Two Types of Options

FIN 673 Binomial Option Pricing Professor Robert B.H. Hauswald Kogod School of Business, AU Two Types of Options An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today. Calls versus Puts Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you call in the asset. Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you put the asset to someone. 1/4/11 Option Pricing - Robert B.H. Hauswald

Positions in Options The seller (or writer) of an option has an obligation. The purchaser of an option has an option. Option profits Option ($) profits ($) 6 4 1-1 - -4 1 3 4 5 Stock price ($) 6 Buy a call Write a put 7 8 9 Buy a put Write a call 1-6 1/4/11 Option Pricing - Robert B.H. Hauswald 3 Stock Options 1/4/11 Option Pricing - Robert B.H. Hauswald 4

Options Contracts: Intrinsic Value Call vs. Put In-the-Money The exercise price is less than the spot price of the underlying asset. At-the-Money The exercise price is equal to the spot price of the underlying asset. Out-of-the-Money The exercise price is more than the spot price of the underlying asset. Market jargon: money-ness 1/4/11 Option Pricing - Robert B.H. Hauswald 5 Option Theory: Building Blocks Options are priced relative to other assets their payoffs can be exactly replicated by a portfolio of risk free bonds and the underlying asset (e.g. stock) options can be priced by arbitrage methods! Options confer a right whose value depends on contingencies: use probability theory to evaluate them prerequisite: a model of the underlying asset s price distributional assumption: log-normal price changes Investors risk attitudes irrelevant for pricing options are priced relative to bonds and underlying their risk and expected return can be substantial 1/4/11 Option Pricing - Robert B.H. Hauswald 6

Value Components Intrinsic Value The difference between the exercise price of the option and the spot price of the underlying asset. Speculative Value The difference between the option premium and the intrinsic value of the option. Option Premium Intrinsic Value + Speculative Value 1/4/11 Option Pricing - Robert B.H. Hauswald 7 Determinants of Option Pricing Determinants of Relation to Relation to Option Pricing Call Option Put Option Stock price Positive Negative Strike price Negative Positive Risk-free rate Positive Negative Volatility of the stock Positive Positive Time to expiration date Positive Usually Positive 1/4/11 Option Pricing - Robert B.H. Hauswald 8

Pricing Terminology Three price elements: current price of underlying asset: stock, real asset (factory) strike (exercise): price at which transaction occurs (option) premium: the option s price itself Intrisic value: conceptually, if exercised now, what would the option be worth? Price location: at/in/out-of-the-money options at: current spot strike in: option profitable if exercised immediately out: option could not be profitably exercised intrinsic value: extent to which an option is in-the-money 1/4/11 Option Pricing - Robert B.H. Hauswald 9 Option-Pricing Formulae We will start with a binomial option pricing formula to build our intuition replicating portfolio risk-neutral pricing Wall Street workhorse Then we graduate: approximation of the binomial model by the normal one option valuation elegant but nonintuitive popularized options 1/4/11 Option Pricing - Robert B.H. Hauswald 1

Binomial Option Pricing Model Suppose a stock is worth $5 today and in one period will either be worth 15% more or 15% less, i.e., S $5 today and in one year S 1 is either $8.75 or $1.5.; if risk-free rate is 5%, price an at-the-money call option S S 1 $8.75 $5 $1.5 1/4/11 Option Pricing - Robert B.H. Hauswald 11 Long Call: Payoff at Expiry 3 Buy a call Option payoffs ($) 1-1 - 5 1 15 5 3 35 4 45 Stock price ($) 5-3 Exercise price $5 (at the money) 1/4/11 Option Pricing - Robert B.H. Hauswald 1

Call Option Pricing at Expiry At expiry, an American call option is worth the same as a European option with the same characteristics. If the call is in-the-money, it is worth: S T - E If the call is out-of-the-money, it is worthless: ; hence, C at C et Max[S T - E, ] Where S T is the value of the stock at expiry (time T) E is the exercise price. C at is the value of an American call at expiry C et is the value of a European call at expiry 1/4/11 Option Pricing - Robert B.H. Hauswald 13 Binomial Option Pricing Model 1. A call option on this stock with exercise price of $5 will have the following payoffs:. We find the value of the call working backwards from the payoffs after extracting probabilities q S S 1 C 1 $8.75 $3.75 $5 $1.5 $ 1/4/11 Option Pricing - Robert B.H. Hauswald 14

