Hedging Complex Barrier. Options Broadway, 6th oor 545 Technology Square. New York, NY Cambridge, MA Current Version: April 1, 1997

Transcription

1 Hedging Comlex Barrier Otions Peter Carr Andrew Chou Morgan Stanley MIT Comuter Science 1585 Broadway, 6th oor 545 Technology Square New York, NY 136 Cambridge, MA 139 (1) (617) Current Version: Aril 1, 1997 Any errors are our own.

2 Summary Page Abstract We show how several comlex barrier otions can be hedged using a ortfolio of standard Euroean otions. These hedging strategies only involve trading at a few times during the otion's life. Since rolling, ratchet, and lookback otions can be decomosed into a ortfolio of barrier otions, our hedging results also aly to them. 1

3 1 Introduction Hedging Comlex Barrier Otions Barrier otions have become increasingly oular in many over-the-counter markets 1. This oularity is due to the additional exibility which barrier otions confer uon their holders. In general, barrier otions allow investors to revise their vanilla otion ositions costlessly at the rst time one or more critical rice levels are reached. For examle, an out-otion with a constant rebate allows an investor to eectively sell the vanilla otion at a xed rice at the rst hitting time of the barrier. Similarly, an in-otion allows an investor to eectively buy a vanilla otion at the rst hitting time at no cost beyond the initial remium. Used in combination, ortfolios of barrier otions can change the strike or maturity of a vanilla otion at the rst hitting time of one or more critical rice levels. The seminal aer by Merton[17] values a down-and-out call otion in closed form. The valuation relies on the ability to erfectly relicate the ayos to the barrier otion using a dynamic strategy in the underlying asset. A series of aers (see [4, 6, 7, 8, 9]) shows how to alternatively relicate the ayos of barrier otions using a static osition in vanilla otions. Since the economic assumtions underlying the static relication are the same as for the dynamic one, the resulting static valuation matches the dynamic one. However, the ecacy of the hedge in static models is likely to be more robust to violations of standard assumtions such as continuously oen markets, constant volatility, and no transactions costs. The urose of this aer is to extend these static hedging results for single barrier otions to more comlex barrier otions. In articular, we will examine the following tyes of barrier otions: 1. Partial Barrier Otions: For these otions, the barrier is active only during an initial eriod. In other words, the barrier disaears at a rescribed time. In general, the ayo at maturity may be a function of the sot rice at the time the barrier disaears.. Forward Starting Barrier Otions: For these otions, the barrier is active only over the latter eriod of the otion's life. The barrier level may be xed initially, or alternatively, may be set at the forward start date to be a secied function of the contemoraneous sot rice. The ayo may again be a function of the sot rice at the time the barrier becomes active. 1 For a descrition of exotic otions in general and barrier otions in articular, see Nelken[18] and Zhang[].

4 3. Double Barrier Otions: Otions that knock in or out at the rst hitting time of either a lower or uer barrier. 4. Rolling Otions: These otions are issued with a sequence of barriers, either all below (for roll-down calls) or all above (for roll-u uts) the initial sot rice. Uon reaching each barrier, the otion strike is lowered (for calls) or raised (for uts). The otion is knocked out at the last barrier. 5. Ratchet Otions: These otions dier from rolling otions in only two ways. First, the strike ratchets to the barrier each time a barrier is crossed. Second, the otion is not knocked out at the last barrier. Instead, the strike is ratcheted for the last time. 6. Lookback Otions: The ayo of these otions deends uon the maximum or the minimum of the realized rice over the lookback eriod. The lookback eriod may start before or after the valuation date but must end at or before the otion's maturity. As shown in [4] and [7], the last three categories above may be decomosed into a sum of single barrier otions. Consequently, rolling, ratchet, and lookback otions can be staticly hedged using the results of the foregoing aers. Furthermore, the decomosition into barrier otions is model-indeendent. Thus, as new static hedging results for single barrier otions are develoed, these results will automatically hold for these multile barrier otions. The structure of this aer is as follows. The next section reviews revious results on static hedging. The next six sections examine the static relication of the six tyes of claims described above. The last section reviews the aer. Three aendices contain technical results. Review of Static Hedging.1 Static Hedging of Path-Indeendent Securities Breeden and Litzenberger[5] showed that any ath-indeendent ayo can be achieved by a ortfolio of Euroean calls and uts. In articular, Carr and Chou[6] showed that any twice dierentiable ayo f(s) can be written as: f(s) = f(f )+(S?F )f (F )+ Z F f (K)(K?S) + dk+ Z 1 F f (K)(S?K) + dk: (1) where F can be any xed constant, but will henceforth denote the initial forward rice. Thus, any such ayo can be uniquely decomosed into the ayo If f () is not twice dierentiable, then (1) still holds if f )() and f () are interreted as generalized functions such as Heaviside ste functions and Dirac delta functions. 3

