曼彻斯特商学院课件－期权

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1. ReadingHull Chapter 1: Perhaps re-read the section on options which is in sections 1.5 – 1,10. Hull Chapter 7: This starts with the basics of options and then goes on into the specific details of trading. Read up to 7.4 for the crucial details and the rest if you’re still interested. Hull Chapter 8: This is quite interesting but perhaps a little technical at this point in time. The most important part as regards the lecture is 8.4 and perhaps 8.5. 8.1 is interesting to see the crucial factors in determining the price of an option and will be useful for future reference (i.e. the last two weeks). Hull Chapter 9: Quite a nice chapter explaining options strategies, I’m going to work through a sample of these in class so it may be useful to famililiarise yourselves with the principal strategies.
2. ReadingHBR Article: Stock options have had a very negative press of late and here we have Nobel prize winner Robert Merton and other high-profile academics discussing some of the issues and argue for stock options to be put on the balance sheet. This will actually come into force at the start of 2005.
3. Futures contracts are contracts which cost nothing to enter. This is because if you are in the long position you gain when the spot price of the underlying asset, S, rises, you also lose when the price goes down (and vice versa with the short position). An option is similar in many ways to a futures contract in that you have the right to buy (or sell) at a certain price in the future (this time it is called the exercise price). However, the principal difference is that you don’t have the obligation to buy at this price – you have the option as whether or not you choose to buy (or sell) at this price. Clearly you will have to pay some premium for this privilege. How much? Options: Introduction
4. Why options?Options can again be used for three main purposes: hedging, speculating and arbitraging. An option enables you to hedge against adverse market movements (e.g. price of oil, price of corn, stock market value, interest rate changes) without, necessarily, limiting your potential gain should the market actually move in your favour. Just like with a futures contract an option enables you to leverage your position and hence make it easier to speculate on market movements. Again with options as the financial instruments become more complex then there are more arbitrage opportunities to exploit.
5. Options: definitionsAn option gives the holder of the option the right, but not the obligation to buy an economic good (usually called the underlying asset), at a specific point in time (the expiry date), for a specific price (the exercise price). The party who bought the option is said to be in the long position, the party who sold (or wrote) the option is said to be in the short position. A Call Option gives the holder of the option the right, but not the obligation, to buy the underlying asset. A Put Option gives the holder of the option the right, but not the obligation, to sell the underlying asset.
6. Options: TypesThere are two main types of options: European options: European options can only be exercised on the expiry date. American options: American options can be exercised at any time between the date of purchase and the expiry date. These names have no geographical significance, hence it is possible to buy American options on European options exchanges. Most traded options are American options, although we will start our discussion by assuming that all options are European.
7. Options: ExampleConsider both call and put options on Microsoft shares, expiring on March 20, 2004. Both have exercise prices of $25 and each option enables the holder to buy or sell the Microsoft shares at this price. The current value of Microsoft shares is $27.03. Call options are currently priced at $2.20 and put options at $0.13. Microsoft current option prices Options Product Specifications The payoffs from these options for different values of the Microsoft shares are as follows:
8. Options: exampleValue of Microsoft Shares on March 20 ($)Payoff to buyer of call option ($)Payoff to buyer of put option ($)1001515010200525003050351004015045200
9. Options: exampleClearly is the value of a Microsoft share exceeds the exercise prices then the holder of the call option will exercise his right to exercise and make a profit of the share price less the exercise price. If it is less then he will not, in this case the option expires worthless. In the case of the put option the if the Microsoft share price drops below the exercise price then she will exercise and receive the difference between these prices. If it is more then the option expires worthless. From this we note that options must always have positive or zero value as the least they can ever be worth is zero.
10. Options: leverageOptions on individual shares are actually 100 shares. Notice that if the value of Microsoft shares at expiry at $30 then the holder of the call option makes a profit of $500 – $220 = $280 or a return of 280/220 = 127%. If instead he would have spent this money on buying the shares then the return would have been only (30 – 27.03)/30 = 9.8%. This is an example of the leverage an investor can obtain from buying options rather than the underlying assets themselves. Notice however that is the share price is only $25, then the investor has made returns of –100% (as the option expires worthless) compared to a loss of –7.5% from buying the shares.
