FUTURES AND OPTIONS. Professor Craig Pirrong Spring, 2007

Transcription

1 FUTURES AND OPTIONS Professor Craig Pirrong Spring, 2007

2 Basics of Forwards and Fuures A forward conrac is an agreemen beween a buyer and a seller o ransfer ownership of some asse or commodiy ( he underlying ) a an agreed upon price a an agreed upon dae in he fuure. A forward conrac is a promise o engage in a ransacion a some laer dae. The forward conrac specifies he characerisics of he underlying. For example, for a commodiy, i specifies he ype of commodiy (e.g., silver), he qualiy of he commodiy (e.g., 99.9 percen pure silver), he locaion of delivery, he ime of delivery, and he quaniy o be delivered. The primary use of a forward conrac is o lock in he price a which one buys or sells a paricular good in he fuure. This implies ha he conrac can be uilized o manage price risk. Mos forward conracs are raded in he over he couner (OTC) marke. Some forward conracs are raded on organized exchanges such as he Chicago Board of Trade or he New York Mercanile Exchange. These exchange raded conracs are called fuures conracs. Forward conracs raded OTC can be cusomized o sui he needs of he ransacing paries. Exchange raded conracs are sandardized. This enhances liquidiy. Performance on fuures conracs are guaraneed by hird paries (brokers and he clearinghouse.) Performance on OTC forwards is no guaraneed. The qualiy of he conracual promise is only as reliable as he firm making i.

3 Forwards and fuures are raded on a wide variey of commodiies and financial asses. Commodiy fuures and forwards are raded on agriculural producs (corn, soybeans, whea, cale, hogs, pork bellies); precious meals (silver, gold, plainum, palladium); indusrial meals (copper, lead, zinc, aluminum, in, nickel); fores producs (lumber and pulp); and energy producs (crude oil, gasoline, heaing oil, naural gas, elecriciy). Financial fuures and forwards are raded on sock indices (S&P 500, Dow Jones Indusrials, foreign indices); governmen bonds (US Treasury bonds, US Treasury noes, foreign governmen bonds); and ineres raes (Eurodollars, EuroEuros). More recenly, forward/fuures rading has begun on weaher and credi risk. These are (no pun inended) he hoes areas in derivaives developmen.

4 The Uses of Derivaive Markes Derivaives markes serve o shif risk. Hedgers use derivaives o reduce risk exposure. For insance, a refiner can lock in coss and revenues by buying crude oil fuures and selling oil and gasoline fuures. Speculaors use derivaives o increase risk exposure in he anicipaion of making a profi. Thus, derivaives markes faciliae he shifing of risk from hose who bear i a a high cos (he risk averse) o hose who bear i a a low cos (he risk oleran). Speculaors perform a valuable service by absorbing risk from hedgers. In reurn, hey receive a reward a risk premium. The risk premium is he expeced profi on a derivaives ransacion. Speculaors may win or lose in any given rade, bu on average speculaors expec o profi. The risk premium is also he cos of hedging.

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6 Commodiy Derivaives: Precious Meals Cash and Carry Arbirage Call S he spo price of a commodiy (silver, for insance) a ime. Moreover, F, T is he fuures price a for delivery a ime T, r is he riskless ineres rae beween ime and T, and s is he cos of sorage beween and T. Assume ha here is no benefi of holding invenories of he commodiy (i.e., he renal/lease rae is zero). Consider he following se of ransacions. Buy he spo commodiy a, sore i unil T. Finance his purchase and sorage wih borrowing. Sell he fuures conrac for delivery a T. Consider he cash flows from hese ransacions a daes and T. TRANSACTION Dae Dae T Buy spo S S T r( T Borrow S + s e ) [ S + s] Sell Fuures 0 F S Pay sorage -s 0, T r( T Ne Cash Flows 0 F e ) [ S + s], T T

7 All of he prices ha deermine ne cash flows a T are known and fixed as of. Therefore, his ransacion is riskless. Since his ransacion involves zero invesmen a, in order o avoid he exisence of an arbirage opporuniy i mus be he case ha he cash flows a T are idenically 0. Therefore, in order o preven arbirage, he following relaion beween spo and fuures prices, ineres raes, and sorage charges mus hold a : r( T F = e ) [ S + s], T

8 Reverse Cash and Carry Arbirage Cash and carry arbirage involves buying he spo, borrowing money, and selling fuures. Reverse cash and carry arbirage involves selling he spo shor, invesing he proceeds, and buying fuures. In order o shor sell he spo asse, an invesor borrows he asse and sells i on he spo marke. The asse borrower mus purchase he asse a T in order o reurn i o he lender. The cash flows from his ransacion are: TRANSACTION Dae Dae T Sell spo S s + S T r Lend ( S + s) e ( T ) [ S + s ] Buy Fuures 0 S F T, T r( T ) Ne Cash Flows 0 e [ S + s] F Noe ha his implies he same arbirage relaion as cash and carry arbirage. Noe: This analysis assumes ha he arbirageur holds invenories of he commodiy. If he doesn, he does no save on sorage charges by selling he commodiy. This drives a wedge (equal o he cos of sorage) beween he cash-and-carry and reverse-cashand-carry arbirage resricions., T

9 Implied Repo Raes Arbirage resricions define relaions beween spo and fuures prices and ineres raes ha mus hold in order o preven raders from earning riskless profis wih no invesmen. Since here are many fuures markes, in order o idenify arbirage opporuniies in several markes simulaneously, i is convenien o conver spofuures price relaions ino a common variable. Since for a given rader he same ineres rae should apply o any arbirage ransacion, regardless of wheher he ransacion is in gold, silver, Treasury bonds or corn, he obvious common variable is an ineres rae. Therefore, raders use spo and fuures price o calculae an implied ineres rae. This is commonly called he implied repurchase ("repo") rae, because he repo rae represens he rae a which mos large raders can borrow or lend. To calculae an implied repo rae, ake naural logarihms of he basic arbirage expression. This implies: ln F = r( T ) + ln[ S + s], T Simplifying his expression, and recognizing ha he difference beween wo logs equals he log of heir raio, produces: F, T ln[ ] / ( T ) = r S + s i

10 This is he ineres rae implied by spo and fuures prices. If his implied repo rae is lower han he acual repo rae a which a rader can borrow/lend, he rader can borrow cheaply hrough he fuures marke, and lend hrough he repo marke. Conversely, if he implied repo rae exceeds he rae a which he rader can borrow/lend hrough he money marke, he should lend hrough he fuures marke and borrow hrough he repo marke. Moreover, by comparing implied repo raes from fuures on differen commodiies (e.g., gold vs. S&P 500) a rader can idenify cheap borrowing and rich lending opporuniies.

11 The Pricing of Energy and Indusrial Meal Fuures: Spread Relaions Precious meal fuures are prey sraighforward. Usually hey sell a near full carry (i.e., in conango). Lease raes are ypically small, and here is an acive marke for leasing precious meal invenories. Pricing energy and indusrial meal fuures is a lile more complicaed. In paricular, he relaions beween spo and fuures prices, and beween nearby and deferred fuures prices are much more complex han is he case wih precious meals. For example, someimes oil, heaing oil, or copper rade a nearly full carry. Oher imes, he fuures prices are far below spo prices- -ha is, he marke is in backwardaion. Moreover, he marke can move from conango o backwardaion o conango very quickly. This raises he quesion: wha deermines he spreads beween fuures and spo prices for energy producs and indusrial meals? The so-called heory of sorage provides he answer o his quesion. In essence, his heory saes ha fuures prices serve o ensure ha consumpion of hese commodiies is disribued efficienly over ime.

12 Consider wo siuaions: Case 1. Supplies of he commodiy (e.g., copper) are abundan. Delivery warehouses are full. Producers are operaing wih excess capaciy. Under hese condiions, i makes sense o sore some of he commodiy. Abundan supplies are on hand, and o consume hem all would glu he marke oday, and perhaps leave us exposed o a shorage in he fuure. Traders will sore he commodiy (i.e., hold invenories) if and only if his is profiable. By soring (raher han selling) he commodiy, he rader gives up he spo price (which he could capure by selling he commodiy), and incurs ineres and warehousing expenses. Sorage is profiable only if fuures/forward prices are above he spo price by he coss of ineres and sorage. Tha is, sorage is profiable only if he marke is a a carry/conango. If sores are huge, he marke mus be a full carry. Case 2. Curren supplies are very scarce relaive o expeced fuure producion and supplies. Tha is, here is a emporary shorage. Producers are operaing a or near capaciy. In hese circumsances, i is foolish o sore he commodiy-- i is scarce oday relaive o expeced fuure supplies. Therefore, we should consume available supplies and hold close o zero invenories. (This exhausion of invenories is someimes called a sock-ou. ) Since he commodiy is scarce oday relaive o wha we expec in he fuure, he curren spo price may be above he forward price. Tha is, he spo price mus rise as high as necessary o raion he limied supplies. Moreover, here is no need o reward sorage. Therefore, he marke need no exhibi a carry. Thus, backwardaion is possible.

13 There are some implicaions of his analysis: Implicaion 1. The spread beween spo and fuures prices, adjused for carrying coss, should depend upon he socks held in invenory. When socks are low, he marke should be in backwardaion; when socks are high, he marke should be a nearly full carry. The indusrial meals markes illusrae his clearly. The heory also has implicaions for he volailiy of prices, and he correlaion beween spo and forward/fuure prices. This is of grea imporance in risk managemen and opion pricing applicaions. Implicaion 2. When socks are shor prices should be more volaile. For hose who are familiar wih supply and demand analysis, a shorage implies ha he supply curve is inelasic, i.e., seep. The seeper he supply curve, he more volaile spo prices. Since shorages are associaed wih backwardaion, we expec very volaile prices in an invered marke (i.e., a marke wih backwardaion), and less volaile prices when he marke shows a carry.

14 Implicaion 3. Again calling upon supply and demand analysis, we know ha shor run supply curves are less elasic han long run supply curves. This implies ha spo prices should be more volaile han forward/fuures prices. Moreover, he difference in shor run and long run supply elasiciies should be greaes, he shorer are supplies oday. Thus, he difference beween spo volailiy and forward volailiy should rise as he marke moves owards backwardaion. Furhermore, his difference should be greaer, he longer he ime o delivery on he deferred conrac. Implicaion 4. When a sockou occurs, he cash-and-carry arbirage link beween spo and forward prices is broken. Under hese condiions, spo and forward prices can move independenly. When socks are abundan, arbirage ensures ha spo and fuures prices move ogeher--he fuures-spo spread equals he cos of carry. Thus, spo and forward/fuures prices should be highly correlaed when he marke is a a carry, bu may exhibi very low correlaion when he marke is in backwardaion. Moreover, since a sockou is more likely, he longer he ime o expiraion of he deferred forward, he correlaion beween spo and forward/fuure prices should be lower, he greaer he mauriy of he forward/fuure.

15 Squeezes, Hugs, and Corners Anoher facor which affecs spo-fuures spreads is a manipulaion. A manipulaion--someimes referred o as a squeeze or corner, or a hug for a mild manipulaion--occurs when a single rader accumulaes a long fuures/forward posiion ha is larger han he physical supplies ha can be delivered economically agains hese conracs. Raher han incur large coss o acquire deliverable supplies, as conrac expiraion approaches shors are willing o buy back heir conracs from he long a a premium. This causes a unique paern in prices. The price of he expiring fuure/forward ha is being cornered rises--someimes precipiously--relaive o he deferred fuures/forward price. Since he expiring fuure/forward and he spo price mus converge during he delivery period, his means ha he spo price rises relaive o he deferred forward/fuure oo. As soon as shors close ou heir posiions, he spo price collapses. The effec of final liquidaion of a corner on spo prices is someimes called he burying he corpse effec. A sharp increase in shipmens o delivery warehouses also occurs. Squeezes/corners are no unknown in commodiy derivaive markes. Exchanges and governmens aemp o preven or deer hem, bu hey sill occur from ime o ime. The experience of he zinc marke in 1989 provides a good illusraion of he effecs of a squeeze on spreads. Moral of he sory: When rading commodiies, if you are shor you mus always be aware of he possibiliy of a manipulaion. Monior fuures marke and cash marke aciviy coninuously o make sure ha you are no unexpecedly caugh in a squeeze.

16 Non-Sorable Commodiies There are also fuures and forward conracs raded on nonsorable commodiies. These include: elecriciy, weaher, bandwidh, live animals (e.g., hogs and cale). Non-sorabiliy has a big impac on price dynamics and forward pricing. Sorage miigaes price volailiy invenories are accumulaed when demand is low and supply is high, hereby reducing he magniude of price declines under hese condiions, and are drawn from when demand is high and supply is low, hereby miigaing he magniude of price increases. Wihou sorage, invenories canno smooh he effecs of supply and demand shocks. This implies ha prices of non-sorables noably elecriciy can be exremely volaile.

17 Forward Prices for Non-Sorables Due o non-sorabiliy, cash-and-carry arbirage is impossible you can hold an invenory of elecriciy from he nighime o he afernoon. This conribues o considerable inra-day variaion in prices for non-sorables wih sysemaic inra-day variaion in demand or supply. I also makes cash-and-carry arbirage pricing mehods preference free pricing echniques impossible for nonsorables. Therefore, esimaing forward prices (and forward curves) for non-sorables mus ake a differen ack. Forward price=expeced spo price + risk premium

18 Elecriciy Forward Pricing For elecriciy, for markes in which good spo price and demand (load) daa are available, such as PJM, California, or Ausralia, can esimae expeced spo prices using radiional saisical echniques. Saisical disribuion of demand can be esimaed accuraely. Use observed spo-price/load daa o esimae a funcion ha relaes prices o load. Combine demand disribuion and spo price/load relaion o esimae an expeced spo price. Esimaing risk premium is rickier need o use marke forward price daa. Essenial o ake risk premium ino accoun when pricing power forwards because his risk premium is huge, especially on-peak.

19 Power Derivaives Markes Power derivaives markes have grown, albei somewha slower han had been anicipaed in he mid-1990s. Virually all power forward and opions rading done overhe-couner. Exchange markes have languished. Heerogeneiy of elecriciy (especially locaional differences) have impeded developmen of liquid power forward markes heerogeneiy fragmens liquidiy. Credi issues (noe effecs of 1998 Midwes price spike, California crisis) have also impeded marke developmen. Finally, lack of inegraion of financial and physical markes has impeded developmen. Unil hese issues are resolved, power derivaives rading may prove reacherous.

20 Weaher Derivaives Weaher derivaives are a new fronier in derivaives rading. Weaher derivaives rading almos exclusively OTC, alhough here are exchange-lised conracs on he CME. Typical weaher derivaive produc is based on heaing- or cooling degree days. One heaing degree day occurs when he low emperaure is one degree below 65 o for one day. A cooling degree day occurs when he high emperaure is one degree above 65 o for one day. Mos weaher derivaives are heaing or cooling degree day opions. For insance, you could have a conrac ha pays $1 per cooling degree day imes he difference beween oal cooling degree days in Chicago in July, 2002 and 250, if ha difference is posiive and zero if i is no. Weaher derivaives can be used o manage quaniy risk. For insance, he quaniy of power or naural gas sold by an energy company depends on weaher condiions. So does he price. Can use weaher derivaives o manage his risk. Reailers can use hem o manage risks due o weaher. Reail sales for cerain producs are very sensiive o weaher.

21 Alhough degree-day opions are he mos common, weaher derivaives can be based on rainfall, snowfall, or any oher measurable weaher variable. Like power derivaives, due o lack of sorabiliy arbirage-based pricing of weaher derivaives no possible. Need o uilize hisorical daa on weaher (correced for global warming???) o esimae expeced payoffs, hen adjus by a risk premium.

22 Deermining he Risk Premium For sorables, our arbirage analysis shows ha he risk premium is irrelevan o deermining he relaion beween spo and forward prices. Tha is, for hese goods we can use risk preference free pricing o deermine hese relaive prices. However, even for sorables he risk premium affecs he level of fuures and spo prices, and heir average movemens hrough ime. The earlies heory of he risk premium is due o Keynes. Keynes posied ha hedgers are ypically shor fuures. Tha is, hey are ypically holders of invenories of a commodiy (e.g., corn) and hey sell fuures as a hedge. Since hedgers are ne shor, speculaors mus be ne buyers in equilibrium (since oal buys=oal sells). Speculaors will no absorb risk unless hey are rewarded by profiing on average. Buying fuures is profiable on average if fuures prices rise on average. Tha is, speculaors will ener he marke only if he fuures price is below he spo price expeced a conrac expiraion. In his heory, he fuures price ends o drif up over ime he speed of he drif measures he risk premium.

23 In Keynes heory, fuures prices are downward biased. In Keynes heory, his downward bias makes shor hedging cosly (since shor hedgers lose on average) so hey will end o hedge less han 100 percen of heir risk. If long hedgers ounumber shor hedgers (a a fuures price equal o he expeced spo price) fuures prices mus be upward biased o arac speculaive ineres. Thus, ne hedging ineres deermines wheher fuures prices are upward or downward biased. For some markes (exchange raded fuures) here is daa on ne hedging ineres in he form of he CFTC s Commimen of Traders Repors available on he www.

24 The Magniude of he Risk Premium The foregoing implies ha he size of he risk premium depends on (a) ne hedging demand a a price equal o he expeced spo price, and (b) he risk aversion of speculaors. The greaer he hedging imbalance, he greaer he risk premium (in absolue value). The more risk averse he speculaors, he greaer he risk premium (in absolue value). The risk aversion of speculaors depends on (a) how well he fuures marke is inegraed wih he broader financial markes, and (b) he correlaion beween he fuures price and movemens in he marke porfolio. Usually i is he case ha risk premia will be smaller if he fuures marke is well inegraed wih broader financial markes because inegraion makes i possible for diversified speculaors o paricipae in he marke; diversificaion reduces speculaor risk exposure. CAPM-ype models imply ha he higher he bea beween he fuures price and he overall marke, he greaer he upward drif in he fuures price.

25 Esimaing he Risk Premium The risk premium is he difference beween he fuures price and he expeced spo price. Also, he risk premium affecs he expeced change in he fuures price. Thus, esimaion of he risk premium requires eiher esimaion of he expeced spo price, or esimaion of he drif in fuures prices. Since he risk premium affecs boh he coss of hedging and he benefis of speculaion, boh require esimaion of expeced spo prices. Thus, spo price forecasing is an imporan par of hedging and speculaion. For mos goods and commodiies i is hard o esimae expeced spo prices wih accuracy. Elecriciy and weaher may be excepions.

26 Credi Derivaives Credi derivaives are anoher new ype of financial conrac. This is a rapidly growing marke. Credi derivaives allow firms o hedge he risks of defaul on loans. For example, a bank ha loans money o a cusomer loses if ha cusomer defauls. The bank may be able o sell-off some porion of he credi risk wihou selling he loan, by enering ino a credi derivaives ransacion. Almos all credi derivaives are raded over he couner. There are a variey of credi derivaive producs in wide use.

27 Credi Even Derivaives A credi even derivaive beween pary A and pary B involves (a) a periodic paymen from A o B, (b) he definiion of a credi even, and (c) a paymen from B o A if a credi even occurs. A credi even may be a bankrupcy or raings downgrade, for insance. Alhough defining a credi even seems sraighforward, i is no. The evens in Russia in 1998 provide an illusraion of he difficulies of defining a credi even. A may be a bank ha wans o reduce is exposure o defaul by a paricular borrower. B may be an insurance company or oher financial firm ha is willing o bear defaul risk. The size of he paymen made in he even of a credi even is relaed o he impac of he even on he value of he loan. The size of periodic paymen depends on he price he marke charges o bear credi risk. Yield spreads beween bonds of differing credi risk measure he marke price of credi risk. Thus, he periodic paymen should be relaed o yield spreads.

28 Toal Reurn Swaps Toal reurn swaps are anoher common ype of credi derivaive. In a oal reurn swap, A and B exchange paymens, where he magniude of he paymens swapped depend on he oal reurns on insrumens of differen credi-worhiness. For insance, he swap may involve A paying B he oal reurn (ineres plus capial gain/loss) on a BBB bond, and B paying A he LIBOR rae. This would make sense for A if he owned BBB bonds and didn wan o bear he credi risk. In essence, his deal passes he credi risk o B wihou selling he acual bond.

29 Why Use Credi Derivaives? I seems somewha weird ha firms would aler credi risk exposures hrough credi derivaives why don hey jus rade he underlying loans? Tha is, why would a bank ha wans o reduce credi exposure ener a oal reurn swap insead of jus selling off credi-risky loans? Taxes, accouning, and regulaory arbirage. Sales of loans may have adverse impac on axes or repored earnings or he balance shee. For insance, a gain or loss mus be recognized on sale for ax or accouning purposes, bu may no be recognized if credi risk is ransferred hrough a credi derivaives ransacion. Also, some inermediaries may operae under regulaions ha limi heir abiliy o purchase below-invesmen grade bonds, bu ha do no limi heir abiliy o use credi derivaives. Liquidiy. Credi derivaives may be more liquid han he underlying securiies. This is ypically he case since mos credi derivaives have shorer mauriies (e.g., one year) han he underlying securiies (e.g., five years). Informaion asymmeries are plausibly smaller for shorer erm insrumens, making hem more liquid

30 Eurodollar Fuures and heir European Cousins Eurodollar fuures conracs raded on he CME are he mos heavily raded fuures conracs in he world. The conrac is cash seled--shors don deliver an ED deposi a expiraion. Insead, on he second London business day before he hird Wednesday of he conrac monh (March, June, Sepember, or December) he CME calculaes he average LIBOR rae. The mark-o-marke price of he conrac on his day is se equal o: V = 10000( R) where R is he annualized LIBOR rae a selemen. The.25 muliplying he LIBOR rae convers he annual rae ino a quarerly rae. Noe ha he conrac price flucuaes inversely wih ineres raes (jus like a bond price). A one bp increase in he ineres rae leads o a $25 decrease in conrac value. Prior o expiraion, he conrac price is: 10000[ ( 100 F )] where F is he fuures price on he day in quesion. This price is used o deermine margin paymens. Noe ha every.01 change in he fuures price resuls in a $25.00 change in conrac value. This is he price value of a basis poin (PVBP) for he EDF. There are currenly relaed counracs in Europe on shor erm German, French, Briish, Spanish, and Duch ineres raes.

31 Eurodollar Fuures Arbirage A conrac expiraion a T, he ED fuures price is 100( 1 4r T T, T + 90 ) annualized, bu quarerly. Prior o T, he ED fuures price is 100( 1 4r *) where r* is he ineres rae currenly embedded in he fuures price. Lend k dollars, wih, where his ineres rae is no Dae Dae T Dae T+90 mauriy T+90 -k 0 k( 1+ r, T + 90 ) Borrow k, wih mauriy T. k k( + ) 1 0 r, T T Shor ED fuures 0 r r T Reinves ED profis 0 -(r r T, T + 90 * 0 T T 90 ) ( 1+ rt, T+ 90)( rt, T + 90 r*) T, T + * Borrow 1 a T wih 0 1 ( 1+ r T T, T + 90) mauriy T+90

32 We choose k such ha profis a T+90 are approximaely equal o zero. We can' make profis riskless (i.e., we can' make hem idenically equal o 0 because we don' know he rae a which we will reinves our fuures profis/losses. Formally, afer doing a lile algebra, profis a T+90 are: T T T T k( 1 + r ) + r r * + r r r r * 1 r T, T + 90 T, T + 90 T, T + 90 T, T + 90 T, T + 90 T, T+ 90 The wo erms in he middle of his expression represen he difference beween he produc of wo ineres raes. This is small relaive o he level of ineres raes, so we will ignore i. Noe, however, ha his difference will seldom equal zero, and is unpredicable as of ime. Therefore, we canno consruc a perfec arbirage because of he reinvesmen risk on our ED fuures. Thus, we ge: k( 1 + r ) = 1+ r *, T+ 90 Solving for k gives: k = 1+ r * 1 + r, T + 90 Now urn o period T. Our ne cash flows hen are:

33 1 + r * r (, T ) 1 + r, T + 90 Noe his cash flow is riskless as of. In order for hese cash flows o equal 0, he following expression mus hold: 1+ r = ( 1+ r )( 1+ r*), T + 90, T Noe ha by he definiion of a forward rae of ineres, his expression implies ha r* is he forward rae of ineres a for he period T o T+90. Tha is, r * = r T, T +90

34 This is he basic no-arbirage expression for ED fuures because by subsiuing for r* we ge: F = 100( 1 4rT T+ 90 ), T, In essence, he arbirage sraegy involves long erm lending financed by back-o-back shor erm borrowing. The ED fuures posiion locks in he rae a which I can borrow over he period T o T+90. If I can borrow back-o-back over he period o T+90 a a lower rae han I can lend over his period, here is an arbirage opporuniy. The no-arb expression essenially ses he borrowing and lending coss equal.

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37 An example is useful here. Assume ha dae is 10/15/01. Dae T is 12/15/01. Dae T+90 is 3/15/02. Assume ha he annualized quarerly compounded ineres rae over he period 10/15-12/15 is 4 percen, and he annualized quarerly compounded ineres rae over he period 10/15/01-3/15/02 is 6 percen. Then r.04 = (1+ ) = , T 90 4 The erm muliplying he ineres rae in his expression reflecs he fac ha here are 61/90'hs of a quarer from o T. Similarly Thus 151 (1 +.06) r = =, T = r = T, T This implies F = 100[1 (4)(.01837)] =, T If he acual ED fuures price is higher han his value, one should shor fuures, lend from o T+90, and borrow from o T and T o T+90. If he acual ED fuures price is lower han his value, execue he opposie ransacions. Inuiively, if he EDF price is oo high, one can lock in a low borrowing rae for he period T o T+90. Conversely, if he EDF price is oo low, one can lock in a high lending rae from T o T+90.

38 Fuures Raes, Forward Raes, and Convexiy The preceding analysis ignored one imporan insiuional feaure of fuures conracs: Marking-o-marke. The arbirage able assumed ha all cash flows on he fuures conrac occurred a expiraion. This is rue of forward conracs, bu no fuures conracs. Gains and losses on fuures conracs are realized on a daily basis. If he fuures price goes down (up) on a given day, he long (shor) pays ou his amoun and he shor (long) receives i. This process of daily selemen is a way of reducing defaul risk on fuures conracs. The aggregae amoun of gain/loss incurred on a marked-o-marke fuures posiion equals he amoun assumed in he arbirage able, bu he iming is differen. This iming difference is imporan if here is a correlaion beween he fuures price and ineres raes. This iming difference can cause a difference beween he noarbirage fuures price, and he no-arbirage forward price (which is wha we acually derived in he ED fuures analysis).

39 Case 1. Ineres rae changes are negaively correlaed wih fuures price changes. This is he case relevan for ineres rae fuures such as he ED fuure. Le s compare he cash flows of a shor posiion in a fuures conrac wih hose of a shor posiion in a forward conrac on he same underlying wih he same expiraion dae. Le he fuures price rise. Because ineres rae changes are negaively correlaed wih fuures price changes, his fuures price increase is ypically associaed wih a fall in ineres raes. Thus, alhough he shor mus pay ou losses (because he fuures price rose) he can finance hese a a low ineres rae. Now consider a fall in fuures prices. This is ypically associaed wih a rise in ineres raes under he proposed scenario. Thus, he shor receives gains and can inves hese a a higher ineres rae. The siuaion for he long is reversed--he can inves his mark-omarke gains a a low ineres rae and mus borrow a a high rae o finance his mark-o-marke losses. The holder of a forward conrac receives no mark-o-marke cash flows, so he correlaion beween price changes and ineres rae changes doesn maer o him regardless of wheher he is long or shor. This implies ha in his case here is an advanage (disadvanage) o being shor (long) a fuures conrac when fuures price changes and ineres rae changes are negaively correlaed. Compeing o exploi his advanage, shors bid down he fuures price so ha i is below he forward price. Tha is, shors are willing o sell fuures a a lower price han hey are willing o sell forwards. Thus, given he negaive correlaion, fuures prices should be below forward prices.

40 Case 2. Posiive correlaion beween fuures price changes and ineres rae changes. The analysis here is exacly he opposie of ha in case 1. There is a benefi o long posiions arising from marking-o-marke. Thus, longs are willing o buy fuures a a higher price han hey are willing o buy forwards. Thus, given a posiive correlaion beween fuures price changes and ineres rae changes, fuures prices should be above forward prices. This difference beween fuures and forward prices resuling from marking-o-marke is essenially aribuable o convexiy. The size of he difference beween forward and fuures prices depends upon a) he size of he correlaion beween fuures price changes and ineres rae changes, and b) he volailiies of fuures price changes and ineres rae changes. Holding volailiies consan, he bigger he correlaion (in absolue value) he greaer he difference because he pary benefiing from he correlaion can do so more ofen. Holding correlaion consan, he bigger he volailiies, he greaer he difference because he pary benefiing from he correlaion can inves bigger gains a beer ineres raes. (Remember, convexiy is more valuable, he greaer is volailiy.) The effec may be small for some commodiies. For example, he correlaion beween oil price changes and ineres rae changes is abou.01 or less--here here is no big difference beween fuures and forward prices. For ED fuures, however, he effec can be large as he correlaion is usually above.95.

41 How o Value he Effec of Convexiy Bias This correlaion/convexiy effec is imporan for ED fuures pricing and hedging. I is also imporan for swap pricing. This is rue because he ED fuures (and oher shor erm ineres rae fuures) are a very ransparen, real ime source of informaion abou he erm srucure, bu one canno use hese prices direcly o deermine a forward yield curve because of he convexiy bias: ED fuures prices ell you he fuures ineres raes, bu for pricing FRAs or swaps you need o know he forward ineres raes. Fuures ineres raes implied by ED fuures are higher han forward ineres raes because he fuures price is biased downward by convexiy. To derive a forward yield curve from EDF prices you mus adjus for his convexiy. How can you do his? Follow hese seps: 1. Assume ha here are T days o fuures expiraion. Then i is possible o show heoreically ha he fuures price bias is equal o: T B = σ ( ) σ ( ) ρ cov( ln F( ), ln z(, T)) = 0 F z( T ) F, z( T ) where σ F ( ) is he volailiy of he fuures price a ime, σ z ( T ) ( ) is he volailiy on day of a zero coupon bond expiring a ime T, and ρ F, z( T ) is he correlaion beween percenage changes in he fuures price and percenage changes in he price of he zero. (Noe ha volailiy means he daily sandard deviaion of percenage price changes.) The cov erm is he covariance beween daily percenage fuures price changes and daily percenage zero coupon price changes.

42 2. Noe ha all of he parameers affecing bias should changes as ime o expiraion nears. For example, ED fuures prices should become more volaile as ime o expiraion nears (i.e., T- becomes small) because forward raes end o be less volaile han spo raes. Also, zero coupon bond price volailiy changes as T- changes because a) spo raes are more volaile han forward raes, and b) he duraion of he zero falls as expiraion nears. The correlaion should also change as ime passes. Given his sysemaic change in parameers, o esimae he bias break down he inerval beween oday and T ino subperiods. For example, for a ED fuures conrac wih six monhs o expiraion, break he six monh period ino wo hree monh periods. (You can break up periods more finely if you wish.) Noing ha he erm in he summaion is equivalen o he covariance beween % fuures price changes and % zero price changes, use hisorical daa o esimae he daily covariance beween percenage price changes of EDFs wih beween 6 and 3 monhs o expiraion and he percenage price changes of zero coupon bonds wih beween 6 and 3 monhs o expiraion. Call his cov1. Nex esimae he daily covariance beween percenage price changes of EDFs wih beween 3 and 0 monhs o expiraion and he percenage price changes of zero coupon bonds wih beween 3 and 0 monhs o mauriy. Call his cov2. Call 1 he number of days beween oday and hree monhs from oday. Call 2 he number of days beween hree monhs from oday and conrac expiraion. Your esimae of oal bias is: B = cov1 + cov Noe ha as ime passes 1 ges smaller, and your esimae of bias falls as a resul.

43 Ineres Rae Swaps Swaps are over-he-couner agreemens beween wo companies o exchange cash flows in he fuure according o a pre-arranged formula. Mos ineres rae swaps are so-called "plain vanilla" swaps. One pary of he swap ("A") makes a fixed paymen periodically o he oher pary company B. Company B makes paymens periodically. These paymens vary wih ineres raes. If ineres raes rise, B pays more o A; if ineres raes fall, B pays A less. In order o economize on paymens, cash flows are need. Tha is, if A owes B 10, and B owes A 7, A simply pays B 3. The "floaing" side of mos swaps is linked o he LIBOR rae. For example, pary B (who pays floaing) may be obligaed o pay A he LIBOR rae x some noional principal value every six monhs. For purposes of illusraion, make he following assumpions: 1. The nominal principal value of he swap is Q. Noe: his principal never changes hands in an ineres rae swap; i is merely used o deermine he ineres paymens made by A and B. 2. A agrees o pay B k every six monhs for 5 years. 3. Every six monhs, B agrees o pay A he 6 monh LIBOR rae a -.5 imes Q. Thus, if he LIBOR rae (annualized) a ime -.5 is 10 percen, and Q is $10 mm, hen B pays A (.5)(.1)($10 mm)=$50,000 a. Call R.5 he annualized 6 monh LIBOR rae a -.5.

44 Thus, every 6 monhs he payoff o B (who pays floaing) is: k R Q Noe ha his is like he payoff o a long 2k ineres rae forward conrac. If ineres raes fall below a -.5 hen B receives a cash inflow a ; if ineres raes rise above his level, B pays cash ou o A. Since A and B swap cash flows in his fashion every six monhs for 5 years, he oal swap conrac is equivalen o a bundle of 10 forward conracs. Given he yield curve a he ime he swap is iniiaed (say, ime 0), he fixed paymen k is se in order o make he value of he bundle of forward conracs equal o 0. Formally, if R. 5, is he forward semi-annually compounded rae of ineres, as of ime 0, over he period -.5 o, and r is he coninually compounded ineres rae used o discoun cash flows received a as of ime 0, hen he presen value of he swap o B (he floaing payer) is: i= 10 i= 2 r V = ( k. R Q ) e + ( k. 5R Q ) e 0 Q iri. 5( i 1),. 5i When iniiaing he swap, he paries o he swap hen choose k o se his expression o equal 0. (The value of he swap o A is -V.)

45 You can use he following logic o deermine he fixed rae k a he iniiaion of a swap. Consider a swap wih N paymen daes. These paymens occur M imes per year--if pays occur semiannually, for insance, M=2. Firs, use he forward yield curve o calculae he discoun facors corresponding o each of he paymen daes. Call D j he discoun facor corresponding o paymen dae j. Then he fair or marke k is such ha he presen value of he cash flows on one DM (or dollar) of noional principal of he swap equals one DM. Formally: 1 = k( D + D D ) + ( 1)( D ) 1 2 (The las erm reflecs paymen of he 1 DM of noional principal.) Thus, N N k = 1 DN D1 + D D N To conver his o an annual rae, muliply k by M. Tha is, i = Mk. This is he same as he par yield on a bond/noe wih he same mauriy as he swap. Swap raes are ofen quoed as a spread over some oher ineres rae. For example, USD ineres rae swaps are usually quoed LIBOR vs. he US Treasury rae plus a spread. Thus, he fixed rae in a 5 year swap may equal he 5 year US T-noe rae plus 75 bp. The credi spread is usually prey sable wihin he day, so his quoing convenion allows he quoed swap rae o respond coninuously o ineres rae movemens.

46 Marking Swaps o Marke The valuaion of ousanding swaps is sraighforward. Break he swap up ino 2 pars--he fixed rae par and he floaing rae par. You can value he fixed par like a bond. Consider a swap wih N remaining fixed paymens of k. The price of a zero coupon bond mauring a he dae of paymen i equals D i. Then he value of he fixed side of he swap is: V Fix = N i = 1 kd Q i The floaing side is similarly easy o value. Noe ha on a paymen dae, he pary receiving floaing is indifferen beween receiving he pre-deermined paymen on ha dae plus eiher a) receiving he floaing paymen over he remaining life of he swap, or b) receiving he noional principal oday, and paying ou he noional principal a he erminal dae of he swap. Par b) is rue because he pary can inves he noional amoun a he floaing rae unil mauriy. Thus, on he paymen dae he value of he floaing side mus equal he value of alernaive b): V = R + Q( 1 D ) Floa where R is he floaing paymen due on ha paymen dae. Thus, prior o he paymen dae, he floaing side of he swap wih N paymens remaining is: VFloa = D ( R + Q ) 1 DNQ N The value of a shor swap is simply V Fix V. Floa

47 LONG TERM INTEREST RATE FUTURES The Forward Price of a Coupon Paying Bond Call S he spo price (he full price including accrued ineres) of a bond a ime. Moreover, F, T is he fuures price a for delivery of his bond a ime T, and r is he riskless ineres rae beween ime and. The bond pays a coupon equal o D a D. Also assume ha ineres raes are consan over ime. (This ensures you can reinves coupons a a known rae.) Consider a cash and carry arbirage sraegy: TRANSACTION Dae Dae D Dae T Buy spo S 0 S T Borrow S 0 e r( T ) Sell Fuures 0 0 F S Dividend Paymen 0 D 0, T r( T Inves Dividend 0 -D e D ) D This analysis implies ha ne cash flows a he iniial dae are equal o 0, and a dae T hey equal: r( T ) r( T F e S e D ) + D, T Arbirage hen implies T S

48 r( T ) r( T F e S e D ) = D, T A similar expression is relevan for a fuures/forward on any asse ha pays a cash flow. For example, for a dividend paying sock D can be inerpreed as he dividend and S as he curren price of he sock. This analysis implies ha he fuures price equals he fuure value of he underlying price ne of he fuure value of he cash flows he asse pays prior o conrac expiraion. This deducion of inerim cash flows is necessary because he underlying price embeds he value of hese cash flows, bu he buyer of he forward conrac does no receive hem.

49 Noe ha he fuures price exceeds he spo price by less han he ineres rae facor. One can calculae an implied repo rae from his expression. The implied repo rae is he shor erm ineres rae ha mus prevail in he marke o make arbirage unprofiable. r i = r( T ) D F, T + e D ln[ ] / ( T ) S (Noe ha his expression is somewha inconsisen because one has o pu include an ineres rae wihin he log expression. In order o solve for a single implied ineres rae, one could use numerical echniques, bu his is beyond he scope of his course.) If he acual shor erm ineres rae differs from he implied repo rae, an arbirage opporuniy exiss. If he implied rae is below he acual rae, borrow hrough he fuures and bond marke by selling bonds spo, buying fuures, and invesing he proceeds a he repo rae. If he implied rae is above he acual rae, borrow a he repo rae and lend hrough he fuures marke by buying bonds spo and selling fuures. There are also some complicaions when he erm srucure of ineres raes (i.e., he relaion beween ineres raes and mauriy) is no "fla." This expression mus be modified if he erm srucure slopes up or down because he rae a which a rader can re-inves he coupons is differen from he iniial rae.

50 T-Bond Fuures: Conversion Facors and Delivery Opions The Treasury Bond fuures conracs raded on he CBoT allow shors o deliver any bond wih greaer han 15 years o mauriy (or call) agains he conrac. Since bonds can have very differen prices due o differences in coupons or mauriy, he exchange has devised a sysem o make i more economical o deliver a wide variey of bonds. In essence, he shor receives higher proceeds if he delivers a high priced (high coupon, long mauriy) bond han if he delivers a low priced one. The exac sysem works as follows. Upon delivery of a given bond i, he shor receives he following amoun of money, called he invoice price: Pi = CFi F + AIi Here CF is he conversion facor of bond i, F is he fuures selemen price on he day he shor delivers, and AI gives he accrued ineres on he bond. The conversion facor is equal o he value of one dollar of face amoun of he bond as of he firs day of he delivery monh under he assumpion ha he bond yields 6 percen (semi-annually compounded) o mauriy. Shors have he opion o choose which bond hey will deliver. They will choose o deliver he bond for which he ne proceeds (invoice price ne of he full price of he bond) are larges. Tha is, hey choose he bond i for which he following expression is larges: P CF F B i i i = where B i is he fla price of he bond.

51 The bond ha has he larges delivery value is called he cheapeso-deliver ( CTD ) bond. Noe ha if he foregoing expression were sricly posiive for any bond, hen an arbirage opporuniy would exis. Therefore, a he expiraion of he fuures conrac he fuures price equals: F = B T i (, ) CF i where i is he CTD bond and Bi (, T) is he forward price of he bond a for delivery a T (fuures expiraion). Prior o expiraion, i is possible o figure ou he bond for which Bi / CFi is smalles. This is he bond ha is currenly CTD. However, here is some possibiliy ha some oher bond may be CTD a expiraion. This can occur because he relaive prices of bonds change randomly over ime. Tha is, prior o expiraion, he fuures conrac is acually an opion because he shor can choose which bond o deliver. This opion is an exoic--a call on he minimum of he N deliverable bonds, wih a srike price of zero. I is possible o show ha due o his opion, he bond fuures price is: N F w B T i (, ), T = i i= 1 CFi where he w i < 1are weighs ha add up o a number less han one. These weighs are funcions of all he bond prices and conversion facors, he volailiies of all he bond reurns, and he correlaions beween bond reurns. Noe ha he fuures price mus be less han he smalles B / CF because of he delivery opion. i i

52 This opion affecs he naure of he fuures conrac as a hedging vehicle. If you wan o mach PVBPs, you need o know he PVBP of he fuure. This, in urn, depends upon he PVBPs of all he deliverable bonds. Specifically: PVBP F = N i= 1 w i PVBPi ( ) CF i Noe ha he PVBP of he fuure can change over ime because a) he PVBPs of individual bonds change, and b) he weighs in he expression change. This can cause some funny hings o happen. On one day, a high duraion bond may receive a large weigh and a low duraion bond a low weigh; a couple of days laer, he low duraion bond may have a high weigh and he high duraion bond a low weigh. Thus, you need o monior he PVBP of he fuure fairly closely o make sure ha your hedge raios remain correc. There are oher opions embedded in he T-bond conrac. The mos imporan, and unusual, is he so-called Wildcard opion. This opion exiss because he shor can deliver up o 8pm a a selemen price se a 2pm on ha day. Thus, if bond prices fall beween 2pm and 8pm, i can be profiable o wai o deliver unil 8pm. This opion is very complex o value, especially due o is ineracion wih he qualiy opion (i.e., he opion o deliver he CTD bond).

