Introduction to Financial Derivatives

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1 Introduction to Financial Derivatives Week of October 28, 213 Options Where we are Previously: Swaps (Chapter 7, OFOD) This Week: Option Markets and Stock Options (Chapter 9 1, OFOD) Next Week : Binomial Tree Approach to Option Valuation (Chapter 11, OFOD) Final Exam Final Exam: Dec 17 th ; 9:am Noon Mergenthaler Assignment Assignment For This Week (October 28 th ) Read: Hull Chapter 9-1 Options & Option Markets Problems (Due Today, Oct 28 th ) Chapter 7: 1, 3, 5,6, 9, 12, 18; 22, 23 Chapter 7 (7e): 1, 3, 5, 6, 9, 12, 18; 2, 21 Problems (Due November 4 th ) Chapter 1: 7, 14, 15, 18, 19; 23 Chapter 9(7e): 7, 14, 15, 18, 19; 23 Look at DerivaGem problems: 1.21 & 1.26 (7e) 9.21 & For Next Week (November 5 th ) Read: Hull Chapter 11 (Binomial Tree Analysis) Problems (Due November 7 nd ): Chapter 1: 7, 14, 15, 18, 19; 23 Chapter 9(7e): 7, 14, 15, 18, 19; 23 Look at DerivaGem problems 1.21 & 1.26 (7e) 9.21 & 9.26 Problems (Due November 12 th ): Chapter 11: 1, 5, 6, 11; 2 Chapter 12(7e): 1, 5, 6, 11;

2 Plan for This Week Review of Option Types Options Review of Options and Mechanics Option Markets, Terminology and Exchanges Option Factors for Pricing Upper and Lower Bounds for Prices Key Arbitrage Arguments Put-Call Parity Details: Early Exercise, Dividends and American vs European style What s Ahead: Binomial Trees (12); Wiener Process & Ito Lemma (13); Black-Scholes-Merton Model (14); BSM for Options on Indexes, Currencies & Futures (16-17); The A Call is an option (the right, not the obligation) to Buy A Put is an option (the right) to Sell A European option can be Exercised only at the End of its life An American option can be Exercised at Any time Greeks (18) Option Positions Long call Long put Short call Short put Long Call on ebay Profit from buying one ebay European call option: option price = $5, strike price = $ Profit ($) Terminal stock price ($)

3 Short Call on ebay Profit from writing one ebay European call option: option price = $5, strike price = $1 Long Put on IBM Profit from buying an IBM European put option: option price = $7, strike price = $ Profit ($) Terminal stock price ($) Profit ($) Terminal stock price ($) Short Put on IBM Profit from writing an IBM European put option: option price = $7, strike price = $ Profit ($) Terminal stock price ($) Payoffs from Options Long Call Long Put K = Strike price, S T = Price of asset at maturity Payoff K max (S T K, ) Payoff K max ( K S T, ) S T S T Short Call Short Put Payoff Payoff K K -max (S T K, ) = min (K S T, ) S T -max ( K S T, ) = min (S T K, ) S T

4 Assets Underlying Exchange-Traded Options Stocks (focus for rest of this unit) Foreign Currency (E/A Philly) Stock Indices (CBOE A x OEX: SP1) Futures (on futures exchange) Commodities Indexes Interest Rates Currencies 1.13 Specification of Options Expiration date Jan, Feb, and Mar cycles (front months) Last trading day 3 rd Friday of the delivery month Expiration is the next day - Saturday Strike price Spaced $2.5, $5, or $1 apart Depends on stock price - $5 to $25, $25 to $2, and above $2 European or American Call or Put 1.14 Terminology Moneyness : At-the-money option In-the-money option Out-of-the-money option Terminology (continued) Option class: All option contracts of the same Type (call or put) and Style (American or European) that cover the same underlying security. Option series: All option contracts of the same class that also have the same unit of trade, expiration date and strike price (IBM 7 Oct Calls). Intrinsic value: The value of an option if it were exercised immediately. Time value: The portion of the option premium that is attributable to the amount of time remaining until the expiration of the option contract - whatever value the option has in addition to its intrinsic value