11. Payoff from a call optionIt is possible to depict the payoffs from this call option graphically:25Share price at expiryCall option value at expiry (PAYOFF)
12. Payoff from a put option…and also the put option:25Share price at expiryPut option value at expiry (PAYOFF)
13. Profit diagramsNotice that these graphs do not include the premium which has been paid for this option, if we include the amount paid then what we see is the following graphs:2525-$2.20-$0.13Profit from buying the Microsoft call option for $2.20Profit from buying the Microsoft put option for $0.13
14. Writing (or selling) optionsWe have seen what happens when you buy call or put options. What about for the person who has sold (or written) these options? If you write a call option then it means that, first, you receive the option premium. At the expiry of the option then you are obliged to sell the underlying asset at the exercise price – if the holder wishes to buy it. In this case you lose out by the difference in the prices, if however, they don’t exercise their option then you simply keep the premium. Note that these transactions are zero sum games, whatever the holder gains, the writer loses. It is similar for a put option only you have to buy at the exercise price if the holder wants to sell.
15. Payoff when writing optionsReturning to the Microsoft example, the payoffs are as follows:Value of Microsoft Shares on March 20 ($)Payoff to writer of call option ($)Payoff to writer of put option ($)100-15150-10200-5250030-5035-10040-15045-200
16. Payoffs from writing optionsIn a similar way we can graphically depict the payoffs from writing call and put options on Microsoft.2525Share price at expiryShare price at expiryPayoff at expiry from writing a call optionPayoff at expiry from writing a put option
17. Profit from writing optionsIt is also to see the profit from writing the Microsoft call and put options: 2525Share price at expiryProfit at expiry from writing a call optionProfit at expiry from writing a put option$2.20$0.13
18. Option principlesClearly buying and writing (selling) options is a zero sum game. Everything made by the buyer is lost by the writer and vice versa. Note that both buying calls and selling puts are bets that the market will go up. Buying puts and writing calls are bets that the market will go down. This option was treated as if it were a European option abut in reality this option would be American thus exercisable at any time. This would make the analysis slightly different as you would have to consider if there are optimal times to exercise the option.
19. Options: detailsOn a regulated exchange then the contract between parties is again secured by the exchange or by the Option Clearing Corporation (OCC). In fact if you buy an option you don’t have a counterparty, rather you actually buy it from the exchange who it turn buy options from parties who want to sell. It is possible to terminate your option position before expiry simply by selling (if you had originally bought) or buying (if you had originally sold), you will make (or lose) the difference between the price you bought or sold at and the price today.
20. Options more terminologyNear-the-money or at-the-money are options where the exercise price is close to the current share price. Out-of-the-money options are options where immediate exercise would produce a negative payoff (i.e the share price is higher than the exercise price for a put option). In-the-money options are options where immediate exercise would produce a positive payoff (i.e. the share price is higher than the exercise price for a call option). LEAPS are long term options on individual shares or stock indices. FLEX options are options where the investors determine the exercise price and expiry dates (See press release document in lecture).
21. Option trading strategiesThere are many ways of combining different types of options contracts to suit an investor’s particular belief about the market or to hedge a particular risk. We will work through three main types (although there are many more especially when you include exotic options), these are: Protective positions. Spreads (vertical call, butterfly) Straddles. These will all be explained using Cisco shares as the underlying asset (although the prices are not actually real prices)
22. The protective put: definitionAssume that you own Cisco shares but you are very worried about a large loss. Its current share price is $92.875. To protect against a severe drop in the share price you buy a put option expiring in July with an exercise price of $95. This currently costs $10.25 bringing your total investment to $103.125. The payoff is given on the next page. Notice that you limit your downside loss to 7.9%, but you pay for this with decreasing returns should the price go up significantly (e.g. 26.1% rather than 40% if the share price reaches $130).
23. Protective put: resultsStock price at option expiry ($)Payoff to put option ($)Total portfolio value% return on portfolio% return to stock alone130013026.1%40.0%120012016.4%29.2%11001106.7%18.4%1000100-3.0%7.7%95095-7.9%2.3%90595-7.9%-3.1%801595-7.9%-13.9%702595-7.9%-24.6%603595-7.9%-35.4%Returns based on cost of $92.875 for share plus a put costing $10.25 = $103.125.