5 from a static osition in f(f ) unit discount bonds, f (F ) initially costless forward contracts 3, and the continuum of initially out-of-the-money otions. To revent arbitrage, the initial value V of the ayo must be the cost of this relicating ortfolio: Z F Z 1 V = f(f )B + f (K)P (K)dK + f (K)C(K)dK; () F where B is the initial value of the unit bond, and P (K); K F and C(K); K F are the initial values of out-of-the-money forward uts and calls resectively. Note that the second term in (1) does not aear in () since the forward contracts held are initially costless. In what follows, we will be roviding athindeendent ayos which lead to values matching those of ath-deendent ayos. We will leave it to the reader to use (1) to recover the static relicating ortfolio and to use () to recover its value.. Static Hedging of Single Barrier Claims A single barrier claim is one that rovides a secied ayo at maturity so long as a barrier for the underlying rice has been hit (in-claim) or has not been hit (out-claim). This subsection shows that one can relicate the ayo of any single barrier claim with a ortfolio of vanilla Euroean otions. The ortfolio is static in the sense that we never need to trade unless the claim exires or its underlying asset hits a barrier. Our static hedging results all rely on Lemma 1 in Carr and Chou[6], which is reeated below and roven in Aendix : Lemma 1 In a Black-Scholes economy, suose that X is a ortfolio of Euroean otions exiring at time T with ayo: X(S T ) = n f(st ) if S T (A; B), otherwise. For H >, let Y be a ortfolio of Euroean otions with maturity T and ayo: Y (S T ) =? STH f(h =S T ) if S T (H =B; H =A), otherwise where the ower 1? (r?d) and r; d; are the interest rate, dividend rate and volatility rate resectively. Then, X and Y have the same value whenever the sot equals H. 3 Note that since bonds and forward contracts can themselves be created out of otions, the sectrum of otions is suciently rich so as to allow the creation of any suciently smooth ayo, as shown in Breeden and Litzenberger[5]. 4

6 The ayo of Y is the reection 4 of the ayo of X along axis H. Note that A or B can be assigned to be or 1 resectively. This lemma is model-deendent in that it uses the Black-Scholes assumtions. The lemma can be used to nd the relicating ortfolio of any single barrier claim. For examle, consider a down-and-in claim which ays f(s T ) at T rovided a lower barrier H has been hit over [; T ]. From the revious section, we know that a ortfolio of vanilla otions can be created which rovides an adjusted ayo, dened as: ( if ST > H, ^f(s T )? S f(s T ) + TH f H if S T < H. (3) If the lower barrier is never hit, then the vanilla otions exire worthless, matching the ayo of zero from the down-and-in. If the barrier is hit over [; T ], then Lemma 1 indicates that at the rst hitting time, the value of the? S TH f term matches the value of a ayo f(s T )1 ST >H, where 1 E denotes an indicator function of the event E. Thus, the otions roviding the ayo? S TH f can be sold o with the roceeds used to buy otions delivering the ayo f(s T )1 ST >H. Consequently, after rebalancing at the hitting time, the total ortfolio of otions delivers a ayo of f(s T ) as required. It follows that whether the barrier is hit or not, the ortfolio of Euroean otions roviding the adjusted ayo f relicates the ayos of the down-and-in claim. By in-out arity 5, the adjusted ayo corresonding to a down-and-out claim is: ^f(s T ) S T ( f(st ) if S T > H,?? S TH f H S T if S T < H. The reection rincile imlicit in Lemma 1 can also be alied to u-barrier claims. The adjusted ayo corresonding to an u-and-in security is: ^f(s T ) (? S f(s T ) + TH f H S T if S T > H, if S T < H. H S T H S T (4) (5) Similarly, an u-and-out security is accociated with the adjusted ayo: ^f(s T ) (? S? TH f H S T if S T > H, f(s T ) if S T < H. (6) Note that all of the above adjusted ayos can be obtained in a simle manner if one already has a ricing formula, either from the literature or from 4 The reection is geometric and accounts for drift. 5 In-out arity is a relationshi which states that theayos and values of an in-claim and an out-claim sum to the ayos and values of an unrestricted claim. 5

7 dynamic relication arguments. In this case, the adjusted ayo is the limit of the ricing formula V (S T ; ) as the time to maturity aroaches zero (after removing domain restrictions such as S > H). 3 Partial Barriers ^f(s T ) = lim # V (S T ; ); S T > : A artial barrier otion 6 has a barrier that is active only during art of the otion's life. Tyically, the barrier is active initially, and then disaears at some oint during the otion's life. One could also imagine the oosite situation, where the barrier starts inactive and becomes active at some oint. We denote these otions as forward-starting otions and discuss them in section 4. We will resent two dierent hedging strategies in this section. In the rst method, we will rebalance when the barrier disaears. This method is very general, in that the nal ayo of the otion can deend uon the sot rice at the time the barrier disaears. Usually, the ayo is not a function of this rice and deends only on the nal sot rice. In this case, we can aly a second hedging method, which is suerior to the rst method in that it does not require rebalancing when the barrier disaears. We will examine down-barriers and leave it to the reader to develo the analogous results for u-barriers. Consider a artial barrier otion with maturity T, which knocks out at barrier H. Let T 1 (; T ) denote the time when the barrier exires. At time T 1, either the otion has knocked out or else it becomes a Euroean claim with some ayo at time T. This ayo may deend uon the sot rice at time T 1, which we denote by S 1. Using risk-neutral valuation (see Aendix 1), we can always nd the function V (S 1 ) relating the value at T 1 of this ayo to S 1. Dene the adjusted ayo at time T 1 as: V (S1 ) if S ^f(s 1 ) =? 1 > H, S? 1 H V (H =S 1 ) if S 1 H: Thus, our relicating strategy is as follows: 1. At initiation, urchase a ortfolio of Euroean otions that gives the adjusted ayo ^f(s 1 ) at maturity date T 1.. If the barrier is reached before time T 1, liquidate the ortfolio. From Lemma 1, the ortfolio is worth zero. 3. At time T 1, if the barrier has not been reached, use the ayo from the exiring otions to urchase the aroriate ortfolio of Euroean otions maturing at time T. 6 Heynen and Kat[13] rovide analytic valuation formulas for artial barrier otions. 6

