Chapter 16 Option Valuation Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

16-2 Call Option Price Boundaries 1. Basic boundaries revisited –C t ≥ 0,Why? – –C t ≥ S t – X,Why? –Thus C t  Max (0, S t – X) where: C t =Price paid for a call option at time t. t = 0 is today, T =Immediately before the option's expiration. S t =Stock price at time t. X = Exercise or Strike Price (X or E)

16-3 Call Option Price Boundaries 2. A tighter boundary Suppose we consider two different portfolios: Portfolio 1: Long position in stock at S 0 Portfolio 2: Buy 1 at the money call option (C 0 ) and buy a T-bill with a face value = X.

16-4 Call Option Price Boundaries Possible values of the two portfolios at contract expiration: STST X + X + T-Bill S T - X0CallSTST STST S T > XS T < XS T > XS T < X Portfolio 2 (C 0 + PV of X)Portfolio 1 (S 0 ) At expiration, the value of Portfolio 2 is always ___ Portfolio 1 so initially the value of Portfolio 2 must also be ____ the value of Portfolio 1. Thus: Finally:   C 0 + PV of X  S 0 OR C 0  S 0 - PV of X C 0  Max[0, (S 0 - X(e -rT ))] =

16-5 Value of a Call Option X Stock Price t Value $0 Prior to expiration Value at expiration or “Exercise” or “Intrinsic” Value 6 mo 2 mo Difference is the “Time Value” of the option What is the time value of a call option? The time value of a call incorporates the probability that S will be in the money at period T given S 0, time to T,  2 stock,X, and the level of interest rates 

16-6 Binomial Call Option Pricing In a binomial framework a stock’s price can move either up or down by a fixed amount over a given time period. Let U, D = % price move upward a stock’s value may take, Suppose S 0 = ____ and E = $90 U = 15% and D = -15% $100 ? Problem with finding C 0 ?

16-7 Aside) Replicating Portfolio, No Arbitrage Pricing, Riskless Portfolio Security A pays $3 if sunny tomorrow(t=1), $0 if not, Security B pays $0 if sunny, $2 if not. If I want to have $6 tomorrow for sure, what can I do? You can create a replicating portfolio with 2A+3B package. If A and B are priced today(t=0) at $1.40 and $0.95 respectively, what should be the cost of such a portfolio? $5.50. If such a portfolio package is traded for $5.70, you are not equilibrium as you can make an arbitrage profits by doing BLSH. The equilibrium can be achieved only when the price of the package to be $5.50 (if the prices of A and B do not change) where no arbitrage exists. This portfolio is riskless. Why? If the risk-less T-Bill interest rate (one day) is 7%, are you in equilibrium? Since PV of $6 tomorrow = $6/1.07 = $5.61, you would apply BL T-Bill at $5.61 and SH the package (2A+3B and deliver $6 regardless of the weather tomorrow) for $6.70 and make arbitrage profits of $0.09. To have no arbitrage, the price of the package has to be $5.61 or the interest has to be 6.95%.

16-8 Binomial Call Option Pricing Suppose we buy shares of stock and write one call. The risk free rate is 10%. $ Call Payoff $ 0-$ 25 Value of Written Call $70.833$ Value of Stock position.8333 x Stock Price $85$115Stock Price SDSD SUSU Value today of this portfolio must be the present value of the riskless payoff discounted at the risk free rate of 10% Letting H = _______ then HS 0 – C 0 = ______ C 0 = S 0 – C 0 _______ = Hedge ratio that makes the portfolio riskless, call it “H” $ / 1.10 = $ $64.39 $ $64.39 = $ (0.8333)($100) - C 0 = $64.39

16-9 Summing Up $18.94 The call value is > the exercise value of the call option. Call value today = Call Intrinsic value or Exercise value = Time Value of the call = $18.94 $10.00 $ 8.94

16-10 Generalizing the Two-State Approach S 0 = 100 SUSU S UU SDSD S DD S UD We could break this up further to three subintervals, four, etc. Break up the year into two intervals

16-11 A Few More Assumptions Add three more assumptions: – The risk free rate is constant. –The  2 Stock is constant over the life of the option. –Any cash dividends will not lead to early exercise And we arrive at the Black-Scholes Formula for option pricing:

16-12 C 0 = Current call option value. X (or E) = Exercise Price S 0 = Current stock price,  = Annual dividend yield on the stock e = , the base of the natural log r = Risk-free interest rate (annualize with continuous compounding the return on a T-bill with the same maturity as the option) T = Time until expiration (not a point in time) in years,  = Annual standard deviation of continuously compounded stock returns N(d) = probability that a random draw from a normal distribution will be less than d. Black-Scholes Option Valuation To convert a regular return to a continuously compounded return take Ln (1 + return)

16-13 The components of the model The exercise value of the call: However if the call will not be exercised early the value today is S 0 – the present value of X: S 0 – X(e -rT ) The cash dividend yield term  reflects that a dividend will reduce the stock price thus hurting the value of the call: d 1 comes from our assumptions about how stock prices move in continuous trading: – E(r) = (r +  2 /2)T when returns are lognormally distributed – Ln (S 0 / X) measures the continuous return needed for the stock to finish in the money –Roughly speaking the d 1 numerator is a measure of the return needed to finish in the money, the denominator measures this relative to the standard deviation of the returns. S 0 – X, S 0 e -  T – X(e -rT )

