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1 Exotic Options Chapter 19. 2 EXOTIC OPTIONS So far we studied and analyzed options strategies that included a variety of calls puts and short or long.

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Presentation on theme: "1 Exotic Options Chapter 19. 2 EXOTIC OPTIONS So far we studied and analyzed options strategies that included a variety of calls puts and short or long."— Presentation transcript:

1 1 Exotic Options Chapter 19

2 2 EXOTIC OPTIONS So far we studied and analyzed options strategies that included a variety of calls puts and short or long positions in the underlying asset. All the options that we studied were standard European or standard American style options. We now turn to study another class of options that are non standard options. They are also labeled exotic options.

3 3 EXOTIC OPTIONS These options are non standard in the sense that one or several of the usually standardized options contractual stipulations are replaced with conditions that are tailored to suit the buyer and seller specific needs.

4 4 EXOTIC OPTIONS: EXAMPLES: Bermudan options – American options with a predetermined set of possible exercise dates. Asian options – Options whose position at expiration is determined by an average of the underlying asset price during a pre specified period. Barrier options – Options that come to existence or cease to exist if the underlying asset price reaches a predetermined threshold level.

5 5 Collars(19.1) Often, investors buy the underlying asset and purchase protective European puts at some exercise price K 1 ; K 1 < S. In order to finance the purchase of the protective puts, the investor may short European calls with K 2 ; K 1 < S < K 2 for the same expiration, T. At times, the investor chooses the exercise prices such that the call premium is equal to the put premium: c(S, T-t, K 1 ) = p(S, T-t, K 2 ).

6 6 Collars ATEXPIRATION StrategyICF S T K 2 Buy stock-S S T S T S T Buy put(K 1 )-p K 1 – S T 0 0 Sell call(K 2 ) c 0 0 K 2 – S T TOTAL-S K 1 S T K 2 P/L K 1 – S S T - S K 2 – S

7 7 Collars The self financing Collar guarantees that the asset, which was purchased for S, will sell for K 1 or better, up to K 2. Given that the probability of S T to exceed K 2 is very low, the possibility of losing the upper side of the asset’s price distribution is close to zero. This strategy guarantees a specific price range for the asset’s selling price at T. =  a Range forward contract.

8 8 Collars Several variations of collars are possible with or without holding the asset and with the put and call prices not equal. These strategies depend on whether the investor wishes to guarantee a selling price or a purchasing price for the asset at T and whether the investor wishes to open a self financing strategy or not. In all of the above situations the valuation of the strategy is based on the Black and Scholes valuation of the call and the put.

9 9 Forward start options(19.3) These are at-the-money options that will begin on a specified future date, T 1, say and will expire at T 2. The value of such an option at its writing time, say 0, is the NPV of the option’s value at its initial date of existence, T 1, with expiration at T 2. It can be easily shown that it has the same value of an at-the-money option with T = T 2 - T 1.

10 10 Forward start options(19.3) In general, however, the option may begin with the exercise price set at X =  S. If  = 1 then the option is at-the-money. Otherwise, it will be out or in the money, depending on  being greater or less than 1 and the type of the option: call or put. Moreover, one may face a sequence of Forward start options where the i+1st option begins at the expiration of the i-th option and its exercise price is set at  times the asset price at the expiration of the i-th option.

11 11 Forward start options Suppose that we face n such options, the value of the entire strategy is

12 12 Forward start options The assumption here is that  remains the same throughout the n periods. An Example: Consider an employee that, as part of his/her compensation package, receives a call with forward start three months from now. The options parameters are: S = 60;  =1.1; r =.08; q =.04;  =.30; T 1 =.25 and T 2 = 1. Substituting these parameters into the formula with only one option we obtain the call value:C = $

13 13 Compound options(19.4) or Options on Options The underlying asset of a compound option is an option. Thus, upon exercise of a compound option, the holder will either receive or deliver another option. The holder of the compound option pays a premium on an option with K 1 that expires on T 1. If exercised, the holder will buy or sell for K 1 an option with K 2 that expires T 2 time periods from today.

14 14 Compound options: Four possibilities: The payoff 1.A call on a callmax{0, c(S, K 2,T 2 ) – K 1 )} 2.A put on a callmax{0, K 1 - c(S, K 2,T 2 )} 3.A call on a putmax{0, p(S, K 2,T 2 ) – K 1 )} 4.A put on a putmax{0, K 1 – p(S, K 2,T 2 )}. c and p are the Black and Scholes values with exercise price K 2 and time to expiration T 2. K 1 is the exercise price of the option on the underlying option, with T 1 time to expiration.

