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## Exotic Options Chapter 19

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**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.**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.**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.**Collars(19.1)**Often, investors buy the underlying asset and purchase protective European puts at some exercise price K1; K1< S. In order to finance the purchase of the protective puts, the investor may short European calls with K2; K1< S < K2 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, K1) = p(S, T-t, K2).**Collars**AT EXPIRATION StrategyICF ST< K1 K1<ST< K2 ST> K2 Buy stock -S ST ST ST Buy put(K1) -p K1 – ST 0 0 Sell call(K2) c 0 0 K2 – ST TOTAL -S K1 ST K2 P/L K1– S ST - S K2 – S**Collars**The self financing Collar guarantees that the asset, which was purchased for S, will sell for K1 or better, up to K2. Given that the probability of ST to exceed K2 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.**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.**Forward start options(19.3)**These are at-the-money options that will begin on a specified future date, T1 , say and will expire at T2 . 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, T1, with expiration at T2. It can be easily shown that it has the same value of an at-the-money option with T = T2 -T1.**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.**Forward start options**Suppose that we face n such options, the value of the entire strategy is**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; T1 =.25 and T2 = 1. Substituting these parameters into the formula with only one option we obtain the call value: C = $4.4064.**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 K1 that expires on T1. If exercised, the holder will buy or sell for K1 an option with K2 that expires T2 time periods from today.**Compound options: Four possibilities:**• The payoff • A call on a call max{0, c(S, K2 ,T2) – K1)} • A put on a call max{0, K1 - c(S, K2 ,T2)} • A call on a put max{0, p(S, K2 ,T2) – K1)} • A put on a put max{0, K1 – p(S, K2 ,T2)}. • c and p are the Black and Scholes values with exercise price K2 and time to expiration T2. • K1 is the exercise price of the option on the underlying option, with T1 time to expiration.**Compound options**Example 1: A put on a call. The underlying call is the following call on the index: c(S=500; K2=520; T2=.5) The put option on this call is: p[c(S=500;K2=520;T2=.5), K1=50; T1=.25] The payoff on this compound option is: Max{0, K1 – c(S, K2,T2)} = Max{0, 50 - c(500, 520, .5)}.**Compound options**The value of this put on call when r = .08; q = .03 and underlying stock index volatility = .35 is: $21.1965. 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.**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 T1 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 T2. The firm is clearly exposed to exchange rate risk and may wish to hedge this risk.**Compound options**Time line: 0……………. T1 ………………….…… T2 BID ACCEPTED……… PAYMENT and or REJECTED DELIVERY If the foreign currency depreciates against The USD, the USD amount equivalent to M will be smaller.**Compound options**The firm could purchase a protective put on the foreign currency for T2 and pay the full premium, ignoring the fact that it may not win the bid. INSTEAD**Compound options**The firm could buy a call for T1, 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 T1. 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.**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.**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.**Compound options**If the firm decides to purchase a protective put on the foreign currency for T2 = 49 days and pay the full premium, ignoring the fact that it may not win the bid, we use: S = USD.6303/CD; K1 = USD.6303/CD. p = p(.6303; .6303; 49/365; r=.0404; =.028) =USD.2669/CD 0r, a total of: p = USD.2669/CD[CD10M] = USD26,690.**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; K2=.6303; T2=49/365) • The call option on this put is: • c[p(S=.6303;K2=.6303;T2=49/365), K1=?; • T1=25/365;=.028]**Compound options**The payoff on this compound option depends on K1. 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.**Compound options**Note: The put value will increase when the CD depreciates against the USD, thus, increasing the compound option value.