Business 494 Final Exam Review PDF

Business 494 -- Derivatives and Options Final Exam Review Chapter 9-11 **[Chapter 9 -- Mechanics of Options]** ***Options vs. Forwards/Future*** - Options give the holder the option to do something with it and does not have to exercise this right but it does cost the trader an upfront...

Business 494 -- Derivatives and Options Final Exam Review Chapter 9-11 **[Chapter 9 -- Mechanics of Options]** ***Options vs. Forwards/Future*** - Options give the holder the option to do something with it and does not have to exercise this right but it does cost the trader an upfront payment to purchase it - In a forward or future , the two parties have committed themselves to some action. Initially costs a trader nothing (except the margin/collateral) to enter into a forward or futures contract ***Types of Options*** - Call Option: gives the holder the right to buy an asset by a certain date (expiration date) for a certain price (Strike price) - Put Option: give the holder the right to sell an asset by a certain date for a certain price - A call is an option to buy - A put is an option to sell - A European option can be exercised only at the end of its life - An American option can be exercised at any time ***Option positions*** - An investor can be: - Long call - Long put - Short call - Short put ***Call option (European and long position)*** - An investor buys a call option (i.e. right to purchase 100 shares) - **Strike Price = \$100** - **Current Stock Price = \$98** - **Price of an Option to buy 1 share = \$5** - **Initial Investment is 100 x \$5 = \$500** - At the expiration of the option the stock price is \$115. At this time, the option is exercised for a gain of: - \$(115-\$100) x 100 shares = \$1,500 - When the initial cost of the option is taken into account, the net gain is \$1,500-\$500 = \$1,000 - No mystery but at \$108 the investor would have a profit at (\$108-\$100) \* 100 - \$500 = \$300 - At \$105 the investor would be at break even or \$0 because the gain on the call option (\$105-\$100)\*100 = \$500 and the cost would be -\$500 - And at \$102, the investor would have a loss of (\$102-\$100)\*100-\$500 = -\$300, but would still exercise because walking away without exercising would be a loss of -\$500 - at \$100 or below the investor would not exercise the option and lose -\$500 or -\$5 an option - Therefore, the maximum loss is capped at \$-500 while the investor can participate in the up side or profit potential, The graph of this is on the next slide: ***Profit From a Long call*** - Profit from buying one European Call option: - Option price = \$5 - Strike Price = \$100 - Note from previous example, there is no free lunch, the strike price was \$100 and the current stock price was \$98 so the underlying asset MUST increase to at least \$102 for you to be "in the money" during the time frame of the option - American Option can be exercised at any time - European Option exercised at end of contract ***Long PUT Option (long and European)*** - An investor buys a put option (i.e. right to sell 100 shares) - Strike Price = \$70 - Current Stock Price = \$65 - Price of an Option to buy 1 share = \$7 - Initial Investment is 100 x \$7 = \$700 ***Option Positions*** - There are two sides to every option contract. On one side is the investor who has taken the long position (i.e., has bought the option). On the other side is the investor who has taken a short position (i.e., has sold or written the option). The writer of an option receives cash up front, but has potential liabilities later. The writer's profit or loss is the reverse of that for the purchaser of the option. - The biggest difference between a long and a short from an economic standpoint is that in a long (you bought a call or put) you pay an upfront premium and your maximum loss is capped. If you are short (you sold or wrote the call of put) you RECEIVE the upfront premium but your losses are not capped and could theoretically be infinite, therefore you are putting a cap on your maximum gain while taking downside risk. (Chapter 11 is with the graphs of the positions -- make sure to understand and to know which is which) ***Foreign Currency Options:*** - Done over-the-market (means to intermediary such as a bank), but also some exchange trading - Most are European style ***Index options*** - U.S options (S&P 500 SPX, S&P 100 OEX, and NASDAQ-100 NDX, and Dow Jones DJX) can exercise at any time - European options can only exercise on the last day ***Future Options*** - Futures Contract: agreement to buy or sell a specific commodity, currency at a predetermined price on a future date - Future Options: the underlying asset is a future contract. The option gives the holder the right to enter into a futures contract at a specific price (the strike price) before the option expiration - The life of a future option (the period during which it can be exercised\_ typically ends a short time before the expiration - Call Option (the right to buy): When a call option on a future contract is exercised, the holder earns the difference between the future price and the strike price, if the futures price is higher. The holder can then into long futures positions at the strike price - Actual Price \> Strike So \> K - Put Option (right to sell): When a put is exercised, the holder earns the difference between the strike price and the futures price, if the strike price is higher. The holder can enter into a short position at the strike price - Strike \> Actual Price K \> So ***Expiration Dates:*** - Usually 1,2,3,6 month cycles - Long dated future contract would be called Long term Equity Anticipated Securities or LEAPS that can last up to 39 months ***Strike Prices:*** - Typical spacing between strike prices at option is around \$2.50 when the stock price is between \$5 and \$25 - \$5 apart when stock is between \$25 and \$200 - \$10 apart when over \$200 Specifications of Stock options S is the stock price K is the Strike price - In the Money - Call option in the money is when S \> K - Put Option S \< K - At the Money - Call Option S = K - Put Option S = K - Out of the Money - Call Option S \< K - Put option S \> K - An option will ONLY be exercised when it is in the money - In absence of transaction costs, an in-the-money option **will always be exercised on the expiration date** if it has not be previously exercised - Intrinsic Value of an option is: the maximum of zero and the payoff from the option if it were exercised immediately - Call Max (S-K, 0) - Put Max (K-S, 0) - CANNOT go below zero ***Dividends*** - OTC options use to be dividend protected - If declared by a company, the strike price for options on the company's stock was reduced on the ex-dividend date by the amount of the dividend. This ensured the value of the option was not affected by the dividend - Exchange-traded Options do not adjust for dividends - On the ex-dividend date, the stock price drops by the dividend amount, which can reduce the value of the call option or increase the value of the put option - Example: Consider a put option to sell 100 shares of a company for \$15 per share - Suppose the company decreases a 25% stock dividend. This is equivalent for a 5-for-4 stock split. The terms have changed so it give the holder the right to sell 125 shares for \$12. ***Stock Splits*** - They do result in adjustments to options contracts. This is because the underlying stock price and the number of shares per contract change, but the overall value of the contract remains the same - The strike price is adjusted to reflect the new, lower stock price - The number of shares per option contract is increased to match the stock split ratio - Example: the 3-for-1 stock splits should cause the stock price to go down 1/3 of its previous value. - Before split a stock had a strike price of \$60 for 100 shares - The 3-for1 split: Strike price is adjusted to \$20 (60/3), and the contract now covers 300 shares instead of 100 shares ***Position Limit:*** - defines the maximum number of option contracts that an investor can hold on on-side of the market ***Exercise Limit:*** - usually equals the position limit - the maximum number of contracts that can be exercised by an individual in a period of 5 consecutive business days - options that are very large and frequently traded stock have position limit of 250,000 contracts \*Position limits and exercise limits are designed to prevent the market from being unduly influenced by the activities of an individual investors to group of investors ***Market makers:*** - an individual who will quote both a bid and an offer price on the option - The bid price is the price at which the market maker is prepared to buy - The offer or asked price is the price at which the market maker is prepared to sell - The offer is always higher than the bid (the amount that exceeds is referred to as the bid-offer spread - Market markers are there to add liquidity to the market - They make money off the bid-offer spread ***Offsetting orders*** - Long Position: When an investor has bought an option can close out that position by issuing a off-setting order to sell the same position - Short Position: Also, the investor who has written an option to close out the position by issuing an offsetting order to buy the same position - This is a common practice to neutralize the position the investor is in ***Open Interest in Trading*** - Open interest in options is the total number of active contracts that have not been settled It's a key metrics for understanding market liquidity and trader interest and help traded gauge market trends - Increasing open interest: more people that enter, could strengthen the current trend - Decreasing open interest: Traders are closing their positions, and the trend may be weakening ***Commissions*** - A hidden cost in option trading and stock trading is the market's makers bid offer spread ***\*\*\*Margin Requirements*** - When call or put