Chapter 9 Properties of Stock Options Outline n

Chapter 9 Properties of Stock Options

Outline n Factors affecting option price n Assumptions and notation n Upper and lower bounds for option price n Put-call parity n Early exercise : calls on a nondividends-paying stock n Early exercise : puts on a non- 1

Factors affecting option prices n There are six factors affecting the price of a stock option: 1. The current stock price, S 0 2. The strike price, K 3. The time to expiration, T 4. The volatility of the stock prive, 5. The risk-free interest rate, r 6. The dividends expected during the 2

Effect of Variables on Option Pricing (Table 9. 1, Page 206) Summary of the effect on the price of a stock option of increasing one variable while keeping all others fixed. Variable S 0 K T r D c + – ? + + – p – +? + – + C + – + + + – P – + + + – + 3

Assumptions and Notation(1/2) We assume that there are some market participants, such as large investment banks, for which the following statements are true: 1. There are no transactions costs. 2. All trading profits (net of trading losses) are subject to the same tax rate. 4

Assumptions and Notation(2/2) S 0 : Stock price today ST : Stock price at option maturity K : Strike price T : Life of option p : European put option price : Volatility of stock price c : European call option price r : Risk-free interest rate P : American Put option price C : American Call option price D : Present value of dividends during option’s life 5

American vs European Options An American option is worth at least as much as the corresponding European option C c P p 6

Upper Bounds for Options -- No Dividends Ø Call option can never be worth more than the stock; Put option can never be worth more than the strike price. Hence, the stock price is an upper bound to the option price. American C≦S 0 P≦K European c≦S 0 (c≦C) p≦ke -r. T (p≦P) 7

Lower Bounds for European Call (1/2) -- No Dividends n A lower bound for the price of a European call option on a nondividend-paying stock is S 0 –Ke –r. T n We first look at a numerical example and then consider a more formal argument. ¡ Example 8

Lower Bounds for European Call (2/2) -- No Dividends Option is a right without obligation ; hence , it has positive value: c 0 For a more formal argument, we consider the following two portfolios: Protfolio A : c + Ke –r. T Protfolio B : S 0 Portfolio A is always worth no less than B! c max( S 0 –Ke –r. T, 0 ) (Equation 9. 1, p. 211) 9

Lower Bounds for European Put (1/2) -- No Dividends A lower bound for the price of a European put option on a non-dividend-paying stock is Ke –r. T – S 0 We first look at a numerical example and then consider a more formal argument. Example S 0 =$37 , K=$40 , r = 5% per annum , T = 0. 5 year 10

Lower Bounds for European Put (2/2) -- No Dividends Option is a right without obligation ; hence , it has positive value: p 0 For a more formal argument, we consider the following two portfolios: Protfolio C : p + S 0 Protfolio D : Ke –r. T Portfolio C is always worth no less than D! p max( Ke -r. T–S 0 , 0 ) (Equation 9. 2, p. 212) 11

Put-Call Parity : No Dividends (1/2) 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(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -r. T = p + S 0 (Equation 9. 3, p. 211) 12

Put-Call Parity : No Dividends (2/2) Portfolio A c(K) Ke -r. T Total ST≧K ST －Ｋ Ｋ Portfolio C p(K) S 0 Total ST≧K 0 ST ST＜K 0 K K ST＜K K－ST ST K 13

Example - Arbitrage Opportunities n Suppose that c=3 T = 0. 25 K = 30 n S 0 = 31 r = 10% D =0 What are the arbitrage possibilities when p = 2. 25 ? p=1? 14

Should Calls be Exercised Early ? n n (p. 216) Usually there is some chance that an American option will be exercised early This should never be exercised early ∵ c≧S 0－Ke -r. T and C≧c ∴ C≧c≧S 0－Ke -r. T≧S 0－K (exercise value) -r. T 15

Reasons for Not Exercising Call Early ( No Dividends ) n No income from the stock is sacrificed n Payment of the strike price is delayed n Holding the call provides insurance against stock price falling below strike price 16

Should Puts be Exercised Early? (p. 217) Are there any advantages to exercising an American put when ∵ p≧Ke -r. T －S 0 and P≧p ∴ P≧p ≧ Ke -r. T －S 0 ＞ K －S 0 But P must be larger than K S 0 because it is always possible to exercise before maturity 17

The Impact of Dividends on Lower Bounds to Option Prices (Page 218) ØWe will use D to denote the present value of the dividends during the life of the option. In the calaulation of D, a dividend is assumed to occur at the time of its ex-dividend date. 18

Extensions of Put-Call Parity American options; D = 0 (Equation 9. 4, p. 215) S 0 - K < C - P < S 0 - Ke –r. T European options; D > 0 (Equation 9. 7, p. 219) c + D + Ke -r. T = p + S 0 American options; D > 0 (Equation 9. 8, p. 219) S 0 - D - K < C - P < S 0 - Ke -r. T 19