17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value of other securities. Options are traded both on organized exchanges and OTC. The Option Contract: Calls 17-3 The Option Contract: Puts 17-4 A call option gives its holder the right to buy an asset: At the exercise or strike price On or before the expiration date Exercise the option to buy the underlying asset if market value > strike. A put option gives its holder the right to sell an asset: At the exercise or strike price On or before the expiration date Exercise the option to sell the underlying asset if market value < strike.
The Option Contract 17-5 Example 17.1 Profit and Loss on a Call 17-6 The purchase price of the option is called the premium. Sellers (writers) of options receive premium income. If holder exercises the option, the option writer must make (call) or take (put) delivery of the underlying asset. A January 2010 call on IBM with an exercise price of $130 was selling on December 2, 2009, for $2.18. The option expires on the third Friday of the month, or January 15, 2010. If IBM remains below $130, the call will expire worthless. Example 17.1 Profit and Loss on a Call 17-7 Example 17.2 Profit and Loss on a Put 17-8 Suppose IBM sells for $132 on the expiration date. Option value = stock price-exercise price $132- $130= $2 Profit = Final value Original investment $2.00 - $2.18 = -$0.18 Option will be exercised to offset loss of premium. Call will not be strictly profitable unless IBM s price exceeds $132.18 (strike + premium) by expiration. Consider a January 2010 put on IBM with an exercise price of $130, selling on December 2, 2009, for $4.79. Option holder can sell a share of IBM for $130 at any time until January 15. If IBM goes above $130, the put is worthless.
17-9 17-10 Example 17.2 Profit and Loss on a Put Market and Exercise Price Relationships Suppose IBM s price at expiration is $123. Value at expiration = exercise price stock price: $130 - $123 = $7 Investor s profit: $7.00 - $4.79 = $2.21 Holding period return = 46.1% over 44 days! In the Money - exercise of the option would be profitable Call: exercise price < market price Put: exercise price > market price Out of the Money - exercise of the option would not be profitable Call: market price < exercise price. Put: market price > exercise price. At the Money - exercise price and asset price are equal American vs. European Options American - the option can be exercised at any time before expiration or maturity European - the option can only be exercised on the expiration or maturity date In the U.S., most options are American style, except for currency and stock index options. 17-11 Different Types of Options Stock Options Index Options Futures Options Foreign Currency Options Interest Rate Options 17-12
Payoffs and Profits at Expiration Calls Notation Stock Price = S T Exercise Price = X Payoff to Call Holder (S T -X) if S T >X 0 if S T <X Profit to Call Holder Payoff - Purchase Price 17-13 Payoffs and Profits at Expiration Calls Payoff to Call Writer -(S T -X) if S T >X 0 if S T <X Profit to Call Writer Payoff + Premium 17-14 Figure 17.2 Payoff and Profit to Call Option at Expiration 17-15 Figure 17.3 Payoff and Profit to Call Writers at Expiration 17-16
Payoffs and Profits at Expiration Puts Payoffs to Put Holder 0 if S T > X (X - S T ) if S T < X 17-17 Payoffs and Profits at Expiration Puts Payoffs to Put Writer 0 if S T >X -(X - S T ) if S T < X 17-18 Profit to Put Holder Payoff - Premium Profits to Put Writer Payoff + Premium Figure 17.4 Payoff and Profit to Put Option at Expiration 17-19 Option versus Stock Investments 17-20 Could a call option strategy be preferable to a direct stock purchase? Suppose you think a stock, currently selling for $100, will appreciate. A 6-month call costs $10 (contract size is 100 shares). You have $10,000 to invest.