Risk-Neutral Derivatives Valuation q S(U), V(U) S(), V() 1- q S(, V( We could value any derivative instrument V at t, V(), as the value of an appropriate replicating portfolio; but an equivalent method is risk-neutral valuation q V ( U ) + (1 q) V ( V () (1 + r ) 1/4/11 Option Pricing - Robert B.H. Hauswald f 15 The Risk-Neutral Approach S(U), V(U) q q is the risk-neutral S(), V() probability of an up move. S() is the value of 1- q the underlying asset S(, V( today. S(U) and S( are the values of the asset in the next period following an up move and a down move, respectively. V(U) and V( are the derivative values in the next period following an up move and a down move, respectively. 1/4/11 Option Pricing - Robert B.H. Hauswald 16

Risk-Neutral Probabilities S(), V() q 1- q S(U), V(U) V () S(, V( q V ( U ) + (1 q) V ( (1 + ) r f The key to finding q is to note that it is already impounded into an observable security price: the value of S() q S( U ) + (1 q) S( S() (1 + r f ) (1 + rf ) S() S( A minor bit of algebra yields: q S( U ) S( 1/4/11 Option Pricing - Robert B.H. Hauswald 17 Risk-Neutral Call Valuation Example Suppose a stock is worth $5 today and in one period will either be worth 15% more or 15% less if the risk-free rate is 5%, what is the value of an at-themoney call option? the binomial tree would look like this: $ 8.75 $5 (1.15) q $8.75,C( $5,C() $ 1.5 $5 (1.15) 1- q $1.5,C( 1/4/11 Option Pricing - Robert B.H. Hauswald 18

Finding the RN Probability q The next step is to compute the risk neutral probabilities $5,C() (1 + rf ) S() S( q S( U ) S( (1.5) $5 $1.5 q $8.75 $1.5 /3 $5 $7.5 $8.75,C( 3 $1.5,C( 1/4/11 Option Pricing - Robert B.H. Hauswald 19 The No-Arbitrage Restriction on q Fundamental result: risk-neutral probabilities do not change for underlying assets and their derivatives After that, find the value of the call in the up state and down state C( U ) $8.75 $5 /3 $8.75, $3.75 $5,C() C( max[$5 $8.75,] $1.5, $ 1/4/11 Option Pricing - Robert B.H. Hauswald

Risk-Neutral Valuation of the Call Finally, we find the value of the call at time : q C( U ) + (1 q) C( C() (1 + ) r f 3 $3.75 + (1 3) $ C() (1.5) $.5 C( ) $.38 (1.5) $5,$.38 $5,C() /3 $8.75,$3.75 $1.5, $ 1/4/11 Option Pricing - Robert B.H. Hauswald 1 Three Period Binomial Option Pricing Example There is no reason to stop with just two periods: generalize to three, four, periods using XLS The principles are the same: find q construct the underlying asset value lattice working forward construct the option value working backward Find the value of a three-period at-the-money call option written on a $5 stock that can go up or down 15 percent each period when the risk-free rate is 5% 1/4/11 Option Pricing - Robert B.H. Hauswald

Three Period Binomial Process: 3 Stock Prices $ 5. (1.15) $ 5. (1.15) 8.75 /3 $5 1.5 $ 5. (1.15) $ 5. (1.15) 1/4/11 Option Pricing - Robert B.H. Hauswald 15.35 3 33.6 /3 $ 5. (1.15)(1.15) 4.44 /3 $ 5. (1. 15) 18.6 /3 38. $5. (1.15) (1.15) /3 8.1 $ 5. (1.15) (1.15).77 /3 $ 5. (1.15) 3 Lattices: Binomial Call Option Pricing C3( U, U, U) max[$38. $5,] 38. 3 $13. + (1 3) $3.1 C( U, U ) /3 13. (1.5) C1( U ) C3( D, U, U ) 33.6 3 $9.5 + (1 3) $1.97 C3( U, D, U ) C3( U, U, /3 9.5 (1.5) max[$8.1 $5,] C ( U, C( D, U ) 8.75 8.1 /3 3 $3.1 + (1 3) $ 6.5 /3 3.1 C1( (1.5) C (,, ) $5 3 U D D 3 $1.97 + (1 3) $ 4.44 C3( D, U, C3( D, D, U ) 4.5 (1.5) /3 1.97 max[$.77 $5,] C( D, 1.5.77 3 $ + (1 3) $ 1.5 /3 (1.5) C,, ) 18.6 3( D D D 3 $6.5 + (1 3) $1.5 max[$15.35 $5,] C (1.5) 15.35 1/4/11 Option Pricing - Robert B.H. Hauswald 4

