Options

1. Options Dr. Lynn Phillips Kugele FIN 338

2. Options Review • Mechanics of Option Markets • Properties of Stock Options • Valuing Stock Options: • The Black-Scholes Model

3. Mechanics of Options Markets

4. Option Basics Option = derivative security Value “derived” from the value of the underlying asset Stock Option Contracts Exchange-traded Standardized Facilitates trading and price reporting. Contract = 100 shares of stock

5. Put and Call Options Call option Gives holder the right but not the obligation to buy the underlying asset at a specified price at a specified time Put option Gives the holder the right but not the obligation to sell the underlying asset at a specified price at a specified time

6. Options on Common Stock Identity of the underlying stock Strike or Exercise price Contract size Expiration date or maturity Exercise cycle American or European Delivery or settlement procedure

7. Option Exercise American-style Exercisable at any time up to and including the option expiration date Stock options are typically American European-style Exercisable only at the option expiration date

8. Option Positions Call positions: Long call = call “holder” Hopes/expects asset price will increase Short call = call “writer” Hopes asset price will stay or decline Put Positions: Long put = put “holder” Expects asset price to decline Short put = put “writer” Hopes asset price will stay or increase

9. Option Writing The act of selling an option Option writer = seller of an option contract Call option writer obligated to sell the underlying asset to the call option holder Put option writer obligated to buy the underlying asset from the put option holder Option writer receives the option premium when contract entered

10. Option Payoffs & Profits Notation: S0 = current stock price per share ST = stock price at expiration X = option exercise or strike price C = American call option premium per share c = European call option premium P = American put option premium per share p = European put option premium r = risk free rate T = time to maturity in years

11. Payoff to Call Holder (S- X) if S >X 0 if S< X Profit to Call Holder Payoff - Option Premium Profit =Max (S-X, 0) - C Option Payoffs & ProfitsCall Holder = Max (S-X,0)

12. Payoff to Call Writer - (S - X) if S > X = -Max (S-X, 0) 0 if S < X = Min (X-S, 0) Profit to Call Writer Payoff + Option Premium Profit = Min (X-S, 0) + C Option Payoffs & ProfitsCall Writer

13. Payoff & Profit Profiles for Calls Payoff: Max(S-X,0) -Max(S-X,0) Profit: Max (S-X,0) – c -[Max (S-X, 0)-p]

14. Payoff & Profit Profiles for Calls Payoff Call Holder Profit Profit 0 Call Writer Profit Stock Price

15. Payoffs to Put Holder 0 if S > X (X - S) if S < X Profit to Put Holder Payoff - Option Premium Profit = Max (X-S, 0) - P Option Payoffs and Profits Put Holder = Max (X-S, 0)

16. Payoffs to Put Writer 0 if S > X = -Max (X-S, 0) -(X - S) if S < X = Min (S-X, 0) Profits to Put Writer Payoff + Option Premium Profit = Min (S-X, 0) + P Option Payoffs and Profits Put Writer

17. Payoff & Profit Profiles for Puts Payoff: Max(X-S,0) -Max(X-S,0) Profit: Max (X-S,0) – p -[Max (X-S, 0)-p]

18. Payoff & Profit Profiles for Puts Profits Put Writer Profit 0 Put Holder Profit Stock Price

19. Option Payoffs and Profits CALLPUT Holder: Payoff Max (S-X,0) Max (X-S,0) (Long) Profit Max (S-X,0) - C Max (X-S,0) - P “Bullish” “Bearish” Writer: Payoff Min (X-S,0) Min (S-X,0) (Short) Profit Min (X-S,0) + C Min (S-X,0) + P “Bearish” “Bullish” S = P = Value of firm at expiration X = Face Value of Debt

20. Long Call Call option premium (C) = $5, Strike price (X) = $100. Profit ($) 30 20 10 Terminal stock price (S) 70 80 90 100 0 110 120 130 -5 Long Call Profit = Max(S-X,0) - C

21. Properties of Stock Options

22. Notation c= European call option price (C = American) p= European put option price (P = American) S0 = Stock price today ST=Stock price at option maturity X= Strike price T= Option maturity in years  = Volatility of stock price r=Risk-free rate for maturity Twith continuous compounding

23. American vs. European Options An American option is worth at least as much as the corresponding European option Cc Pp

24. Factors Influencing Option Values

25. Effect on Option Values Underlying Stock Price (S) & Strike Price (K) • Payoff to call holder: Max (S-X,0) • As S , Payoff increases; Value increases • As X , Payoff decreases; Value decreases • Payoff to Put holder: Max (X-S, 0) • As S , Payoff decreases; Value decreases • As X , Payoff increases; Value increases

26. Option Price QuotesCalls

27. Option Price QuotesPuts

28. Effect on Option Values Time to Expiration = T • For an American Call or Put: • The longer the time left to maturity, the greater the potential for the option to end in the money, the grater the value of the option • For a European Call or Put: • Not always true due to restriction on exercise timing

29. Option Price Quotes

30. Effect on Option Values Volatility = σ • Volatility = a measure of uncertainty about future stock price movements • Increased volatility increased upside potential and downside risk • Increased volatility is NOT good for the holder of a share of stock • Increased volatility is good for an option holder • Option holder has no downside risk • Greater potential for higher upside payoff

31. Effect on Option Values Risk-free Rate = r • As r : • Investor’s required return increases • The present value of future cash flows decreases = Increases value of calls = Decreases value of puts

32. Valuing Stock Options: The Black-Scholes Model

33. BSOPMBlack-Scholes (-Merton) Option Pricing Model • “BS” = Fischer Black and Myron Scholes • With important contributions by Robert Merton • BSOPM published in 1973 • Nobel Prize in Economics in 1997 • Values European options on non-dividend paying stock

34. Concepts Underlying Black-Scholes Option price and stock price depend on same underlying source of uncertainty A portfolio consisting of the stock and the option can be formed which eliminates this source of uncertainty (riskless). The portfolio is instantaneously riskless Must instantaneously earn the risk-free rate

35. Assumptions Underlying BSOPM • Stock price behavior corresponds to the lognormal model with μ and σ constant • No transactions costs or taxes. All securities are perfectly divisible • No dividends on stocks during the life of the option • No riskless arbitrage opportunities • Security trading is continuous • Investors can borrow & lend at the risk-free rate • The short-term rate of interest, r, is constant

36. Notation c and p = European option prices (premiums) S0 = stock price X = strike or exercise price r = risk-free rate σ = volatility of the stock price T = time to maturity in years

37. Formula Functions • ln(S/X) = natural log of the "moneyness" term • N(d) = the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x • N(d1) and N(d2) denote the standardnormal probability for the values of d1 and d2. • Formula makes use of the fact that: N(-d1) = 1 - N(d1)

38. The Black-Scholes Formulas

39. BSOPM Example Given: S0 = $42 r = 10% σ = 20% X = $40 T = 0.5

40. BSOPMCall Price Example d1 = 0.7693 N(0.7693) = 0.7791 d2 = 0.6278 N(0.6278) = 0.7349

41. BSOPM in Excel • N(d1): =NORMSDIST(d1) Note the “S” in the function “S” denotes “standard normal” ~ Φ(0,1) =NORMDIST() → Normal distribution Mean and variance must be specified ~N(μ,σ2 )