24. Vertical call spread: definitionSpreads consist of positions where you own portfolios entirely in options. In the vertical call spread you buy a call (with exercise price X) and write a call with a higher exercise price Y). The spread is called a X–Y vertical call spread. Consider the Cisco July 95-100 spread. Purchase a call with exercise price 95 (cost of $11.25) and write a call with exercise price 100 (recouping $9.125) which costs you $2.125. It is actually possible to simply buy the spread rather than entering into both the positions. It is also possible to do the same with puts if you have a bearish view of the share.
25. Vertical call spread payoffStock price at expiry ($)Payoff to long 95 call ($)Payoff to writing 100 call ($)Portfolio value ($)% return on investment12025-205135.3%11015-105135.3%100505135.3%9940488.2%9830341.2%97202-5.9%96101-52.9%95000-100.0%90000-100.0%80000-100.0%Return based upon costs of $11.25 - $9.125 = $2.125.
26. 95-100 Vertical spread payoff95Share price at expirySpread value at expiry (PAYOFF)1005
27. Butterfly spread: definitionA butterfly spread consists only of options. They involve four option positions with the same maturity but three different exercises prices X, Y and Z (where X < Y < Z). The strategy is to buy a call with exercise price X, write two calls with exercise price Y, and buy a call with exercise price Z. In our example, we will construct a July 90 – 95 – 100 butterfly spread. So, we: Buy one call with exercise price $90, for $13.75. Sell two calls with exercise price $95, for -$11.25 each. Buy one call with exercise price $100, for $9.125. Total cost: $0.375.
28. Butterfly spreads payoffsShare price at expiry ($)Payoff to long call at 90 ($)Payoff to writing two calls at 95 ($)Payoff to long call at 100 ($)Total portfolio payoff ($)% Return to portfolio12030-2*25 = -50200-100.0%11020-2*15 = -30100-100.0%10010-2*5 = -1000-100.0%999-2*4 = -801166.7%988-2*3 = -602433.3%977-2*2 = -403700.0%966-2*1 = -204966.7%9550051233.3%944004966.7%933003700.0%922002433.3%911001166.7%900000-100.0%800000-100.0%700000-100.0%Returns based upon investment costs of $13.75 – 2*$11.50 + $9.125 = $0.375.
29. Butterfly spread payoff90Share price at expirySpread value at expiry (PAYOFF)100595
30. Straddle: definitionA straddle consists of buying both a call and a put option both at the same exercise price. In our example, we buy a call with an exercise price of $95 at a cost of $11.25 and a put option at a cost of $10.25, so a total cost of $21.50.
31. Straddle payoffShare price at expiry ($)Payoff from the long call ($)Payoff from the long put ($)Total portfolio payoff ($)% return1303503562.8%1202502516.3%11015015-30.2%100505-76.7%95000-100.0%90055-76.7%8001515-30.2%700252516.3%600353562.8%
32. Straddle payoff: graph25Share price at expiryStraddle payoff at expiry25Share price at expiry
33. Option pricingIt will be very tough to obtain an exact value for the options but it will be possible to establish some important bounds and relationships. First note that both call and put options must have positive value, i.e. no-one should have to pay you to take ownership of a call or put option. This is clearly true as the least an option is worth at expiry is 0. Call > 0 Put > 0 American options must be worth at least as much as European options because American options can be exercised at any time – i.e. it is a European option with extra flexibility. American call European call American put European put
34. Put-Call parityThe most useful simple relationship between call and put option prices is the put call parity. Consider, for the sake of an example, European put and call options on a share with current price $50 (paying no dividends), both options have exercise prices of $55 and the same maturity. Construct two portfolios: A: Buy one European put option and also buy the share. B: Buy one European call option and invest an amount of cash which, at expiry, will have value equal to the exercise price ($55). The following table demonstrates that they both have the same value at the maturity of the option.
35. Put-call parity: cash flowsShare Price ($)First PortfolioSecond PortfolioPut + Share = TotalCall + Safe Asset = Total70070701555706506565105565600606055555550555505555505505505555451045550555540154055055553520355505555Portfolio payoffs to show Put-call parity, exercise price is 55
36. Put-call parityNotice that the first portfolio is often termed the insurance portfolio as the portfolio can never have a value less than the exercise price, $55 in this case, which means that the holder of this portfolio can always sell the share for $55 regardless of its price. Clearly regardless of what the share price is then these two portfolios are equal in value at the maturity of the options (regardless of how long this actually is). If this is true then the present value of the two portfolios must also be the same, otherwise there will be an arbitrage opportunity….
