Introduction to Financial Derivatives

Transcription

1 Introduction to Financial Derivatives Weeks of November 19 & 6 th, 1 he Black-Scholes-Merton Model for Options plus Applications Where we are Previously: Modeling the Stochastic Process for Derivative Analysis (Chapter 13, OFOD) Last Week: Black-Scholes-Merton Model (Chapter 14, OFOD) his Week: Black-Scholes-Merton Model for Options (continued) (Chapter 14, 16-17, OFOD) Last Day of Class: Wednesday, Dec 5 th, 1 Final Exam: hursday, Dec th ; 9:am Noon Assignment Assignment For Week of Nov 19 th Read: Hull Chapter 14 (Black-Scholes-Merton Model) Problems (Due November 19 th ) Chapter 13: 3, 5, 9, 11; 13 Chapter 1(7e): 3, 5, 9, 11; 1 Problems (Due by November 8 th ) Chapter 14: 4, 5, 11, 13, 15, 17; 8 Chapter 13 (7e): 4, 5, 11, 13, 15, 17; For Week of Nov 6 th Read: Hull Chapter 14 (Black-Scholes-Merton) Read: Hull Chapter (Options on Indices & Currencies + Futures Options) Problems (Due by November 8 th ) Chapter 14: 4, 5, 11, 13, 15, 17; 8 Chapter 13 (7e): 4, 5, 11, 13, 15, 17; 8 Problems (Due by December 3 rd ) Chapter 16 (15, 7e): 4, ; 3 Chapter 17 (16, 7e): 7, 8, 15;

2 Plan Black-Scholes-Merton Differential Equation and Risk- Neutral Valuation Black-Scholes Formula for Non-Dividend paying Stock For Dividend paying stock Applications of Black-Scholes-Merton heory (Chapter 14, 16-17) Derivatives of a Stock Paying a Dividend Yield (& Index Options) Derivatives on Currency exchange (FX) Derivatives on Futures (and Commodities) 11.5 Plan for his Week Modeling the Stochastic Process for Derivative Analysis Ito Process & Ito s Lemma with the Stock Price Process Lognormal Property of Stock Price Process Expected Return Volatility Black-Scholes-Merton Differential Equation Risk Neutral Valuation What s Ahead: Option Analysis for Stock Indices, Currencies, and Futures (Chapter OFOD) 11.6 he Stock Price Assumption Consider a stock whose price is S and (as we have seen) follows the Ito Process, ds S dt S dz where is expected return and is volatility (per year) In a short period of time of length t, the return on the stock can be characterized as S StS t with ~ (,1), the standardized normal distribution So it follows that ΔS/S is normally distributed as well S ~ t, t S 11.7 Itô s Lemma If we know the stochastic process followed by x, Itô s lemma tells us about the stochastic process followed by some function G (x, t ) Since a derivative security is a function of the price of the underlying and time, Itô s lemma plays an important part in the analysis of derivative securities 11.8

3 Application of Ito s Lemma to a Stock Price Process When the stock price process is ds S dt S dz hen for a function G of S and t Ito s Lemma gives that G G G G dg S ½ S dt S dz S t S S 11.9 Applications of Ito s Lemma r Let F S e F( S,) be the forward price of term on a non-dividend paying stock r ( t) Or more generally FSt (, ) Se We can use Ito s Lemma to determine the process for F where the process for S is dss dt S dz F r ( t) F F r ( t) and we find e ; ; rse S S t r ( t) r ( t) r ( t) So df e S rse dt e Sdz ( rfdt ) Fdz Like S, the forward price process F follows an Ito Process, but has growth rate μ r rather than μ he growth rate in F is the excess return of S over the risk-free rate 11.1 Applications of Ito s Lemma Suppose G = ln S where S is the stock price process, ds Sdt Sdz then dg dt dz a generalized Wiener process hus, the stock price process is a lognormal process If a variable has a process whose natural logarithm is normally distributed, the process is said to be a lognormal process Clearly this is so for our stock process (as we will see next) 1 ln S ln S ~ [( ), ] 1 ln S ~ [ln S ( ), ] Lognormal Property of Stock Prices From Ito s Lemma, letting G = ln S, gives dg dt dz So the change in ln S between and some future is normally distributed with mean (µ - σ /) and variance σ Which we can express as ln S ln S ~, or ln S ~ ln S, Since ln S is normal, S is lognormally distributed And S ln ~, S

