Some Remarks on Derivaives Markes (hird ediion, 03) Elias S. W. Shiu. The parameer δ in he Black-Scholes formula The Black-Scholes opion-pricing formula is given in Chaper of McDonald wihou proof. A raher unrealisic assumpion on dividend paymens is needed o derive formula (.). The assumpion is given in he firs senence of he las paragraph on page 9: dividends are paid coninuously a a rae ha is proporional o he sock price. More precisely, for each share of he sock, he amoun of dividends paid beween ime and ime +d is assumed o be S()δd, where S() denoes he price of one share of he sock a ime, 0. (Noe ha he book also wries S() as S. The symbol S in formula (.) is he value of he sock price a ime 0, while S in oher formulas, such as hose in Appendix.B, may mean he value of he sock price a ime.) The dividend yield, δ, is a nonnegaive consan. The equaion ha defines he sock price process is (0.), which can be rewrien as ds() + S()δd = S()αd + S()σdZ(). The oal reurn on a sock has wo componens: price change and dividend income. The wo erms on he lef-hand side of his equaion are he wo componens of he oal reurn on he sock beween ime and ime +d. I is poined ou on page 30 ha, if all dividends are re-invesed immediaely, hen one share of he sock a ime 0 will grow o e δt shares a ime T, T 0. Here is an alernaive derivaion. Le n() denoe he number of shares of he sock held a ime under he reinvesmen policy. Thus, n(0) =. Insead of receiving dividends, he invesor receives addiional shares of he sock: n()s()δd = [n(+d) n()]s(). Cancelling S() yields n()δd = dn(). Hence, for T 0, T T d n () δ d = 0, 0 n () or δ T= ln[ nt ( )] ln[ n(0)] = ln[ nt ( )], which means n(t) = e δt. Thus, if we wan one share of he sock a ime T, we can buy e δt shares a ime 0 and reinves all dividends beween ime 0 and ime T. This gives a meaning o he quaniy Se δt in formula (.). More generally, if we buy e δ(t ) shares a ime, < T, and reinves all dividends beween ime and ime T, we ge one share of he sock a ime T. Hence, e δ(t ) S() = F, T ( S), () he ime- prepaid forward price for delivery of one share of he sock a ime T. Wih = 0, his is formula (5.4) on page 30. 3/9/04
. Variaions of he Black-Scholes formula. Formula (.5) I is saed a he end of he firs paragraph on page 354 ha formula (.5) is ineresing because he dividend yield and he ineres rae do no appear explicily; hey are implicily incorporaed ino he prepaid forward prices. Formula (.5) can be used o price opions on a sock ha pays discree dividends. In his case, he sock price process, {S()}, canno be a geomeric Brownian moion, because here mus be a downward ump in he sock price immediaely afer each dividend is paid. In paricular, he logarihm of he sock price canno be a sochasic process wih a consan sandard deviaion per uni ime. So, wha is he σ in formula (.5)? I urns ou ha formula (.5) follows from he assumpion ha he sochasic process of he prepaid forward price for delivery of one share of he sock a ime T, { F, T ( S) ; 0 T}, is a geomeric Brownian moion, wih σ being he sandard deviaion per uni ime of is naural logarihm. If he sock pays dividends coninuously a a rae proporional o is price, hen formula () holds. In his case, he prepaid forward price process, { F, T ( S) }, is a geomeric Brownian moion if and only if he sock price process, {S()}, is a geomeric Brownian moion; boh sochasic processes have he same parameer σ. Formula (.) is a consequence of (.5), bu he converse is no rue because (.) is no applicable for pricing opions on socks wih discree dividends.. Exchange opions I is poined ou in Secion 4.6 ha ordinary calls and pus are special cases of exchange opions. As hined a in he foonoe on page 354, formula (.5) can be generalized o price European exchange opions. For =,, le S () denoe he price of asse a ime, 0. Consider a European exchange opion whose payoff a ime T is max( S( T) S( T),0). () If {ln[ FT, ( S )] ; 0 T} and {ln[ FT, ( S )] ; 0 T} are a pair of correlaed Brownian moions, hen i can be shown ha he ime- price of he European exchange opion is F, T ( S ) ln[ F S F S N, T ( ) /, T ( )] + σ T σ T F, T ( S ) ln[ F S F S N, T ( ) /, T ( )] σ T, σ T 0 < T. (3) Here, Var(ln [ F, T ( S) / F, T ( S)] ) = σ, 0 T. To emphasize he simpliciy of formula (3), le us wrie ν = σ T, and F = F, ( S ), for =,. Then, (3) becomes T 3/9/04