53

54 Foreign Currency Cash and Carry Arbirage In deermining he relaion beween foreign exchange spo and fuures prices, i is necessary o expand he arbirage able somewha because we have o keep rack of borrowing and lending in wo currencies. Moreover, we have o ake ino accoun ha we can borrow or lend in he foreign currency a rae r f. Call S he spo price of Deuschmarks ("DM") in dollars. Thus, a spo price of.6471 means ha each DM buys.6471 dollars. Time Time T USD DM USD DM Buy e r ( T ) f DM e r ( T ) f S e r ( T ) f 0 0 Lend DM 0 ( ) f 0 1 e r T r ( T ) f Borrow USD e S r 0 r f e ( )( T ) S 0 Sell DM Fuures 0 0 F, T -1 f Ne Cash Flows 0 0 F e S, T ( r r )( T ) 0

55

56 To preven arbirage, he following expression mus herefore hold: f F = e S, T ( r r )( T ) Noe ha he fac ha he foreign currency can be borrowed/len a ineres depresses he fuures price relaive o he spo price. Moreover, if he foreign ineres rae is larger han he domesic ineres rae, hen he fuures price is below he spo price. Also recognize ha you can use fuures markes o borrow and lend foreign currencies. To borrow DM, underake he following ransacions: buy DM spo, borrow USD, sell DM fuures. You can also use inermarke spreads o borrow dollars hrough one FOREX fuures marke, and lend dollars hrough anoher. For example, if you borrow DM and sell hem for dollars, and buy DM fuures, his is equivalen o borrowing dollars. You can hen lend dollars hrough, say, he JY fuures marke by using he USD you earned from he shor sale of he DM by buying JY spo, lending JY, and selling JY fuures. This is equivalen o borrowing DM, selling

57 hem for JY, lending he JY, buying DM fuures, and selling JY fuures. This can be a profiable ransacion if he implied USD repo raes differ beween he JY and DM markes.

58

59 OPTIONS BASICS 1. A call opion gives he owner he righ, bu no he obligaion, o buy he underlying asse (e.g., a sock, a bond, a currency, a fuures conracs) a a fixed price. This fixed price is called he "srike price." Call opions have a fixed expiraion dae. Time o expiraion can range beween days and years. 2. A pu opion gives he owner he righ, bu no he obligaion o sell he underlying asse a a fixed price. 3. There are wo basic ypes of opions: European and American. The holder of a European opion can exercise i only on he expiraion dae. Tha is, here is no early exercise for European opions. In conras, he holder of an American opion can exercise i any ime prior o he expiraion dae as well as on he expiraion dae iself. Mos exchange raded opions are American. Many over he couner opions are European. 4. Opions are raded on socks a he Chicago Board Opions Exchange ("CBOE"), he NYSE, he American Sock Exchange, he Pacific Sock Exchange and he Philadelphia Sock Exchange. CBOE is he oldes and larges of he opions exchanges. 5. Opions on fuures are raded on mos fuures exchanges. 6. Opions on currencies are raded on he Philadelphia Exchange.

60 7. OTC opions markes are also imporan. 8. Many financial insrumens have opions embedded in hem. For example, a morgage gives he morgagee he opion o prepay a any ime. A converible bond embeds a call opion.

61 ARBITRAGE RESTRICTIONS Opions prices mus obey cerain arbirage resricions. 1. Pu-call pariy. Consider European opions on a non-dividend paying sock wih price S. A posiion consising of a call sruck a K and a shor pu wih srike K provides he same payoffs as a forward conrac wih a forward price equal o K. Boh opions expire a ime T. We know ha he fair (i.e., no arbirage) forward r( T ) price equals e S. Thus, he presen value of a forward r T conrac wih a forward price of K equals S e ( ) K. Since he long call-shor pu posiion gives he same payoff as a forward conrac, i mus be he case ha he value of his posiion equals he value of he forward conrac. Tha is: r( T ) c( S, K,, T ) p( S, K,, T ) = S e K This is called he pu-call pariy relaion. 2. Given pu and call prices, he sock price, he srike price, and he ime o expiraion, i is possible o solve he pu-call pariy expression for an implied ineres rae in order o deermine wheher hese prices presen an arbirage opporuniy.

62 3. If he sock pays dividends prior o he expiraion of he opions, i is sraighforward o modify he pu-call pariy expression o reflec his fac. Recall ha in deriving pu-call pariy, we simply equaed he value of he opion posiion o he value of a forward conrac wih forward price K. Therefore, all ha is necessary o adjus he formula for dividends is o use our no-arbirage formula for a forward price on an asse paying a dividend. Here, he presen value of such a forward conrac is r( ) r( T ) S e D e K D where D is when he dividend is paid. Therefore, r( ) r( T ) c( S, K,, T) p( S, K,, T) = S e D e K D

63 EARLY EXERCISE OF OPTIONS Early exercise has one clear disadvanage: by exercising an opion, a rader gives up he value of he opion over is remaining life. This value mus be posiive. Therefore, early exercise is desirable if and only if here is some off-seing benefi. There is only one possible source of such a benefi. Namely, here is a poenial value o early exercise if his allows he owner of he opion o receive cash flows earlier. Case 1. Call on a non-dividend paying sock. Assume he owner of a call wih srike price K and ime o expiraion T exercises he opion a <T, and borrows money in order o pay he srike price. Then, a T, he wealh of he rader is r( T ). If he rader had no exercised he opion, his ST e K wealh a T would equal max[ 0, S K]. I is clear ha he rader's wealh is greaer if he does no exercise early. This is rue for wo reasons, firs, if he price of he sock falls below he srike price beween and T, he rader won' exercise he opion a T, and hus isn' "suck" wih a less valuable sock. Second, by deferring exercise, he call owner saves he ineres on he srike price over he period o T. Conclusion: Never exercise a call on a nondividend paying sock early. T Case 2. Call on a dividend paying sock.

64 Assume he firm pays a dividend equal o D a <T. If he call owner exercises a, his wealh a T equals: r T S e ( ) ( K D ) T This may (bu may no) exceed max[ 0, ST K] because by exercising he call early, he owner receives he dividend. Therefore, unlike he case wih a call on a non-dividend paying sock, we can' say for cerain ha he payoffs o early exercise are always lower han he payoffs from exercise a expiraion. If his dividend is large enough, i may exceed he ineres on he srike price and he opion value foregone from early exercise. Thus, you may exercise a call on a dividend paying sock early. If you do exercise early, you will only do so immediaely before he paymen of a dividend (i.e., on he cum dividend dae). If he fuure value of he dividend is smaller han he value of he ineres paid on he srike price from o T, early exercise will r( T ). If no be opimal. The value of his ineres equals e K K r e ( T ) r T K K e ( > ) D hen r( T S e ) ( K D) < S K < max[ 0, S K] T and herefore early exercise is no profiable. Inuiively, his means ha if he dividend received by exercising early is no large enough o compensae he opion T T

65 holder for he ineres incurred on he srike price due o early exercise, early exercise canno be opimal. Case 3. Pu on a non-dividend paying sock. Consider a rader who owns a pu and a share of he underlying sock. If he exercises he pu a <T and invess he r( T ) srike proceeds, his wealh a equals e K. There are some values of he sock price such ha his exceeds he rader's wealh a T if he does no exercise he pu a, max[ 0, K S ] + S. By exercising early, he pu owner receives cash flows (i.e., he proceeds from exercise) earlier; he ineres earned on he srike price over his inerval of ime. This may compensae he rader for he gains he would earn by holding ono he sock if he price of he sock were o rise above he srike price beween and T. Early exercise of a pu can be opimal a any ime prior o expiraion. Case 4. Pu on a dividend paying sock. All else equal, dividend paymens cause he price of he sock o decline. Therefore, if he pu is no proeced agains dividend paymens, dividends end o induce he holder of a pu o defer exercise in order o ake advanage of his predicable price decline. I sill may be he case, however, ha he advanages of receiving he proceeds from exercise early (i.e., he ineres earned on he srike price) offses his effec. Case 5. Pus and calls on fuures conracs. When he holder of an opion on a fuures conrac exercises, she receives a cash paymen equal o he difference beween he T T

66 srike price and he fuures price a he ime of exercise and a fuures posiion. For example, if he fuures price a exercise equals F, he holder of a pu receives a cash paymen equal o K F and a shor fuures posiion a he marke price. The value of he shor posiion equals 0. Early exercise of eiher a pu or a call herefore leads o an acceleraion of cash flows. Therefore, one canno rule ou early exercise of fuures opions. Case 6. Calls on foreign currency. Foreign currencies can be invesed a ineres; recall ha his is equivalen o a coninuous dividend yield. Therefore, here may be advanages o exercising a call on a foreign currency early in order o receive his cash flow over a longer period of ime.

67 Early exercise requires a modificaion in he pu-call pariy expression. In paricular, we can no longer derive an equaliy resricion, bu only an inequaliy resricion. Firs consider he case of American pus and calls on non-dividend paying socks. We know ha he value of an American call on a non-dividend paying sock equals he value of a European call on he same sock (wih he same srike and ime o expiraion). Moreover, we know ha an American pu may be exercised prior o expiraion. This opion o exercise early has value, so an American pu mus be more valuable han a European pu. Therefore, r( T ) C( S, K,, T ) P( S, K,, T ) < S e K Also remember ha dividends reduce he value of an American call, and increase he value of an American pu. Thus, his expression mus hold for opions on dividend paying socks oo.

68 A BINOMIAL MODEL OF STOCK PRICE MOVEMENTS In order o derive a formula for sock opion prices, we mus firs specify a model ha describes how sock prices move. Our firs model is a so-called binomial model because a any poin in ime, i is assumed ha he percenage change in he sock price- -he sock "reurn"--can ake only wo values, u>1 (an "up" move) or d<1 (a "down" move). The probabiliy of an up move equals q, and he probabiliy of a down move equals 1-q. We will see ha hese acual probabiliies are irrelevan o he pricing of opions. The binomial model divides ime beween he presen and he expiraion dae of an opion ino discree inervals of equal lengh. Each inerval is in lengh. By allowing more inervals beween now and expiraion, his inerval becomes shorer. In he limi, wih an infinie number of inervals, he lengh of an inerval becomes vanishingly small, and equal o d. Tha is, if he sock price equals S oday, i may equal eiher us > S or ds < S a he end of he nex inerval of ime. I is possible o show ha as he lengh of a ime inerval becomes vanishingly small, he disribuion of sock prices is lognormal. We will uilize his fac in deriving he Black-Scholes formula for pricing opions.