8 We can also nd a relicating strategy for an in-barrier claim. Consider an exotic claim maturing at T with no barrier, but with a ayo that deends uon S 1. Standard techniques such as risk-neutral valuation allow us to identify the function V (S 1 ) giving the value at time T 1. Therefore, by in-out arity, the adjusted ayo at time T 1 is: if S1 > H, ^f(s 1 ) =? S V (S 1 ) + 1 H V (H =S 1 ) if S 1 H: Our relicating strategy is as follows: 1. At initiation, urchase a ortfolio of Euroean otions that ays o ^f(s 1 ) at time T 1.. If the barrier is reached before time T 1, then rebalance the ortfolio to have ayo V (S 1 ) at time T 1 for all? S 1. By single barrier techniques, the S value of the adjusted ayo term 1 H V (H =S 1 ) exactly matches the value of the ayo V (S 1 )1 S1>H. 3. At time T 1, if the barrier has not been reached, our ayo is zero. For this hedging method, the ossible rebalancing oints are the rst assage time to the barrier and time T 1. We now resent a second method that only requires rebalancing at the rst assage time. However, we require the ayo at time T to be indeendent of S 1. Our relicating ortfolio will use otions that exire at both T 1 and T. Let the artial barrier claim ayo at time T be g(s ), where S is the sot rice at time T. From the last section, we know that we can form a ortfolio of Euroean otions with ayo g(s ) at time T. Let V (S 1 ) denote the function relating its value at time T 1 to S 1. Now, suose our artial barrier claim is a down-and-out. Then, the desired ayo at time T 1 is: V (S1 ) if S 1 > H, f(s 1 ) =?? S 1 H V (H =S 1 ) if S 1 H: Unfortunately, our current ortfolio of otions maturing at T has a value at T 1 of only V (S 1 ). Thus, we must add a ortfolio of Euroean otions maturing at T 1 to make u this dierence. These otions rovide the following adjusted ayo at time T 1 : if S1 > H, ^f(s 1 ) =? S?V (S 1 )? 1 H V (H =S 1 ) if S 1 H. Our relicating strategy is as follows: 1. At initiation, urchase a ortfolio of Euroean otions that: rovides ayo g(s ) at maturity T, and 7

9 rovides ayo ^f(s 1 ) at maturity T 1.. Uon reaching the barrier before time T 1, liquidate all otions. From Lemma 1, our ortfolio will be worth zero. 3. If the barrier is not reached before time T 1, our ayo will be g(s ) at time T as desired. Note that it is imossible for the otions maturing at time T 1 to ay o without the barrier being reached. Interestingly, the otions maturing at T 1 never nish in-the-money. If the barrier is reached, they are liquidated. Otherwise, they exire out-of-the-money at time T 1. Thus, our only rebalancing oint is the rst assage time to the barrier. For a down-and-in claim, we can aly in-out arity. Our relicating ortfolio is simly a ortfolio of Euroean otions that ays o? ^f(s 1 ) at maturity date T 1. If the barrier is not hit by T 1, these otions exire worthless as desired. If the barrier is hit before T 1, the value of this ortfolio matches the value of a ortfolio of Euroean otions aying o g(s ) at time T. Thus, the otions maturing at T 1 can be sold o with the roceeds used to buy the otions maturing at T. Using this second method, one only needs to rebalance at the rst assage time to the barrier, if any. To illustrate both methods, consider a down-and-out artial barrier call with strike K, maturity T, artial barrier H, and barrier exiration T 1. Using the rst hedging method, our initial relicating ortfolio will have maturity T 1 and ayo (see Figure 1): C(S1 ) if S 1 > H, ^f(s 1 ) =? S? 1 H C( H S 1 ) if S 1 < H, (7) where C(S 1 ) is the Black-Scholes call ricing formula for a call with sot S 1, strike K, and time to maturity T? T 1. The initial value of the artial barrier call is just the discounted exected value 7 of ^f at time T 1. The ayo of this otion is indeendent of S 1, so we can also aly the second hedging method. The ortfolio of otions maturing at T reduces to a single call struck at K. The ortfolio of otions maturing at T 1 has the ayo (see Figure ): if S1 > H, ^f(s 1 ) =? S?C(S 1 )? 1 H C(H =S 1 ) if S 1 H: The initial value of the barrier otion is given by the sum of the initial values of the otions maturing at T 1 and T. 7 See Aendix 3 for a closed form solution for this value. 8

10 5 Adjusted Payoff for a Partial Barrier Call 15 1 Adjusted Payoff Sot when Partial Barrier ends (r = :5, d = :3, = :15, H = 9, K = 1, T? T 1 = :5) Figure 1: Adjusted ayo for a Partial Barrier Call Using First Hedging Method. 4 Forward Starting Barrier Otions For forward-starting otions, the barrier is active only over the latter eriod of the otion's life. As we shall see, forward-starting barrier otions are very similar to artial barrier otions. Again, we will resent two dierent relicating methodologies. The rst method is more general and can be alied to cases where the barrier and/or ayo deend uon the sot rice when the barrier becomes active. This method ossibly requires rebalancing when the barrier aears and at the rst assage time to the barrier. The second method requires that the barrier and ayo be indeendent of the sot rice when the barrier aears, but only requires rebalancing at most once. Consider a forward-starting claim maturing at T, and let the barrier aear at time T 1. At time T 1, the exotic becomes identical to a single barrier claim. Using the static hedging techniques described in subsection., we can rice 8 the exotic at time T 1 as V (S 1 ). Our rst hedging method is create a ortfolio of Euroean otions that ays o V (S 1 ) at time T 1. At time T 1, the ayo from these otions will be used to buy a ortfolio of otions maturing at T which relicates the ayos of a 8 Note that at the forward start date, we also know how to rice a artial barrier claim using the results of the last section. Consequently, one can use the techniques of this section to rice a barrier claim, where the barrier is active only over a eriod occurring strictly within the claim's life. 9