Black-Scholes-Merton The Black-Scholes-Merton Model is C S N( d1) Ee rt N( d) where C the value of a European option at time t r the risk-free interest rate. d d 1 σ ln( S / E) + ( r + ) T σ T d 1 σ T N(d) Probability that a standardized, normally distributed random variable will be less than or equal to d. 1/4/11 Option Pricing - Robert B.H. Hauswald 5 BSM Example: MSFT Call Find the value of a six-month call option on the Microsoft with an exercise price of $15 The current value of a share of Microsoft is $16 The interest rate available in the U.S. is r 5%. The option maturity is 6 months (half of a year). The volatility of the underlying asset is 3% per annum. Before we start, note that the intrinsic value of the option is $1 our answer must be at least that amount. 1/4/11 Option Pricing - Robert B.H. Hauswald 6

MSFT Call Option: Parameters First calculate d 1 and d ln( S / E) + ( r +.5σ ) T d1 σ T d ln(16 /15) + (.5 +.5(.3).3.5 ).5 1.58 Then, d d1 σ T.5815.3.5.316 1/4/11 Option Pricing - Robert B.H. Hauswald 7 MSFT Call Option: Parameters Recall the BSM formula C S N( d1) Ee rt N( ) d d 1.58 d.316 N(d 1 ) N(.5815).713 N(d ) N(.316).641 C C $16.713 15e $.9.5.5.641 1/4/11 Option Pricing - Robert B.H. Hauswald 8

Conclusions Option theory highlights the following principles valuable to wait to exercise (resolution of uncertainty): the longer the time to expiration, the higher the value you only exercise in good states, you save your exercise money if you wait and the bad (low price) state results the cost of waiting on a stock option is the lost profits (dividends) that would be yours from owning the stock Results hold independent of distributional assumptions (continuous, binomial, jump process) Can you show that equity is a call option: implications for risk taking behavior by stockholders? 1/4/11 Option Pricing - Robert B.H. Hauswald 9 Appendix: More on Option Pricing Black-Scholes-Merton Model: the classics options on dividend-paying stock: IMPORTANT for applications to real investments (real options) Generalizations three and more periods path-dependent option payoffs Applications three-period stock option pricing repricing executive stock options lookback options 1/4/11 Option Pricing - Robert B.H. Hauswald 3

Assume S $5, X $45, T 6 months, r 1%, and σ 8%, calculate the value of a call and a put. ( 5.8 ln ) +.1.5 45 + d 1.884.8.5 d.884.8.5.686 From a standard normal probability table, look up N(d 1 ).81 and N(d ).754 (or use Excel s normsdist function) C 5e Another BSM Example (.5) P $8.3 $5 + (.81) 45e $45e.1(.5).1(.5) (.754) $8.3 $1.15 1/4/11 Option Pricing - Robert B.H. Hauswald 31 Black-Scholes-Merton Intuition Interpretation: final payoff S - K weighted by discount factor: time value of money probability of price realizations: expected values c t S t N(d 1 ln where d 1 ) e rt ( d σ T ) -rt ( St /Ke ) 1 + T σ N(x): standard normal distribution function, σ std. dev. of firms stock return, Τ: option s time to maturity, S t current stock price This formula applies to what asset class? Where is it clearly inappropriate? 1/4/11 Option Pricing - Robert B.H. Hauswald 3 T KN 1 σ

Assumptions of Black-Scholes No restrictions on short selling Transactions costs and taxes are zero European option No dividends are paid Process describing stock price return is continuous Market has continuous trading Short-term interest rate is known and constant Stock returns are log-normally distributed 1/4/11 Option Pricing - Robert B.H. Hauswald 33 Black-Scholes with Dividends Dividends are a form of asset leakage if dividend are paid repeatedly, adjust B-S-M to allow for constant proportional dividends: c Se t S δ where and -δt N(d N(d S δ 3 3 3 ) Ke Se δt rt ln (S d ) Ke rt δ N(d N(d 3 σ 3 σ σ /K) + [r + σ / ]t and δ is a constant t t ) t ) dividend yield 1/4/11 Option Pricing - Robert B.H. Hauswald 34

Generalizing the Binomial Model The binomial option pricing model is an alternative to the Black-Scholes option pricing model especially given the computational efficiency of Excel in some situations, it is a superior alternative. Path dependency in option payoff require the use of lattice (binomial) option pricing models path dependency: how you arrive at a price (the path you follow) for the underlying asset is important for payoff example of a path dependent security: the no regret call option where the exercise price is the lowest price of the stock during the option life 1/4/11 Option Pricing - Robert B.H. Hauswald 35 Executive Stock Options 1/4/11 Option Pricing - Robert B.H. Hauswald 36