69 USING THE BINOMIAL MODEL TO PRICE AN OPTION Consider a porfolio of a shor posiion in a single call on a sock, and shares of he sock underlying he call. Assume he value of he call in unis of ime equals c u if he sock price rises over his inerval, and equals c d if he sock price falls. Thus, a + he value of he porfolio equals us c u if he sock price rises, and equals ds if he sock price falls. c d Noe ha we can choose o make he value of he porfolio he same regardless of wheher he sock price rises or falls. Formally, choose such ha or us c = ds c u d = cu cd S( u d) Since he value of he porfolio doesn' depend upon he sock price change, he porfolio is riskless. Therefore, he reurn on he porfolio mus equal he risk free rae. Tha is, if he value of he call oday equals c, hen: r e ( S c) = u S c = d S c u d

70 Solving for c implies: where r c = e [ pc + ( 1 p) c ] u d p = r e d u d Noe ha he value of he call a is equal o he expeced presen value of he call a +, using p o measure he probabiliy of an up move and (1-p) o measure he probabiliy of he down move. I is essenial o recognize ha p is no equal o q, he rue probabiliy of a sock price increase. Insead, p is he probabiliy of an up move such ha he sock's expeced rae of reurn equals he risk free rae of reurn. This would occur in a marke where raders are risk neural. This analysis implies ha we can price opions as if we are in a risk neural world! Pu differenly, we don' need o know he expeced reurn on a sock o price opions on ha sock. This is rue because we can form a porfolio consising of he sock and he opion o eliminae all risk.

71 In order o deermine he price of he call, we work backwards from he end of he binomial ree because a he end of he ree we know he values of he opion. This is bes illusraed by an example. The following pages presen a wo sage binomial ree which assumes ha he iniial value of he sock price is 20, u=1.1, d=.9, and r=.12 (annualized, coninuously compounded rae). Each ime inerval is a quarer of a year. The firs page oulines wha he values of he sock can ake afer 2 periods. Given hese values, we know ha p= Consider a call expiring in wo periods wih a srike price of 21. The value of he call in wo periods afer wo up moves equals =3.2. The value of he call afer one up move and one down move equals 0 because 19.8<21. Similarly, he value of he call afer wo down moves equals 0. Consider he value of he call afer one period if a single up move has aken place. Here c u = 3. 2 and c d = 0. Thus, afer one up move, he value of he call equals: c. 12(. 25 = e ) [ ] = I is sraighfoward o recognize ha he value of he call afer one period equals zero if he sock has moved down during ha inerval.

72 Now move back o he iniial ime, when he value of he sock equals 20. Here c u = and c d = 0. Thus, he value of he call a his ime equals: c. 12(. 25 = e ) [ ] =

73 In sum, we use backward inducion repeaedly o value an opion on an underlying sock. We go backwards because we know he payoffs o he opion a he expiraion dae, and can herefore apply our formula repeaedly by proceeding from he end of ime o he beginning. The main choice you mus make in esablishing a binomial ree is for u and d. We choose hese parameers such ha he heoreical value of he variance of he sock given by he binomial model equals he acual value of he variance of he sock we are ineresed in. Remember ha he variance of a sock's reurn equals he expeced value of he squared deviaion beween he realized reurn and he expeced reurn. The expeced reurn in our risk neural model equals r. The reurn in an upmove equals u and he reurn in a down move equals d. Therefore, he variance equals: r 2 r 2 p( u e ) + ( 1 p)( d e ) = 2 2 p( 1 p)( u d) = σ where σ is he acual sandard deviaion of he sock's reurn. The firs equaliy follows from he fac ha r Se = pus + ( 1 p) ds

74 We have already found he relaion beween p and u and d. Thus, we have one equaion in wo unknowns. We eliminae one unknown by choosing d=1/u. If we solve all of his for u, we ge: u = e σ if is small.

75 Firs, define a Then: DERIVATION OF THE FORMULA FOR p. = e r. a S u S + c = ac u Subsiuing for implies: a( cu cd ) u( cu cd ) + cu = u d u d ac Gahering erms wih cu and cd: cu u d a u u d cd ( + ) + ( u d u a ) = ac a d u d c u a u d c ac u + d = Define p = ( a d) / ( u d) hen noe ha u a a d u d = 1 u d = 1 p

76 Thus, ac = pc + ( 1 p) c u Dividing boh sides by a produces he expression presened earlier. I is imporan o noe ha if he probabiliy of an upmove in he sock price equals p, hen he expeced reurn on he sock equals he risk free rae of reurn. To see why, noe ha if he probabiliy of an upmove equals p, he expeced value of he sock nex period equals: S[ pu+ ( 1 p) d] = S[ p( u d) + d] = S a [ d ( u d u d ) + d ] = as This implies ha he sock earns an expeced reurn equal o he risk free rae of ineres. d

77 Use of he Binomial Model A major advanage of he binomial model is ha i can be used o value American opions for which early exercise is possible, and hence valuable. An example illusraes his. Firs consider he example conained on he following page: an American pu on a sock. (This is example 14.1 from Hull.) In his example, σ=.40, T-=.4167, r=.1, S=50, K=50. If we divide he ime unil expiraion ino 5 segmens, we ge =.4167/5= Given hese values, we can deermine p=.5076, and 1-p= Also, a=1.0084, and e r = 1 / = The imporan hing o do when "folding back" he binomial ree is o deermine a each node wheher early exercise is opimal. Firs consider node E. The sock price a his node equals 50, so he proceeds from early exercise equal 0. The value of he opion if we don' exercise equals [(.5076)(0)+(.4924)(5.45)]/1.0084=2.66. If we do exercise a node E, we ge less han if we wai. So we don' exercise early. Things are differen a node A, when he sock price equals Here he proceeds from early exercise are = If we don' exercise, he opion is worh:

78 [(.5076)(5.45)+(.4924)(14.64)]/1.0084=9.89. A his node, we ge more if we exercise early han if we don'. Early exercise is herefore opimal in his case. The imporan hing o remember is ha we now use as he value of he opion a node A when we are calculaing he opion price a earlier poins in he ree. This demonsraes how we can use he binomial opion o calculae American opion prices. Analyically, he binomial model is sraighforward. In order o increase he accuracy of his approach, however, i is necessary o make fairly small by increasing he number of ime inervals. This can increase he cos of compuing opions prices using he binomial mehod. Thus, when pricing an opion, we face a rade off: he binomial mehod can handle early exercise easily, bu is compuaionally cumbersome. This raises he quesion, is here a more compuaionally racable model? The answer is yes: If we are willing o consider only European opions, i is possible o produce an opion pricing model ha is very easy o use--he Black-Scholes model.

79 THE BLACK-SCHOLES MODEL FOR A NON-DIVIDEND PAYING STOCK The Black-Scholes model is essenially he same as he binomial pricing model when he number of ime inervals approaches infiniy, i.e., as becomes arbirarily close o zero. This is someimes called a "coninuous ime" model in conras o he "discree ime" binomial model because we no longer divide he ime line ino several discree periods, bu insead consider ime as a coninuum. I is possible o show ha as he number of ime seps approaches infiniy, he reurn on he underlying sock obeys a normal disribuion. The normal disribuion is simply he well known bell shaped curve. Recall ha he reurn on a sock equals he percenage change in price on he sock. Also noe ha over a very small ime inerval d he reurn on a sock equals: ln S + d ln S Thus, if he reurn on he sock in he coninuous ime world is normally disribued, hen he sock price in he fuure is lognormally disribued. I is also essenial o remember ha in valuing opions we can assume ha he expeced reurn on he sock equals he risk free

80 rae of ineres. Again, his is because we can consruc a porfolio including he sock and he opion ha is riskless. This can be represened formally. If he sock price a T (which may be he expiraion dae of an opion) is lognormally disribued, hen we can wrie: S T = S e ( r. 5σ 2 )( T ) + σ T Z where Z is a normally disribued variable wih expeced value (i.e., mean) equal o 0. (You can check ha his expression is correc by aking he naural logs of boh sides. You will find ha he difference in he logs is normally disribued because Z is normally disribued.) We can now value a European opion ha expires a T. Recall ha he value of he opion a is he expeced presen value of he payoffs of he opion a T, where we use he riskless ineres rae o discoun hese payoffs. Also remember ha any expeced value is he sum of he possible payoffs muliplied by he probabiliy of receiving a given payoff. In our analysis, he size of he payoff depends upon he realizaion of Z because Z deermines he sock price a T. Moreover, because Z is normally disribued, he probabiliy of observing any given Z is n( Z) =. e Z 2Π 5 2

81 Noe ha he Π in his expression is he mahemaical consan Pi=

82 We can use his informaion o value an opion. Consider a call opion wih srike price K. We know ha he opion's payoff is posiive if and only if: r T T Z S = S e (. σ )( ) + σ K T 5 2 There is a value of Z, call i Z*, such ha his expression holds wih equaliy. This is he "criical value" of Z: For larger Z, he call is in he money a expiraion, for smaller Z, he opion is ou of he money. Therefore: ln S + ( r. 5σ 2 )( T ) + σ T Z* = ln K Solving for Z* implies: Z* = ln( K / S ) ( r. 5σ 2 )( T ) σ T Given his resul, he expeced presen value of his opion's payoffs (and hence is price) is: c = e r( T ) { Z * [ S T ( Z) K ] n( Z) dz + Z* 0n( Z) dz} This expression holds because he payoff o he opion is 0 when Z<Z*.

