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## Chapter 21

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**Option Valuation**Chapter 21**Option Values**• Intrinsic value - profit that could be made if the option was immediately exercised. (Alternatively, the value of the option, if today was its maturity date) • Call: Max(stock price - exercise price,0) • Put: max(exercise price - stock price,0) • Time value - the difference between the option price and the intrinsic value.**Time Value of Options: Call**Option value Value of Call Intrinsic Value Time value X Stock Price**Factors Influencing Option Values: Calls**FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases**Restrictions on Option Value: Call**• Value cannot be negative • Value cannot exceed the stock value • Value of the call must be greater than the value of levered equity C > S0 - ( X + D ) / ( 1 + Rf )T C > S0 - PV ( X ) - PV ( D )**Allowable Range for Call**Call Value Upper bound = S0 Lower Bound = S0 - PV (X) - PV (D) S0 PV (X) + PV (D)**Arbitrage**Arbitrage: • No possibility of a loss • A potential for a gain • No cash outlay • In finance, arbitrage is not allowed to persist. • “Absence of Arbitrage” = “No Free Lunch” • The “Absence of Arbitrage” rule is often used in finance to figure out prices of derivative securities. • Think about what would happen if arbitrage were allowed to persist. (Easy money for everybody)**The Upper Bound for a Call Option Price**Call option price must be less than the stock price. • Otherwise, arbitrage will be possible. • How? • Suppose you see a call option selling for $65, and the underlying stock is selling for $60. • The arbitrage: sell the call, and buy the stock. • Worst case? The option is exercised and you pocket $5. • Best case? The stock sells for less than $65 at option expiration, and you keep all of the $65. • There was zero cash outlay today, there was no possibility of loss, and there was a potential for gain.**The Upper Bound for a Put Option Price**• Put option price must be less than the strike price. Otherwise, arbitrage will be possible. • How? Suppose there is a put option with a strike price of $50 and this put is selling for $60.The Arbitrage: Sell the put, and invest the $60 in the bank. (Note you have zero cash outlay). • Worse case? Stock price goes to zero. • You must pay $50 for the stock (because you were the put writer). • But, you have $60 from the sale of the put (plus interest). • Best case? Stock price is at least $50 at expiration. • The put expires with zero value (and you are off the hook). • You keep the entire $60, plus interest.**The Lower Bound on Option Prices**• Option prices must be at least zero. • By definition, an option can simply be discarded. • To derive a meaningful lower bound, we need to introduce a new term: intrinsic value. • The intrinsic value of an option is the payoff that an option holder receives if the underlying stock price does not change from its current value.**Option Intrinsic Values**Call option intrinsic value = max [S–X,0] In words: The call option intrinsic value is the maximum of zero or the stock price minus the strike price. Put option intrinsic value = max [X-S, 0] In words: The put option intrinsic value is the maximum of zero or the strike price minus the stock price.**Option “Moneyness”**• “In the Money” options have a positive intrinsic value. • For calls, the strike price is less than the stock price. • For puts, the strike price is greater than the stock price. • “Out of the Money” options have a zero intrinsic value. • For calls, the strike price is greater than the stock price. • For puts, the strike price is less than the stock price. • “At the Money”options is a term used for options when the stock price and the strike price are about the same.**Intrinsic Values and Arbitrage, Calls**• Call options with American-style exercise must sell for at least their intrinsic value. (Otherwise, there is arbitrage) • Suppose: S = $60; C = $5; X = $50. • Instant Arbitrage. How? • Buy the call for $5. • Immediately exercise the call, and buy the stock for $50. • In the next instant, sell the stock at the market price of $60. • You made a profit with zero cash outlay.**Intrinsic Values and Arbitrage, Puts**• Put options with American-style exercise must sell for at least their intrinsic value. (Otherwise, there is arbitrage) • Suppose: S = $40; P = $5; K = $50. • Instant Arbitrage. How? • Buy the put for $5. • Buy the stock for $40. • Immediately exercise the put, and sell the stock for $50. • You made a profit with zero cash outlay.**Lower Bounds for Options (con’t)**• As we have seen, to prevent arbitrage, option prices cannot be less than the option intrinsic value. • Otherwise, arbitrage will be possible. • Note that immediate exercise was needed. • Therefore, options needed to haveAmerican-style exercise. • Using equations: If S is the current stock price, and X is the strike price: Call option price max [S-X,0] Put option price max [X-S,0]**Pricing Bounds Summary**Calls: Upper Bound: Stock Price, S Lower Bound: Intrinsic Value: max [S–X, 0] Puts: Upper Bound: Exercise Price, X Lower Bound: Intrinsic Value max [X-S,0]**How would you price an option?**• Step 1. Project the distribution of ending stock price (based on the stock price volatility) • Step 2. Compute the payoff for that distribution • Step 3. Using the correct risk adjusted discount rate, discount the expected payoffs Nice idea – but although Step 1 and 2 are “easy” to do, what is the RADR for an option? Option problem took more than 75 years to solve**Binomial Option Pricing**• Call options pay off when the stock prices rises above the eXercise price. Thus, to price options, we need a model that has variable stock prices • The simplest way to represent an uncertain stock price is to say it can either go up or down a given amount**Binomial Option Pricing Example**Expiration Today Today Expiration 200 75 100 C 50 0 Call Option Value X = 125 Stock Price**Hedge ratio approach**• If the stock goes up in value, so will the call option (as seen in the previous slide). • If you go long in the stock, and short in the call, its easy to set up a risk free future, by carefully choosing the fraction of stock to hold long for each option you write. • Buy d stock for every option you write, so that the payoff in the up state equals the payoff in the down state**Create Equal Expiration Cash Flows**• Upstate: dSu – Cu = d200 – 75 • Downstate: dSd – Cd = d50 – 0 • Set equal, and solve for d (called a d hedge, or hedge ratio) • 200d – 75 = 50d which gives: d = 0.5 • By inspection we can see that the hedge ratio is:**Create Equal Expiration Cash Flows**• CF in up = .5*200 – 75 = 25 = CF down • What’s the PV of receiving $25 for sure? = 25/(1+rf) for rf = 8% = 23.15 • So, the value of 0.5 stocks, minus the value of one Call option is $23.15 today • 0.5(100) – C = 23.15 • Solve to C, and you have = 26.85**Binomial Option Pricing: Alternate Portfolio**Alternative Portfolio Buy 1 share of stock at $100 Borrow enough money to get a zero payout in down state (Owe $50 at expiration, i.e. borrow $46.30 (8% Rate) Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 Expiration 150 53.70 0 Payoff Structure is exactly 2 times the Call**Binomial Option Pricing**150 75 53.70 C 0 0 2C = $53.70 C = $26.85**Implied Probability**• If the stock has a beta = 1.5, and the expected return on the market is 15% (risk free as noted is 8%), then what is the probability the stock will go up to the higher value? • CAPM – • Hpr = {[pSu + (1-p)Sd] – So}/So • Solve for p**RADR for the Call**• What is the discount rate implied on the call (ie RADR) • What is the Beta of the Call?**Price of the put**• Use Put-Call parity to price the put**RADR for the Put**• What is the RADR for the put? • What is the Beta for the put?**Option value if volatility is smaller**• The size of the up or down movement must be set according to the volatility of the stock. • Lets say the upstate would yield a price of 160 and the downstate a price of 62.5. What happens to the value of the put and the call (with this lower volatility)? What happens to the RADR and the implied beta of the call and put**What if we change the eXercise Price?**• If the strike price is $100, what happens to the value of the call and the put, using the revised pricing values?**What if the risk free interest rate decreases**• For the previous example, what happens when we use a 2% risk free rate, rather than an 8% rate?**Generalizing the Two-State Approach**Assume that we can break the year into two six-month segments. In each six-month segment the stock could increase by 10% or decrease by 5%. Assume the stock is initially selling at 100. Possible outcomes: Increase by 10% twice Decrease by 5% twice Increase once and decrease once (2 paths).**Generalizing the Two-State Approach**Note: The size of the up and down jump is determined by the volatility of the stock price and the length of the period 121 110 104.50 100 95 90.25**Delta hedging**• Assume that the eXercise price is $100 • How many stock should you own at the outset? How many stock after the first period • How do I do that? Habit 2 – begin with the end in mind (solve like a dynamic program) • First solve for Su position, then for Sd position, then work back to origination**Compute hedge ratio and option values**• At Su, d = (21 – 4.5)/(121-104.5)=1 • Risk free portfolio = 121-21 = 100 • PV of ptf = 100/(1.04) = 96.1538 • 1*110 – Cu = 96.1538, so Cu = 13.8462 • Hedge ratio for down state is 0.3103 • Cd = 0.316 • What do you notice about hedge ratios in relationship to the stock price versus exercise price?**Wind back to the beginning**• Hedge ratio at beginning: (13.85 – 2.60)/(110-95) = .75 Risk free ptf = Present value ptf = Value of call = If I program a computer to buy or sell stock to keep my hedge in place (pursuing a “delta hedging” strategy), when do I buy, and when do I sell?**Expanding to Consider Three Intervals**• Assume that we can break the year into three intervals. • For each interval the stock could increase by 5% or decrease by 3%. • Assume the stock is initially selling at 100.**Expanding to Consider Three Intervals**S + + + S + + S + + - S + S + - S S + - - S - S - - S - - -**Possible Outcomes with Three Intervals**Event Probability Stock Price 3up 1/8 100 (1.05)3 =115.76 2 up 1 down 3/8 100 (1.05)2 (.97) =106.94 1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79 3 down 1/8 100 (.97)3 = 91.27**Black-Scholes Option Valuation**Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 + (T1/2) where Co = Current call option value. So = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.**Black-Scholes Option Valuation**X = Exercise price e = 2.71828, 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 Standard deviation of annualized cont. compounded rate of return on the stock**Call Option Example**So = 100 X = 95 r = .10 T = .25 (quarter) = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18**Probabilities from Normal Dist**N (.43) = .6664 Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700**Probabilities from Normal Dist.**N (.18) = .5714 Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793**Call Option Value**Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?**Put Value Using Black-Scholes**P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 95e-10x.25(1-.5714)-100(1-.6664) = 6.35**Put Option Valuation: Using Put-Call Parity**P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35**Black-Scholes Model with Dividends**• The call option formula applies to stocks that do not pay dividends. • One approach is to replace the stock price with a dividend adjusted stock price. Replace S0 with S0 - PV (Dividends)**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 (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock.