Questions: Provide a numerical example to show as T increase, the value of an European style put option would decrease;Provide a numerical example to show as T increase, the value of an European style call option would decrease.

Please see attached. The questions are on slide #71, the relevant knowledge relating to this question are from slide #37 to the end.

I appreciate your help.

Derivative Security: A contingent claim (financial contract) whose value is derived from an underlying primary asset. Introduction to Derivatives Derivative security: options, forwards, futures, and swaps. Definition of derivatives 2. Discounted Cash Flow (DCF) valuation 3. Hedge funds and derivatives 1. 1 Tie Su -- FIN422 -- 2016 @ UM 2 Derivative Security: Types of Derivatives: Options: right, not obligation, to enter a transaction. Underlying asset: Stocks, stock indices / bonds, interest rates / currencies 2. Commodities (gold, crude oil, natural gas, corn, orange juice) 3. Another derivative contract: 1. Futures: futures options (options on futures) 2. Swaps: swaptions 3. Options: call on call, call on put, put on call, put on put 4. Other assets: 1. Volatility: VIX derivatives: http://www.cboe.com/micro/volatility/introduction.aspx 2. Weather: http://www.cmegroup.com/trading/weather/; 3. Credit events: http://www.cmegroup.com/trading/cds/index.html; 1. 3 Tie Su -- FIN422 -- 2016 @ UM Tie Su -- FIN422 -- 2016 @ UM Forwards: an obligation to transact, negotiated in the over-the- counter market. Futures: a standardized forward contract, traded on an exchange. Swaps: a portfolio of forward contracts. Other products of financial engineering: Structured products: https://www.google.com/?gws_rd=ssl#q=structured+products 4 Tie Su -- FIN422 -- 2016 @ UM An Example of a Structured Product: Asset Value and Market Price An equity-linked note: invest $1 today and the future payoff is Value is an intrinsic nature of the asset. However, value is based on the next year's return on the S&P 500 Index, rsp500. $1 if rsp500 is negative; $1(1+0.6rsp500) if rsp500 is positive. Equity-linked note (ELN) is a debt instrument, usually a bond, that differs from a standard fixed-income security in that the final payout is based on the return of the underlying equity, which can be a single stock, basket of stocks, or an equity index. Equitylinked notes are a type of structured products. 5 Tie Su -- FIN422 -- 2016 @ UM 6 The DCF Approach unobservable: intrinsic value, fundamental value. Price is from a transaction or negotiation. Price is directly observable. In an efficient market, we hope that price is an unbiased estimator of value. Economists determine price by the intersection of supply and demand curves. Financial economists use the Discounted Cash Flow (DCF) approach to determine value as the present value of expected future cash flows. Tie Su -- FIN422 -- 2016 @ UM Use and Abuse of Derivatives: Nature of derivative contracts: value t 0 E cash flowt 1 rt zero-sum game. Derivative securities shift risks but do not create/destroy wealth. t AIG lost $182 billion, while hedge fund manager John Paulson brought home $3.7 billion in 2007 by betting against sub-prime. His take-home pay totaled to $8.1 billion between 2006-2008. The Discounted Cash Flow (DCF) approach is the most important approach in asset valuation. It focuses on the size, timing, and risk of cash flows. http://www.paulsonco.com/about-paulson/ 7 Tie Su -- FIN422 -- 2016 @ UM 8 Tie Su -- FIN422 -- 2016 @ UM 9 2007: A Great Year to Be a Hedge Fund Manager! 2008: Another Great Year to Be a Hedge Fund Manager! Tie Su -- FIN422 -- 2016 @ UM 10 Hedge Fund Fees: \"2 & 20\" Hedge Fund Fees: \"2 & 20\" HWM = High Water Mark A hedge fund has a \"2 & 20\" fee structure: 2% of management fee plus 20% incentive fee on fund returns in excess of T-Bill rate. In a performance reporting period, a hedge fund yields 22% when the T-Bill rate is 1%. What are the fees to the hedge fund managers and the return to the hedge fund shareholders. AUM = Assets Under Management Lock-up period: 1-3 years RNP = Redemption Notice Period: 45-90 days Fees: 2&20 with HWM Return to HF shareholders = 22% - 6.2% = 15.8% Tie Su -- FIN422 -- 2016 @ UM Entry Trading Fees High ($5m) Lock up, 45-90 days RNP 2&20 with HWM Mutual fund Low ($2500) Daily @ end of day 1% of AUM w/o HWM ETF/ETN Lowest ($200) Intra-day 0.5% of AUM w/o HWM Hedge fund Fees to HF managers = 2% + (22% - 1%)20% = 6.2% 11 Tie Su -- FIN422 -- 2016 @ UM 12 Tie Su -- FIN422 -- 2016 @ UM What Is an Option? Options Markets The right, not obligation, to buy an underlying asset at a pre- specified price, on or before a pre-specified date is called an American style call option. Terminology 2. Profit and payoff profiles 3. Trading strategies 4. Option markets 1. 