83 Doing a lile subsiuion, we ge c = e 2 r( T ) ( r.5σ )( T ) + σz [ Se Z* T e K ].5Z 2 2Π dz Consider he exponenials. We can add he exponenial erms o ge he following exponen: Define a new variable 2 2 ( r. 5σ )( T ) + σz T. 5Z = 2 2 r( T ). 5[ σ ( T ) + Z 2σZ T ] y = Z σ T Noe ha y is normally disribued (because Z is). Moreover, y 2 = σ 2 ( T ) + Z 2 2σZ T In addiion, he call opion is in he money if

84 y = Z σ T Z * σ T y Z * σ T = 2 ln( K / S ) ( r. 5σ )( T ) σ T 2 ln( K / S ) ( r+. 5σ )( T ) y * σ T σ T =

85 Then we can rewrie he firs erm in our inegral as e r ( T ) y* 5 S y* S e 2. y e dy 2Π r( T ) 2.5y e dy 2Π = This is jus equal o N(-y*), where N(x) gives he area under a normal disribuion curve o he lef of x. Thus, he firs erm in our inegral is: S K r T S N[ ln( / ) + ( +. 5σ 2 )( ) ] σ T Now consider he second erm in our inegral:

86 ] ) )(.5 ( ) / ln( [ *) ( ) ( ) ( 2 ) ( ) ( * ) ( * ) ( T T r K S KN e Z KN e dz Z n K e dz Z n K e T r T r Z T r Z T r + = = = σ σ

87 Puing he wo erms ogeher produces he Black-Scholes formula for he call: where r T c = S N d e ( ) ( ) KN( d ) 1 2 d 1 = 2 ln( S / K) + ( r+. 5σ )( T ) σ T and d = d σ T 2 1 We can hen use pu-call pariy o ge he value of a pu. Recall: Then r T c S + e ( ) K = p r T p = S ( N( d ) 1) + e ( ) K( 1 N( d )) Noe ha N( x) = N( x) Therefore, p = e r( T ) KN ( d ) S N( d ) 2 1

88 OPTIONS ON OTHER ASSETS One can rewrie he Black-Scholes formula for a call as follows: c = e r( T ) e r T [ ( ) S N( d ) KN( d )] Noe ha he erm muliplying N( d ) non-dividend paying sock is he forward price of a One can also rewrie he erm inside he normal funcion in he model as: d 1 = r( T ) 2 ln( Se / K) +. 5σ ( T ) σ T Noe ha he numeraor in he logarihm erm is also he forward price of a non-dividend paying sock. In general, i is possible o show ha o find he value of a call on some oher ype of asse, one simply uses he forward price of he asse wherever one finds he forward price of he sock in he Black-Scholes equaion. For example, for a European call on a dividend paying sock, we use d 1 * = r( T ) 2 ln[( Se FVD) / K] +. 5σ ( T ) σ T

89 where FVD is he fuure value of he dividend on he sock. As a resul, we can wrie he price of a European call on a dividend paying sock as: r T r T c = e ( ) {[ e ( ) S FVD] N( d *) KN( d *)} 1 2

90 We can use similar reasoning o price opions on oher asses. For example, consider an opion on a fuures or forward conrac. The forward price of he fuures (or forward) price is jus he fuures (forward) price iself. Therefore, if he fuures price equals F, T hen he price of a call opion on his fuure is: where c = e r( T ) [ F N( d ' ) KN( d ' )], T 1 2 d 1 ' = 2 ln( F / K) +. 5σ ( T ), T σ T This formula is called he Black Model. Finally, consider he value of an opion on a foreign currency. If he foreign ineres rae is r F and he spo price of he currency is S hen where r T r rf T c = e ( ) [ e ( )( ) S N( d ' ' ) KN( d ' ' )] 1 2

91 d ' ' 1 = 2 ln( S / K) + ( r rf +. 5σ )( T ) σ T

92 DYNAMIC HEDGING: DELTA, GAMMA, AND VEGA The Black-Scholes model also describes how he price of he opion changes when cerain underlying variables change. For example: c S = N( d ) 1 The inerpreaion for his "Dela" is exacly he same as he inerpreaion of he in he binomial model: I gives he number of shares of sock o hold wih a shor call posiion in order o form a riskless posiion. To see why, noe ha if one holds shares of sock and one shor call, he change in he value of he porfolio is ( c + S ) S = N( d ) + N( d ) = Noe ha he value of his posiion doesn' change even if he sock price changes. Thus, i is riskless. Pu differenly, if one wans o hedge a single long call, one sells shares of he sock. I is possible o show ha if one changes coninuously in response o changes in he sock price, one's posiion will be

93 riskless. This is called "dynamic hedging." I is necessary o change every insan for his sraegy o work perfecly. Of course, in he real world, i is impossible o make such adjusmens. Moreover, ransacions coss make his sraegy prohibiive. Therefore, i is necessary o make adjusmens less frequenly. I is desirable o know, however, when i is necessary o make changes mos frequenly. The "Gamma" of an opion provides his informaion. Formally, Γ 2 c = = 2 S S Tha is, Γ ells you how rapidly changes when he sock price changes. I quanifies he curvaure of he opion price funcion. This is imporan informaion for a rader managing he risk of an opion posiion. Lef unended, a posiion wih subsanial Γ>0 may become overhedged (i.e., he rader's acual sock posiion becomes subsanially larger han he opimal ) or may become unerhedged (i.e., he raders acual sock posiion becomes subsanially smaller han he opimal ) in response o even small changes in he sock price. Pu differenly, you are likely o need

94 o change a hedge posiion quickly if here is a lo of Gamma. Gamma neural hedgers (i.e., hedgers wih posiions wih Γ=0) can sleep beer han hedgers wih subsanial Gamma (eiher posiive or negaive). The oher imporan variable a hedger needs o rack is Vega. This gives he sensiiviy of he opion price o changes in σ. Tha is, Υ = c σ Remember ha Black-Scholes assumes ha volailiy is consan over ime. In realiy, however, volailiy can change rapidly. If so, a rader wih subsanial Vega is subjec o large gains or losses. Jus as a hedger wans o know his vulnerabiliy o sock price moves, he should also rack his risk o volailiy changes. These "Greeks" are imporan for anoher reason. Consider a rader who wans o purchase or sell an opion ha is no raded on an organized exchange, or on he OTC marke. (For example, an individual migh wan o buy a 5 year call on he S&P 500). Even hough he rader canno rade he opion, he can replicae he payoffs of he opion synheically by dynamically buying or selling he asse underlying he opion. Tha is, a every insan of ime, he rader holds unis of he asse underlying he opion, where gives he dela of he opion he

95 wans o replicae. If he rader can adjus he posiion coninuously and coslessly, such a sraegy will produce cash flows a opion expiraion ha are exacly idenical o hose of he opion. You may also wan o replicae an opion ha is raded. For example, if you wan o buy a pu, bu you hink pus are overpriced in he marke (e.g., he implied volailiy of raded pus is greaer han your forecas of volailiy), you may wish o replicae he pu hrough a dynamic rading sraegy. Similarly, you can arbirage he marke by buying (selling) underpriced (overpriced) opions, and dynamically replicaing off-seing posiions. In realiy, i is impossible (and very cosly!) o adjus he porfolio coninuously in his fashion. This is paricularly rue if he opion is near he money, so has a lo of Gamma. Moreover, he dynamic hedger is vulnerable o changes in volailiy. The problems faced by "porfolio insurers" (raders who aemped o replicae calls on he sock marke hrough dynamic hedging echniques) on 10/19/87 provides a graphic illusraion of he poenial problems. In order o address hese problems, he rader can add raded opions o his porfolio. The rader's objecive is o mach he Dela, Gamma, and Vega of his porfolio o he Dela, Gamma, and Vega of he opion he rader desires o replicae. This reduces he amoun of adjusmen needed, and provides some insurance agains volailiy shocks.

96 For example, consider replicaing an opion wih = * and Γ=Γ*. You can use he underlying sock and an opion wih = ** and Γ=Γ** o consruc your replicaing porfolio. To ge a dela and gamma mach, you have wo equaions and wo unknowns. The unknowns are N S, he number of shares of sock o rade, and N o, he number of opions o buy or sell as par of your hedge porfolio. Formally, solve: and * = N + N ** S o Γ* = N o Γ ** You could also consruc a dela, gamma, and vega mached posiion. To do so require wo opions and he sock. You have o solve hree equaions in hree unknowns here.

97 Noe ha even a gamma mached or gamma and vega mached posiion mus be adjused over ime. Recenly, researchers have developed sraegies ha allow you o creae fire-and-forge hedges. Tha is, using hese echniques, you can replicae an opion by consrucing a porfolio a he iniiaion of he rade, and never adjusing he porfolio unil he expiraion dae of he opion you wan o replicae. As you migh guess, his requires you o rade in a large number of opions.

98 FORMULAE FOR GAMMA, VEGA, AND THETA (The Π in he formulae is he mahemaical consan ) The Gamma of a Euro call or pu equals: Γ =. d e Sσ 2π( T ) The Vega of a Euro call or pu equals: Λ = S T e d 2π The Thea of a Euro call equals: d Se σ Θ = 2 2π( T ) 1 2 rke r( T ) N( d ) 2 The Thea of a Euro pu equals:. 5d Se σ Θ = + r( T ) rke N( T d2 ) 2 2π( ) 1 2 Noe ha Thea is unambiguously negaive for a Euro call, bu may be posiive for a Euro pu.

99 For a call, Gamma, Dela, and Thea are relaed by his equaion: rc = Θ + rs +.5σ 2 S 2 Γ A similar expression holds for a pu.

100 FORMULAE FOR GAMMA, VEGA, AND THETA On FORWARDS (The Π in he formulae is he mahemaical consan ) The Dela of a Euro fuures call equals: ) ( ' 1 ) ( d N e T r = The Dela of a Euro fuures pu equals: ) ( ' 1 ) ( d N e T r = Gamma of a Euro fuures call or pu equals: ) ( ) ( T F e e d T r Γ = π σ The Vega of a Euro call or pu equals: 2π ) ( d T r e e T F = Λ

101 The Thea of a Euro fuures call equals: Θ = e r( T ) 2.5d [ 1 Fe σ rfn( d1 ') rkn ( d 2 2π ( T ) 2 ' )] The Thea of a Euro fuures pu equals: Θ = e r( T ) 2.5d [ 1 Fe σ rfn( d1') + rkn ( d ')] 2 2π ( T ) 2 Noe ha Thea is unambiguously negaive for a Euro call, bu may be posiive for a Euro pu. For a call, Gamma, Dela, and Thea are relaed by his equaion: 2 rc = Θ +. 5σ F A similar expression holds for a pu. 2 Γ

102 Using Black-Scholes Mos of he inpus o he Black-Scholes model are observable. These include: he srike price, he underlying price, he ineres rae, and ime o expiraion. One key inpu is no observable he volailiy. Good opion values depend on good volailiy values. Where do volailiy values come from? There are wo sources: hisorical daa and implied values. Hisorical volailiies are calculaed using hisorical daa on reurns. For insance, o calculae he volailiy for MSFT sock, one would plug hisorical daa on MSFT reurns ino a spreadshee and calculae he sandard deviaion (adjused for he daa frequency). If he B-S model were sricly correc, his approach would be he righ one. Moreover, if he B-S model were sricly correc ha is, if volailiy was ruly consan you would wan o ge as much hisorical daa as is available o ge he mos precise esimaes of volailiy possible. An alernaive approach is o use implied volailiy. In his approach, you find he value of σ ha ses he value of he opion given by he B-S formula equal o he value of he opion observed in he markeplace.