4 Lognormal Property of Stock Prices Lognormal Property of Stock Prices A lognormal variable can take any value from to and has distribution: From our expression for ln S we can show that ES ( ) S e var( S ) S e ( e 1) with a straightforward exercise in integrating the pdf of a normal distribution If we define the continuously compounded rate of return per annum realized between and as x, then or 1 S x S Se x= ln S and as a consequence of the lognormal property of price x ~, So the realized cc rate of return per annum, going forward from t=, is normally distributed As increases, the std dev of x declines We are more certain about the average rate of return, the longer the period, he Expected Returns and Suppose we have daily data for a period of several months {S, S 1, S, } is the average of the returns in each day [=E(S/S)] is the expected return over the whole period covered by the data measured with continuous compounding (or daily compounding, which is almost the same) he Expected Return From the Ito Process stock-price equation, t, is the expected percentage change in a small interval, t his is not the expected continuously compounded return on the stock actually realized over a period of time of length, defined x=(1/ )ln[ S where E[x] = / S] as was shown he expected return on the stock is not We derived that the expected value of the stock price (a consequence of the lognormal price process) as S e aking ln of ES ( ) Se gives ln[ E( S )] ln( S) If ln[ E( S)] E[ln( S)] then we could write E[ln( S )] ln( S ) E[ln( S ) ln( S )] E[ln( S / S )] But Jensen s Inequality tells us that in fact ln[ E( S )] E[ln( S )]

5 he Expected Return he Volatility Indeed, consider the two representations ES ( ) Se ln S ~ ln S, We can show ln[ ES ( )] E[ln( S)] as in Jenson s Inequality and as expected by the nonlinearity of ln (. ) ln[ E( S )] ln( S ) E[ln( S )] ln( S) So ln[ ES ( )] E[ln( S)] ln( S) ln( S) he volatility is defined as the standard deviation of the continuously compounded rate of return in 1 year he standard deviation of the return in time t is t If a stock price is $5 and its volatility is 5% per year the standard deviation of the price change in one day is From S St S t We can see that the standard deviation of the price change is.5 (5) sqrt(1/5) = $.79 Similarly, the standard deviation of the % change in stock price in one day is 5 x sqrt(1/5) = 1.575% Estimating Volatility from Historical Data 1. ake observations S, S 1,..., S n at intervals of years (the interval is best if daily, =.397 or.74 ). Calculate the continuously compounded return in each interval as: S i u i ln Si1 3. Calculate the standard deviation, s, of the u i s n n n s ( ui u) ui ( ui) n1 i1 n1 i1 n( n1) i1 s 4. he historical volatility estimate is: ˆ with error ˆ / n Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in trading days not calendar days when options are valued 5 days per year vs 365 or 36 1 day is (1/5) year or.397 (or.74 or.78) year 11. 5