15 15 Compound options Example 1: A put on a call. The underlying call is the following call on the index: c(S=500; K 2 =520; T 2 =.5) The put option on this call is: p[c(S=500;K 2 =520;T 2 =.5), K 1 =50; T 1 =.25] The payoff on this compound option is: Max{0, K 1 – c(S, K 2,T 2 )} = Max{0, 50 - c(500, 520,.5)}.

16 16 Compound options The value of this put on call when r =.08; q =.03 and underlying stock index volatility  =.35 is: $ CONCLUSION: You pay $21.20 for a put on a call on the index. If the put is exercised, you will receive $50 for selling a call on the index with exercise price of 520 and a time to expiration of.25 yrs from then.

17 17 Compound options Example 2: A call on a put. The following is a very common situation for foreign multinational firms: A foreign firm submits a bid for selling equipment in the U.S.A. for a fixed amount, M, of foreign currency. At time T 1 the firm will find out if it won or lost the bid. If it did win the bid, it will sell the equipment and receive the USD equivalent of M on date T 2. The firm is clearly exposed to exchange rate risk and may wish to hedge this risk.

18 18 Compound options Time line: 0……………. T 1 ………………….…… T 2 BID ACCEPTED………PAYMENT and or REJECTEDDELIVERY If the foreign currency depreciates against The USD, the USD amount equivalent to M will be smaller.

19 19 Compound options The firm could purchase a protective put on the foreign currency for T 2 and pay the full premium, ignoring the fact that it may not win the bid. INSTEAD

20 20 Compound options The firm could buy a call for T 1, which will give the firm the right to purchase the FORX put in case it won the bid and the call is in the money at T 1. In this way, the firm will have two payments. The call premium, will typically be smaller than the premium on the outright put. The put premium payment will occur if the firm wins the bid and the call is in the money. The sum of these two payments may or may not exceed the outright put premium.

21 21 Compound options EXAMPLE 2 : a call on a put: A Canadian firm submits a bid to sell equipment in the U.S.A. for CD10M. The firm will find whether it won the bid or not in 25 days. If the bid was won, it will deliver the equipment and be paid in full 24 days later. The payment will be in USD. The current exchange rate is USD.6303/CD.

22 22 Compound options Had the deal been done today the firm would have received USD6.303M. However, if the firm wins the bid and the CD depreciates against the USD, the firm will realize a smaller amount. The firm may buy protective puts.

23 23 Compound options If the firm decides to purchase a protective put on the foreign currency for T 2 = 49 days and pay the full premium, ignoring the fact that it may not win the bid, we use: S = USD.6303/CD;K 1 = USD.6303/CD. p = p(.6303;.6303; 49/365; r=.0404;  =.028) =USD.2669/CD0r, a total of: p = USD.2669/CD[CD10M] = USD26,690.

24 24 Compound options INSTEAD: A call on a put. The underlying put option is a put on the CD with the following parameters: p(S=.6303; K 2 =.6303; T 2 =49/365) The call option on this put is: c[p(S=.6303;K 2 =.6303;T 2 =49/365), K 1 =?; T 1 =25/365;  =.028]

25 25 Compound options The payoff on this compound option depends on K 1. In order to decide on the exercise value of the compound call, the firm calculates the value of an at-the- money put with 24 days to maturity: p = USD.185/CD and in order to compare the compound option strategy with the outright protective put strategy, the firm calculates the compound option value for a range of striking prices:.16,.18,.20,.22 and.24 cents per CD.

26 26 Compound options Note:The put value will increase when the CD depreciates against the USD, thus, increasing the compound option value.

