BARRIER OPTIONs Down-and-out call 39.33% Maturity Barrier % Strike Type European % Down-and-in call Maturity % Barrier 8 Strike Type European % On the bond 22.89% Maturity Coup. Rate 10% % % % % % % % % Knock-out 7.35 Knock-in Bond % Fwrd rate 2.00% % % % % % % % % % % % % Options50 Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Valuation of Exotics Options51

25 Options/25 Option Valuation: Applications Warrant bonds Convertibles Callable bonds Corporate valuation Options52 What s a Company Worth? Required returns Types of Models Balance sheet models Dividend discount models Corporate cash flow models Price/Earnings ratios Estimating Growth Rates Application IBM Options66

26 Options/26 What s a Company Worth? Alternative Models The options approach Option to expand Option to abandon Creation of key resources that another company would pay for Patents or trademarks Teams of employees Customers Examples? Lycos Messageclick.com Options67 What s a Company Worth? The Options Approach Value of the Firm or project Present Value of Expected Cash Flows if Option Excercised Options68

27 Options/27 The Option to Expand PV of Cash Flows from Expansion Additional Investment to Expand Firm will not expand in this section Expansion becomes attractive in this section Present Value of Expected Cash Flows on Expansion Options74 An Example of a Corporate Option J&J is considering investing \$110 million to purchase an internet distribution company to serve the growing on-line market. A financial analysis of the cash flows from this investment suggests that the present value of the cash flows from this investment to J&J will be only \$80 million. Thus, by itself, the corporate venture has a negative NPV of \$ 30 million. If the on-line market turns out to be more lucrative than currently anticipated, J&J could expand its reach a global on-line market with an additional investment of \$ 150 million any time over the next 2 years. While the current expectation is that the cash flows from having a worldwide on-line distribution channel is only \$100 million, there is considerable uncertainty about both the potential for such an channel and the shape of the market itself, leading to significant variance in this estimate. This uncertainty is what makes the corporate venture valuable! Options75

28 Options/28 Valuing the Corporate Venture Option Value of the underlying asset (S) = PV of cash flows from purchase of on-line selling venture, if done now =\$100 Million Strike Price (K) = cost of expansion into global on-line selling = \$150 Million We estimate the variance in the estimate of the project value by using the annualized variance in firm value of publicly traded online marketing firms in the global markets, which is approximately 20%. Variance in Underlying Asset s Value = 0.20 Time to expiration = Period for which venture option applies = 2 years 2-year interest rate: 6.5% Options76 Black-Scholes Option Valuation Call value = S o N(d 1 ) - Xe -rt N(d 2 ) d 1 = [ln(s o /X) + (r + σ 2 /2)T] / (σ T 1/2 ) d 2 = d 1 - (σ T 1/2 ) where S o = Current stock price X = Strike price, T = time, r = interest rate N(d) = probability that a random draw from a normal distribution will be less than d. Options77

29 Options/29 Valuing the Corporate Venture Option Value of the underlying asset (S) = PV of cash flows from purchase of on-line selling venture, if done now =\$100 Million Strike Price (X) = cost of expansion into global on-line selling = \$150 Million We estimate the variance in the estimate of the project value by using the annualized variance in firm value of publicly traded online marketing firms in the global markets, which is approximately 20%. Variance in Underlying Asset s Value = 0.20 Time to expiration = Period for which venture option applies = 2 years 2-year interest rate: 6.5% Call Value= 100 (0.7915) -150 (exp(-0.065)(2) (0.3400) = \$ 52.5 Million Options78 Summary Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Valuation of Exotics Applications to corporate securities and corporate valuation Options79

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