Lecture 3: Option Trading and Arbitrage RelationsNeal Galpin FNCE30007, 2018 1 Last Week • Arbitrage Pricing and Payoff Tables • We used Tables like this to ask “Is there an arbitrage opportunity?” t=0 t=1 Buy Bond A -100 +105 Sell Bond B +101 -105 • We could have asked “What price of Bond B does not permit arbitrage?” t=0 t=1 Buy Bond A -100 +105 Sell Bond B +X -105 Neal Galpin FNCE30007, 2018 2 Today • Options, same approach as before: Ø Today we study strategies and relative prices of puts and calls Ø From next week on we’ll look at pricing options relative to the underlying • What can we do with options? Ø Place specific bets on future events Ø Insure against future outcomes Ø Use as building blocks in complex securities àWeek 10! • How do prices of Puts and Calls relate to each other? • What is the value of an American option relative to a European option? Neal Galpin FNCE30007, 2018 3 Payoff vs Profit Diagrams • Long Stock Short Stock Profit Profit S(T) S(T) • Call option Put option Profit Profit S(T) S(T) Neal Galpin FNCE30007, 2018 4 In Words/Math • A long call option will give you 0 if you don’t exercise or S-X if you do Ø You exercise only when S-X > 0 Ø The payoff is S-X or 0; whichever is bigger Ø We write that as max(S-X,0) • A long put option will give you 0 if you don’t exercise or X-S if you do Ø You exercise only when X-S > 0 Ø The payoff is X-S or 0; whichever is bigger Ø We write that as max(X-S,0) • Shorts get the negative…Well...sort of Neal Galpin FNCE30007, 2018 5 Zero Sum Game… • Suppose at time T, we can buy a share at $10 or sell at $9; consider a long call with a strike price of $9.50 • If the long exercises, they will pay $9.50 for a share that costs $10 but they can only sell for $9.00 Ø If they want to keep the share, that’s a good deal (bought at $9.50, would have paid $10) Ø If they don’t want to keep the share, that’s a bad deal (bought at $9.50, can only sell at $9.00) • If the short has to deliver because the long exercises: Ø the short position has to buy to deliver, the short receives $9.00 - $10.00 (sold for $9, had to buy for $10) Ø the short position already has the share, the opportunity cost is $9.00 - $9.50 (sold for $9, could have sold for $9.50) • Conclusion: Yuck transaction costs; we will work with the case that neither the long or short wants the share Neal Galpin FNCE30007, 2018 6 DISCLAIMER • The following is not investment advice. • All of the strategies on display are risky, with some of them possibly leading to a loss larger than your initial investment. • Passing your derivatives exam is a necessary condition to understand the potential and the risks, but not automatically a sufficient condition. • At the time of writing, I did not have a position in any option contract or any of the stocks featuring in these slides. Neal Galpin FNCE30007, 2018 7 Place Specific Bets Neal Galpin FNCE30007, 2018 8 Place Specific Bets • Suppose you have family living in Hunts Point (WA) • You went to visit them in early August 2013. On a morning jog, you run into • He tells you about his exciting new running plans, which he will start next year when he will have plenty of time on his hands • Hearing that, what do you do? A. You take a selfie with this guy B. You nod politely and finish your tour C. You spend the rest of your run trying to get rid of the echo of his voice in your ear D. … Neal Galpin FNCE30007, 2018 9 Microsoft CEO • WSJ MoneyBeat blog (2013-08-12): 15 Neal Galpin FNCE30007, 2018 10 What happened in the stock and options market? Data: Thomson Reuters Tick History MATLAB: L4MSFT_Calls.m Neal Galpin FNCE30007, 2018 11 Zooming in Data: Thomson Reuters Tick History MATLAB: L4MSFT_Calls.m Neal Galpin FNCE30007, 2018 12 What happened to the stock price following the trade? Source: Yahoo! Finance Neal Galpin FNCE30007, 2018 13 Microsoft CEO (cont’d) • WSJ MoneyBeat blog (2013-08-23) Neal Galpin FNCE30007, 2018 14 Directional Strategies • Bullish strategies: pay off when the underlying increases in value Ø Buying the stock Ø Buying a call Ø ……………………… • Bearish strategies: pay off when the underlying decreases in value Ø ……………………… Ø ……………………… Ø ……………………… Neal Galpin FNCE30007, 2018 15 Vertical Bull Spread • We can combine different options in a spread • Note: this profit diagram does not account for time value of money! Neal Galpin FNCE30007, 2018 16 Vertical Bear Spread • Neither does this one! Neal Galpin FNCE30007, 2018 17 Option Price Data used in Figures and Examples • Option data for the fictitiously named company OPSY: Ø Current stock price S is $22.50. Ø Time to maturity T is six months. Ø Strike prices K are $17.50, $20, $22.50, and $25. Ø Continuously compounded, risk-free interest rate r is 5 percent per year à B = e-0.05*0.5 = 0.9753 Ø European options prices: Strike Price Call Price Put Price K0 = $17.50 $5.50 $0.10 K1 = 20.00 3.50 0.50 K2 = 22.50 2.00 1.50 K3 = 25.00 1.00 3.00 Neal Galpin FNCE30007, 2018 18 Non-directional Bets • Reuters (2013-09-18) • Suppose you share this view, how can you profit using options? Neal Galpin FNCE30007, 2018 19 Butterfly Spread • Butterfly spread Profit Long call (K0 = 17.5) 7 Long call (K2 = 22.5) 17.5 22.5 2 0 –0.5 S(T) –2 Butterfly call spread –5.5 20 • Which part of the article does this cover? Neal Galpin FNCE30007, 2018 20 Straddles & Strangles • Straddle / Strangle • For which event would we want to use these strategies? Neal Galpin FNCE30007, 2018 21 Combination Strategies • Combination strategies combine options of different types on the same underlying stock and expiring on the same date, where the options are either both purchased or both written. Ø Buying a call and a put with the same strike price creates a straddle (called a put-to-call strategy in London). Ø Buying a call and a put with different strike prices creates a strangle. • Straddles and strangles are volatility plays. Ø Bet on events that can have positive or negative outcomes. Ø Sometimes used in expectation of neutral markets. Neal Galpin FNCE30007, 2018 22 Option Relations Neal Galpin FNCE30007, 2018 23 Option Relations • We explore three sets of options price relations assuming no-arbitrage but without assuming a stock price evolution Ø Put–call parity for European options. Ø Option price bounds. Ø The early exercise of American options. Neal Galpin FNCE30007, 2018 24 Put-Call Parity • Remember the forward? Value Maturity (T) Today Long 0 S(T) – F Forward • Now consider a portfolio of a long call and short put, both with strike K = F = S(1+R) Value Maturity (T), if Maturity (T), if Today S(T) < F S(T) > F Long call c Short put -p Total c–p • What is the value of the c-p portfolio? Neal Galpin FNCE30007, 2018 25 Put-Call Parity (cont’d) • Repeat the long call, short put strategy with K≠F Value Maturity (T), if Maturity (T), if Today S(T) < K S(T) > K Long call c Short put -p Total c–p • What is the sign of (c-p) if K<F? And what if K>F? Neal Galpin FNCE30007, 2018 26 Put-Call Parity (cont’d) • Let the discount factor between now and T equal B • Portfolio pays S(T)-K for sure at time T, so the current value of the portfolio must be • General result: Put-Call Parity (PCP) for European options on a non-dividend paying stock c – p = S – BK ↔ -S + BK + c - p = 0 (16.2) Neal Galpin FNCE30007, 2018 27 Put-Call Parity (cont’d) • Another rewrite -S + BK + c - p = 0 ↔ c + BK = p + S Payoff Payoff S(T) S(T) Neal Galpin FNCE30007, 2018 28 Put-Call Parity (cont’d) • What option pricing model did we use? • Any other assumptions needed? Neal Galpin FNCE30007, 2018 29 PCP Violations for Palm Shares • March 2, 2000: 3Com did an equity carve-out of its subsidiary Palm. Ø 3Com sold 5 percent of Palm shares to the general public in an IPO. § 3Com planned to distribute the remaining shares in a spin-off to its existing shareholders (who would get 1.525 shares of Palm for every share of 3Com they held). Ø Palm stock closed at the end of the day at $95.06. Ø This implied that 3Com’s stock price was -$63 per share. Predictably, Palm started declining after the IPO. • A similar thing happened with Alibaba & Yahoo in 2014 (see link) Neal Galpin FNCE30007, 2018 30 PCP Violations for Palm Shares (cont’d) • March 16, 2000: Options began trading on Palm stock. At the time Ø S = $55, K = $55, T = 1 month, B = $0.995, Ø c = $5 and p = $9. • PCP gives (p = c + BK – S) Ø Synthetic put price = Call price + Present value of strike - Stock price Ø $4.725 = $5 + 0.995 ´ $55 – $55 • How can you make an arbitrage profit from Palm’s option prices? • What could have prevented you from implementing the strategy? Neal Galpin FNCE30007, 2018 31 Put-Call Parity Adjusted for Dividends • We can modify the basic PCP to incorporate dividends. Start from c + BK = p + S • Assuming that a fixed-dollar dividend div is paid on a known future date t1 and B1 is today’s price of a zero maturing on the ex-dividend date, put–call parity for European options (with a known dollar dividend) can be written as Neal Galpin FNCE30007, 2018 32 PCP Adjusted for Dividends (cont’d) • If the underlying pays dividends at a continuous rate d, then put–call parity for European options (with a known dividend yield) can be written as c = p + e-dTS - e-rTK (2) • This is again using the infinite cash flow idea Neal Galpin FNCE30007, 2018 33 American versus European Options Neal Galpin FNCE30007, 2018 34 American Options are worth more than European Options • European options can only be exercised at expiration, whereas American options can be exercised anytime. Ø