17-21 17-22 Option versus Stock Investments Option versus Stock Investment Strategy A: Invest entirely in stock. Buy 100 shares, each selling for $100. Investment Strategy Investment Strategy B: Invest entirely in at-the-money call options. Buy 1,000 calls, each selling for $10. (This would require 10 contracts, each for 100 shares.) Strategy C: Purchase 100 call options for $1,000. Invest your remaining $9,000 in 6-month T-bills, to earn 3% interest. The bills will be worth $9,270 at expiration. Equity only Buy stock @ 100 100 shares $10,000 Options only Buy calls @ 10 1000 options $10,000 Leveraged Buy calls @ 10 100 options $1,000 equity Buy T-bills @ 3% $9,000 Yield Strategy Payoffs 17-23 Figure 17.5 Rate of Return to Three Strategies 17-24
17-25 17-26 Strategy Conclusions Protective Put Conclusions Figure 17.5 shows that the all-option portfolio, B, responds more than proportionately to changes in stock value; it is levered. Portfolio C, T-bills plus calls, shows the insurance value of options. C s T-bill position cannot be worth less than $9270. Some return potential is sacrificed to limit downside risk. Puts can be used as insurance against stock price declines. Protective puts lock in a minimum portfolio value. The cost of the insurance is the put premium. Options can be used for risk management, not just for speculation. Covered Calls 17-27 Table 17.2 Value of a Covered Call Position at Expiration 17-28 Purchase stock and write calls against it. Call writer gives up any stock value above X in return for the initial premium. If you planned to sell the stock when the price rises above X anyway, the call imposes sell discipline.
Figure 17.8 Value of a Covered Call Position at Expiration 17-29 Straddle 17-30 Long straddle: Buy call and put with same exercise price and maturity. The straddle is a bet on volatility. To make a profit, the change in stock price must exceed the cost of both options. You need a strong change in stock price in either direction. The writer of a straddle is betting the stock price will not change much. Table 17.3 Value of a Straddle Position at Option Expiration 17-31 Figure 17.9 Value of a Straddle at Expiration 17-32
Spreads 17-33 Table 17.4 Value of a Bullish Spread Position at Expiration 17-34 A spread is a combination of two or more calls (or two or more puts) on the same stock with differing exercise prices or times to maturity. Some options are bought, whereas others are sold, or written. A bullish spread is a way to profit from stock price increases. Figure 17.10 Value of a Bullish Spread Position at Expiration 17-35 Collars 17-36 A collar is an options strategy that brackets the value of a portfolio between two bounds. Limit downside risk by selling upside potential. Buy a protective put to limit downside risk of a position. Fund put purchase by writing a covered call. Net outlay for options is approximately zero.
Put Call Parity The call-plus-bond portfolio (on left) must cost the same as the stock-plus-put portfolio (on right): X C S0 P (1 r ) T f 17-37 Put Call Parity Disequilibrium Example Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity = 1 yr X = 105 X C S0 P (1 r ) T f 117 > 115 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative 17-38 17-39 17-40 Table 17.5 Arbitrage Strategy Option like Securities Callable Bonds Convertible Securities Warrants Collateralized Loans
18-42 Option Values Option Valuation 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 Figure 18.1 Call Option Value before Expiration 18-43 Table 18.1 Determinants of Call Option Values 18-44
Restrictions on Option Value: Call 18-45 Figure 18.2 Range of Possible Call Option Values 18-46 Call value cannot be negative. The option payoff is zero at worst, and highly positive at best. Call value cannot exceed the stock value. Value of the call must be greater than the value of levered equity. Lower bound = adjusted intrinsic value: C > S 0 - PV (X) - PV (D) (D=dividend) Figure 18.3 Call Option Value as a Function of the Current Stock Price 18-47 Early Exercise: Calls 18-48 The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires. The value of American and European calls is therefore identical. The call gains value as the stock price rises. Since the price can rise infinitely, the call is worth more alive than dead.
Early Exercise: Puts 18-49 Figure 18.4 Put Option Values as a Function of the Current Stock Price 18-50 American puts are worth more than European puts, all else equal. The possibility of early exercise has value because: The value of the stock cannot fall below zero. Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money. 18-51 18-52 Binomial Option Pricing: Text Example Binomial Option Pricing: Text Example 100 Stock Price 120 90 C 10 0 Call Option Value X = 110 Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90-90 Net Payoff 0 30 18.18 30 0 Payoff Structure is exactly 3 times the Call
18-53 18-54 Binomial Option Pricing: Text Example Replication of Payoffs and Option Values 18.18 30 0 3C 30 0 Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged: Stock Value 90 120 Call Obligation 0-30 Net payoff 90 90 3C = $18.18 C = $6.06 Hence 100-3C = $81.82 or C = $6.06 18-55 18-56 Hedge Ratio Expanding to Consider Three Intervals In the example, the hedge ratio = 1 share to 3 calls or 1/3. Generally, the hedge ratio is: range of call values C H range of stock values us u 0 Cd ds 0 Assume that we can break the year into three intervals. For each interval the stock could increase by 20% or decrease by 10%. Assume the stock is initially selling at $100.
