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Overview of the Options Market No Arbitrage Bounds on Option Prices Put-Call Parity Early Exercise of American Options Topic 4: Option Basics and Put-Call Parity Finance 353: Derivatives Hengjie Ai The Fuqua School of Business, Duke University Daytime MBA 2009 1 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market No Arbitrage Bounds on Option Prices Put-Call Parity Early Exercise of American Options Outline of the Lecture Overview of the Options Market No Arbitrage Bounds on Option Prices Put-Call Parity Optimal Exercise of American Options Required Reading Before Class: Section 9.1, 9.2, 9.3 of textbook 2 / 23 Finance 353 Topic 4: Option Basics Some Market Statistics Option Trading Volume by Type of Contracts and by Exchange Some Market Statistics Option Trading Volume by Type of Contracts and by Exchange Some Market Statistics Equity Option Trading Volume and Index Option Trading Volume Some Market Statistics Equity Option Trading Volume and Index Option Trading Volume Some Market Statistics Futures Option Trading Volume Overview of the Options Market Some Market Statistics No Arbitrage Bounds on Option Prices Reading Newspaper Quotes Put-Call Parity Floor-Based Trading and Electronic Trading Early Exercise of American Options Reading Newspaper Quotes Quote from WSJ 6 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Some Market Statistics No Arbitrage Bounds on Option Prices Reading Newspaper Quotes Put-Call Parity Floor-Based Trading and Electronic Trading Early Exercise of American Options Floor-Based Trading and Electronic Trading Floor-based trading Market makers Floor brokers Order book o¢ cials and exchange o¢ cials Margin Requirements Electronic Trading The primary market maker The competitive market maker The electronic access member 7 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Some Market Statistics No Arbitrage Bounds on Option Prices Reading Newspaper Quotes Put-Call Parity Floor-Based Trading and Electronic Trading Early Exercise of American Options Floor-Based Trading and Electronic Trading Floor-based trading Market makers Floor brokers Order book o¢ cials and exchange o¢ cials Margin Requirements Electronic Trading The primary market maker The competitive market maker The electronic access member 7 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Upper Bounds on the Value of Call and Put Options An Upper Bound on the Price of Call Options Price of a call must be less than the current price of the stock: c S0 Intuition: A call allows you to buy the stock, however, at a cost (strike price). A no arbitrage argument: How would you make an arbitrage if c > S0 ? Portfolio Strategy Cash Flow Cash Flow @ Date T @ Date 0 ITM OTM Long Stock S0 ST ST Sell Call c ST + K 0 Total c S0 K ST 8 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Upper Bounds on the Value of Call and Put Options An Upper Bound on the Price of Call Options Price of a call must be less than the current price of the stock: c S0 Intuition: A call allows you to buy the stock, however, at a cost (strike price). A no arbitrage argument: How would you make an arbitrage if c > S0 ? Portfolio Strategy Cash Flow Cash Flow @ Date T @ Date 0 ITM OTM Long Stock S0 ST ST Sell Call c ST + K 0 Total c S0 K ST 8 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Upper Bounds on the Value of Call and Put Options An Upper Bound on the Price of Put Options Price of a put is never higher than the strike price rT p e K Intuition: With a put option, in order to obtain the strike price, you must give up the stock, which is of positive value. A no arbitrage argument: How would you make an arbitrage if p > e rT K ? Answer: Write the put and invest the money The upper bounds for option prices derived here are very general It does not depend on whether the option is European or American It does not depend on the dividend policy of the stock 9 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Lower Bounds for the Price of Put Options A Lower Bound for Put Options on Non-Dividend Paying Stocks For non-dividend-paying stocks: rT p e K S0 A no arbitrage proof: Portfolio Strategy Date 0 Cash Flow Date T Cash Flow C Long 1 put p max fK , ST g Long 1 stock S0 D Invest Ke rT Ke rT K Conclusion: Portfolio C has higher payo¤ than D, therefore must have higher cost. In general, [PV (ST ) + p ] PV (K ) 10 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Lower Bounds for the Price of Put Options A Lower Bound on Put Options on Stocks with Dividend For stocks with continuous dividend: rT δT p e K e S0 A no arbitrage proof: Portfolio Strategy Date 0 Cash Flow Date T Cash Flow C0 Long 1 put p max fK , ST g Long e δT stock e δ T S0 D0 Invest Ke rT Ke rT K Similarly, for stocks with discrete dividends: ! n p e rT K S0 ∑e rti Di i =1 11 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Lower Bounds for the Price of Call Options Non-Dividend Paying Stocks For Non-dividend Paying Stocks: rT c0 S0 e K A no arbitrage proof: Portfolio Strategy Date 0 Cash Flow Date T Cash Flow A Long 1 call c max fK , ST g Invest Ke rT in bond e rT K B Long 1 stock S0 ST Conclusion: There cannot be an asset that always beats the stock. In general, c + PV (K ) PV (ST ) 12 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Upper Bounds on the Value of Call and Put Options No Arbitrage Bounds on Option Prices Lower Bounds for the Price of Put Options Put-Call Parity Lower Bounds for the Price of Call Options Early Exercise of American Options Lower Bounds for the Price of Call Options Dividend Paying Stocks For stocks with continuous dividend: δT rT c e S0 e K A no arbitrage argument: Portfolio Strategy Date 0 Cash Flow Date T Cash Flow A0 Long 1 call c max fK , ST g Invest Ke rT in bond e rT K B0 Long e δT stock e δ T S0 ST Similarly, for stocks with discrete dividends: ! n c S0 ∑e rti Di e rT K i =1 13 