5 Dividends & Stock Splits Suppose you own N options with a strike price of K : No adjustments are made to the option terms for cash dividends When there is an n-for-m stock split, the strike price is reduced to mk/n the number of options is increased to nn/m Stock dividends are handled in a manner similar to stock splits Dividends & Stock Splits Consider a call option to buy 1 shares for $2/share How should terms be adjusted: for a 2-for-1 stock split? ½ the strike; 2 times the number of options for a 5% stock dividend? A 21 for 2 split 2/21 the strike; 21/2 the number of options Market Makers Most exchanges use market makers to facilitate options trading A market maker quotes both bid and ask (or offer ) prices when requested The market maker does not know whether the individual requesting the quotes wants to buy or sell Commissions Usual complication with fixed plus variable component and maximums/minimums Options can usually be closed-out with offsetting transaction commission is paid again, in full though, bid/offer spread is also paid If Option is Exercised, then commission for buy/sell is paid on stock transaction

6 Margin For Options with maturities less than 9-months, all purchases are with cash no borrowing / margin For Options greater than 9-months, a buyer can borrow up to 25% of the option value on margin For the Option writer, funds must be maintained in a margin account as collateral against default The amount depends on the position Margin When a naked option is written the initial margin is the greater of: 1. A total of 1% of the proceeds of the sale plus 2% of the underlying share price (15% for an index) less the amount (if any) by which the option is out of the money; 2. Or A total of 1% of the proceeds of the sale plus 1% of the underlying share price for call writer A total of 1% of the proceeds of the sale plus 1% of the exercise (strike) price for put writer Strategies have variations on the margin requirement CBOE has a document covering all cases (CBOE Margin Manual) Strategies = covered call, straddle, etc Margin Example: Write 4 naked calls (each call on 1 shares) for $5 each; strike = $4; share price is $38 Option is $2 out of money; 1. goes as 4 x (5 +.2 x 38 2) = $4,24 2. goes as 4 x (5 +.1 x 38) = $3,52 Initial Margin is the greater: $4,24 If the options had been for a puts; 1. goes as (option is in the money) 4 x (5 +.2 x 38) = $5,4 2. goes as 4 x (5 +.1 x 4) = $3,6 Initial Margin is the greater: $5, Warrants Warrants are options that are issued by a corporation or a financial institution Whereas options are written on existing shares of stock; corporations issue new shares when warrants are exercised Warrants are sometimes issued with bond offerings to reduce the cost of borrowing; these warrants are often stripped and sold separately by bond investors

7 Executive Stock Options Executive stock options are a form of remuneration issued by a company to its executives They are usually at the money when issued When options are exercised the company issues more stock and sells it to the option holder for the strike price 1.25 Executive Stock Options They become vested after a period of time (usually 1 to 4 years) They cannot be sold They often last for as long as 1 or 15 years Accounting standards now require the expensing of executive stock options 1.26 Convertible Bonds Convertible bonds are regular bonds that can be exchanged for equity at certain times in the future according to a predetermined exchange ratio Very often a convertible is callable The call provision is a way in which the issuer can force conversion at a time earlier than the holder might otherwise choose 1.27 Notation (for Stock Options; other options are similar) c : European call option price p : European put option price S : 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 continuous compounding

8 Effect of Variables on Option Pricing (all others fixed) Variable S K T r D c p C P ?? American vs European Options An American option is worth at least as much as the corresponding European option C c P p Increasing the value of the Variable Effect on Price Upper Bounds for Option Prices The Price of the underlying stock always bounds the value of a Call c S and C S Otherwise, sell the call and buy the shares for a riskless profit at expiration 1.31 Upper Bounds for Option Prices The Strike always bounds the value of a Put p K and P K The p (P) axis intercept is at K as stock price goes to Otherwise, sell the put and bank K to buy shares on expiration; sell shares into market, book profit For a European Put, the value is also bound by K at expiration; and it cannot be worth more than the PV of K (as no early exercise) p Ke rt