103 Implied volailiy can be esimaed wih exreme accuracy using he Solver funcion of Excel. There are very good approximaions ha you can use o deermine he implied volailiy for an a-he-money opion. Obviously, if one uses he implied volailiy, one is assuming ha (a) he model is correc, and (b) he opion is correcly priced. Thus, implied volailiy canno be used o deermine wheher a paricular opion is mis-priced. So wha is i good for? If he B-S model is correc, all opions should have he same implied vol. Therefore, comparing implied volailiies across opions allows one o deermine wheher opions on he same underlying are mis-priced relaive o one anoher. Buy he opion wih low volailiy, sell he opion wih high volailiy. Also, you can use he implied volailiy o ge more accurae measures of he Greeks. In pracice, mos opions are quoed using an implied volailiy (ha is hen plugged ino he B-S formula). Therefore, mos opions raders hink in vol erms raher han in price erms, and opion rading is essenially volailiy rading.

104 How Well Does he Black-Scholes Model Work? The Black-Scholes model is he mos influenial heory ever inroduced in finance, and perhaps in all of he social sciences. Despie his success, i has is weaknesses. Praciioners have learned how o pach he holes in Black- Scholes. We know ha he B-S model is no sricly correc. Firs, conrary o he model s predicions, here are sysemaic differences in implied volailiy by srike price and ime o expiraion. Mos imporan, deep in-he-money and deep ou-of-hemoney opions have consisenly higher volailiies han ahe-money opions. This is referred o as he volailiy smile. For equiy index opions, his is more of a smirk because opions wih low srike prices have far higher implied vols han ATM or high srike opions. This implies ha σ canno be consan. This is confirmed by empirical evidence. This evidence shows ha volailiy varies sysemaically over ime. Moreover, volailiy clumps big price moves end o follow big price moves, and small price moves end o follow small price moves.

105 Dealing Wih he Smile There are wo basic approaches o he smile. The firs approach posis ha volailiy is a funcion of he underlying price and ime. Tha is, σ ( S, ). Using very advanced echniques, one can find such a funcion ha fis a variey of opions prices. (Many praciioners use a crude approach ha gives a very unreliable and unsable esimae of his funcion). The second approach is o pos ha volailiy is random and o wrie down a specific sochasic process for volailiy. The second approach is probably more realisic, bu raises many difficulies. Mos imporan, volailiy risk canno be hedged using he underlying. Consequenly, any pricing formula mus include a volailiy risk premium. Some praciioners and academics assume ha his risk premium is zero. This resuls in convenien pricing formulae, bu is inconsisen wih a grea deal of evidence. Mos noably, opions hedged agains moves in he underlying price earn a posiive risk premium his wouldn happen if vol risk premia equal zero. Thus, he smile is a knoy issue ha mos praciioners address in an ad hoc manner.

106 How Exoic So far we have considered vanilla opions basic pus and calls. Oher kinds of opions exoics are raded in he OTC marke. Alhough he variey of exoics is limied only by he imaginaion of raders, some exoics are more common han ohers. Someimes exoics are bundled implicily in securiies.

107 Digial or Be Opions A digial opion pays a fixed amoun of money if a cerain even occurs. For insance, a digial call on Microsof sruck a $75 ha pays $10, pays $10 if he price of MSFT a expiraion exceeds $75, regardless of wheher he price a expiraion is $75.01 or $175. Similarly, a digial pu sruck a $75 ha pays $10, pays $10 if he price of he underlying a expiraion is $74.99 or $0. The value of a digial is easy o deermine. A digial call r( T ) value is e PN ( d 2 ) where P is he payoff and d 2 is he same as in he B-S formula. Similarly, a digial pu value r( T ) is e PN ( d2). Digials are raded in he OTC marke, bu perhaps he mos (in)famous example of digial opions was embedded in bonds bough by Orange Couny CA in he early 1990s. To raise yields he Orange Cy reasurer bough bonds ha had embedded shor digial opions on ineres raes. A one ouch opion is an American digial. I is called a one ouch because i is opimal o exercise as soon as he underlying price his ( ouches ) he srike price (do you know why?)

108 Knock-Opions Knock opions come in several varieies. Knock-in opions. These are opions wih an underlying opion ha comes ino exisence only if some condiion is me. There are up-and-in and down-and-in varieies. For example, an up-and-in call wih a srike price of $75 and a knock barrier of $100 has a payoff a expiraion if-andonly-if he underlying price hi or exceeded $100 some ime prior o expiraion. Knock-ou opions. These are opions ha go ou of exisence if some condiion is me. For example, a downand-ou call sruck a $75 wih a knock barrier of $50 expires worhless if he underlying price his or is less han $50 a any ime during he opion s life. Thus, even if he sock price is $100 a expiraion, he opion is worhless if he sock price hi $50 prior o expiraion. Knock-opions are pah dependen because heir payoff depends no only on he value of he underlying a expiraion, bu he pah ha he underlying price follows prior o expiraion. Manipulaion can be an issue wih hese barrier opions.

109 Asian Opions An Asian opion has a payoff ha depends on he average price of he underlying over some ime period. For insance, an Asian call on crude oil wih a monhly averaging period has a payoff ha depends on he average price of crude oil over a one-monh-long period. There are average price opions where he srike price is se prior o expiraion, and he payoff is based on he difference beween an average price and he pre-agreed srike price. There are also average srike opions where he srike price is he average price on he underlying during some averaging period, and he payoff is based on he difference beween he value of he underlying a expiraion and his srike price. There are no closed form soluions for Asian opions wih arihmeic averaging. Various numerical echniques can be used o price hem.

110 Some Oher Exoics Compound opions. These are opions on an opion, such as a pu on a call, or a call on a call. There are formulae for valuing such opions given Black-Scholes assumpions. Exchange opions. These give you he righ, bu no he obligaion o exchange one asse for anoher. There are formulae for valuing such opions given Black-Scholes assumpions. COD (cash-on-delivery) opions. Here you only pay he premium if hey wind up in he money. These opions can have a negaive payoff. Quanos. Cross-currency opions are he mos common example. An example is a conrac on he Nikkei Index wih he payoff convered o USD. This conversion can occur a eiher an exchange rae se when he opion is wrien or he exchange rae prevailing a expiraion. The payoff o he opion is Xmax[S-K,0] where X is he exchange rae, S is he value of he Nikkei, and K is he srike price (in JY). Here here are wo sources of risk he Nikkei index (in yen), and he JY-USD exchange rae. Compos. These are opions wih payoffs max[xs-k,0] where he srike price is in he domesic currency.

111 Lookbacks. Lookbacks are pah dependen opions because he payoff depends on he maximum (or minimum) price reached by he underlying during he opion s life. For insance, a lookback call wih a srike price K has a payoff max[max(s)-k,0] where max(s) is he maximum priced achieved by he underlying during he life of he opion. Obviously his lookback is more valuable han a vanilla call.

112 Issues Wih Exoics Exoics frequenly presen serious hedging difficulies, especially when hey are near he money and close o expiraion. For example, a digial opion s gamma is posiive when i is ou of he money and negaive when i is he money. Righ before expiraion, he opion s gamma is posiive infiniy a a price immediaely below he srike price and negaive infiniy a a price immediaely above he srike price. This behavior makes i hard o hedge. As anoher example, a knock-ou opion has an infinie gamma when he underlying price is a he knock-ou boundary. Compound opions have high gammas when hey are a he money because hey are opions on opions he convexiy in he underlying opion compounds he convexiy in he op opion. Recall ha convexiy also has ramificaions for he sensiiviy opion value o volailiy. Hence, some exoics may have subsanial vol (vega) risk.

113 Ge Real Hereofore we have discussed opions ha are financial claims. The world is full of non-financial real opions. In paricular, virually all business invesmen and operaional decisions involve choice i.e., opionaliy. Opion o defer invesmen. Time-o-build opion build a projec in sages wih he choice o abandon afer each sage. Opion o increase or reduce invesmen scale. Opion o abandon. Opion o swich inpus or oupus (operaional flexibiliy). Growh opions (R&D, leases on propery or resources). Combinaions of he above opions.

114 Valuing Real Opions Tradiional capial budgeing approaches ignore real opions. This is unforunae as real opions may have a large affec on he profiabiliy of invesmen. Firms may underinves, overinves, or inves a he wrong ime if hey ignore real opions. Increasingly firms are using opion pricing echniques o evaluae invesmen projecs. Given he inense informaional demands, i is ofen no pracical o use hese echniques o ge precise esimaes of real opion value. Noneheless, he use of opions valuaion echniques forces firms o analyze he characerisics of heir invesmens more rigorously.

115 An Example Consider a firm ha has he opion o defer invesmen in drilling a gas well. I can inves oday, spend $104, and ge a well worh $100 (i.e., which generaes $100 in ne presen value). From an NPV perspecive, his projec is a urkey. However, he invesmen is risky. In one year, if good news abou he value of gas arrives, he well is worh $180. If bad news arrives, i is worh $60. The ineres rae is 8 percen per year. In erms of our binomial model, u=1.8 and d=.6. The pseudo-probabiliy is ( )/(1.8-.6)=.4. The opion o delay he projec one year is herefore [.4( )+.6(0)]/1.08=28.15 Exercise: Show ha he value o delay he projec for up o wo years is worh $31.82, and he value o delay he projec for up o hree years is worh $45.01.

116 Inuiion The inuiion here is prey sraighforward he opion o delay allows you o wai for he arrival of informaion, namely wheher gas prices are going o rise or fall. Since beer informaion means beer invesmen decisions, you he opion o delay is valuable. The simple example here implies ha you will never inves if you always have he opion o delay. This is an American call opion. Under he assumpions of he analysis (wih no inermediae cash flows) sandard analysis implies ha deferral is always he bes choice. In he real world, invesmens hrow off benefis only if he firm invess in he asse. These benefis are like dividends and can induce he firm o exercise he call opion early. In commodiy markes, for insance, prices end o be mean revering ha is, if he price is higher (lower) han average oday, i is expeced o fall (rise). This mean reversion can make i opimal o inves now raher han waiing.

117 Implicaions Real opions have imporan implicaions. Under some circumsances, he value of invesmen opions is increasing in uncerainy (i.e., he volailiy of he value of he asse). An increase in uncerainy induces firms o defer invesmen. Policy uncerainy (e.g., uncerainy in ax, regulaory or moneary policy) can affec firm invesmen sraegies. One would expec o see greaer levels of invesmen in sable jurisdicions (counries, saes) han unsable ones

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