6 he Concepts Underlying the Black- Scholes-Merton Model he option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty he portfolio is instantaneously riskless and must instantaneously earn the risk-free rate his leads to the Black-Scholes-Merton differential equation 11.1 Derivation of the Black-Scholes-Merton Differential Equation We start with the Stock Price Process and Ito s Lemma for a function of the price process hen form the candidate riskless portfolio S S t S z ƒ ƒ ƒ ƒ ƒ S ½ S t S z S t S S A portfolio consisting of 1 derivative ƒ + shares S 11. Derivation of the Black-Scholes-Merton Differential Equation he value Π of the portfolio is given by ƒ ƒ S S he change in its value in time t is given by ƒ f S S Substituting the stock price equation for S, and using the Ito result for f, gives ƒ ƒ 1 ƒ ƒ ƒ ( S S ) t Sz ( StSz) S t S S S ƒ 1 ƒ ( S ) t t S 11.3 Derivation of the Black-Scholes-Merton Differential Equation As equation for Π does not involve z, the portfolio is riskless during time t, and must return the risk-free rate r t Substituting for Π and Π gives ƒ 1 ƒ f ( S ) t r( f S) t t S S Which results in the Black-Scholes-Merton differential equation ƒ ƒ ƒ rs ½ S rƒ t S S

7 Black-Scholes-Merton Differential Equation It has many solutions corresponding to all the different derivatives that can be defined with S, the stock price, as the underlying variable he particular derivative that is obtained when the equation is solved depends on the boundary conditions that are used. For a European Call, the key boundary condition is f = max (S K, ) when t = For a European Put, the key boundary condition is f = max (K S, ) when t = 11.5 Black-Scholes-Merton Differential Equation For a Forward Contract on a non-dividend paying stock, with delivery price K, the key boundary condition is ƒ = S K when t = In general, we know that the value of a forward contract, f, at any time t[, ] is given in terms of the stock price S at time t by ƒ = S K e r ( t ) Substituting into the BSM differential equation ƒ ƒ ƒ r ( t) rs ½ S rke rs 1 ½ S t S S r ( t) rke rs rf Showing that the differential equation is indeed satisfied 11.6 Black-Scholes-Merton Differential Equation Any function f(s, t) that is a solution of the BSM differential equation is the theoretical price of a derivative that could be traded. If such a derivative existed, it would not create/allow any arbitrage opportunities Conversely, if a function f(s, t) does not satisfy the BSM differential equation, it cannot be the price of a derivative without creating arbitrage opportunities for traders If f(s, t) = e S, it does not satisfy BSM so it is not a candidate for being the price of a derivative dependent on the stock price rt If f( S, t) 1 e S, it does satisfy BSM, and so could be the price of a tradable derivative One that pays off 1/S at t= 11.7 Risk-Neutral Valuation he concept of risk-neutral valuation was introduced in the context of the binomial model Single most important result for derivative analysis he Black-Scholes-Merton differential equation is independent of all variables affected by risk preference Only variables are S, t, σ, and r no expected return, µ he solution to the differential equation is therefore the same in a risk-free world as it is in the real world his leads to the principle of Risk-Neutral Valuation Assume the expected return of the underlying asset is r, i.e. µ = r Calculate the expected payoff from the derivative Discount the expected payoff at the risk-free rate, r

8 Risk-Neutral Valuation Valuing a Forward Contract with Risk-Neutral Valuation he principle of risk-neutral valuation is the same as we saw for the binomial model Provides an opportune expedience for obtaining solutions to the Black-Scholes-Merton differential equation Such solutions are still valid in worlds where investors are risk-averse (real world) When we move from a risk-neutral world to a risk-averse world two things happen he expected growth rate in the stock price changes he discount rate for payoffs from the derivative changes hese two changes always offset each other exactly 11.9 Consider a long forward contract on a non-dividend paying stock, S, that matures at with delivery price K he payoff at maturity is S K Denoting the value of the forward contract at time zero r by f, means that f e Eˆ S K under Risk-Neutral Valuation, where we also discount the expected payoff in a risk neutral world Expected return on the underlying asset is r so E ˆ( S ) r Se Present value of expected payoff is r ˆ r r r r f e ES Ke e Se Ke r S Ke 11.3 Plan for his Week Look-back at Black-Scholes-Merton Differential Equation and Risk-Neutral Valuation Black-Scholes Formula for Non-Dividend paying Stock For Dividend paying stock Applications of Black-Scholes-Merton heory (Chapter 15-16) Derivatives of a Stock Paying a Dividend Yield (& Index Options) Derivatives on Currency exchange (FX) Derivatives on Futures (and Commodities) 1.3 In Summary Black-Scholes-Merton Differential Equation Derived from the Stock Price process and Ito s lemma, knowing the derivative is a function of the Stock Price process Form riskless portfolio: short one derivative & long stock Gives the differential equation f f 1 f rs S rƒ t S S With boundary conditions, it s solution describes the derivative Risk-Neutral Valuation applies as the BSM Diff Eq equation is independent of any variable affected by risk preference only S, t, σ, and r no expected return, µ Even though the return, μ, was in the model for the stock price! Principle of Risk-Neutral Valuation Assume the expected return of the underlying asset is r, i.e. µ = r Calculate the expected payoff from the derivative Discount the expected payoff at the risk-free rate, r