9 Lower Bound for European Call Option Prices (No Dividends) Consider the two portfolios Portfolio A: One European Call option plus cash = Ke rt Portfolio B: One Share cmax S Ke rt, For PF A at expiration, T If the cash is invested, it will grow to K at T If S T > K, then the option is exercised; PF A is worth S T We pay for the share with the cash and have a share worth S T If S T < K, then the option is worthless; PF A is worth K Hence PF A is worth max (S T, K) 1.33 Lower Bound for European Call Option Prices (No Dividends) PF A is worth max (S T, K) at expiration PF B at expiration is worth S T At expiration, PF A has value at least as great as PF B In the absence of arbitrage, this must also be true today Hence c + Ke -rt S or c S Ke -rt In addition, since the worst that can happen is that the call will expire worthless, cmax S Ke rt, 1.34 Calls: An Arbitrage Opportunity? Suppose that c = 3 S = 2 T = 1 r = 1% K = 18 D = What is the arbitrage opportunity? rt.1 S Ke 2 18e c Which violates the lower bound which says c S Ke -rt Hence, sell the stock, buy the call, CF = 2 3 = 17 At expiration, have 17e.1 = If stock is >18, exercise call; deliver into short, pocket.79 If stock is <18, forget call; buy stock, deliver into short, pocket > Lower Bound for European Put Option Prices (No Dividends) Consider the two portfolios Portfolio C: One European Put option plus One Share Portfolio D: Cash = Ke rt rt pmax Ke S, For PF C at expiration, T If S T < K, then the option is exercised; PF C is worth K We deliver the share and receive K If S T > K, then the option is worthless; PF C is worth S T Hence PF C is worth max (S T, K)

10 Lower Bound for European Put Option Prices (No Dividends) PF C is worth max (S T, K) at expiration PF D at expiration is worth K At expiration, PF C has value at least as great as PF D In the absence of arbitrage, this must also be true today Hence p + S Ke -rt or p Ke -rt S Again, since the worst that can happen is that the put expires worthless, rt pmax Ke S, 1.37 Put: An Arbitrage Opportunity? Suppose that p = 1 S = 38 T =.25 r = 1% K = 4 D = What is the arbitrage opportunity? rt.25 Ke S 4e p Which violates the lower bound which says p Ke -rt S Hence, buy the stock, buy the put, borrow 39 At expiration, owe 39e.25 = If stock is < 4, exercise put; deliver stock for 4, pay loan, pocket.2 If stock is > 4, forget put; sell stock, pay loan, pocket > Put-Call Parity (No Dividends) Put-Call Parity (No Dividends) Consider the two portfolios Portfolio A: One European Call option plus cash = Ke rt Portfolio C: One European Put option plus One Share PF A at expiration, T If the cash is invested, it will grow to K at T If S T > K, then the option is exercised; PF A is worth S T We pay for the share with the cash and have a share worth S T If S T < K, then the option is worthless; PF A is worth K Hence PF A is worth max (S T, K) 1.4 PF A is worth max (S T, K) at expiration PF C at expiration, T If S T < K, then the option is exercised; PF C is worth K We deliver the share and receive K If S T > K, then the option is worthless; PF C is worth S T Hence PF C is worth max (S T, K) At expiration, PF A has value equal to PF C In the absence of arbitrage, this must also be true today Hence c + Ke -rt p + S This is known as Put-Call Parity