11 5 Adjusted Payoff When Partial Barrier Ends 5 Adjusted Payoff at Maturity Adjusted Payoff 5 5 Adjusted Payoff Sot When Partial Barrier Ends Sot at Maturity (r = :5, d = :3, = :15, H = 9, K = 1, T? T 1 = :5) Figure : Adjusted ayos for a Partial Barrier Call Using Second Hedging Method. single barrier claim maturing at T. Thus, our hedging strategy always requires rebalancing at time T 1. The subsequent single barrier relication may require an additional rebalancing. An imortant secial case arises if V (S 1 ) may be written as S 1 n(), where n() is indeendent of S 1. This situation arises for barrier otions where the strike and barrier are both roortional to S 1. In this case, the hedge is to buy n()e?dt1 shares at time and re-invest dividends until T 1. The shares are then sold and the roceeds are used to buy otions roviding the aroriate adjusted ayo at T. We now discuss the second method, which is alicable when the barrier and ayo are indeendent of S 1. As before, we will examine down-barriers and leave it to the reader to aly the same techniques to u-barriers. Consider a forward-starting knockout with ayo g(s ) at time T and barrier H active over [T 1 ; T ]. At T 1, the security is identical to a single barrier claim, so we would like our adjusted ayo at time T to be: g(s ) if S > H, ^g out (S ) =?? S H g(h =S ) if S H. Let V (S 1 ) denote the value at T 1 of this adjusted ayo. To relicate the 1

12 value of the forward starting claim, we need our ortfolio at time T 1 to be worth: f(s 1 ) = V (S1 ) if S 1 > H, if S 1 H. The ayo of zero below the barrier arises because our forward-starting otion is dened to be worthless if the stock rice is below the knockout barrier when the barrier is activated. Thus, we will add otions maturing at time T 1 with ayo: ^f out (S 1 ) = if S > H,?V (S 1 ) if S H. Our relicating strategy is: 1. At initiation, urchase a ortfolio of Euroean otions that: rovides ayo ^g out (S ) at maturity T, and rovides ayo ^f out (S 1 ) at maturity T 1.. If the sot rice at time T 1 is below H, the exotic has knocked out, so liquidate the ortfolio. 3. Otherwise, the ortfolio is held at T 1. If the barrier is hit between time T 1 and T, the ortfolio is liquidated. Otherwise, the ortfolio ays o g(s ) at T. Note that whenever the ortfolio is liquidated before maturity, it has zero value by construction. For knock-in claims, one can aly in-out arity. Our relicating ortfolio consists of otions maturing at T with ayo: if S > H, ^g in (S ) = g(s ) +? S H and otions maturing at time T 1 with ayo: g(h =S ) if S H, ^f in (S 1 ) = if S1 > H, V (S 1 ) if S 1 H, where V (S 1 ) was dened reviously as the time T 1 value of the ayo ^g out at time T. To see why this ortfolio relicates the ayos of a forward-starting knockin claim, note that if S 1 > H at time T 1, then the ^f in (S 1 ) relicas exire worthless. The remaining otions relicate the ayos of a a single barrier knockin, as required. On the other hand, if S 1 H at time T 1, then the otions maturing at T have a value at T 1 equal to that of an in-barrier claim, while the otions maturing at T 1 have a ayo at T 1 equal to the value of an out-barrier claim. By 11

13 1.5 Adjusted Payoff for Forward Starting No touch Binary 1.5 Adjusted Payoff Sot at Forward Start Time (r = :5, d = :3, = :15, H = 1, T? T 1 = :5). Figure 3: Adjusted ayo for Forward Starting No-touch Binary Using First Hedging Method. in-out arity, the sum of the two values is that of a vanilla claim aying ^g(s ) at T, as required. To maintain the hedge to T, the ayo from the otions maturing at T 1 is used to buy the aroriate osition in otions maturing at T. Thus, in contrast to the rst method, at most one rebalancing is required. To illustrate both methods, consider a forward-starting no-touch binary 9 with down barrier H, maturity T, and barrier start date T 1. Using the rst method, the ortfolio of otions with maturity T 1 has ayo (as shown in Figure 3): NT B(S1 ) if S f(s 1 ) = 1 > H, if S 1 < H, where NT B(S 1 ) is the Black-Scholes rice of a no-touch binary 1 with sot S 1, time to maturity T? T 1, and barrier H. Since the barrier and ayo are indeendent of S 1, we can also aly the second method. The ortfolio of otions with maturity T has ayo (see Figure 4): 1 if S > H, ^g out (S ) =?? S H if S H, 9 A no-touch binary ays one dollar at maturity if the barrier has not been hit. 1 See Reiner and Rubinstein[19] for the formula. 1