9 Valuing a Forward Contract with Risk-Neutral Valuation Consider a long forward contract on a non-dividend paying stock, S, that matures at with delivery price K he payoff at maturity is S K Denoting the value of the forward contract at time zero r by f, means that f e Eˆ S K, discounting the the expected payoff in a risk neutral world Expected return on the underlying asset is r so E ˆ( S ) r Se Present value of expected payoff is r ˆ r f e ES Ke r r r e Se Ke r S Ke 1.34 he Black-Scholes Formulas Black-Scholes Formulas for the Present Value of a European Call, c (Put, p ) with expiration and strike K on a non-dividend paying stock with price S r cs Nd ( 1) Ke Nd ( ) r p K e N( d ) S N( d ) where and ln( S / K) ( r / ) d d 1 1 ln( S / K) ( r / ) d he Black-Scholes Formulas he function N(x) is the cumulative probability distribution function for a standardized normal distribution with density function, ϕ(,1), where generally for mean µ and standard deviation σ and 1 (, ) e (,1) ( x ) / 1 e x / Where the resulting cumulative distribution function he Black-Scholes Formulas he cumulative probability distribution function, N(x), for a standardized normal distribution, ϕ(,1), where generally for mean µ and standard deviation σ x 1 u F( x;, ) exp du x N and where for the standardized case x 1 u N( x) F( x;,1) exp du and tables for N(x) are found in the back of the book

10 Deriving the Black-Scholes Formulas Deriving the Black-Scholes Formulas Risk-neutral approach continued Let mln Eˆ S ˆ / ln E S w / where w is the std dev of ln S Qwm Define a new variable Q(ln S m) / w where w and S e 1 Q So Q (,1) and / pq ( ) e Consider a European Call hen he expected value at maturity in a risk-neutral world Eˆ E ˆ max S K, S Kg( S) ds max S, K K Qwm e K p( Q) dq since ln S (ln )/ Qwm Km w By the risk-neutral valuation argument, the European Call price is r ce E ˆ max S, Qwm K e p( Q) dqk p( Q) dq (ln Km)/ w (ln Km)/ w Looking at the expected value we get w m Qwm Q / Qw / Eˆ max S K, S Kg( S) ds and e e / exp Q w m w / / / e e K where ln(s ) is normally distributed (recall, but note risk-neutrality) mw / e p( Qw) dq K ( ) (ln K m)/ w p Q dq (ln Km)/ w ln S ln, ln ˆ S r E S /, mw / e 1 N(ln K m) / wwk1 N(ln K m) / w Define N( x) as probability that a random variable, (,1), is less than x One way of deriving the BS formula is by solving the BSM differential equation subject to the boundary conditions Another approach is to use risk-neutral valuation (vs. BSM) Deriving the Black-Scholes Formulas Risk-neutral approach continued From the last page, using properties of N(x) and substituting for m where mln EˆS ˆ / ln E S w / & w is the std dev of ln S We find ˆ mw / Emax S K, e 1 N(ln K m) / ww K1 N(ln K m) / w mw / e N ln K m/ wwkn ln K m/ w mw / e N ln K ln S ( r / ) / KN ln S ln K ( r / ) / mw / w w r and e exp ln ES ESSe r Se Nln S K ( r / ) / KN ln S K) ( r / ) / r Se Nd ( ) KNd ( ) Deriving the Black-Scholes Formulas Risk-neutral approach continued So finally ˆ r Emax S K, Se N ln S K ( r / ) / KN ln S K ( r / ) / r Se N( d1) KN( d) d1 ln S K( r / ) / d ln S K( r / ) / And discounting back to today at the risk-free rate gives r ˆ r ce Emax S K, SNd ( 1) Ke Nd ( )