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Put-Call Parity for Stock Options No Arbitrage Bounds on Option Prices An Alternative Formulation of the Put-Call Parity Put-Call Parity Put-Call Parity for Other Options Early Exercise of American Options Put-Call Parity for Stock Options Non-Dividend Paying Stocks For non-dividend paying stocks: rT c p = S0 e K A no arbitrage argument: Portfolio Strategy Date 0 Cash Flow Date T Cash Flow A Long 1 call c max fK , ST g Invest Ke rT in bond e rT K C Long 1 put p max fK , ST g Long 1 stock S0 Conclusion: law of one price implies portfolio A and B must have the same price at date 0: c + e rT K = p + S0 14 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Put-Call Parity for Stock Options No Arbitrage Bounds on Option Prices An Alternative Formulation of the Put-Call Parity Put-Call Parity Put-Call Parity for Other Options Early Exercise of American Options Put-Call Parity for Stock Options Dividend Paying Stocks For stocks with continuous dividend at annual rate δ: δT rT c p=e S0 e K Proof: compare portfolio A 0 and C 0 For stocks with (known) discrete dividend payments: ! n c p= S0 ∑e rti Di e rT K i =1 where the dividend payments D1 , D2 , , Dn are scheduled at t1 , t2 , , tn . 15 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Put-Call Parity for Stock Options No Arbitrage Bounds on Option Prices An Alternative Formulation of the Put-Call Parity Put-Call Parity Put-Call Parity for Other Options Early Exercise of American Options An Alternative Version of the Put-Call Parity An alternative version: c p = PV (stock ) PV (Exercise Price ) This encompasses the formula developed above. Verify it. Implications of the Put-Call Parity If we can price a call, we will be able to price a put, and vice versa. We can create synthetic options, synthetic stocks, and synthetic T-bills using the formula: Call Put = Stock TBill It does NOT apply to American options 16 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Put-Call Parity for Stock Options No Arbitrage Bounds on Option Prices An Alternative Formulation of the Put-Call Parity Put-Call Parity Put-Call Parity for Other Options Early Exercise of American Options Put-Call Parity for Other Options Consider European call and put options on the Euro: with strike price K ($/e) and maturity date T . Current exchange rate is x0 ($/e). The continuously compounded interest rate for $ is r , and r for e. The put-call parity is: r T rT c p=e x0 e K Put-Call Parity for Stock Exchange Options strike c p = PV (ST ) PV ST 17 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options Why is it important? Question: Should an American option have a higher price than a European option with the same strike price and maturity? Answer: It depends. We cannot understand the valuation of an American option without understanding how to exercise it. When is early exercise optimal? 18 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options American Call on Non-Dividend Paying Stock (I) The Short Answer It is NEVER optimal to exercise before maturity. Why? The short answer: You make more money by selling the option than exercising it cAmerican c St e r (T t ) K > St K If you exercise it, the payo¤ is St K . If the option is out of the money, then you do not want to exercise either. 19 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options American Call on Non-Dividend Paying Stock (II) The Long Answer What if you cannot sell the option? Compare the two strategies: Strategy A: Exercise now, and sell the stock at time T Strategy B: Delay exercise until maturity Strategy Date t Date T Cash Flow Cash Flow In The Money Out of the Money A K ST ST B 0 ST K 0 K e r (T t ) K i h e r (T t ) K Total K S T + e r (T t) 1 K e r (T t)K > ST 20 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options American Call on Non-Dividend Paying Stock (III) What if you expect the stock price will fall? You will short the stock, in stead of exercise the call Compare the two strategies: Strategy A: Exercise at time t, and sell the stock immediately Strategy B: Delay exercise until maturity, short the stock at time t Assume interest rate is 0 for simplicity Strategy Date t Date T Cash Flow Cash Flow In The Money Out of the Money A St K 0 0 B 0 ST K 0 K e r (T t ) K e r (T t ) K St h ST i ST Total St K e r (T t) 1 K e r (T t)K ST > 0 21 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options American Call on Dividend Paying Stocks Lessons from the last example If you have an opinion about the underlying asset and want to t speculate on that opinion, trade that asset directly. Don’ do it by exercising the option. Even if you cannot trade the asset directly, sell the option rather than exercise it. Advantages of delaying exercise of American calls You pay the strike price K at a later date (time value of money) You can decide later on after uncertainty is resolved However, if the stock pays dividend, you lose the dividend payment if you delay exercise. Conclusion: If dividend payment is small, then late exercise is probably optimal It might be optimal to exercise early if dividend is large, especially when you are anticipating a large dividend payment. 22 / 23 Finance 353 Topic 4: Option Basics Overview of the Options Market Why is it important? No Arbitrage Bounds on Option Prices American Call on Dividend Paying Stocks (Optional Reading) Put-Call Parity Early Exercise of Put Options (Optional Reading) Early Exercise of American Options Early Exercise of Put Options Disadvantages: By exercising early, we sell the stock for $K today, yet the stock price might go down in the future (optionality) By exercising early, we lose the dividend payment Advantages We receive $K early (time value of money) Conclusion: The only reason to exercise early is the time value of money Not optimal to exercise early if the dividend payment is large, or if stock price is volatile 23 / 23 Finance 353 Topic 4: Option Basics