27 27 Compound options Again. The underlying put option is a put on the CD with the following parameters: p(S=.6303; K 2 =.6303; T 2 =49/365) The call option on this put is c[p(S=.6303;K 2 =.6303;T 2 =49/365), K 1 =?; T 1 =25/365;  =.028] Calculation of the compound option shows an increasing compound option value with a decreasing exercise price: K 1 c[p(S, K 2,T 2 ); K 1 =?; T 1 =25/365;  =.028].24USD.1096/CD  USD10,960.22USD.1184/CD  USD11,840.20USD.1277/CD  USD12, USD.1377/CD  USD13,770.16USD.1485/CD  USD14,850

28 28 Compound options A call on a put. CONCLUSION: The Canadian firm will pay ICF for the compound option ( call on a put) today. If it wins the bid in 25 days and the call ends up in-the-money, it will pay the exercise price of the call in order to purchase the put. This will lead to the following possibilities: WIN BID and K 1 ICFp >.6303TOTAL.24USD10,960USD24,000USD34,960.22USD11,840USD22,000USD33,840.20USD12,770USD20,000USD32, USD13,770USD18,000USD31,770.16USD14,850USD16,000USD30,850

29 29 Compound options EXAMPLE 3: A call on a put. An American firm submits a bid for a project in Germany for EUR100M. The firm will find whether it won the bid or not in three months. If the bid was won, the project will begin in 91 days (immediately upon winning the bid) and will be completed and paid for in six months. The payment will be in EURs that will be exchanged into USDs and deposited in USDs immediately. The current exchange rate is USD.9/EUR. Had the deal been done today the firm would have received USD90M. However, if the firm wins the bid and the USD depreciates against the EUR, for example to USD.8/EUR the firm will realize a smaller amount. Lets analyze the two hedging alternatives:

30 30 Compound options Example 3 continued: A protective put: If the firm decides to purchase a protective put on the foreign currency for T 2 =.5yrs and pay the full premium, ignoring the fact that it may not win the bid, we use: S = USD.9/EURK 1 = USD.9/EUR This put will cost: p = p(.9;.9;.5; r usd =.06; r eur =.03;  =.01) = USD.0188/EUR 0r, a total of: p = USD.0188/EUR[EUR100M] p = USD1,880,000. AGAIN:The outright purchase of the six months protective put ignores the possibility that the American firm will lose its bid.

31 31 Compound options Example 3 continued: The underlying put option is a put on the EUR with the following parameters: p(S=.9; K 2 =.9; T 2 =.5) The call option on this put is c[p(S=.9;K 2 =.9;T 2 =.5), K 1 =?; T 1 =.25;  =.01]

32 32 Compound options The payoff on this compound option depends on K 1. In order to decide on the exercise value of the compound call, the firm calculates the value of an at-the-money put with 3 months to maturity: p = USD.0146/EUR. Thus the American firm decides to set K 1 =USD.0146/EUR and calculates the compound option value. Whatever this value is, the result is that three months from now, the firm will know whether it won or lost the bid. If it lost the bid then the cost is limited to the compound option value, which is considerably less than the cost of the outright six-month put.

33 33 Compound options Example 3 continued: If, on the other hand the call ends up in-the- money, the put is then purchased for USD.0146/EUR[EUR100M] = USD1,460,000. If the call is out-of-the money, it is not exercised and the put premium is saved.

34 34 The value of unprotected American calls: an application of compound options: When a stock pays out cash dividends, the stock price falls by the dividend amount. This price fall causes the premiums of calls on this stock to to decrease. The exchanges do not compensate call holders for the lost value caused by cash dividend payments. Hence the title: unprotected American calls. One may argue that investors, being aware of the expected cash dividend payments, take this into account and that market prices adjust accordingly. While this is true, nonetheless, we still face the following problem: Suppose that the stock will pay a known cash dividend, D, at a known future date. What is the call (fair) market value?

35 35

36 36 Unprotected American calls It follows that on xd, the call holder faces the: Max{S xd - K + D; c(S xd, K, T-xd)}. On that day, the put-call parity implies that: c(S xd, K,T-xd) = p(S xd, K, T-xd)+ S xd –Ke - r(T – xd). Substitute the put-call parity into the option value to obtain: S xd + D – K + Max{0, p(S xd, K, T-xd) – [D - K(1-e - r(T – xd) )]}.

37 37 Unprotected American calls The current value of this cash flow is: S - Ke - r(xd - t) + the compound option value: call on a put. The Call is for expiration at xd and with exercise price: D - K(1-e - r(T – xd) ); If exercised, the call holder will buy a put that expires on T and with exercise price K.

38 38 Unprotected American calls What is the meaning of exercising the compound option? It means that you pay the call exercise price: D - K(1-e - r(T – xd) ) and receive the put, a total cash flow of: S xd + D – K+p(S xd, K, T-xd)–[D–K(1-e - r(T – xd) )]. But Upon substitution of p(S xd, K,T-xd) = c(S xd, K,T-xd) - S xd + Ke - r(T – xd).