Economics of Stock Options Executive Stock Options exist to align the interests of shareholders and managers. Executive Stock Options are call options (technically warrants) on the employer s shares Inalienable Typical maturity: 1 years; typical vesting period: 3 years Most include implicit reset provision to preserve incentive compatibility Executive Stock Options include an important tax break: grants of at-the-money options are not taxable income: taxes are due if the option is exercised 1/4/11 Option Pricing - Robert B.H. Hauswald 37 Valuing Executive Compensation FASB allows firms to record zero expense for grants of at-the-money executive stock options. However the economic value of a long-lived call option is enormous, especially given the propensity of firms to reset the exercise price after drops in the price of the stock. Due to the inalienability, the options are worth less to the executive than they cost the company. The executive can only exercise, not sell his options: captures only the intrinsic, not the speculative value This dead weight loss is overcome by the incentive compatibility for the grantor 1/4/11 Option Pricing - Robert B.H. Hauswald 38

Top Stock Option Grants Company CEO Stock Option Award Citigroup, Inc. Sanford Weill $351,319, American Express Harvey Golub $134,1, Cisco Systems, Inc. John Chambers $13,1, Bank of America Hugh McColl Jr. $14,3, Honeywell Inc. Michael Bosignore $11,496, ALCOA Paul O Neill $96,353, 1/4/11 Option Pricing - Robert B.H. Hauswald 39 Valuation of a Lookback Option When the stock price falls due to the stock market as a whole falling, the board of directors tends to reset the exercise price of executive stock options To see how this reset provision adds value, let us price that same three-period call option (exercise price initially $5) with a reset provision if the stock price falls, boards lower exercise price rationale? Notice that the exercise price of the call will be the smallest value of the stock price depending upon the path followed by the stock price to get there 1/4/11 Option Pricing - Robert B.H. Hauswald 4

Three Period Binomial Asset Price Process: Lookback 38. 33.6 8.1 8.75 8.1 4.44.77 $5 8.1 4.44 1.5.77.77 18.6 1/4/11 Option Pricing - Robert B.H. Hauswald 15.35 41 Resetting Strike Prices C3( U, U, U) max[$38. $5,] 38. 13. C 33.6 3 ( U, U, max[$8.1 $5,] 3.1 8.1 $3.1 8.75 C3 ( U, D, U ) max[$8.1 $4.44,] 3.66 8.1 4.44 $3.66 $5 C3 ( U, D, max[$.77 $4.44,] C3 ( D, U, U) max[$8.1 $1.5,] 6.85.77 8.1 $6.85 4.44.77 1.5 C3 ( D, U, max[$.77 $1.5,].77.71 C 18.6 3 ( D, D, U) max[$.77 $18.6,].71 C (,, ) max[$15.36 18.6,] 15.35 3 D D D 1/4/11 Option Pricing - Robert B.H. Hauswald 4

Lookback Call Option Prices 3 $13. + (1 3) $3.1 C( U, U ) (1.5) 8.75 C ( 33.6 9.5 8.1 3 $3.66 + (1 3) $ U, (1.5) 4.44 38. 8.1 13. $3.1 $3.66.33.77 $5 C ( 3 $6.85 + (1 3) $ D, U ) (1.5) 8.1 $6.85 4.44 4.35.77 1.5.77.71 18.6 3 $.71+ (1 3) $ C ( D, 1.7 (1.5) 15.35 1/4/11 Option Pricing - Robert B.H. Hauswald 43 Lookback Call Option Prices C1( U ) 3 $9.5 + (1 3) $.33 (1.5) 38. 33.6 9.5 8.1 8.75 6.61 8.1 4.44 13. $3.1 $3.66 $5 5.5 C1( 3 $4.35 + (1 3) $1.7 (1.5).33 4.44.77 8.1 $6.85 C 4.35.77 1.5 3.31 3 $6.5 + (1 3) $1.5.77.71 18.6 (1.5) 1.7 15.35 1/4/11 Option Pricing - Robert B.H. Hauswald 44

Excel Applications of the BOPM The BOPM is easily incorporated into Excel 14% s $ 8.75 spreadsheets 1 Maturity $ 5. 1 n $ 3.75 1 D t q $ 5. S $ 5. X Stock Price $ 5. 5% r f Exercise Price $ 5. 1.15 u Ordinary Call $.38.85 d 1.5 a 1- q 66.67% Risk Neutral Prob $ 1.5 33.33% 1- R.N. Prob $ 5. $ - 1/4/11 Option Pricing - Robert B.H. Hauswald 45