**Compound options**Again. The underlying put option is a put on the CD with the following parameters: p(S=.6303; K2=.6303; T2=49/365) The call option on this put is c[p(S=.6303;K2=.6303;T2=49/365), K1=?; T1=25/365;=.028] Calculation of the compound option shows an increasing compound option value with a decreasing exercise price: K1 c[p(S, K2,T2); K1=?; T1=25/365;=.028] .24 USD.1096/CD USD10,960 .22 USD.1184/CD USD11,840 .20 USD.1277/CD USD12,770 .18 USD.1377/CD USD13,770 .16 USD.1485/CD USD14,850**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 K1ICFp > .6303TOTAL .24 USD10,960 USD24,000 USD34,960 .22 USD11,840 USD22,000 USD33,840 .20 USD12,770 USD20,000 USD32,770 .18 USD13,770 USD18,000 USD31,770 .16 USD14,850 USD16,000 USD30,850**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:**Compound options**Example 3 continued: A protective put: If the firm decides to purchase a protective put on the foreign currency for T2 = .5yrs and pay the full premium, ignoring the fact that it may not win the bid, we use: S = USD.9/EUR K1 = USD.9/EUR This put will cost: p = p(.9; .9; .5; rusd=.06; reur=.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.**Compound options**Example 3 continued: The underlying put option is a put on the EUR with the following parameters: p(S=.9; K2=.9; T2=.5) The call option on this put is c[p(S=.9;K2=.9;T2=.5), K1=?; T1=.25;=.01]**Compound options**The payoff on this compound option depends on K1. 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 K1=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.**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.**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?**Unprotected American calls**It follows that on xd, the call holder faces the: Max{Sxd - K + D; c(Sxd, K, T-xd)}. On that day, the put-call parity implies that: c(Sxd, K,T-xd) = p(Sxd, K, T-xd)+ Sxd –Ke- r(T – xd). Substitute the put-call parity into the option value to obtain: Sxd + D – K + Max{0, p(Sxd, K, T-xd) – [D - K(1-e- r(T – xd))]}.**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.**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: Sxd + D – K+p(Sxd, K, T-xd)–[D–K(1-e- r(T – xd))]. But Upon substitution of p(Sxd, K,T-xd) = c(Sxd, K,T-xd) - Sxd + Ke- r(T – xd).**Unprotected American calls**The value received upon exercising the compound option is: c(Sxd, 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: Sxd + D – Kexercise the American call.**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 ST and therefore, the cash flow to the stock holders at T can be summarized as follows: CF = max{0, ST – F}.**Compound options**CF = max{0, ST – 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.**Chooser options(19.5)**The option is traded now, at time 0, determining a future time T1 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 T1 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.**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(S1, T2, K2) and p = p(S1, T3, K3). 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(S1,T,K) = c(S1,T,K) - S1e- q(T- T1) + Ke- r(T- T1) Into Max{c(S1,T,K), p(S1,T,K) }: Max{c, c - S1e- q(T- T1) + Ke- r(T- T1)},**Chooser options**which can be rewritten: c +Max{0, Ke- r(T- T1) - S1e- q(T- T1)} or c + e- q(T- T1)Max{0, Ke- (r-q)(T- T1) - S1}. 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 T1 with exercise price Ke- (r-q)(T- T1) .**Chooser options**Example: Stock XYZ is trading for $125.9375/share. A Straddle with K = 125, T = 35 days, r = .0446 and = .83 will cost: c = $13.21 + p = $12.09 = $25.30. The Chooser option with T1 =20 days will be worth c = $13.21 plus the put value with T = 15 days = .0411yrs; p = 7.80. And the Chooser option costs $21.01. It costs less than the Straddle because there is a possibility that the payoff at expiration will be zero.**Chooser options**Example: Stock XYZ is trading for $50/share. A chooser option with a decision time at T1 = .25yrs and expiration T = .5yrs is with K = 50, r = .08, q = 0, and = .25. The Chooser option premium is $6.11/share.**Chooser options**Case II The call and the put are for different exercise prices and times to expiration. c = c(S1, T2, K2) and p = p(S1, T3, K3). To evaluate this option, let S* be the underlying asset price at which today’s premium of a call with K2 that expires at T2 is equal to the T1 value of a put with K2 and T2. c0(S*, K2, T2) = p(S*, K3,T3). 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:**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.**Chooser options**Example: At time T1 = .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.**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.