options with maturities less than nine months are purchased, the option price must be paid in full. - Often investors are not allowed to buy (long) positions on margin because options already contain a substantial leverage and buying on a margin would raise this leverage to an unacceptable level - For maturities greater than nine months, investors can buy on margin, borrowing up to 25% of the option value - A trader who write (short) options is required to maintain funds in a margin account. - The amount of margin required depends on the trader's position ***Naked Options*** - Seller of the option does not hold a corresponding position in the underlying asset, they also have not taken an proactive measures to hedge against potential losses - Not combined with a offsetting position in the underlying stock (covered option, so the loss and gains can be large An investor writes 4 naked call options of a stock. The option price is \$5, the strike price is \$40 and the stock price is \$38. The option is \$2 out of the money. Writing a Naked CALL Option: 1. A total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount if any by which the option is out of the money a. 4 contracts \* 100 shares \* (\$5 + 20% \* \$38 -2) = \$4,240 2. A total of 100% of the option proceeds plus 10% of the underlying share price b. 4 contracts \* 100 shares \* (\$5 + 10% \* \$38) = \$3,530 An investor writes 4 naked PUT options of a stock. The option price is \$5, the strike price is \$40 and the stock price is \$38. The option is \$2 out of the money. Writing a Naked PUT option 1. Total of 100% of the proceeds of the sale plus 20% of the underlying share price less the amount if any by which the option is out of the money a. 4 contracts \* 100 shares \* (\$5 + 20% \* \$38) = \$5,040 2. A total of 100% of the option proceeds plus 10% of the exercise price b. 4 contracts \* 100 shares \* (\$5 + 10% \*\$40) = \$3,600 ***Over the Counter Market Options*** - Derivative dealers trade directly with other financial institutions, corporations, and fund managers - OTC on foreign exchange and interest rates are particularly popular - The disadvantage is the option writer mat default. This means the purchases is subject to some credit risk. To overcome some of it, collateral is increasingly required - The instruments traded in the over-the-counter markets are often structured by financial institutions to meet the price needs of their clients. Sometimes this involves choosing exercise dates, strike prices, and contract sizes that are different from those affected by an exchange. These are exotic options **[Chapter 10 -- Factors affecting Option Prices]** There are six factors that affect the price of a stock option: 1. The current stock price -- So 2. The strike Price -- K 3. The time to expiration -- T 4. The volatility of the stock price -- 5. The risk-free interest rate -- r 6. The dividends that are expected to be paid ***[Upper Price Bounds]*** ***Upper bond of a call option*** - The option can never be more than the stock - Therefore the stock price is an upper bound to the option price c \< So and C\< So - If these relationship were not true, an arbitrageur could easily make a riskless profit by buying the stock and selling the call option ***American Put*** - Give the holder the right to sell one share of a stock for K (strike price) - No matter how low the stock price become, the option can never be worth more than K - P \< K ***European Put*** - The maturity option cannot be more than K - It also cannot be worth more than the present value of K today - P \< Ke-rT ***[Lower Bounds of Non-Dividend Stock]*** ***European Call*** So-Ke-rt Example: Stock has a current price of \$20, the strike price is \$18, the risk free rate with continuous compounding is 10% and time to expiry is 1 year. What is the low price boundary? Summary: - Buy the option for \$3 - Short the stock to realize \$20 (net 20-3 - Invest for 1 year at 10% ***Lower bounds of a non-dividend stock*** - The worst that can happen for a call option is that it expire worthless, it value cannot be negative. Lowest value can only be \$0 - c \> MAX (so-ke\^-ert, \$0) ***Lower Boundary of a call*** - Example: Consider a European option on a non-dividen paying stock when stock price is \$51, the strike price is \$50, the time to maturity is six months, and the risk free rate of interest is 12% per annum. ***Lower Boundary of a Put Option (European, non dividend paying)*** - Example: Consider a European put option on a non-dividend-paying stock when the stock price is \$38, the exercise price is \$40, the time to maturity is three months, and the risk-free rate of interest is 10% per annum. What is the lower bound for the put - **p = Ke^-rT^- So** - p ≥ max(**Ke^-rT^- So, \$0)** - this theory is when options have the same strike price and time to maturity - Assume you have two investment portfolio options: - **Portfolio A** -- 1 European call option and a zero-coupon bond that provides