Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.1 Properties of Stock Option Prices Chapter 7

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.2 Effect of Variables on Option Pricing (Table 7.1, page 169) cpCP Variable S0S0 X T  r D ++ – + ?? – + – – –– + – + – +

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.3 Assumptions There are no transaction costs All trading profits are subject to the same tax rate Borrowing and lending at the risk-free interest rate is possible

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.4 Notation c : European call option price p :European put option price S 0 :Stock price today X :Strike price T :Life of option  :Volatility of stock price C :American Call option price P :American Put option price S T :Stock price at time T D :Present value of dividends during option’s life r :Risk-free rate for maturity T with cont comp

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.5 American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.6 Upper Bounds Stock price is an upper bound to the American or European call c ≤ So and C ≤ So, if this would not hold, an easy arbitrage profit would be available by buying the stock and selling the call option. Strike price is an upper bound to the American or European put p ≤ X and P ≤ X For European put

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.7 Lower bound European calls on non-dividend-paying stock

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.8 Calls: An Arbitrage Opportunity? Suppose that c = 3 S 0 = 20 T = 1 r = 10% X = 18 D = 0 Is there an arbitrage opportunity?

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.9 What if the price of a European Call is and arbitrageur buys the call and shorts the stock and invests on r for a year

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.10 If the stock price is greater than 18, arbitrageur exercises the option, closes short and makes a profit of $0,79. If the stock price is less than 18, (for ex. 17) the stock is bought in the market and a short position is closed out with even greater profit of 1,79.

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.11 Portfolio A: 1 Eur Call + Xe -rT Portfolio B: 1 share of stock Cash in A is invested in r and will grow to X in time T If S T >X, c is exercised and A =S T If S T <X, c is worthless and A =X At time T portfolio A is worth max (S T, X)

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.12 Portfolio B is worth S T at time T Portfolio A is worth at least as portfolio B at time T Hence, A must be worth at least as B today

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.13 Lower Bound for European Call Option Prices; No Dividends ( Equation 7.1, page 172) Since c must be non negative i.e. c ≥ 0

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.14 Lower bound European puts on non-dividend-paying stock

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.15 Puts: An Arbitrage Opportunity? Suppose that p = 1 S 0 = 37 T = 0.5 r =5% X = 40 D = 0 Is there an arbitrage opportunity?

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.16 What if the price of a European Put is and arbitrageur borrows $38 for six months and buys a put and the stock at the end of six months arbitrageur will have to pay

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.17 If the stock price is bellow 40, arbitrageur exercises the option, Sells the stock and makes a profit of $1,04. If the stock price is greater than 40, (for ex. 42) the option is discarded and the stock is sold in the market with even greater profit of 3,04.

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.18 Portfolio C: 1 Eur Put +1 share Portfolio D: cash of Xe -rT If S T <X, p is exercised at T and C =X If S T >X, p is worthless and C =S T At time T portfolio C is worth max (S T, X)

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.19 Portfolio C is worth at list as portfolio D at time T Hence, C must be worth at least as D today Assuming cash is invested at r, D is worth X at T

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.20 Lower Bound for European Put Prices; No Dividends (Equation 7.2, page 173) Since p must be non negative i.e. p ≥ 0 p  max (Xe -rT - S 0, 0)

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.21 Put-Call Parity; No Dividends (Equation 7.3, page 174) Consider the following 2 portfolios: –Portfolio A: European call on a stock + PV of the strike price in cash –Portfolio C: European put on the stock + the stock Both are worth MAX( S T, X ) at the maturity of the options They must therefore be worth the same today –This means that c + Xe -rT = p + S 0 This relationship is known as put-call parity

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.22 Arbitrage Opportunities Suppose that c = 3 S 0 = 31 T = 0.25 r = 10% X =30 D = 0 What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.23 Early Exercise Usually there is some chance that an American option will be exercised early An exception is an American call on a non-dividend paying stock This should never be exercised early

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.24 For an American call option: S 0 = 50; T = 0.083; X = 40; D = 0 Should you exercise immediately? What should you do if 1You want to hold the stock for the next 3 months? 2You do not feel that the stock is worth holding for the next 3 months? An Extreme Situation

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.25 Reasons For Not Exercising a Call Early (No Dividends ) No income is sacrificed We delay paying the strike price Holding the call provides insurance against stock price falling below strike price

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.26 Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0.25; r =10% X = 100; D = 0

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.27 The Impact of Dividends on Lower Bounds to Option Prices (Equations 7.5 and 7.6, page 179)

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 7.28 Extensions of Put-Call Parity American options; D = 0 S 0 - X < C - P < S 0 - Xe -rT Equation 7.4 p. 178 European options; D > 0 c + D + Xe -rT = p + S 0 Equation 7.7 p. 180 American options; D > 0 S 0 - D - X < C - P < S 0 - Xe -rT Equation 7.8 p. 180