13 Tie Su -- FIN422 -- 2016 @ UM The right, not obligation, to sell an underlying asset at a pre- specified price, on or before a pre-specified date is called an American style put option. 14 Basic Terminology: Tie Su -- FIN422 -- 2016 @ UM Option Classes: Exercise price, strike price, striking price (X) Call options (C) Adjusted for stock splits, stock dividends, special cash dividends, but not adjusted for regular cash dividends. The U.S. exchangetraded options are \"dividend unprotected\". Put options (P) Expiration date (T), every Friday in the near term and the third Fridays of distant months. 15 Tie Su -- FIN422 -- 2016 @ UM 16 Tie Su -- FIN422 -- 2016 @ UM Option Series: Option Style: American-style options: options that can be exercised at any Different exercise prices (say 10 different exercise prices) time on or before maturity date. In the U.S., all stock options and S&P 100 Index (OEX) options are American-style options. Different expiration months (say 6 different time to maturities) One underlying asset derives (106=60) calls and 60 puts. European-style options: options that can be exercised on only the day of expiration, but not before expiration. In the U.S., all stock index (except OEX) options are European-style options. 17 Tie Su -- FIN422 -- 2016 @ UM 18 Buyers of an American-style option: Continue to hold the long position in the option Sell the option Exercise the option Writers of an American-style option: Continue to hold the short position in the option Buy back the option Fulfill the option's buyer's exercise of the option Buyers of a European-style option: Continue to hold the long position in the option Sell the option Exercise the option at option expiration Writers of a European-style option: Continue to hold the short position in the option Buy back the option Fulfill the option's buyer's exercise of the option at option expiration Option Style: Bermudan-style options: options that have a deferred American- style feature: They cannot be exercised in the first portion of their lives, but they can be exercised at any time beyond a certain time period. e.g., a five-month option that is exercisable at any time during the last month before maturity. Firms issuing callable bonds generally carry a Bermudan-style call option against their bondholders. 19 Tie Su -- FIN422 -- 2016 @ UM Tie Su -- FIN422 -- 2016 @ UM 20 Tie Su -- FIN422 -- 2016 @ UM Moneyness of Options: Intrinsic Values (IV) of American-Style Options: In-the-money Call option: S > X IV(call) = Max(S-X, 0) Put option: S X 21 Tie Su -- FIN422 -- 2016 @ UM Option Premium: the Value of an Option Option premium = intrinsic value (IV) + time value (TV) Intrinsic (parity, exercise) value: the value of the option if it were exercised right away. Time (speculative) value: the value of the ability to wait. Time value = option premium - intrinsic value Early exercise premium = option premium of an Americanstyle option - option premium of an otherwise identical Europeanstyle option = CA - CE or PA - PE. It measures the value of the ability to exercise the option prior to its maturity. 22 Tie Su -- FIN422 -- 2016 @ UM Apple Inc. (AAPL) stock price was at $98.53 on Monday, 1/11/2016, up $1.57 from the previous trading day. CALL OPTIONS: Strike Contract Name Expire at close Friday, October 21, 2016 Last Bid Ask Change %Change Volume Open Interest Implied Volatility 95.00 AAPL161021 C00095000 11.35 11.75 11.90 0.26 2.34% 69 186 29.50% 97.50 AAPL161021 C00097500 10.50 10.45 10.60 0.15 1.48% 91 484 29.24% 100.00 AAPL161021 C00100000 9.45 9.25 9.40 0.75 8.62% 217 14910 28.99% 105.00 AAPL161021 C00105000 7.15 7.15 7.30 0.40 5.93% 1616 1470 28.54% http://finance.yahoo.com/q/op?s=AAPL&date=1477008000 23 Tie Su -- FIN422 -- 2016 @ UM 24 Tie Su -- FIN422 -- 2016 @ UM Apple Inc. (AAPL) stock price was at $98.53 on Monday, 1/11/2016, up $1.57 from the previous trading day. PUT OPTIONS: Strike Contract Name Trading commission is $10+$2 per contract. How much do you collect when you write 6 put contracts? Expire at close Friday, October 21, 2016 Last Bid Ask Change %Change Volume Open Interest Implied Volatility 95.00 AAPL161021 P00095000 9.25 8.85 9.00 -0.20 -2.12% 232 2625 31.40% 97.50 AAPL161021 P00097500 10.20 10.05 10.25 -0.50 -4.63% 