a payoff of K at time T - **Portfolio B** -- 1 European put option plus 1 share of the underlying stock - **c + Ke^-rT^ = p + So** Example: Suppose that the stock price is \$31, the exercise price is \$30 , the risk-free interest rate is 10% per annum, the price of a three-month European call option is \$3 , and the price of a three-month European put option is \$2.25. You would need to value each side using put-call parity \-\-\-- The **call side** c + Ke^-rT^ = \$3 +\$30e^-.10\*.25^ = \$32.26 (portfolio A) \-\-\-- The put side p + So = \$2.25 + \$31 = \$33.25 (portfolio C) Portfolio C is overpriced relative to Portfolio A so an arbitrageur can buy (long) the assets is portfolio A (European call option and a zero coupon bond) and short the assets of portfolio C (short the put and the stock) ***Put-Call Parity*** ***American CALLS on a dividend paying stock*** - there is never an optimal time to exercise an American call option on a non-dividend paying stock before the expiration date if it does not pay a dividend - Because the value of an American call Option is greater than a European call option, and the European call option is defined as c = S~0~-Ke^-rt^ - the American Call Price C is always greater than the Option's intrinsic value prior to maturity. If it were optimal to exercise at a particular time prior to maturity, C would equal the option's intrinsic value at that time. It follows that it can never be optimal to exercise early. - It should not be exercised early because of the insurance that is provided. A call option, when held instead of the stock itself, in effect insures the holder against the stock price falling below the strike price. Once the option has been exercised and the strike price has been exchanged for the stock price, this insurance vanishes. - Another reason it should not be exercised early is the aspect of time value of money. From the perspective of the option holder, the later the strike price is paid out the better. - Because an American call option is theoretically never exercised early, the lower bounds are the same as a European Call. Lower Bound **C = So- Ke^-rT^** - ***American PUTS on a no Dividend Paying stock*** - A put option should always be exercised early if it is sufficiently deep in the money - Can also provide insurance like a call option. - A put option is different from a call option in that it may be optimal for an investor to forgo this insurance and exercise early in order to realize the strike price immediately. - In general, the early exercise of a put option becomes more attractive as S~0~ decreases, as r increases, and as the volatility decreases. - **Max (K-S~0~,0)≤P** is the lower bound and - **P≤K** is the Upper bound ***Effects of Dividends*** - To determine lower bounds of Calls and Puts - Portfolio A = one European Call Option plus an amount of cash **D + Ke^-rT^** - Portfolio B = one share of the underlying stock - Lower Bound of a Call **c≥max(S~0~-D-Ke^-rT^, 0)** - **Lower Bound for a Put p≥max(D+Ke^-rT^-S~0~, 0)** ***Early Exercise with the effect of dividends*** - Sometimes it will be optimal to exercise an American call immediately prior to an ex-dividend date. It is never optimal to exercise a call at other times - **c+D+Ke^-rT^ = p + S~0~** assuming European option - **S~0~-D-K ≤C-P ≤S~0~ - Ke^-rT^** assuming American Option **[Chapter 11 -- Trading Strategies ]** ***Principle Protected Notes -- PPN's*** - For the retail market - More for conservative investors ***PPN payoff of investor*** - Say if you invested \$1,000 in a PPN, if the portfolio increases the investor will get the \$1,000 plus the gain. - If the value of the portfolio goes down, the option has no value - The payoff from the zero-coupon bond of \$1,000 ensures that the investor receives the original \$1,000 principle invested - The attraction of a principal-protected note is that an investor is able to take a somewhat risky position without risking any principal. - The worst that can happen is that the investor loses the chance to earn interest, or other income such as dividends, on the initial investment for the life of the note. ***PPN Bank Perspective*** - There are a number of ways the bank can still create a viable 3-year product. - For example, the strike price of the option can be increased so that the value of the portfolio has to rise by, say 15%, before the investor makes a gain; - the investor's return could be capped; - the return of the investor could depend on the average price of the asset instead of the final price; a knockout barrier could be specified. ***Strategies Involving a Single Option and a Stock*** ***Spreads*** - a spread trading strategy involves taking a position in two or more options of the same type (i.e. you take two or more call options) ***Bull Spread*** - Most common form of spread - Can be created from calls (address this first) and puts (second way) - An investor who enters into a bull spread is hoping that the stock price will increase - In a Bull Spread an example would be buying a European Call Option (long) on a stock price with a Strike Price and selling (short) a European call option on the same stock with a higher strike price. Both options would have the same expiration date. - A Bull spread when created from calls will require an initial investment since the call price should decrease as the strike price increases - If the stock price does well and is greater than the higher strike price the payoff is the difference between the two strike prices - K~2~ --K~1~ - If the strike price on the expiration date lies between the two strike Prices (K~1~ and K~2~) the payoff is - S~t~ --K~1~ - If the stock price on the expiration date is below the lower strike price (K1) the payoff is zero. A table with text on it Description automatically generated Example: An investor buys for \$3 a three-month call with a strike price of \$30 and sells for \$1 a three-month call with a strike price of \$35. The payoff from this bull spread strategy is \$5 if the stock price is above \$35 and zero if it is below. If the stock price is between \$30 and \$35, the payoff is the amount by which the stock price exceeds \$30. The cost of the strategy is \$3-\$1 = \$2 The profit is therefore as follows:  ***Bull Spread -- Why Do it?*** - Limits the investor upside as well as downside risk - The investor has a call option with a strike price equal to K~1~ and has chosen to give up some upside potential by selling a call option with strike price K~2~ where (K~2~\>K~1~) - In return for giving up the upside potential, the investor gets the price of the option with strike price K~2~ ***Bull Spread Strategy -- Three Types*** **- Three types of bull spread can be distinguished:** ***Bull Spreads with Puts*** - Bull spreads can also be created by buying a European put (long) with a low strike price and selling a European put (short) with a high strike price ***Bear Spreads*** - Hoping the stock price will decline - Can be created by buying a European put with one strike price and selling a European put with another strike price - The strike price of the option purchased is greater than the strike price of the option sold ***Bear Spread Payoff*** - Suppose that K~1~ is the strike price of the call option bought, K~2~ is the strike price of the call option sold, and S~T~ is the stock price on the expiration date of the options. - If the stock price does well and is greater than the higher strike price, the payoff is the difference between the two strike prices, or **K~2~ - K~1~** - If the stock price on the expiration date lies between the two strike prices, the payoff is **S~T~-K~1~** - If the stock price on the expiration date is below the lower strike price, the payoff is zero. ***Butterfly Spreads*** - Involves positions in option using three different strike prices - It can be created by buying a European call option with a relatively low strike price, **K~1~** , buying a European call option with a relatively high strike price, **K~3~**, and selling two European call options with a strike price, **K~2~** , halfway between **K~1~** and **K~3~** - Generally is close to the current stock price. - Can also be created with put options - A butterfly spread leads to a profit if the stock price stays close to **K~2~**, but gives rise to a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. - The strategy requires a small investment initially. ***Butterfly Spreads Using Put (reverse strategy***) - Options are sold with strike prices of **K~1~** and **K~3~** and two options with the middle strike price **K~2~** are purchased. This strategy produces a modest profit if there is a significant movement in the stock price.  ***Payoff from a Butterfly Spread*** Example Suppose a certain stock is currently at \$61. Investor feels that no movement in price is likely in next 6 months. A call option with a strike price of \$55 will cost \$10, one with a strike price of \$60 is \$7 and one with a strike price of \$65 will be \$5. - Buy one call with \$55 Strike Price - Buy one call with \$65 Strike Price - And then sell two calls with a \$60 strike price - The cost would be \$10 +\$5 -- (\$2 \* \$7) = \$1 - If the stock price in 6 months is greater than \$65 or less than \$55 , the total payoff is \$0 and the investor incurs a net loss of \$1 - But is the stock price is between \$56 and \$64, a profit is made ***Straddle*** - A combination is an option trading strategy that involves taking a position in both calls and puts on the same stock. - which involves buying a European call and put with the same strike price and expiration date. - If the stock price is close to this strike price at expiration of the options, the straddle leads to a loss. However, if there is a sufficiently large move in either direction, a significant profit will result. - A straddle is appropriate when an investor is expecting a large move in a stock price but does not know in which direction the move will be.