39 39 Unprotected American calls The value received upon exercising the compound option is: c(S xd, K,T-xd)  not to exercise the American call Moreover, if the compound option is not exercised, the value in the investor’s hand is: S xd + D – K  exercise the American call.

40 40 Compound options Finally, another application of compound options: Consider a firm with equity and debt. For simplicity, assume that the entire debt issue is a pure discount bond maturing T time periods hence. At T, stock holders must pay bond holder the face value of the debt, F, or else, bond holders take over the firm. Assume that stock holders wish to wait until T, pay back the debt and liquidate the firm. The firm value at T is S T and therefore, the cash flow to the stock holders at T can be summarized as follows: CF = max{0, S T – F}.

41 41 Compound options CF = max{0, S T – F}, is the cash flow of a call on the value of the firm, given by the bond holders to the stock holders. Here comes the surprising conclusion: An option on the firm’s stock is a compound option. Upon its exercise, the holder buys or sells the firm’s stock; I.e., buys or sells the right to buy the firm back from the bond holders at time T.

42 42 Chooser options(19.5) The option is traded now, at time 0, determining a future time T 1 at which the option holder must decide whether the option will be a call or a put. Let c and p denote the options underlying the Chooser option, then, at T 1 the value of the option is: Max{c,p}. Suppose that an investor expects that the market will make a strong swing but is not sure whether it will be a down or an up swing. The standard strategy is a straddle – buy a call and a put in order to capture the expected volatility of the underlying asset.

43 43 Chooser options The chooser option is an alternative to a straddle, an alternative whose premium is lower than the straddle premium. The straddle premium includes the put premium and the call premium. The chooser premium is Max{c,p}. In general c = c(S 1, T 2, K 2 ) and p = p(S 1, T 3, K 3 ). Case I. Both options are for the same expiration date T and the same exercise price, K. In this relatively simple case, substitute the put call parity for European options: p(S 1,T,K ) = c(S 1,T,K) - S 1 e - q(T- T1) + Ke - r(T- T1) Into Max{c(S 1,T,K), p(S 1,T,K ) }: Max{c, c - S 1 e - q(T- T1) + Ke - r(T- T1) },

44 44 Chooser options which can be rewritten: c +Max{0, Ke - r(T- T1) - S 1 e - q(T- T1) } or c + e - q(T- T1) Max{0, Ke - (r-q)(T- T1) - S 1 }. From the last expression we see that the Chooser option is: a Call, expiring at T with exercise price K, plus e - q(T- T1) puts, expiring at T 1 with exercise price Ke - (r-q)(T- T1).

45 45 Chooser options Example: Stock XYZ is trading for $ /share. A Straddle with K = 125, T = 35 days, r =.0446 and  =.83 will cost: c = $ p = $12.09 = $ The Chooser option with T 1 =20 days will be worth c = $13.21 plus the put value with T = 15 days =.0411yrs; p = And the Chooser option costs $ It costs less than the Straddle because there is a possibility that the payoff at expiration will be zero.

46 46 Chooser options Example: Stock XYZ is trading for $50/share. A chooser option with a decision time at T 1 =.25yrs and expiration T =.5yrs is with K = 50, r =.08, q = 0, and  =.25. The Chooser option premium is $6.11/share.

47 47 Chooser options Case II The call and the put are for different exercise prices and times to expiration. c = c(S 1, T 2, K 2 ) and p = p(S 1, T 3, K 3 ). To evaluate this option, let S* be the underlying asset price at which today’s premium of a call with K 2 that expires at T 2 is equal to the T 1 value of a put with K 2 and T 2. c 0 (S*, K 2, T 2 ) = p(S*, K 3,T 3 ). By definition, chose the call for S > S* Chose the put for S < S*. The payoff to the Chooser option, given S*, can be written as:

48 48 Chooser options This payoff is equivalent to the payoff of two compound options: A call on a call with zero strike price Plus A call on a put with zero strike price.

49 49 Chooser options Example: At time T 1 =.25yrs, the option holder must chose between a call, c(S = 50, K = 55, T = 6 months) and a put, p(S = 50, K = 48, T = 7 months). r =.1,  =.35 and q =.05. The Chooser option’s premium is: $6.05/share.