111 755 31.24% 100.00 AAPL161021 P00100000 11.35 11.35 11.55 -0.50 -4.22% 531 1867 30.97% 105.00 AAPL161021 P00105000 14.82 14.25 14.40 0.07 0.47% 36 985 30.39% How much do you pay when you buy 8 call contracts? http://finance.yahoo.com/q/op?s=AAPL&date=1477008000 25 Tie Su -- FIN422 -- 2016 @ UM 26 Tie Su -- FIN422 -- 2016 @ UM What are the intrinsic value and time value of the call option? IV(Call) = max(0, S-X) = max(0, 98.53 - 97.50) = $1.03 (in-the-money) TV(Call) = C - IV(Call) = 10.50 - 1.03 = $9.47 Does the call option have a large time/speculative premium? List all factors to support your answer. What does a call option buyer expect? List all factors that would make a call option buyer profitable. What does a call option writer/seller expect? List all factors that would make a call option writer profitable. Consider the AAPL October 2016 $97.50 call and put options: Today = 1/11/2016 Option expiration = 10/21/2016 Option time to maturity = T = 284 days = 284/365 years Stock price = S = $98.53 Option exercise price = X = $97.50 Call option premium = C = $10.50 Put option premium = P = $10.20 Both call and put option are American-style options. AAPL is expected to pay a $0.52 dividend on 2/7, 5/7, and 8/7/2016. http://finance.yahoo.com/q/hp?s=AAPL&a=11&b=12&c=1980&d=00&e=11&f=2016 &g=v Current six-month T-Bill yields 0.42% per year, r = 0.42%; http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/ 27 Tie Su -- FIN422 -- 2016 @ UM 28 Tie Su -- FIN422 -- 2016 @ UM Factors that affect option time value: 1. Stock return volatility 2. Level of stock price 3. Option time to maturity Call option buyer expects: Call option value to increase Stock return volatility to increase Stock price to increase Call option writer/seller expects: Call option value to decrease Stock return volatility to decrease Stock price to decrease Option time to maturity to decrease 29 Tie Su -- FIN422 -- 2016 @ UM What are the intrinsic value and time value of the put option? IV(Put) = max(0, X-S) = max(0, 97.50 - 98.53) = $0 (out-of-the-money) TV(Put) = P - IV(put) = 10.20 - 0 = $10.20 Does the put option have a large time/speculative premium? List all factors to support your answer. What does a put option buyer expect? List all factors that would make a put option buyer profitable. What does a put option writer expect? List all factors that would make a put option writer profitable. 30 Put option buyer expects: Put option value to _________ Stock return volatility to _________ Stock price to _________ Put option writer/seller expects: Put option value to _________ Stock return volatility to _________ Stock price to _________ Option time to maturity to _________ 31 Tie Su -- FIN422 -- 2016 @ UM Tie Su -- FIN422 -- 2016 @ UM Example: Insurance (protective) put: buy a stock and a put option on the stock. 32 Strategy Insurance put Positions S0 + P0 ST > $1 million What is the value of this American-style put? Tie Su -- FIN422 -- 2016 @ UM 56 Tie Su -- FIN422 -- 2016 @ UM Example 4b: Early Exercise of American-Style Put Options. Example 5a: Early Exercise of American-Style Call Options and Dividend Payments. S = $100 S = $100 (end of day price) X = $1,000,000 X = $95 American-style put option: T = 2,000 years dividend = $15 ex-dividend date = tomorrow What is the value of this American-style put? At 4% interest rate, the opportunity cost (interest lost) on the intrinsic value ($999,900) is $109.58 per day, much higher than the remaining time value of the put option. The put should be exercised right away! The value of the put is its intrinsic value, or $999,900. When an American option should be early exercised, its time value becomes zero. 57 Tie Su -- FIN422 -- 2016 @ UM American style call option: T = 1 day What is the value of this American-style call? 58 Example 5a: Early Exercise of American-Style Call Options and Dividend Payments. Example 5a: Early Exercise of American-Style Call Options and Dividend Payments. S = $100 (end of day price) S = $100 (end of day price) X = $95 X = $95 dividend = $15 dividend = $15 ex-dividend date = tomorrow ex-dividend date = tomorrow American style call option: T = 1 day American style call option: T = 1 day What is the value of this American-style call? What is the value of this American-style call? a) $4.75 b) $5.00 c) $5.25 b) a) c) Tie Su -- FIN422 -- 2016 @ UM $4.75: American option cannot be sold below its IV. $5.00 $5.25: Everyone would write the option and profit. Would you early exercise the call, or hold it till maturity? Early exercise the call. When an American option should be early exercised, its time value becomes zero. Would you early exercise the call, or hold it till maturity? 