11 Interpretation of Black-Scholes Formula he formula r 1 r ce Eˆ max S K, r r e Se N( d1) KN( d) SNd ( ) Ke Nd ( ) Shows that N(d ) is the probability the call will be exercised in a riskneutral world and KN(d ) is the strike price times the probability the strike will be paid; Similarly, S N(d 1 )e r is the (risk neutral) expected value of a variable that is equal to S if S >K and zero otherwise So we have the call is the present value of the expected payoff 1.4 Properties of Black-Scholes Formula In the extreme, the options looks like forward contracts he call a long forward he put a short forward As S becomes very large c tends to S Ke -r and p tends to zero he call is almost certain to be exercised so it looks like a forward contract In the BS equations, d 1 and d become very large so N(d 1 ) and N(d ) are both close to one Complementing, N(-d 1 ) and N(-d ) tend toward zero As S becomes very small c tends to zero and p tends to Ke -r S 1.43 Properties of Black-Scholes Formula As volatility approaches zero, the stock is virtually riskless and will grow to S e r at time he payoff from a Call is max( S e r K, ) Its value today is e -r max( S e r K, ) = max( S Ke -r, ) o show this is consistent with Black-Scholes When S > Ke -r : ln( S /K ) + r > and as σ tends to zero, d 1 and d tend to infinity, so N(d 1 ) and N(d ) tend to one he call becomes c = S Ke -r When S < Ke -r : ln( S /K ) + r < and as σ tends to zero, d 1 and d tend to negative infinity, so N(d 1 ) and N(d ) tend to zero; c = he Call price is therefore always max(s Ke -r, ) Similarly for the Put: max( Ke -r S, ) when volatility tends to zero 1.44 Implied Volatility he implied volatility of an option is the volatility for which the Black-Scholes price equals the market price here is a one-to-one correspondence between prices and implied volatilities raders and brokers often quote implied volatilities rather than dollar prices to remove the transient nature of a quote on the option being coherent with the current market price for the underlying

12 An Issue of Warrants & Executive Stock Options - Dilution When a regular call option is exercised the stock that is delivered must be purchased in the open market When a warrant or executive stock option is exercised new reasury stock is issued by the company If little or no benefits are foreseen by the market the stock price will reduce at the time the issue is announced After the options have been issued it is not necessary to take account dilution when they are valued Before they are issued we can calculate the cost of each option as N/(N+M) times the price of a regular option with the same terms where N is the number of existing shares and M is the number of new shares that will be created if exercise takes place Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes Only dividends with ex-dividend dates during life of option should be included he dividend should be the reduction in the stock price expected often only 8% of declared amount because of tax reasons American Calls An American call (w/expiration ) on a non-dividend-paying stock should never be exercised early An American call on a dividend-paying stock should only be exercised at t i, immediately prior to an ex-dividend date Indeed, if it is, the investor receives S(t i ) - K If it is not exercised, the stock price drops by the dividend and the r ( t value of the Call is, i ) C S( ti) Di Ke ( ) If St ( ) r t i i Di Ke r ( ti) r ( ti) St ( i) K or Di Ke K or Di K[1 e ] it is not optimal to exercise early (keep the call, forego exercise) r ( t) If D [1 i i K e ] it is optimal to exercise early Let ex-dividend dates be at times t 1, t,, t n = t n+1 Early exercise is sometimes optimal at time t i if the dividend at that r time is greater than, ( ti1 ti K[1 e ) ] Kr( ti 1 ti) Rarely except for i = n and the last dividend is close to expiration 1.48 Black s Approximation for Dealing with Dividends in American Call Options Set the American price equal to the maximum of two European prices: 1. he 1st European price is for an option maturing at the same time as the American option. he nd European price is for an option maturing just before the final ex-dividend date