50 50 Chooser options Example: An multinational American firm’s division in the UK receives and pays cash flows in British sterling, £, on a regular basis. The UK division, is required by the parent firm to exchange the £ into USD upon it receipt. Thus, the firm is exposed to exchange rate risk. For instance, during the next 100 days the firm will receive £ payment 68 days hence and will pay £ 100 days hence. Clearly the exposure could be hedged against by purchasing a sterling put for 68 days and buying a sterling put for 100 days.

51 51 Chooser options Suppose that the firm expects an announcement by the UK central bank in 30 days. This announcement is believed to be making some long term impact on the FORX market. In this case, the firm may chose to buy Chooser option for 30 days on a call, c($1.58/£, T=70days, K = $1,60/£) and a put, P($1.58/£,T = 38days, K = $1.56£). The cost of this Chooser option = $.0548/£.

52 52 Barrier options(19.6) Barrier options are a modified form of standard option. The strike price determines the payoff at expiration. But if the underlying asset’s market price crosses or does not cross a predetermined BARRIER price the option may or may not exist. Knock in option – the option becomes a standard option if the barrier was crossed some time before expiration. It will then pay if it ends up in-the-money. Knock out option – the option is a standard option as long as the barrier is not crossed. It ceases to exist once the barrier is crossed. The barrier may be crossed from below or from above and therefore, we can categorize barrier options as follows:

53 53 Barrier Barrier effect on payoff Option TypeLocation Crossed Not crossed CallDown-and-out BS Worthless Standard Up-and-in B>S Standard Worthless PutDown-and-out BS Worthless Standard Up-and-in B>S Standard Worthless In principle, barrier options may pay a rebate to the option holder when the barrier is crossed. In practice only put option pay a rebate as a consolation prize when the option is knocked out.

54 54 Barrier options Barrier options are traded for several reasons: 1.As a buyer you eliminate paying for scenarios you think are unlikely. As a seller, you may enhance your income by shorting a barrier option that pays off on scenarios that you think are improbable. Example: You expect the stock price to rise next year by some 105% of spot. However, even though you believe that the market will rise, if it will not rise by a support level of 95% it will then decline. Buy a down-and-out call with K = (1.05)S and a barrier at (.95)Market, or Sell a down-and-in call with K = (1.05)S and gets knocked in only if the market fall below 95%.

55 55 Barrier options 2.Barrier options may match hedging needs more closely than similar standard options. Example: You wish to protect the value of a stock that you just bought. You intend to sell the stock if its price rises by 10% or more. By purchasing a protective put you pay a premium that protects you even when the price rise 10%, 15% etc. Instead, you may buy an up-and-out put with the same exercise price but with a barrier set at (1.1)S. This way you have the same protection on the down side and no protection if S increases by more than 10% and you sell the stock.

56 56 Barrier options 3.Barrier options premiums are generally lower than those of standard options. Since you pay for the option only if it is knocked in and you do not pay for the option if it is knocked out, the premium is paid for specific scenarios: Down-and-out call – its value is close to the standard call value, because it gets knocked out for low stock values where the standard call has little value Down-and-out put – worth much less than a standard put, because it gets knocked out at low stock values where the standard put is deep in the money.

57 57 Barrier options Example: S = $100;K = $100;B = $90;T = 1; q = 5%;r = 10%;  = 15%. Standard European Down-and-out Call$7.84$7.22 Put$3.75$0.28

58 58 Barrier options Down-and-in call – worth much less than the standard call, because it gets knocked in only when the stock price has made a large and unlikely down move. Down-and-in put – its value is close to the standard put value, because it gets knocked in for low stock values where the put is deep in the money. Example:S = $100;K = $100;B = $90; T = 1; q = 5%; r = 10%;  = 15%. Standard European Down-and-in Call$7.84$0.62 Put$3.75$3.46

59 59 Barrier options Up-and-out call – worth only a fraction of the standard call value, because it gets knocked out for high stock values where the standard call is deep in the money. Up-and-out put – worth almost the same as much as the standard put, because it gets knocked out at high stock values where the standard put is out of the money. Example:S = $100;K = $100;B = $120;T = 1; q = 5%;r = 10%;  = 25%. Standard European Up-and-out Call$11.43$0.66 Put$7.34$6.70