37. Put call parity: arbitrageAssume that the maturity of the options is one year and that the current share price is $50 and the 1-year risk free rate is 1%. The current price of the put option is $7, and the current value of the risk free asset is 55/(1.01) = $54.46. In this case the call option should have value $2.54 (i.e. $7 + $50 - $54.46 = $2.54). What happens if call options are actually $4? Well the value of portfolio A is $57 and the value of portfolio B is $58.46. You buy low sell high i.e. buy the put and the share and sell the call and borrow $54.46 for 1 year. This gives you a profit today of $2.54. What is your cash flow at the end of the year?
38. Put call parity: arbitrageBuy share + Buy put + Write call + Repay loan = Total700-15-550650-10-550600-5-5505500-5505050-55045100-55040150-55035200-550Payoff from buying portfolio A and selling portfolio B, with exercise price of $55 and maturity of 1 year.
39. Put-call parity: resultSo you are receiving $2.54 now for no cost, hence this is an arbitrage opportunity. As such these portfolios must be equal in value at any point prior to maturity giving the following result (where X is the exercise price). European Put + Share = European Call + Present value of X This can be rearranged to give the arbitrage portfolio: Put + Share – Call – Present value of X = 0 Also it can allow you to determine the value of call and put options given the value of the other: Put = Call + Present value of X – Share Call = Put + Share – Present value of X
40. Put-call parity: exampleGiven our Microsoft data, does put-call parity hold? We assume that the current risk-free rate is 1.5% p.a, there are 250 trading days in the year and given that the current share price is $27.03 and exercise price is $25. The current value of the put option is $0.125 giving Call = $0.125 + $27.03 – $25e-0.015*30/250 Call = $2.20 Thus with these assumptions put-call parity holds. Note that there is a choice of how you calculate the present value, you can use discrete and continuous compounding but with these time scales then it is probably easier to use continuous compounding.
41. Put-call parity:American optionsNote that put call parity also provides bounds for the value of European call and put options. As we know that both must be worth at least zero then this gives: European Call Share – Present value of X European Put Present value of X – Share e.g. in the Microsoft example (assuming the same time and risk-free rate) then Call $27.03 - $24.96 = $2.07 This provides us with a very interesting result, as the American call option is worth at least as much as the European call then American call Share – Present value of X
42. Put-call parity: American optionsHowever, with an American option it is always possible to exercise early and receive the difference between the current share price and the exercise price Share Price – X But, we know that the American call is always worth more than Share Price – Present value of X which is larger than Share price – X. Thus, If the share pays no dividends then it is never optimal to exercise an American call option. As such, it has the same price as a European Call option.
43. Put-call parity: dividendsWhen we consider shares which pay out dividends then the put-call parity changes to the following: Euro. Put + Share = Euro. Call + PV(X) + PV(Dividends) This again can be used to spot arbitrage opportunities or to value a call given the value of a put or vice versa. Note that the inclusion of a dividend may well mean that an American call option may be exercised before expiry thus making it more valuable than a European call. Note that we still have not arrived at a method for calculating the value of a European call or put option…
44. Option pricing: Black-ScholesIt is actually a very complicated procedure to come up with the price of an option – even a simple European style option. In fact the value of an option is described by the Black-Scholes equation: Where V is the value of the option, S is the value of the asset, t is the time from expiry, r is the risk-free rate and s the volatility of the underlying asset. The value of European Call and Put options are:
45. Option pricing: Black-ScholesWhere, and, Unfortunately there do not exist ‘closed form’ or formulae expressions for the value of American options, thus one must use numerical techniques (e.g. binomial trees, finite-difference methods, Monte-Carlo methods or quadrature).
46. ConclusionsWe have defined what an option is and seen the payoff from both writing and buying call and put options. We have also seen that unlike with futures contracts the buyer of the option must pay a premium to buy the option. We have been through some basics option strategies showing how flexible options are and how they can be used to both hedge and speculate on market movements. Although options are very difficult to price we have been able to derive some general results mainly by using the put-call parity which describes the relationship between call options, put options and the underlying asset. From this we were able to deduce that for underlying assets paying no dividends American call options are worth the same as European call options.

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