13 Option Results for a Stock paying a known Dividend Yield When a stock goes ex-dividend, the price of the stock is decremented by the dividend amount If the stock pays a dividend yield, q, the growth rate of the stock, μ, will be the growth rate, (if it paid no yield), decremented by the dividend yield, q With a dividend yield, q, the stock price S today grows to S at ( q) time : S Se Se In the absence of the dividend yield the stock price S today grows to S Se at time Alternatively, with a dividend yield, we could express the stock price S e -q today growing to S at time in the absence of dividends as q ( q) S ( Se ) e Se Se 1.51 Option Results for a Stock paying a known Dividend Yield So, it is the same probability distribution for S at time when he stock starts at price S and provides a dividend yield q, or he stock starts at price S e -q and pays no dividend We may draw the following conclusion as a rule: When valuing a European option of term on a stock paying a known dividend yield q, reduce the current stock price from S to S e -q and then value the option as though the stock had no dividend, but were starting with the price S e -q 1.5 Option Results for a Stock paying a known Dividend Yield Derivation of the Black-Scholes Differential Equation w/dividend Yield Extend earlier results to a stock paying a dividend yield q q r Lower Bound for European calls: cmax( Se Ke,) r q Lower Bound for European puts: pmax( Ke Se,) r q Put-Call Parity: c Ke p S e Extending Black-Scholes formulas to a stock paying a dividend yield q (replace S by S e -q ) where We start with the Stock Price Process and Ito s Lemma for a function of the price process hen form the candidate riskless portfolio S S t S z f f 1 f f f S S q r t S z c Se N( d1) Ke N( d) S t S S r q p Ke N( d) Se N( d1) A portfolio consisting of ln( S q / K) ( r q / ) Se q d since 1 ln ln Se ln K 1 derivative K f q ln( S / K) ( r q / ) ln S ln Kln e + shares d S S ln q K

14 Derivation of the Black-Scholes Differential Equation w/dividend Yield he value Π of the portfolio is given by f f S S he change in its value in time t is given by f f S S In the same time t, the holder of the portfolio also ƒ earns dividends on the stock position of qs t S as well as the capital gains Π Define W as the change in wealth of the holder of the portfolio in time t, then f f W f S qs t S S 1.55 Derivation of the Black-Scholes Differential Equation w/dividend Yield Substituting the stock price equation for S, and using (as before) the Ito result for f, gives f f 1 f f W ( S S ) t Sz S t S S f f ( StSz) qs t S S f 1 f f ( S qs ) t t S S 1.56 Derivation of the Black-Scholes Differential Equation w/dividend Yield As equation for W does not involve z, the portfolio Π is riskless during time t, and must return the risk-free rate W r t Substituting for W and Π gives f 1 f f f ( S qs ) t r( f S) t t S S S Which results in the Black-Scholes-Merton differential equation for a stock with dividend yield as f 1 ( r q) S f S f rf t S S 1.57 Risk-Neutral Valuation redux for Stock paying a Dividend Yield he concept of risk-neutral valuation is the single most important result for derivative analysis he Black-Scholes equation is again found independent of all variables affected by risk preference Only variables are S, t, σ, and r no expected return, µ he solution to the differential equation is therefore the same in a risk-free world as it is in the real world his leads to the principle of Risk-Neutral Valuation Assume the expected total return of the underlying asset is r he dividends provide a return of q, the stock price growth rate is r-q When the growth rate is r-q, the expected stock price at is S e (r-q) ( ) hat is: ES ˆ( ) r q Se Calculate the expected payoff from the derivative Discount the expected payoff at the risk-free rate, r