14 1.5 Adjusted Payoff at Forward Start Time 1.5 Adjusted Payoff at Maturity Adjusted Payoff Adjusted Payoff Sot at Forward Start Time Sot at Maturity (r = :5, d = :3, = :15, H = 1, T? T 1 = :5). Figure 4: Adjusted ayos for Forward Starting No-touch Binary Using Second Hedging Method. and the ortfolio of otions with maturity T 1 has ayo: ^f out (S 1 ) = if S1 > H,?NT B(S 1 ) if S 1 H, where we extend the NT B() formula to values below H. 5 Double Barriers A double barrier claim is knocked in or out at the rst assage time to either a lower or uer barrier. Double barrier calls and uts have been riced analytically in Kunitomo and Ikeda[16] and Beaglehole[1], and using Fourier series in Bhagavatula and Carr[]. In analogy with the single barrier case, our goal is to nd a ortfolio of Euroean otions, so that at the earlier of the two rst assage times and maturity, the value of the ortfolio exactly relicates the ayos of the double barrier claim. In order to do this, we will need to use multile reections. Consider a double knockout with down barrier D, u barrier U, and maturity date T. We begin by dividing the interval (; 1) into regions as in Figure 5. 13

15 We can succinctly dene the regions as: k U Region k = D; D k U U! : D To secify the adjusted ayo for a region i, we will use the notation ^f (i) (S T ): We begin with ^f () (S T ) = f(s T ): Region -3 Region - Region -1 Region Region +1 Region + Region +3 D 3 =U D =U D U U =D U 3 =D Sot Figure 5: Dividing (; 1) into regions. From Lemma 1, we see that for a reection along D, the region k (eg. k =?) would be the reection of region?k? 1 (eg.?k? 1 = +1). Similarly, for reection along U, region k would be the reection of region?k + 1. It is useful to dene the following two oerators: R D ( ^f(s T )) =? ST D It follows that: and ^f(d =S T ) and R U ( ^f(s T )) =? ST U ^f (k) (S T ) = R D ( ^f (?k?1) (S T )); for k < ^f (k) (S T ) = R U ( ^f (?k+1) (S T )); for k > : ^f(u =S T ): Note that R U and R D bijectively ma between the corresonding regions. Also, we are taking the negative of the reection, so that the valuation of the ayos will cancel. By induction, we can comletely determine the entire adjusted ayo as: ^f (k) (S T ) = 8 R >< D R U R D : : : {z } >: f(s T ) for k =, (f(s T )) for k <, k oerators R U R D R U : : :(f(s T )) for k >. {z } k oerators 14

16 A ortfolio of Euroean otions that delivers the above adjusted ayo relicates the ayo to a double barrier claim. If we never touch either barrier, then the adjusted ayo from region matches the ayo of the original exotic. Uon reaching a barrier, the values of the ayos above the barrier are cancelled by the values of the ayos below the barrier. Therefore, our ortfolio is worth zero at either barrier at which oint we can liquidate our osition. To nd the adjusted ayo for a double knockin claim, we aly in-out arity. The adjusted ayo is given by: ^f (k) (S T ) = 8 >< >: for k =, f(s T )? R D R U R D : : :(f(s T )) for k <, {z } k oerators f(s T )? R U R D R U : : :(f(s T )) for k >. {z } k oerators As an examle, consider a double knockout binary, which ays one dollar at maturity if neither barrier is hit beforehand. Then, f(s T ) = 1, and the adjusted ayo is (see Figure 6): ^f(s T ) = (? S? TU? D j U in region j + 1,? U D j in region j, where j is an integer and recall = 1? r?d. Two secial cases are of interest. For r = d, we have = 1, and the adjusted ayo become iecewise linear. For r? d = 1, we have =, and the adjusted ayo is iecewise constant. To value the double knocout binary, we simly value the adjusted ayo in each region and sum over all regions. If the current sot rice is S, the value of the ayo in region k is: 8 <? S? U D j h i U e?rt N( ln(x1)?t )? N( ln(x)?t T ) in region k = j + 1, T V (S; k) = :? U j h i D e?rt N( ln(x1)+t )? N( ln(x)+t ) in region k = j, T T where x 1 = SDk?1, x U k = SDk, and = r? d? 1 U k+1. Therefore, the value of the double knockout binary is the sum of the values for each region. NT B(S) = 1X k=?1 V (S; k): Although this sum is innite, we can get an accurate rice with only a few terms. Intuitively, the regions far removed from the barriers will contribute little to the rice. Therefore, we only need to calculate the sum for a few values of k near. In Table 1, we illustrate this fact. 15

17 1.5 Adjusted Payoff for a Double No touch Binary 1.5 Adjusted Payoff Final Sot Price (r = :5, d = :3, = :15, D = 9, U = 11) Figure 6: Adjusted ayo for Double No-touch Binary. 6 Rolling Otions The relication of rolldown calls 11, ratchet calls, and lookback calls was examined by Carr, Ellis, and Guta[7]. In the next three sections, we review their decomosition into single barrier otions and then aly our techniques for barrier otion relication. A rolldown call is issued with a series of barriers, H 1 > H > : : : > H n, which are all below the initial sot. At initiation, the roll-down call resembles a Euroean call with strike K. If the rst barrier H 1 is hit, the strike is rolled down to a new strike K 1 < K. Uon hitting each subsequent barrier H i < H i?1, the strike is again rolled down to K i < K i?1. When the last barrier H n is hit, the otion knocks out. A roll-down call can be decomosed in terms of knockout otions as: X n?1 RDC = DOC(K ; H 1 ) + [DOC(K i ; H i+1 )? DOC(K i ; H i )]: (8) i=1 This relication is model-indeendent and works as follows. If the nearest barrier H 1 is never hit, then the rst otion rovides the necessary ayo, while the terms in the sum cancel. If H 1 is reached, then DOC(K ; H 1 ) and 11 See Gastineau[1] for an introduction to rolling otions. 16

