8.1 Properties of Stock Option Prices Some model-independent results Chapter 8

8.2 Notation c : European call option price p :European put option price S 0 :Stock price today K :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 option maturity D :Present value of dividends during option’s life r :Risk-free rate for maturity T with cont comp

8.3 Effect of Variables on Option Pricing (Table 8.1, page 168) cpCP Variable S0S0 K T  r D ++ – + ?? – + – – –– + – + – +

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

8.5 Pricing bounds

8.6 Calls: An Arbitrage Opportunity? Suppose that c = 3 S 0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity? (Buy the call, sell underlying, deposit PV(K)!)

8.7 Puts: An Arbitrage Opportunity? Suppose that p = 1 S 0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity? (Buy put, buy stock, borrow PV(K)!)

8.8 Put-Call Parity; No Dividends (Equation 8.3, 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, K ) at the maturity of the options They must therefore be worth the same today –This means that c + Ke -rT = p + S 0

8.9 Arbitrage Opportunities Suppose that c = 3 S 0 = 31 T = 0.25 r = 10% K =30 D = 0 What are the arbitrage possibilities and how are they exploited when p = 2.25 ? p = 1 ?

8.10 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

8.11 For an American call option: S 0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if 1You feel confident about the performance of 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

8.12 Reasons For Not Exercising a Call Early, Case 1) (No Dividends ) No income is sacrificed by holding the option We delay paying the strike price Holding the call provides insurance against stock price falling below strike price Everything else is the same => wait, don’t exercise

8.13 Case 2) The investor should rather sell the option He will receive more than the intrinsic value since other investors prefer to hold the stock. Otherwise price could not be S 0.

8.14 Should Puts Be Exercised Early ? Are there any advantages to exercising an American put when S 0 = 60; T = 0.25; r =10% K = 100; D = 0

8.15 The Impact of Dividends on Lower Bounds to Option Prices (Equations 8.5 and 8.6, page 179)

8.16 Extensions of Put-Call Parity American options; D = 0 European options; D > 0 c + D + Ke -rT = p + S 0 American options; D > 0

8.17 Trading Strategies Involving Options Note: Net profit diagrams Chapter 9

8.18 Three Alternative Strategies Take a position in the option and the underlying Take a position in 2 or more options of the same type (A spread) Combination: Take a position in a mixture of calls & puts (A combination)

8.19 Positions in an Option & the Underlying (Figure 9.1, page 186) Profit STST K STST K STST K STST K (a) (b) (c)(d)

8.20 Bull Spread Using Calls (Figure 9.2, page 187) K1K1 K2K2 Profit STST

8.21 Bull Spread Using Puts Figure 9.3, page 189 K1K1 K2K2 Profit STST

8.22 Bear Spread Using Calls Figure 9.4, page 189 K1K1 K2K2 Profi t STST

8.23 Bear Spread Using Puts Figure 9.5, page 190 K1K1 K2K2 Profit STST

8.24 Butterfly Spread Using Calls Figure 9.6, page 191 K1K1 K3K3 Profit STST K2K2

8.25 Butterfly Spread Using Puts Figure 9.7, page 192 K1K1 K3K3 Profit STST K2K2

8.26 Calendar Spread Using Calls Figure 9.8, page 193 Profit STST K

8.27 Calendar Spread Using Puts Figure 9.9, page 193 Profit STST K

8.28 A Straddle Combination Figure 9.10, page 194 Profit STST K

8.29 Strip & Strap Figure 9.11, page 195 Profit KSTST KSTST StripStrap

8.30 A Strangle Combination Figure 9.12, page 196 K1K1 K2K2 Profit STST

8.31 Warning!! Next week (but only Tuesday) our use of quantitative methods will accelerate. Chapter 10 onwards. We need models for uncertainty to price options. We will start using stochastic variables and processes. The fall break is a good opportunity to refresh/review your stat and prob knowledge.