15 Deriving the Black-Scholes Formulas for Stock w/dividend Yield Deriving the Black-Scholes Formulas for Stock w/dividend Yield In originally validating the Black-Scholes formulas, we Risk-neutral approach continued ( Eˆ(ln S ) ln S ( rq ) ) used the approach of risk-neutral valuation Let mln Eˆ S ˆ / ln E S w / where w is the std dev of ln S Qwm Define a new variable Q (ln S m) / w where w and S e We can do the same thing for Stock with a dividend yield 1 Q So Q (,1) and / pq ( ) e Consider a European Call hen he expected value at maturity in a risk-neutral world Eˆ E ˆ max S, max S K, S Kg( S) ds K K Qwm e Kp( Q) dq (ln Km)/ w By the risk-neutral valuation argument, the European Call price is r ce E ˆ max S, Qwm K e p( Q) dqk p( Q) dq (ln Km)/ w (ln Km)/ w Looking at the expected value we get w m Qwm Q / Qw / Eˆ max S K, S Kg( S) ds e e / exp Q w m w / / e e / K where ln(s ) is normally distributed (recall, but note risk-neutral) mw / e ( ) ( ) p Qw dqk p Q dq (ln Km)/ w (ln Km)/ w ln S ln ( ), l n ˆ S rq ES /, mw 1.59 e / 1 N(ln K m)/ wwk1 N(ln K m)/ w 1.6 Define N( x) as probability that a random variable, N(,1), is less than x Deriving the Black-Scholes Formulas for Stock w/dividend Yield Risk-neutral approach continued From the last page, using properties of N(x) and substituting for m where mln EˆS ˆ / ln E S w / & w is the std dev of ln S We find ˆ mw / Emax S K, e 1 N(ln K m) / ww K1 N(ln K m) / w mw / e N ln K m/ wwkn ln K m/ w mw / e N ln K ln S ( r q / ) / KN ln S ln K ( r q / ) / mw / w w ( rq) e exp ln ES ES Se ( rq) Se N ln S K( rq / ) / KN ln S K ( r q / ) / ( rq) 1.61 Se N( d) KN( d ) 1 Deriving the Black-Scholes Formulas for Stock w/dividend Yield Risk-neutral approach continued Where finally ˆ ( rq) Emax S K, Se N ln S K ( r q / ) / KN ln S K ( r q / ) / ( rq) Se N( d1) KN( d) d1 ln S K( rq / ) / d ln S K ( rq / ) / And 1 r ˆ q r ce Emax S K, S e N( d ) Ke N( d )

16 he Binomial Model Applicable to a stock paying a dividend yield, as covered earlier in Chapter 11 o match stock price volatility, set t t u e ; d e Risk-neutral probability of an up move is chosen so the expected return is r-q over a time step of t and So a d p ; a e u d With the derivative value psu (1 p) Sd Se ( rq) t rt f e [ pf (1 p) f ] u d S ƒ ( r q) t S u ƒ u S d ƒ d 1.63 Index Options he most popular underlying indices in the U.S. are he Dow Jones Index times.1 (DJX) he Nasdaq 1 Index (NDX) he Russell Index (RU) he S&P 1 Index (OEX) he S&P 5 Index (SPX) Contracts are on 1 times index Settled in cash OEX is American and the rest are European Valuation: Use the formula for an option on a stock paying a dividend yield Set S = current index level Set q = average dividend yield expected during the life of the option 1.64 Currency Options European-Style Currency Options Currency options (both Euro- and American-style) trade on the Philadelphia Exchange (PHLX) here is also exists an active OC market Currency options are used by corporations to buy insurance when they have an FX exposure Denote the foreign interest rate by r f When a U.S. company buys one unit of the foreign currency it has an investment of S dollars he return from investing at the foreign rate is r f S dollars his shows that the foreign currency provides a dividend yield at rate r f 1.65 Foreign currency is an asset that provides a dividend yield of r f Use the formula for an option on a stock paying a dividend yield with S, the current exchange rate, and q = r ƒ Black-Scholes gives where r f r ( 1) ( ) r r f 1 cs e N d Ke N d p Ke N( d ) S e N( d ) d 1 d ln( / ) ( f / ) S K r r S K r r ln( / ) ( f / )