59 Tie Su -- FIN422 -- 2016 @ UM 60 Tie Su -- FIN422 -- 2016 @ UM Example 5b: Early Exercise of American-Style Call Options and Dividend Payments. Example 5b: Early Exercise of American-Style Call Options and Dividend Payments. If the dividend is $0.15, instead of $15, then should you early exercise? If the dividend is $0.15, instead of $15, then should you early exercise? S = $100 (end of day price) S = $100 (end of day price) X = $95 X = $95 dividend = $0.15 dividend = $0.15 ex-dividend date = tomorrow ex-dividend date = tomorrow American style call option: T = 1 day American style call option: T = 1 day What is the value of this American-style call? a) b) c) What is the value of this American-style call? $4.75 $5.00 $5.25 a) b) c) Would you early exercise the call, or hold it till maturity? 61 Tie Su -- FIN422 -- 2016 @ UM $4.75 $5.00 $5.25 Would you early exercise the call, or hold it till maturity? 62 Tie Su -- FIN422 -- 2016 @ UM Example 6a: Time to Maturity and European-Style Call Options. Conclusion: Decision to early exercise an American-style call option is based on the tradeoff between reduction in intrinsic value (dividend payment, not early exercise) and loss of time value (early exercise). S = $1,000,000 X = $100 European-style call option #1: T = 2 days European-style call option #2: T = 2,000 years Tradeoff: choose the smaller of the two: reduction of IV(CA) due to a cash dividend payment if not exercise CA. loss of TV(CA) if exercise CA. 63 Tie Su -- FIN422 -- 2016 @ UM Q: Which European-style call has a higher value? If you get this question right (and get it right not by chance), you may have paid attention in class! 64 Tie Su -- FIN422 -- 2016 @ UM Example 6a: Time to Maturity and European-Style Call Options. Example 6b: Early Exercise of American-Style Call Options. S = $1,000,000 S = $1,000,000 X = $100 X = $100 European-style call option #1: T = 2 days American-style call option: T = 2,000 years European-style call option #2: T = 2,000 years What is the value of this American-style call? This question is conceptually challenging. 2. Should you early exercise the American-style call? 1. Q: Which European-style call has a higher value? A: The #2 call has a higher value because it has more time to maturity. 65 Tie Su -- FIN422 -- 2016 @ UM 66 Example 6b: Early Exercise of American-Style Call Options. Example 6b: Early Exercise of American-Style Call Options. S = $1,000,000 S = $1,000,000 X = $100 X = $100 American-style call option: T = 2,000 years American-style call option: T = 2,000 years 1. 2. What is the value of this American-style call? This question is conceptually challenging. 0, 0, $1 67 Tie Su -- FIN422 -- 2016 @ UM Tie Su -- FIN422 -- 2016 @ UM 0 $1 Should you early exercise the American-style call? No! If you exercise now, you lose $100, which is option's TV. If the stock price becomes $1.1 million tomorrow, the call allows you to benefit from the stock price increase, but $999,900 cash does not. So the value of the call is greater than $999,900. Don't exercise an American-style call just because it is deep in-themoney. A significantly large cash dividend is the only trigger for an early exercise of an American-style call. $1 68 Tie Su -- FIN422 -- 2016 @ UM Early Exercise of American-Style Call Options. Early Exercise of American-Style Put Options. A cash dividend on the underlying asset is the necessary, but not sufficient condition to early exercise an American-style call option on a stock or stock index. To be deep in-the-money is the necessary, but not sufficient condition to early exercise an American-style put option. A cash dividend delays early exercise of American-style put option. It means that without a cash dividend, you never early exercise an American-style call option. 69 Tie Su -- FIN422 -- 2016 @ UM 70 Tie Su -- FIN422 -- 2016 @ UM Homework: . Provide a numerical example to show . Explain why it is not optimal to early exercise an American call if there is no dividend? Explain why it may be optimal to early exercise an American put (regardless dividend)? Provide a numerical example to show Option Pricing Models 1. 2. 3. 4. 5. 71 Tie Su -- FIN422 -- 2016 @ UM 72 Derivation of one-period binomial tree Two-period binomial tree Excel programming The BS model: assumptions, derivation, and applications Option Greeks Tie Su -- FIN422 -- 2016 @ UM