University of Colorado at Boulder Leeds School of Business MBAX-6270 MBAX Introduction to Derivatives Part II Options Valuation

Transcription

1 MBAX-6270 Introduction to Derivatives Part II Options Valuation Notation c p S 0 K T European call option price European put option price Stock price (today) Strike price Maturity of option Volatility of stock price C P S T D r American call option price American put option price Stock price (at option maturity) PV of dividends paid during life of option Risk-free rate for maturity T, with continuous compound.

2 Effect of Variables on Option Pricing Variable c p C P S K + + T +/- (*) +/- (*) r + + D + + (*) Depending on other inputs Option prices Bounds, No-Arbitrage Conditions Consider American Options Consider Dividends

3 Upper Bounds for Option Prices An American or European call option gives the holder the right to buy one share of a stock for a certain price. No matter what happens: and An American put option can be exercised immediately and no matter how low the stock trades: A European put we know that at maturity the option cannot be worth more than K. Therefore: Lower Bound for European Call Prices; No Dividends A European Call option must be worth at least the value of the stock purchased now, minus the PV of the strike paid at time T (the price you would pay if you decided to exercise the option): c S 0 Ke -rt

4 Lower Bound for European Call Prices; No Dividends Consider two portfolios: Portfolio A One European call option with strike=k and maturity=t One zero coupon bond that pays K at time T Portfolio B One share of the stock What are these portfolios worth at time T if the asset is trading at Portfolio A is worth, Portfolio B is worth Lower Bound for European Call Prices; No Dividends Since portfolio A is always worth more than portfolio B at time T it must true now. Thus

5 Lower Bound for European Put Prices; No Dividends A European Put option must be worth at least the PV of the strike price (the money you receive at time T to sell the stock if you decided to exercise the option), minus the value of the stock purchased now: p Ke -rt S 0 Let s prove it Lower Bound for European Put Prices; No Dividends Consider two portfolios: Portfolio A One European put option with a strike of K and a maturity of T. One share of the stock Portfolio B One zero coupon bond that pays K at time T. What are these portfolios worth at time T if the asset is trading at Portfolio A is worth, Portfolio B is worth

6 Lower Bound for European Put Prices; No Dividends Portfolio A cannot be worth less than portfolio B at time T. So it must be true now. Thus Put-Call Parity: No Dividends In deriving the lower bound results we had two portfolios: Portfolio A: European call on a stock + zerocoupon bond that pays K at time T Portfolio B: European put on the stock + the stock At maturity these two portfolios were worth under all market conditions

7 The Put-Call Parity Result Both are worth at the maturity of the options They must therefore be worth the same today. This means that: Payoff K S T American Options - Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a nondividend paying stock This should never be exercised early No income is sacrificed You delay paying the strike price Holding the call provides insurance against stock price falling below strike price Let s take a closer look

8 American Call on non-dividend Paying Stock Assume that interest rates are positive then we know and If we exercise an in-the-money American call option today then we will make: But Therefore if you exercise early you lose money Should You Exercise American Puts Early? Consider an extreme situation Suppose the strike price is $10 and The stock price is $0.01 If you exercise now you immediately make $9.99 If you wait then your maximum gain might be $10, but you might receive less than $10 because negative stock prices are impossible.

9 The Impact of Dividends on Lower Bounds to European Call Prices Consider two portfolios: Portfolio A : One European call option plus an amount of cash equal to Portfolio B : One share Following a similar argument as before we find: (*) Remember: D is the PV of the dividend The Impact of Dividends on Lower Bounds to European Put Prices Consider two portfolios: Portfolio A : One European call option plus one share Portfolio B : One share Following a similar argument as before we find: (*) Remember: D is the PV of the dividend

10 Options Recap on Bounds Call price must be: non-negative less than the stock at least as much as the PV of exercising now Put price must be: non-negative at least as much as PV of exercising now less than the strike for American, and less than PV of strike for European Bounds for European or American Call Options Prices (No Dividends)

11 Bounds for European and American Put Options (No Dividends) Option Pricing The Black-Scholes model

12 Black-Scholes-Merton Model First published in 1973 in a paper by Black and Scholes The Pricing of Options and Corporate Liabilities Robert Merton called this the Black-Scholes option pricing model The model gives a closed formula for European option prices Merton (MIT) and Scholes (Stanford) received the 1997 Nobel Prize in Economics Assumptions No arbitrage opportunities in the market You can borrow/lend at the risk-free rate You can buy and sell the underlying in any quantity (including fractional quantities) You can buy and sell the underlying at any time continuously No transaction costs Constant interest rates Constant volatility No dividends (this one can be relaxed)

13 The Concepts Behind the Model The 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 The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes-Merton differential equation, the solution follows The Black-Scholes-Merton Formulas where Notice the symmetry in the formulas

14 The Black-Scholes-Merton Formulas alternative formulation based on forwards where Notice the symmetry in the formulas The N(x) Function N(x) is the Standard Normal CDF, i.e. the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x See tables at the end of the book p818-9 or Use the Excel function Norm.Dist(x,0,1,True) or Norm.S.Dist(x,True)

15 Normal Distribution: density and N(x) pdf CDF N(x)= CDF= Area(pdf) Options Additional Thoughts

16 Properties of Black-Scholes Formula Q.What happens when becomes very large? tends to and tends to zero Q. What happens when becomes very small? tends to zero and tends to Q. What happens as becomes very large? tends to S and tends to Q. What happens as becomes very large? tends to S and tends to Extensions of Put-Call Parity American options; D = 0 S 0 K < C P < S 0 Ke rt European options; D > 0 c + D + Ke rt = p + S 0 American options; D > 0 S 0 D K < C P < S 0 Ke rt

17 Volatility The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in a short time period time is approximately Q. If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day? Estimating Volatility from Historical Data Take observations S 0, S 1,..., S n at intervals of t years (e.g. for weekly data t = 1/52) Calculate the cont. comp. return as: Calculate the standard deviation, s, of The historical volatility estimate is Caution: historical volatility does not necessarily/ever give you the marketimplied price!

18 Nature of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed Traders will manage this through a variety of mechanisms One might assume that there are 252 trading days in one year rather than 365 actual days One might weight days differently: non-business days have approximately 1/3 the weight of business days Market-Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price. There is a one-to-one correspondence between prices and implied volatilities. Traders and brokers sometimes quote implied (Black-Scholes) volatilities, rather than dollar prices. (Q. why?)