18 Regions Used to Price Price k.8687?1 k 1.671? k.6718?3 k ?4 k ?5 k Month Otion (T = :5) Regions Used to Price Price k.475?1 k ? k.7713?3 k ?4 k ?5 k Year Otion (T = 1) (S = 1, r = :5, d = :3, = :15, U = 11, D = 9) Table 1: Price Convergence of Double Knockout Binary Pricing Formula. DOC(K 1 ; H 1 ) become worthless. We can re-write the remaining ortfolio as: X n?1 RDC = DOC(K 1 ; H ) + [DOC(K i ; H i+1 )? DOC(K i ; H i )]: i= This osition is analogous to the initial one, but with initial strike K 1, and with barriers H ; : : :; H n. If all the barriers are hit, then all the otions knock out as required. It is straightforward to relicate the ayos of a rolldown call with vanilla otions. For each down-and-out call, use (4) to nd the adjusted ayo. By summing the adjusted ayos, we can ascertain our total static hedge. Every time a barrier is reached, we need to reeat the rocedure to nd our new hedge ortfolio. Thus, the maximum number of rebalancings is the number of barriers. As an examle, consider a rolldown call with initial strike K = 1. Suose it has two rolldown barriers at 9 and 8 (ie. H 1 = 9, H = 8). Furhter suose that uon hitting the 9 barrier, the strike is rolled down to the barrier (ie. K 1 = 9). If the sot hits 8, the otion knocks out. Then, our relicating ortfolio is: DOC(1; 9)? DOC(9; 8) + DOC(9; 9): Each of these otions can be statically relicated. The sum of the corresonding adjusted ayos is (see Figure 7): f(s T ) = (S T?1)? + ST 9 + ST 8 + ST? 1 +? 9? 9 S T 8 S T 9 The ortfolio will need to be rebalanced uon hitting the barriers at 9 and 8. 9 S T?

19 Adjusted Payoff for Roll down Call 1 Adjusted Payoff Sot at Maturity (r = :5, d = :3, = :15) Figure 7: Adjusted ayo for Roll-down Call. 7 Ratchet Otions Ratchet calls dier from roll-down calls in only two ways. First, the strikes K i are equal to the barriers H i for i = 1; : : :; n?1. Second, rather than knocking out at the last barrier H n, the otion is ket alive and the strike is rolled down for the last time to K n = H n. As in [7], this feature can be dealt with by relacing the last sread of down-and-out calls [DOC(H n?1 ; H n )? DOC(H n?1 ; H n?1 )] in (8) with a down-and-in call DIC(H n ; H n ): X n? RC = DOC(K ; H 1 ) + [DOC(H i ; H i+1 )? DOC(H i ; H i )] + DIC(H n ; H n ): i=1 Substituting in the model-free results DOC(K; H) = C(K)? DIC(K; H) and DIC(H; H) = P (H) simlies the result to: X n? RC = DOC(K ; H 1 ) + [P (H i )? DIC(H i ; H i+1 )] + P (H n ): (9) i=1 The hedge roceeds as follows. If the forward never reaches H 1, then the DOC(K ; H 1 ) rovides the desired ayo (S T? K ) + at exiration, while the uts and down-and-in calls all exire worthless. If the barrier H 1 is hit, then the DOC(K ; H 1 ) vanishes. The summand when i = 1 has the same value as a DOC(H 1 ; H ) and so these otions should be liquidated with the roceeds 18

20 used to buy this knockout. Thus the osition after rebalancing at H 1 may be rewritten as: X n? RC = DOC(H 1 ; H ) + [P (H i )? DIC(H i ; H i+1 )] + P (H n ): i= This is again analogous to our initial osition. As was the case with rolldowns, the barrier otions in (9) can be relaced by static ositions in vanilla otions. Thus, the relicating strategy for a ratchet call involves trading in vanilla otions each time a lower barrier is reached. 8 Lookbacks A lookback call is an otion whose strike rice is the minimum rice achieved by the underlying asset over the otion's life. This otion is the limit of a ratchet call as all ossible barriers below the initial sot are included. A series of aers have develoed hedging strategies for lookbacks which involve trading in vanilla otions each time the underlying reaches a new extreme. Goldman, Sosin, and Gatto[11] were the rst to take this aroach. They worked within the framework of the Black Scholes model assuming r? d =. Bowie and Carr[4] and Carr, Ellis, and Guta[7] also use a lognormal model but assume r = d instead. Hobson[14] nds model-free lower and uer bounds on lookbacks. This section obtains exact relication strategies in a lognormal model with constant but otherwise arbitrary risk-neutral drift. Our aroach is to demonstrate that lookback calls or more generally lookback claims can be decomosed into a ortfolio of one-touch binaries 1. For each binary, we can create the aroriate adjusted ayos. Thus, we can create the adjusted ayo of a lookback by combining the adjusted ayos of the binaries. This combined adjusted ayo will give us ricing and hedging strategies for the lookback. For simlicity, consider a lookback claim that ays o min(s). Let m be the current minimum rice. At maturity, the claim will ay o: m? Z m bin(k)dk; (1) where bin(k) is the ayo of a one-touch down binary struck at K. Thus, our relicating ortfolio is a zero couon bond with face value m and the continuum of dk one-touch binaries struck at K < m. We can calculate the adjusted ayo of the lookback by adding the adjusted ayos of the bond and binaries. The adjusted ayo of the bond is its face 1 A one-touch binary ays one dollar at maturity so long as a barrier is touched at least once. 19