Expanding to Consider Three Intervals 18-57 Possible Outcomes with Three Intervals 18-58 S + + S + + + Event Probability Final Stock Price 3 up 1/8 100 (1.20) 3 = $172.80 S S + S - S + - S - - S + + - S + - - S - - - 2 up 1 down 3/8 100 (1.20) 2 (.90) = $129.60 1 up 2 down 3/8 100 (1.20) (.90) 2 = $97.20 3 down 1/8 100 (.90) 3 = $72.90 18-59 18-60 Black Scholes Option Valuation Black Scholes Option Valuation C o = S o N(d 1 ) - Xe -rt N(d 2 ) d 1 = [ln(s o /X) + (r + 2 /2)T] / ( T 1/2 ) d 2 = d 1 -( T 1/2 ) where C o = Current call option value S o = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualized, continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of the stock
18-61 18-62 Figure 18.6 A Standard Normal Curve Example 18.1 Black Scholes Valuation S o = 100 X = 95 r =.10 T =.25 (quarter) =.50 (50% per year) Thus: d 1 2.10 5 ln 100 95.5 0.25 0.25 2.43 d 2.43.5 0.25.18 Probabilities from Normal Distribution Using a table or the NORMDIST function in Excel, we find that N (.43) =.6664 and N (.18) =.5714. Therefore: C o = S o N(d 1 ) - Xe -rt N(d 2 ) C o = 100 X.6664-95 e -.10 X.25 X.5714 C o = $13.70 18-63 Call Option Value Implied Volatility Implied volatility is volatility for the stock implied by the option price. Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? 18-64
Black Scholes Model with Dividends The Black Scholes call option formula applies to stocks that do not pay dividends. 18-65 Example 18.3 Black Scholes Put Valuation P = Xe -rt [1-N(d 2 )] - S 0 [1-N(d 1 )] Using Example 18.2 data: 18-66 What if dividends ARE paid? S = 100, r =.10, X = 95, σ =.5, T =.25 One approach is to replace the stock price with a dividend adjusted stock price Replace S 0 with S 0 - PV (Dividends) We compute: $95e -10x.25 (1-.5714)-$100(1-.6664) = $6.35 Put Option Valuation: Using Put Call Parity 18-67 Using the Black Scholes Formula 18-68 P = C + PV (X) - S o = C + Xe -rt - S o Using the example data P = 13.70 + 95 e -.10 X.25-100 P = $6.35 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
Figure 18.9 Call Option Value and Hedge Ratio 18-69 Portfolio Insurance 18-70 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 Figure 18.10 Profit on a Protective Put Strategy 18-71 Figure 18.11 Hedge Ratios Change as the Stock Price Fluctuates 18-72
18-73 18-74 Hedging On Mispriced Options Hedging and Delta Option value is positively related to volatility. If an investor believes that the volatility that is implied in an option s price is too low, a profitable trade is possible. Profit must be hedged against a decline in the value of the stock. Performance depends on option price relative to the implied volatility. The appropriate hedge will depend on the delta. Delta is the change in the value of the option relative to the change in the value of the stock, or the slope of the option pricing curve. Delta = Change in the value of the option Change of the value of the stock Example 18.6 Speculating on Mispriced Options 18-75 Table 18.3 Profit on a Hedged Put Portfolio 18-76 Implied volatility = 33% Investor s estimate of true volatility = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate = 4% Delta = -.453
18-77 18-78 Example 18.6 Conclusions Delta Neutral As the stock price changes, so do the deltas used to calculate the hedge ratio. Gamma = sensitivity of the delta to the stock price. Gamma is similar to bond convexity. The hedge ratio will change with market conditions. When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral. The portfolio does not change value when the stock price fluctuates. Rebalancing is necessary. Table 18.4 Profits on Delta Neutral Options Portfolio 18-79 Empirical Evidence on Option Pricing 18-80 The Black-Scholes formula performs worst for options on stocks with high dividend payouts. The implied volatility of all options on a given stock with the same expiration date should be equal. Empirical test show that implied volatility actually falls as exercise price increases. This may be due to fears of a market crash.