As “more cannot be worth less,” an American option can never be priced less than an otherwise identical European option. • For a call: cA ³ c ³ 0 (16.5a) • For a put: pA ³ p ³ 0 (16.5b) Neal Galpin FNCE30007, 2018 35 Call Price Bounds • An American call’s price is less than or equal to the stock price, that is, cA £ S (16.6) Ø When the stock price is zero, the call price is also zero. § follows by setting S = 0 in (16.6), combined with cA ≥ 0 . • An American call’s price is at least equal to the immediate exercise value (intrinsic value) cA ≥ max(S-K,0) Neal Galpin FNCE30007, 2018 36 Price Bounds for American Calls • Combining Prices Upper Bound Lower Bound Call prices in here 45° 0 K S, stock price • The book shows similar relations for the put option • Can we make these bounds a bit tighter? Neal Galpin FNCE30007, 2018 37 American versus European Options • The value of a European call (and hence an American call) is at least equal to the present value of exercising at maturity c ≥ max(S-e-rTK,0) • Two ways to look at this 1. Put-call parity: 2. Use the OPSY data (changing the price of the $20 call) and an arbitrage table Portfolio Today (Time 0) Expiration date (Time T) Cash flow Cash flow S(T) £ $20 S(T) > $20 Buy $20 strike call –$2 0 S(T) – $20 Short sell stock $22.5 –S(T) –S(T) Buy bonds to lend –0.9753 ´ $20 $20 $20 present value of strike (K = $20, B = $0.9753) Cash flows $0.99 $20 – S(T) 0 Neal Galpin FNCE30007, 2018 38 Early Exercise of an American Call • WSJ MoneyBeat blog (2013-08-12) 15 • WSJ MoneyBeat blog (2013-08-23) Neal Galpin FNCE30007, 2018 39 Early Exercise of an American Call Today, share price $34.70 • Suppose MSFT does not pay dividends, and 1+R = 1.001 • Should you exercise your options if you owned the Ø strike $35 options? Ø strike $32 options? Neal Galpin FNCE30007, 2018 40 Early Exercise of an American Call What are the payoffs? • Strike $35: • Strike $32: Ø Exercise immediately: Ø Sell the option: • A Call offers Insurance! • For a non-dividend paying stock: American call < / = / > European call Neal Galpin FNCE30007, 2018 41 Early Exercise with Dividends • When the stock pays dividends, we face a trade-off in deciding whether to exercise or not. By exercising, we Ø bring forward the payment of the strike price $K, and Ø take possession of the stock and start earning dividends. • This suggests that early exercise may be optimal if Ø Neal Galpin FNCE30007, 2018 42 Early Exercise with Dividends (cont’d) • Given PV(DIV) > K - PV(K) = K*R/(1+R), when would you want to exercise? A. As soon as you can B. Any time before the dividend is paid C. Just before the dividend will be paid • Suppose we have two call options on OPSY, with strikes K1=$20 and K2=$22.50, respectively. Which option is more likely to be exercised early for a given dividend payment? Neal Galpin FNCE30007, 2018 43 Early Exercise Rules-of-Thumb • Want to make him angry? ? • Use these rules-of-thumb Ø Exercise your calls the first time that cA = S - K – PV(DIV) Ø Exercise your puts the first time that pA = K - S – PV(DIV) Neal Galpin FNCE30007, 2018 44 Conclusion • Option strategies Ø Directional § Bullish § Bearish Ø Non-directional § Volatility-based • Options in (re)insurance • Relations between puts and calls Ø Put-Call Parity • American vs European options Ø Early exercise Neal Galpin FNCE30007, 2018 45 Resources • FED/Debt ceiling volatility: Ø http://www.reuters.com/article/2013/09/18/us-usa-options-fed- idUSBRE98H04E20130918 • Berkshire Hathaway: Ø http://www.warrenbuffett.com/warren-buffett-looks-at-mispricing-long-term-options/ Ø http://www.businessinsider.com.au/warren-buffett-q1-equity-index-puts-2013-5 Ø http://www.gurufocus.com/news/50998/a-closer-look-at-berkshire-hathaways-equity- put-options • Options Report: Ø http://blogs.wsj.com/moneybeat/tag/options-report/ Ø http://blogs.wsj.com/moneybeat/tag/options-market/ • Alibaba vs Yahoo: Ø http://ftalphaville.ft.com/2014/09/19/1977892/a-friendly-reminder-that-yahoos-core- business-is-worth-11-6-billion/ Neal Galpin FNCE30007, 2018 46 Resources (cont’d) • Bloomberg have released a series of videos on using their platform for option strategies and pricing: Ø Call strategies, http://bloomberginstitute.us2.list- manage.com/track/click?u=24e64d1d4dd8142171b1be4c6&id=4d73435432 &e=17ebd96668 Ø Puts and Spreads, http://bloomberginstitute.us2.list- manage.com/track/click?u=24e64d1d4dd8142171b1be4c6&id=5bb68cfee3& e=17ebd96668 Ø Put and Call Strategies, http://bloomberginstitute.us2.list- manage2.com/track/click?u=24e64d1d4dd8142171b1be4c6&id=2a795fe561 &e=17ebd96668 • ASX: https://www.asx.com.au/documents/resources/UnderstandingStrategies.pdf • Neal Galpin FNCE30007, 2018 47
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