60 60 Barrier options Up-and-in call – worth almost the same as the standard call, because it gets knocked in only when the stock price is up where the standard call also gets most of its value. Up-and-in put – its value is much less than the standard put value, because it gets knocked in for high stock values where the put is deep out of the money and thus, has very low value. Example:S = $100;K = $100;B = $120;T = 1; q = 5%;r = 10%;  = 25%. Standard European Up-and-in Call$11.43$10.78 Put$7.34$0.64

61 61 Barrier options Example: Down-and-out call In 51 days a firm From Denmark will make a payment of 200M on the principal of a Euro Danish bond. S = USD.18213/ If the payment were to be today the payment would have been USD36,426,000. If, the Danish currency appreciates against the dollar, however, the firm will have to pay more in USD. A standard call option with S =.18213; T = 51 days; K =.1950;  = 14.4%; r USA = 4.18%; r DEN = 5.24% will cost per Danish or USD92,400. This call guarantees that the firm will buy the dollars for no more than USD.195/. A barrier option: a down-and-out call with the barrier set at B = USD.1750 will cost.0453 or USD90,600. New risk: The exchange rate may go down, knock the call out and then rise again.

62 62 Barrier options Example: Up-and-out put. In 40 days an American firm will receive JY100M. The current spot exchange rate is USD /JY. The payment now would be USD1,100,900. If the JY depreciates against the dollar, however, the payment will be less. The firm buys an up-and-out put with a barrier at USD /JY. If the barrier is crossed the put is out and the firm feels that its risk exposure to the JY depreciating is negligible. If the barrier is not crossed, the firm holds a protective put.

63 63 Barrier options Example: Up-and-in call. In 40 days an American firm will pay EUR50M on a Eurobond that matures in 100 days. If the EURO appreciates against the dollar, the firm payment in USD increases. The firm expects the EURO to stay the same or even depreciate over the next 100 days, but it still wishes to protect itself against an appreciating EURO. The firm decides to purchase an Up-and-in call. This way, if the firm’s expectations materialize, the EURO will depreciate against the dollar and the call will not come into existence. If, however, the firm’s prediction turns out to be incorrect, the EURO will appreciate against the dollar, the barrier will be crossed and the call will come into existence, protecting the firm payment in USD.

64 64 Barrier options There are many other types of barrier options. Some are Barrier option for a pre specified time only, others are barriers on BINARY options – to be discussed later. One type of barrier options that is very interesting is a Two-asset barrier option: Consider a Norwegian oil producer. As oil is typically sold for USD/bbl, the producer income in EUROs depends on the USD oil price and on the USD-EURO exchange rate. A standard currency option on the EURO will hedge the FORX risk exposure. Instead, A currency option that is knocked out if the oil price increases beyond a particular level, will give the producer some flexibility between the oil price level and the FORX rate. It will be cheaper than the outright option on the EURO.

65 65 Binary Options (page 441) Cash-or-nothing: pays Q if S > K at time T, otherwise pays 0. Value = e –rT Q N(d 2 ) Asset-or-nothing: pays S if S > K at time T, otherwise pays 0. Value = S 0 N(d 1 )

66 66 Decomposition of a Call Option Long Asset-or-Nothing option Short Cash-or-Nothing option where payoff is K Value = S 0 N(d 1 ) – e –rT KN(d 2 )

67 67 Lookback Options (page 441) Lookback call pays S T – S min at time T Allows buyer to buy stock at lowest observed price in some interval of time Lookback put pays S max – S T at time T Allows buyer to sell stock at highest observed price in some interval of time

68 68 Shout Options (page 443) Buyer can ‘shout’ once during option life Final payoff is either –Usual option payoff, max(S T – K, 0), or –Intrinsic value at time of shout, S  – K Payoff: max(S T – S , 0) + S  – K Similar to lookback option but cheaper How can a binomial tree be used to value a shout option?

69 69 Asian Options (page 443) Payoff related to average stock price Average Price options pay: –max(S ave – K, 0) (call), or –max(K – S ave, 0) (put) Average Strike options pay: –max(S T – S ave, 0) (call), or –max(S ave – S T, 0) (put)

70 70 Asian Options No analytic solution Can be valued by assuming (as an approximation) that the average stock price is lognormally distributed

71 71 Exchange Options (page 445) Option to exchange one asset for another For example, an option to exchange U for V Payoff is max(V T – U T, 0)

72 72 Basket Options (page 446) A basket option is an option to buy or sell a portfolio of assets This can be valued by calculating the first two moments of the value of the basket and then assuming it is lognormal


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