21 value, and the adjusted ayo of a one-touch binary with barrier K is (from (3)): ^f bin(k) (S T ) = if ST > K, 1 + (S T =K) if S T < K; where recall 1? (r?d). Consequently, the adjusted ayo of a lookback otion is: ^f lb (S T ) = m? Z m f bin(k) (S T )dk; (11) where ^f lb () and ^f bin(k) () are the adjusted ayos for the lookback claim and the binary resectively. Note that the adjusted ayo of a binary struck at K is zero for values above K. Therefore: Z m ^f bin(k) (S T )dk = ( R m S T h1 +? S TK i dk for S T < m for S T > m: (1) The integral term deends uon the value of. In articular: Z m ST m? ST + S T ln(m=s T ) for = dk = S T K m? S T + c S T ((m=s T ) c=? 1) for 6= 1; (13) where c = r? d. Assuming 6= 1, the combination of (11), (1), and (13) imlies that the adjusted ayo for the lookback claim is (see Figure 8): ST? ^f lb (S T ) = c S T ((m=s T ) c=? 1) for S T < m m for S T > m: (14) When = (ie. c = ), the above ayo simlies to: ^f lb (S T ) = ST? m for S T < m m for S T > m: (15) In this case, the adjusted ayo is linear. Note that in all cases, the adjusted ayo is a function of m. 8.1 Hedging As shown in (1), relicating the lookback claim involves a continuum of onetouch binaries. The hedging strategy for each binary involves rebalancing at the barrier. Thus, hedging the lookback claim involves rebalancing every time the minimum changes whcih occurs an innite number of times. While this strategy cannot be called static, rebalancing is certainly less frequent than in the usual continuous trading strategy. In fact, the set of oints where the

22 15 Adjusted Payoff for a Lookback 1 95 Adjusted Payoff Final Sot Price Figure 8: Adjusted ayos for Lookback (r = :5, d = :3, = :15, m = 1). minimum changes is almost certainly a set of measure zero 13. We also note that our rebalancing strategy only involves at-the-money otions which have high liquidity. 8. Lookback Variants Lookbacks comes in many variants, and our techniques are alicable to many of them. In this subsection, we give several variants and show how they may be hedged. Let m T denote the minimum realized sot at exiry, and let S T denote the sot rice at exiry. Lookback call. The nal ayo is S T? m T. The relication involves buying the underlying and shorting the lookback claim aying the minimum at maturity. If =, then the adjusted ayo is that of a straddle struck at m: m? ST for S ^f lb (S T ) = T < m (16) S t? m for S T > m: As discussed in Goldman, Sosin, and Gatto[11], the lookback value is always given by the value of a straddle struck at the minimum to date. As new minima are achieved, the strike of the straddle is rolled down via self-nancing trades. 13 In Harrison[1], it is shown that the set of times where the running minimum of a Brownian motion changes value is (almost surely) an uncountable set of measure zero. 1

23 Put on the Minimum. The nal ayo is max(k? m T ; ). Let m denote the current achieved minimum. The relicating ortfolio is: The adjusted ayo is: max(k? m; ) + Z min(m;k) bin(s)ds: ^f ut?on?min = flb (with m = K) if m > K, K? m + ^f lb if m < K; where ^f lb is the adjusted ayo of the lookback claim from (14). In the rst case, we substitute m = K in the formula for the adjusted ayo. Note that the adjusted ayo is xed for m > K. Our hedge is static until the minimum goes below K, after which we need to rebalance at each new minimum. Forward Starting Lookbacks. These lookbacks ay m 1, the minimum realized rice in the window from time T 1 to the maturity date T. In this situation, we can combine the methods from forward-starting otions and lookbacks. At time T 1, we can value the lookback otion with maturity T as LB(S 1 ). At initiation, we urchase a ortfolio of Euroean otions with ayo LB(S 1 ) at time T 1. At time T 1, we use the roceeds of the ayo to hedge the lookback as reviously described. If LB(S 1 ) = S 1 n() where n() is indeendent of S 1, then the initial hedge reduces to the urchase of n()e?dt1 shares. Once again, dividends are re-invested to time T 1 at which oint the shares are sold and the lookback is hedged as before. A similar analysis can be alied to the lookbacks that involve the maximum. We leave it to the reader to solve the analogous roblem. 9 Summary This aer has shown that the ayos of several comlex barrier otions can be relicated using a ortfolio of vanilla otions which need only be rebalanced occasionally. The ossible rebalancing times consist of times at which the barrier aears or disaears and at rst hitting times to one or more barriers. Although all of the comlex otions considered can also be valued by standard techniques, the hedging strategies considered are likely to be more robust uon relaxing the standard assumtions of continuously oen markets, constant volatility, and zero transactions costs.