17 European-Style Currency Options Alternatively, we can simplify the Black-Scholes formulas by using the forward exchange rate, F, for maturity : ( r rf ) F S e Now Black-Scholes gives where r ce [ F N( d ) KN( d )] 1 r pe [ KN( d ) F N( d )] d d 1 1 ln( F / K) / ln( F / K) / 1.67 Mechanics of Futures Options Futures Options are American-Style When a Call futures option is exercised the holder acquires Long position in the futures Cash equal to the excess of the most recent futures settlement price over the strike price When a Put futures option is exercised the holder acquires Short position in the futures Cash equal to the excess of the strike price over the most recent futures settlement price If the futures position is closed out immediately at the futures price F Payoff from Call = F K Payoff from Put = K F 1.68 Put-Call Parity for Futures Options Consider Euro-style Call and Put futures options PF A: A Call plus Cash equal to Ke -r ( c + Ke -r ) PF B: A Put plus a Long Futures contract plus Cash equal to F e -r Where F is the futures price today (at t = ) ( p + F e -r ) PF A at expiration has value max(f, K ) F > K, Call is exercised, with cash K, PF A has value F F < K, Call is worthless, PF has value K PF B at expiration has value max(f, K ) Put has value max( K - F, ) Futures Contract has value F F Cash grows to F So F + (F F ) + max( K - F, ) = F + max( K - F, ) = max(f, K ) r PF have same value at and today cke pfe r 1.69 Futures Option Valuation from the Binomial Approach Form the portfolio: short one derivative and long futures he value at the end of one time period is F u ( Fu F) fu ( Fd F) fd ƒ u when ( fu fd)/( Fu Fd ) F he value of the PF today is ƒ F d r r [( Fu F) fu] e [( Fd F) fd] e f () f ƒ as the long future has no value at inception d Substituting and simplifying gives r f e [ pfu (1 p) fd] where p = (1 d)/(u d) as asserted back in Chapter

18 Futures Prices Drift in a Risk- Neutral World Define F t as the futures price at time t If we enter into a futures contract today its value is zero After a short increment, t, it provides a payoff F t F If r is the t risk-free rate at time, risk-neutral valuation rt gives e E ˆ[ F ] t F, as the contract has no value where Ê denotes expectations in the risk-neutral world hus we have similarly for so concatenating these results Eˆ( F t ) F EF ˆ( ˆ t) F t, EF ( 3t) F t,... where Eˆ( F ) F for any ln( F / K) / d1 ln( F / K) / d d1 ds ( r q) Sdt Sdz df Fdz q r he drift of the futures price in a risk-neutral world is zero From the stock price equation with dividend yield q equal to r Valuing European Futures Options Black-Scholes formula for an option on a stock paying a dividend yield q Set S = current futures price, F Set q = risk-free rate, r, ensuring the drift of F is Results in r ce FN( d1) KN( d) r p e KN( d ) F N( d ) 1 Futures Option Prices vs. Spot Option Prices If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot As futures price is higher than spot, if early exercise futures option pays off better An American put on futures is worth less than a similar American put on spot When futures prices are lower than spot prices (inverted market) the reverse is true If Euro-style call (put) futures option expires before futures contract, it is worth more (less) (in a normal market where futures prices are higher than spot) Summary of Key Results We can treat stock indices, currencies, and futures like a stock paying a dividend yield of q For stock indices, q = average dividend yield on the index over the option life For currencies, q = r ƒ For futures, q = r