16-14 N(d) N(d) terms measure the probability of how far in the money the stock price is likely to be at expiration. The form of the model & N(d 1 ) and N(d 2 )

16-15 Using the Model S 0 =$100; X or E = $95 T = 3 months r = 5.2%  = 0 with no dividends  = 40% $11.23 Cumulative Normal Cumulative Normal Call data

16-16 A Standard Normal Curve

16-17 Spreadsheet version of the model

16-18

16-19 Determinants of Put Option Values Determinants of a put option’s value Greater Dividend payout Lower Interest rate r Greater* Time to expiration T Greater Volatility  Greater Exercise Price (E or X) Lower Stock Price, S … the value of a put option is Ceteris paribus, If this variable is greater

16-20

16-21 Put Call Parity Illustrated A 3 month 45 call option priced at $7.50 is available for a stock priced at $50. The annual risk free rate is 6%. If put call parity holds what must be the cost of a put with similar terms? P 0 = C 0 - S 0 + X(e -rT ) P 0 = $ $50 + $45[e -(0.06)(.25) ] = $1.83 In words: “Owning a put is equivalent to buying a call, shorting the stock and owning a bond.” Black-Scholes Put Model gives same answer

16-22 The Black-Scholes Hedge Ratio (H)  Change in Co relative to Change in So (Sensitivity) cf) Duration This means that the call option’s value will move by approximately N(d 1 ) dollars when the stock’s price moves one dollar. Delta hedging For a non-dividend paying stock

16-23 The Delta Ratio The Black-Scholes Hedge Ratio (H) for a call option H = slope of the call value line As a call moves into the money H approaches +1 As a call moves out of the money H approaches 0

16-24 The Delta Ratio We can use this concept to exploit a call price that appears to be mispriced according to the Black-Scholes model.

16-25 Arbitrage a mispriced call Suppose the call is actually priced at $8.75. How could a trader take advantage of this? (Note that the stock pays no dividends.) Write the overpriced call and acquire N(d 1 )shares of stock. Write one call and acquire shares of stock Let the stock’s price move up a few cents and let the option now be correctly priced.

16-26 Arbitrage a mispriced call C 0 = _____, supposed to be _____; S 0 = _____;S 1 = _____;C 1 = ______ The “Difference” is the gain from the strategy assuming that the option price has corrected? $ 0.70+$34.07-$33.37 $0.054+$42.17-$ Buy N(d1) shares of stock $0.645-$8.105$8.751Write call Diff Net Time 1 Net time 0 # $8.75 $8.05 $77.5$77.6$8.105 Reverse

16-27 More on Delta The sensitivity of a position’s value to a change in stock price is sometimes called the position’s Delta. –If the position is not affected by a change in stock price the position has a delta of ____ and is said to be ‘____________.’ –If a position increases in value when stock price increases (and vice versa) it is ‘____________.’ –If a position increases in value when stock price decreases (and vice versa) it is ‘____________.’ zero delta neutral positive delta negative delta

16-28 Dynamic Hedging Floor Value Portfolio Delta Neutral + Delta StSt Value t $50 M S Min

16-29 Implied Volatility & the B-S Model The stock’s standard deviation is the one variable in the Black-Scholes model that is not easily observable. Apparent mispricings in options may be a result of mis- estimation of the stock’s standard deviation. An option buyer or writer may use the Black-Scholes model to calculate the implied standard deviation of the stock that would be required to make the observed option price match the Black-Scholes calculated price. If an investor believed the implied volatility was high (low) compared to the actual volatility the investor may decide to write (buy) the call.

16-30 Implied Volatility of the S&P 500 (VIX Index)

16-31 Problem 1a Stock B has higher volatility than Stock A, so ceteris paribus, Put B should be worth more than Put A. Because they cost the same, X – S must be larger for A. Given X is the same on both, then Put A must be written on the lower-priced stock.

16-32 Problem 1b Call B. Despite the higher price of stock B, call B is cheaper than call A. This can be explained by a lower time to maturity. At In

16-33 Problem 2 N(d 1 ) increases as the call moves into the money. N(d 1 ) is the call option’s “Delta.”

16-34 Problem 3a: Binomial a. When S U = 130, P U = When S D = 80, P D = The hedge ratio is 0 30 [(P U – P D ) / (S U – S D ) = [(0 – 30) / (130 – 80)] = –3/5 P 0 ≥ Max (0, $110 - S T )

16-35 Problem 3b,c: Binomial b. c. Portfolio cost = 3S 0 + 5P 0 = Therefore P 0 = Present value = $390 / 1.10 = $ Riskless portfolio S D = 80S U = shares puts1500 Total390 $ ($ $300) / 5 = $10.91 P U = $0; P D = $30 = $ P 0

16-36 Problem $8.13

16-37 Problem 5 Using Put Call Parity: $ P 0 = C 0 - S 0 + X(e -rT ) P 0 = $ $50 + $50[e -(0.10)(.5) ] = $5.69

16-38 Problem 6 The call price will decrease by less than $1. The change in the call price would be $1 only if: i.there were a 100% probability that the call would be exercised &; ii.the interest rate were zero.