24 References [1] Beaglehole, D., 199, \Down and Out, U and In Otions", University of Iowa working aer. [] Bhagavatula and Carr, 1995, \Valuing Double Barrier Otions with Time Deendent Parameters", Morgan Stanley working aer. [3] Black, F. and M. Scholes, 1973, \The Pricing of Otions and Cororate Liabilities", The Journal of Political Economy, 81, 637{659. [4] Bowie, J., and P. Carr, 1994, \Static Simlicity", Risk, 7, 45{49. [5] Breeden, D. and R. Litzenberger, 1978, \Prices of State Contingent Claims Imlicit in Otion Prices", Journal of Business, 51, 61{651. [6] Carr P. and A. Chou, 1996, \Breaking Barriers: Static Hedging of Barrier Securities", Morgan Stanley working aer. [7] Carr P., K. Ellis, and V. Guta, 1996, \Static Hedging of Exotic Otions", forthcoming in Journal of Finance. [8] Derman, E., D. Ergener, and I. Kani, 1994, \Forever Hedged," Risk, 7, 139{45. [9] Derman, E., D. Ergener, and I. Kani, 1995, \Static Otions Relication," Journal of Derivatives, Summer,, 4, 78{95. [1] Gastineau, G., 1994, \Roll-U Puts, Roll-Down Calls, and Contingent Premium Otions", The Journal of Derivatives, 4{43. [11] Goldman B., H. Sosin, and M. Gatto, 1979, \Path Deendent Otions: Buy at the Low, Sell At The High", Journal of Finance, 34, 5, 1111{117. [1] Harrison, J., 1985, Brownian Motion and Stochastic Flow Systems, Wiley, New York NY. [13] Heynen R., and H. Kat, 1994, \Partial Barrier Otions", MeesPierson Working Paer. [14] Hobson, D., 1996, \Robust Hedging of the Lookback Otion", University of Bath working aer. [15] Hull, J., 1993, Otions, Futures, and Other Derivative Securities, Prentice-Hall, Englewood Clis NJ. [16] Kunitomo, N. and M. Ikeda, 199, \Pricing Otions with Curved Boundaries", Mathematical Finance, October 199, 75{98. 3

25 [17] Merton, R., 1973, \Theory of Rational Otion Pricing", Bell Journal of Economics and Management Science, 4, 141{183. [18] Nelken I., 1995, Handbook of Exotic Otions, Probus, Chicago IL. [19] Reiner E., and M. Rubinstein, 1991, \Breaking Down the Barriers", Risk. [] Zhang, P., 1996, Exotic Otions: A Guide to the Second Generation Otions, World Scientic, New York, New York. 4

26 Aendix 1: Risk-neutral Valuation In the Black-Scholes model, any Euroean claim with maturity T and ayo f(s T ) can be riced by taking the discounted conditional exected value of the ayo under the risk-neutral measure[15]: V (S) = e?rt E [f(s T )js] Z 1 = e?rt f(s T )(S T ; S)dS T ; where S is the current rice and 1 (S T ; S) S T T ex? (ln(s T =S )? (r? d? 1 )T ) T is the lognormal density function. We denote the interest rate, the dividend rate, and the volatility rate by r, d, and, resectively. Aendix : Proof of Lemma 1 Lemma 1 In a Black-Scholes economy, suose X is a ortfolio of Euroean otions exiring at time T with ayo: X(S T ) = n f(st ) if S T (A; B), otherwise. For H >, let Y be a ortfolio of Euroean otions with maturity T and ayo: Y (S T ) =? STH f(h =S T ) if S T (H =B; H A), otherwise where the ower 1? (r?d) and r; d; are the interest rate, the dividend rate, and the volatility rate resectively. Then, X and Y have the same value whenever the sot equals H. Proof. For any t < T, let = T? t. By risk-neutral ricing, the value of X when the sot is H at time t is: Z B V X (H; t) = e?r f(s T )(H; S T ; )ds T A Z B = e?r 1 f(s T ) S T ex? (ln(s T =H)? (r? d? 1 )) ds T : Let ^S = H S T. Then, ds T =? H ^S d ^S and: Z H V X (H; t) =?e?r =B A H =A f(h = ^S) " 1 ^S ex #? (ln(h= ^S)? (r? d? 1 )) d ^S 5

27 Z H =A = e?r H =B! ^S f(h = ^S) H " 1 ^S ex #? (ln( ^S=H)? (r? d? 1 )) d ^S; where 1? (r?d). By insection, V X (H; t) exactly matches the risk-neutral valuation of Y, namely V Y (H; t). Aendix 3: Pricing Formula for Partial Barrier Call The value of a artial barrier call with sot rice S, strike K, maturity T, artial barrier H, and time of barrier disaearance T 1 can be comuted by taking the discounted exected value under the risk neutral measure of (7) as: e?dt SM(a 1 ; b ; )? e?rt KM(a ; b ; ) S? H e?dt (H =S)M(c 1 ; d 1 ; )? e?rt KM(c ; d ; ) where = 1? (r?d), M(a; b; ) denotes the cumulative bivariate normal with correlation = T 1 =T, and a 1 = ln(s=h) + (r? d + =)T 1 T 1 ; a = a 1? T 1 ; b 1 = ln(s=k) + (r? d + =)T T ; b = b 1? T ; c 1 = ln(h=s) + (r? d + =)T 1 T 1 ; c = c 1? T 1 ; d 1 = ln(h =SK)? (r? d + =)(T? T 1 ) + (r? d + =)T 1 T d = ln(h =SK)? (r? d? =)(T? T 1 ) + (r? d? =)T 1 T : 6