11 Put-Call Parity: An Arbitrage Opportunity? Suppose that c = 3 S = 31 T =.25 r = 1% K = 3 D = What are the arbitrage opportunities when: p=2.25? Then PF A: And PF C: So PF C is overpriced to PF A. Sell PF C, buy PF A. p=1.? Then PF A: And PF C: rt.25 cke 3 3e ps rt.25 cke 33e ps So PF A is overpriced to PF C. Sell PF A, buy PF C Early Exercise Usually there is only a very small chance that an American Call option will ever be exercised early on a non-dividend paying stock A Call owner that thinks a stock is overpriced and otherwise has no long term desire to hold the stock could sell (not exercise) to capture option premium; shorting the stock is another alternative, but still no exercise Couldn t the price always go higher?! As a general rule, never exercise an American call early on a nondividend paying stock Indeed, rt We know that C c S Ke And given that r>, C>S K If it were optimal to exercise early, C= S K, so it is never optimal to exercise early In the absence of external knowledge, never exercise the American Call early on a non-dividend paying stock 1.43 Put-Call Parity (American-Style & No Dividend) Put-Call Parity (American-Style & No Dividend) Equality in Put-Call Parity only holds for European Options Consider the case of American Options on a non-dividend paying stock, P p and C = c : So P c + Ke -rt -S from P-C Parity & relation of A & E options And P C + Ke -rt -S or C - P S -Ke -rt Consider the two portfolios (options have identical strike and expiration) Portfolio A: One European Call option plus cash = K Portfolio E: One American Put option plus One Share The cash in PF A is invested at risk free rate If Put is NOT exercised early, PF E is worth max (S T, K) at T 1.44 At expiration PF A is worth max (S T K, ) + Ke rt = max (S T, K) K + Ke rt PF A is therefore worth more than PF E at expiration Suppose that the put in PF E is exercised early, at α This means that PF E is worth K at time α Even if the Call option were worthless, at t = α, PF A is worth Ke rα Therefore, PF A is always worth at least as much as PF E Hence c + K P + S And since C = c : C + K P + S or C P S K Combining with first result: S K C P S Ke -rt

12 Qualitative Reasons For Not Exercising an American Call Early No income is sacrificed in a non-dividend paying stock Payment of the strike price is delayed Holding the call provides insurance against stock price falling below strike price this comes at a cost, however Remember, insurance cost is commensurate with how much coverage you are buying Should an American Put Be Exercised Early? Like the Call, a Put can be viewed as insurance. But the price of a stock can never go below zero It may be optimal for the investor to forego the purchased insurance in order to realize the strike immediately As with Calls, we know p Ke rt S But even with r>, it may be optimal to exercise if the stock price is rt rt sufficiently low (e.g., if K S Ke or S K(1 e ) ) in this case, P K S (equality may hold, e.g., when S = ) Because there are circumstances when it is desirable to exercise an American Put early, it follows that it is always worth more than the corresponding European Put, so P>p. rt For example, when S =, K P p Ke The Impact of Dividends on Lower Bounds to Option Prices The extension to the bounds for European Options in the case of dividends is simply to adjust the cash positions in PF A or PF D to include the PV of the dividend, D For the Call rt c S D Ke For the Put p D Ke rt S 1.48 Extension of Put-Call Parity for Dividend Paying Stock European options; D >, Include PV of Dividend with Cash in PF A c + D + Ke -rt = p + S American options; D >, P > p and C c We have C P < S Ke -rt D > reduces C & increases P so this relation still holds To refine further, consider the American P-C Parity PFs with dividends (Options have identical strikes and expiration) Portfolio A: One European Call option plus cash = D + K Portfolio E: One American Put option plus One Share Cash is invested at the risk-free rate

13 Extension of Put-Call Parity for Dividend Paying Stock If the put is not exercised early, PF E is worth max (S T, K) + De rt at T PF A is worth, at expiration, max (S T K, ) + (D + K)e rt = max (S T, K) + De rt + Ke rt K Portfolio A is therefore worth more than PF E at expiration Suppose the put in PF E is exercised early, at α This means that PF E is worth De rα + K at time α Even if the Call were worthless, at α PF A is worth (D+K)e rα It follows that PF A is worth more than PF E in all cases, α> Hence, c + D + K > P + S Because C c; C + D + K > P + S or C P > S D K Hence S D K<C P<S Ke -rt

Introduction to Financial Derivatives

55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling: