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## Option valuation

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**Option valuation**FIN 545 Prof. Rogers Spring 2011**Segment 1**Principles of option valuation Associated Audio content = Approx 20 Minutes (15 slides)**Basic Notation and Terminology**• Symbols • S0 (stock price) • X (exercise price) • T (time to expiration = (days until expiration)/365) • r (risk-free rate) • ST (stock price at expiration) • C(S0,T,X), P(S0,T,X)**Principles of Call Option Pricing**• Minimum Value of a Call • C(S0,T,X) ³ 0 (for any call) • For American calls: • Ca(S0,T,X) ³ Max(0,S0 - X) • Concept of intrinsic value: Max(0,S0 - X) • Concept of time value of option • C(S,T,X) – Max(0,S – X) • For example, what is the time value of a call option trading at $5 with exercise price of $20 when the the underlying asset is trading at $22.75?**Principles of Call Option Pricing (continued)**• Maximum Value of a Call • C(S0,T,X) £ S0 • “Right” but not “obligation” can never be more valuable than underlying asset (and will typically be worth less). • Option values are always equal to a percentage of the underlying asset’s value!**Principles of Call Option Pricing (continued)**• Effect of Time to Expiration • More time until expiration, higher option value! • Volatility is related to time (we’ll see this in binomial and Black-Scholes models). • Calls allow buyer to invest in other assets, thus a pure time value of money effect.**Principles of Call Option Pricing (continued)**• Effect of Exercise Price • Lower exercise prices on call options with same underlying and time to expiration always have higher values!**Principles of Call Option Pricing (continued)**• Lower Bound of a European Call • Ce(S0,T,X) ³ Max[0,S0 - X(1+r)-T] • A call option can never be worth less than the difference between the underlying’s value and the present value of the exercise price on the call (or zero, if this difference is negative).**Principles of Call Option Pricing (continued)**• American Call Versus European Call • Ca(S0,T,X) ³ Ce(S0,T,X) • If there are no dividends on the stock, an American call will never be exercised early (unless there are complicating factors…we’ll discuss employee options eventually). • Rather than exercise, better to sell the call in the market. • Options are worth more alive than dead! • If no dividends, the value of the American call and identical European call should be equal. • If dividend is sufficiently large to invoke potential for early exercise, this “early exercise option” is a source of additional value for an American call (vs. the equivalent European).**Principles of Put Option Pricing**• Minimum Value of a Put • P(S0,T,X) ³ 0 (for any put) • For American puts: • Pa(S0,T,X) ³ Max(0,X - S0) • Concept of intrinsic value: Max(0,X - S0)**Principles of Put Option Pricing (continued)**• Maximum Value of a Put • Pe(S0,T,X) £ X(1+r)-T • European put option value must be no more than the present value of the exercise price of the put option. • Pa(S0,T,X) £ X • American put option value is bounded above by the exercise price. • No “present value effect” because of potential for early exercise (more on this shortly).**Principles of Put Option Pricing (continued)**• The Effect of Time to Expiration • Same effect as call options: more time, more value!**Principles of Put Option Pricing (continued)**• Effect of Exercise Price • Raising exercise price of put options increases value!**Principles of Put Option Pricing (continued)**• Lower Bound of a European Put • Pe(S0,T,X) ³ Max(0,X(1+r)-T - S0) • The value of a put option cannot be less than the difference between the present value of the put option’s exercise price and the underlying’s value (or zero if this difference is negative).**Principles of Put Option Pricing (continued)**• American Put Versus European Put • Pa(S0,T,X) ³ Pe(S0,T,X) • Early Exercise of American Puts • There is typically the probability of a sufficiently low stock price occurring that will make it optimal to exercise an American put early. • Dividends on the stock reduce the likelihood of early exercise.**Principles of Put Option Pricing (continued)**• Put-Call Parity • Form portfolios A and B where the options are European. • Portfolio A: Buy share of stock; buy put option on stock with exercise price X, and maturity date T • Portfolio B: Buy call option on stock with exercise price X, and maturity date T; buy risk-free bond with face value X and maturity date T • The portfolios have the same outcomes at the options’ expiration. Thus, it must be true that • S0 + Pe(S0,T,X) = Ce(S0,T,X) + X(1+r)-T • Equation illustrates “put-call parity.” • Equation can be rearranged to offer various interpretations.**Put-Call Parity Example**• Price of underlying asset (S) = $19.50 • Premium for call option on underlying asset with exercise price = $20 and 3 months until expiration = $2.50 • Premium for put option on underlying asset with exercise price = $20 and 3 months until expiration = $1.50 • Risk-free rate = 5% • Does put-call parity hold? • Which option is overpriced? • What would be the trading strategy?**Segment 2**Valuing options with binomial models Associated audio content = approx 38 Minutes (12 slides)**One-Period Binomial Model**• Conditions and assumptions • One period, two outcomes (states) • S = current stock price • u = 1 + return if stock goes up • d = 1 + return if stock goes down • r = risk-free rate • Value of European call at expiration one period later • Cu = Max(0,Su - X) or • Cd = Max(0,Sd - X)**One-Period Binomial Model (continued)**• This is the theoretical value of the call as determined by the stock price, exercise price, risk-free rate, and up and down factors. • The probabilities of the up and down moves are never specified. They are irrelevant to the option price.**One-Period Binomial Model (continued)**• An Illustrative Example • Let S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 • First find the values of Cu, Cd, h, and p: • Cu = Max(0,100(1.25) - 100) = Max(0,125 - 100) = 25 • Cd = Max(0,100(.80) - 100) = Max(0,80 - 100) = 0 • h = (25 - 0)/(125 - 80) = 0.556 • p = (1.07 - 0.80)/(1.25 - 0.80) = 0.6 • Then insert into the formula for C:**Student exercises**• Calculate the option values if the following changes are made to the prior example: • S = 110 • S = 90 • u = 1.40, d = 0.70 • u = 1.15, d = 0.90 • r = 10% • r = 4%**Two-Period Binomial Model**• We now let the stock go up another period so that it ends up Su2, Sud or Sd2. • The option expires after two periods with three possible values:**Two-Period Binomial Model (continued)**• After one period the call will have one period to go before expiration. Thus, it will worth either of the following two values • The price of the call today will be ????**Two-Period Binomial Model (continued)**• An Illustrative Example • Let S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 • Su2 = 100(1.25)2 = 156.25 • Sud = 100(1.25)(0.80) = 100 • Sd2 = 100(0.80)2 = 64 • The call option prices at maturity are as follows:**Two-Period Binomial Model (continued)**• The two values of the call at the end of the first period are**Two-Period Binomial Model (continued)**• Therefore, the value of the call today is**Student exercises**• Calculate the option values if the following changes are made to the prior example: • S = 110 • S = 90 • u = 1.40, d = 0.70 • u = 1.125, d = 0.90, r = 3.5%**Segment 3**Absence of arbitrage and option valuation Associated audio content = approx 24 minutes (10 slides)**The “no-arbitrage” concept**• Important point: d < 1 + r < u to prevent arbitrage • We construct a hedge portfolio of h shares of stock and one short call. Current value of portfolio: • V = hS - C • At expiration the hedge portfolio will be worth • Vu = hSu - Cu • Vd = hSd - Cd • If we are hedged, these must be equal.Setting Vu = Vd and solving for h gives (see next page!)**One-Period Binomial Model (continued)**• These values are all known so h is easily computed. The variable, h, is called “hedge ratio.” • Since the portfolio is riskless, it should earn the risk-free rate. Thus • V(1+r) = Vu (or Vd) • Substituting for V and Vu • (hS - C)(1+r) = hSu - Cu • And the theoretical value of the option is**No-arbitrage condition**• C = hS – [(hSu – Cu)(1 + r)-1] • Solving for C provides the same result as we determined in our earlier example! • Can alternatively substitute Sd and Cd into equation • If the call is not priced “correctly”, then investor could devise a risk-free trading strategy, but earn more than the risk-free rate….arbitrage profits!**1-period binomial model risk-free portfolio example**• A Hedged Portfolio • Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. • Value of investment: V = 556($100) - 1,000($14.02) $41,580. (This is how much money you must put up.) • Stock goes to $125 • Value of investment = 556($125) - 1,000($25) = $44,500 • Stock goes to $80 • Value of investment = 556($80) - 1,000($0) = $44,480 (difference from 44,500 is due to rounding error)**One-Period Binomial Model (continued)**You invested $41,580 and got back $44,500, a 7 % return, which is the risk-free rate. • An Overpriced Call • Let the call be selling for $15.00 • Your amount invested is 556($100) - 1,000($15.00) = $40,600 • You will still end up with $44,500, which is a 9.6% return. • Everyone will take advantage of this, forcing the call price to fall to $14.02**One-Period Binomial Model (continued)**• An Underpriced Call • Let the call be priced at $13 • Sell short 556 shares at $100 and buy 1,000 calls at $13. This will generate a cash inflow of $42,600. • At expiration, you will end up paying out $44,500. • This is like a loan in which you borrowed $42,600 and paid back $44,500, a rate of 4.46%, which beats the risk-free borrowing rate.**2-period binomial model risk-free portfolio example**• A Hedge Portfolio • Call trades at its theoretical value of $17.69. • Hedge ratio today: h = (31.54 - 0.0)/(125 - 80) = 0.701 • So • Buy 701 shares at $100 for $70,100 • Sell 1,000 calls at $17.69 for $17,690 • Net investment: $52,410**Two-Period Binomial Model (continued)**• A Hedge Portfolio (continued) • The hedge ratio then changes depending on whether the stock goes up or down • What is the hedge ratio if “up”? • What is the hedge ratio if “down”? • Describe how you alter your portfolio in each circumstance. • In each case, you wealth grows by 7% at the end of the first period. You then revise the mix of stock and calls by either buying or selling shares or options. Funds realized from selling are invested at 7% and funds necessary for buying are borrowed at 7%.**Two-Period Binomial Model (continued)**• A Hedge Portfolio (continued) • Your wealth then grows by 7% from the end of the first period to the end of the second. • Conclusion: If the option is correctly priced and you maintain the appropriate mix of shares and calls as indicated by the hedge ratio, you earn a risk-free return over both periods.**Two-Period Binomial Model (continued)**• A Mispriced Call in the Two-Period World • If the call is underpriced, you buy it and short the stock, maintaining the correct hedge over both periods. You end up borrowing at less than the risk-free rate. • If the call is overpriced, you sell it and buy the stock, maintaining the correct hedge over both periods. You end up lending at more than the risk-free rate.**Segment 4**Extensions of the binomial model valuation process Associated audio content = approx 33 minutes (16 slides)**Extensions of the binomial model**• Early exercise (American options) • Put options • Call options with dividends • Real option examples**Pricing Put Options**• Same procedure as calls but use put payoff formula at expiration. Using our prior example, put prices at expiration are**Pricing Put Options (continued)**• The two values of the put at the end of the first period are**Pricing Put Options (continued)**• Therefore, the value of the put today is**Pricing Put Options (continued)**• Let us hedge a long position in stock by purchasing puts. The hedge ratio formula is the same except that we ignore the negative sign: • Thus, we shall buy 299 shares and 1,000 puts. This will cost $29,900 (299 x $100) + $5,030 (1,000 x $5.03) for a total of $34,930.**Pricing Put Options (continued)**• Stock goes from 100 to 125. We now have • 299 shares at $125 + 1,000 puts at $0.0 = $37,375 • This is a 7% gain over $34,930. The new hedge ratio is • So sell 299 shares, receiving 299($125) = $37,375, which is invested in risk-free bonds.**Pricing Put Options (continued)**• Stock goes from 100 to 80. We now have • 299 shares at $80 + 1,000 puts at $13.46 = $37,380 • This is a 7% gain over $34,930. The new hedge ratio is • So buy 701 shares, paying 701($80) = $56,080, by borrowing at the risk-free rate.