ln[ F / F ] ν ln[ F / F ] F N + ν F N ν, (4) which is no a difficul formula o remember. To see ha formula (4.6) follows from formula (3), we noe he assumpions for (4.6): for =,, { S () } is a geomeric Brownian moion wih volailiy σ and dividends of amoun S () δ dbeing paid beween ime and ime +d; he correlaion coefficien beween he coninuously compounded reurns, ln[ S ( ) / S (0) ] and ln[ S( ) / S (0) ], is ρ. Thus, similar o (), and F, T ( S ) = σ = Var(ln F ( S ) / F ( )] e ν δ ( T ) S (), =,, (5) [, T, T S ) = Var(ln[ S ()/ S () ]) because of (5) = Var(ln[ S () ] ln[ S () ]) = Var(ln[ S () ]) + Var(ln[ S () ]) Cov(ln[ S () ], ln[ S () ]) = σ + σ ρσσ, which is equivalen o (4.7) on page 44..3 All-or-nohing opions The exchange opion payoff, given by (), can decomposed as he difference of wo all-or-nohing opion payoffs, max( S ( T) S ( T ), 0) = S ( T ) I[ S ( T ) > S ( T )] S ( T ) I[ S ( T ) > S ( T )]. Here, I[.] denoes he indicaor funcion, i.e., I[A] = if he even A happens, and I[A] = 0 if A does no happen. (All-or-nohing opions are discussed in Secion 3..) I is no a surprise ha he ime- price of he firs all-or-nohing opion payoff is he firs erm in (3), and he ime- price of he second payoff is he second erm in (3). I urns ou ha i is sufficien o know only one of hese wo formulas. Suppose ha we know ha he ime- price of he ime-t payoff S ( T ) I[ S ( T ) > S ( T )] (6) is F, T ( S ) ln[ F S F S N, T ( ) /, T ( )] + σ T. σ T Then, by symmery, he ime- price of he ime-t payoff S ( T ) I[ S ( T ) > S ( T )] (7) is ln[ F, ( ) /, ( )] T S FT S FT, ( S) N + σ T. σ T Because (7) can be rewrien as S ( T ) S ( T ) I[ S ( T ) S ( T )], 3 3/9/04
is ime- price is ln[, ( ) /, ( )] FT S FT S FT, ( S ) FT, ( S) N + σ T σ T = F ln[ FT, ( S ) / FT, ( S)], T ( S ) N + σ T σ T = F, T ( S ) ln[ F S F S N, T ( ) /, T ( )] σ T, σ T which is indeed he second erm in (3)..4 In erms of forward prices Because FT, ( S ) FT, ( S) =, F ( S ) F ( S ) T, T, he formulas in he las wo subsecions can be expressed in erms of forward prices. For example, he ime- forward price for ime-t delivery of () is ln[ FT, ( S ) / FT, ( S)] FT, ( S) N + σ T σ T ln[ FT, ( S ) / FT, ( S)] FT, ( S) N σ T, 0 < T. (8) σ T Noe ha if F in (4) denoes he forward price FT, ( S ), insead of he prepaid forward price FT, ( S ), =,, hen formula (4) is (8). In oher words, if he F and F in (4) are forward prices, hen (4) gives he forward price for (); if F and F are prepaid forward prices, hen (4) gives he prepaid forward price (i.e., price) for ()..5 Black s formula for pricing opions on zero-coupon bonds Wih he exchange opion formula (3), one can derive formula (5.54), which is Black s formula for pricing a European call opion on a zero-coupon bond. For T, consider S () = (, T + s) and S () = K (, T). Then, F, T ( S ) = (, T + s) and F, T ( S ) = K (, T). We make he assumpion 4 3/9/04
Var(ln[ (, T + s)/ (, T)]) = σ, 0 T. By (3), he ime- price of he European call opion wih ime-t payoff max[0, T (T, T + s) K] (5.5) is ln[ ( T, + s) /[ K( T, )]] (, T + s) N + σ T σ T ln[ ( T, + s) /[ K( T, )]] K (, T) N σ T σ T, (9) which is (5.54). Following (5.53), we le F T, [(T + s)] denoe he ime- forward price for ime-t delivery of a zero-coupon bond ha pays a ime T+s. Then (9) can be rewrien as ln[ FT, [ T ( + s)]/ K] (, T)[ F T, [(T + s)] N + σ T σ T ln[ FT, [ T ( + s)]/ K] K N σ T ], (0) σ T which is formula (5.55). We can also obain (0) using (8) wih F T, ( S ) = F T, [(T + s)] and F T, ( S ) = K. Since formula (8) gives he forward price, muliplicaion wih (, T) yields (0). We noe ha here are five oher expressions for he forward price F T, [(T + s)], namely, F TT,, + s, (T, T + s), (, T + s)/ (, T), (, T + s)/(, T), F T, [(T, T + s)]. For he firs four expressions, see he second and hird paragraphs on page 75. The fifh expression is used in he second ediion of McDonald; i has appeared in Quesion 4 of he May 009 MFE examinaion. Also, boh K in foonoe 8 on page 4 should be K. 3. The parameers δ and σ in he binomial model The quaniies δ and σ also appear in Chapers 0 and, boh of which relae o binomial models. On page 95, δ is called he coninuous dividend yield. On page 30, σ is called he annualized sandard deviaion of he coninuously compounded reurns on he sock. Because a binomial model is a discree model, i seems srange ha hese coninuous-ime conceps are involved. A moivaion for incorporaing δ and σ in a binomial model is ouched upon in Secion.3. By leing he lengh of each ime period, h, end o zero (and he number of periods, n, end o infiniy), we can obain a geomeric Brownian moion model for sock price movemens wih he dividend yield δ and volailiy σ. Noe ha McDonald has suggesed hree pairs of formulas for α ( h)+σ e h u = and α ( h) σ h d = e. In (0.9), α(h) = (r δ)h. In (.3), α(h) = 0, which means u = /d. In (.4), 5 3/9/04
α(h) = (r δ ½σ )h. In he limi as h 0, opion prices derived in hese hree models are he same. The usual model in McDonald is α(h) = (r δ)h. A binomial ree so consruced is called a forward ree (page 303). In his case, he risk-neural probabiliies are and ( r δ) h ( r δ) h ( r δ) h σ h σ h e d e e e p* = = = = u d e e e e + e σ h ( r δ) h+ σ h ( r δ) h σ h σ h σ h σ h e p* = =. σ h σ h + e + e Because σ > 0, we have p* < ½ < p*, which may be viewed as a buil-in bias in he model. 4. Greeks Greeks are parial derivaives of an opion price. The definiions given on page 356 are numerical approximaions. We need o be careful abou he unis in which changes are measured. For example, i is saed on page 356 ha []hea (θ) measures he change in he opion price when here is a decrease in he ime o mauriy of day. However, he mahemaical definiion for hea is he parial derivaive of he opion price wih respec o. In he Black-Scholes opion-pricing formula, he variable is (usually) in years. Thus i is saed on page 379 ha []o obain a per-day hea, divide by 365. If we measure hea per rading day insead of per calendar day, we divide by 5. For call and pu opions, dela, rho and psi are very easy o derive. Sar wih he Black-Scholes opion pricing formula (.) or (.4). Differeniae wih respec o S, r or δ, while preending d and d are consan. (You can check hese six parial derivaives wih he formulas in Appendix.B.) To see his, denoe formula (4) as V( F, F ). The parial derivaive V( F, F) is F ln[ F/ F] ν ln[ F/ F] ν ln[ F/ F] ν N + FN FN + + F, ν ν ν ν νf he las wo erms of which cancel o zero because of he ideniy ( x+ y) / ( x y) / xy e = e e. (The ideniy above also explains he las formula in foonoe 6 on page 379.) Hence, ln[ F/ F] ν V( F, F) = N +. () F ν Similarly,, Wih F ln[ F / F ] ν V( F, F) = N. ν δ ( T ) =, i follows from he chain rule ha F e S () 6 3/9/04
and F δ ( T ) [ ] = [ ] = [ ] e S F S F F δ ( T ) [ ] = [ ] = [ ] Se ( T ) δ F δ F Hence, for call and pu opions, dela, rho and psi can be derived by preending ha d and d are consan. Applying () and () o (4) yields (, ) (, ) V F F = F V F F + F V ( F, F F ). F This may remind some sudens of he Euler heorem for homogeneous funcions in mulivariae calculus.. 5. Ineres raes In McDonald, ineres raes are usually coninuously compounded raes (forces of ineres). Excepions are Appendix.B, he nonannualized ineres rae R(T, T+s) in Secion 5., and he Black-Derman-Toy Model in Secion 5.4 and Appendix 5.A. 6. Marke price of risk The no-arbirage argumen in Secion 0.5 is an imporan insigh in finance. Le us rewrie (0.6), (0.7) and he equaion in foonoe 8 on page 68 as follows: for =,, ds = ( α δ )d+ υdz, S where δ is he (no necessarily consan) dividend yield of asse, and υ = σ. Noe ha he price dynamics of boh asses are driven by he same Brownian moion {Z()}, and ha α, δ and υ can depend on. In his more general seing, equaion (0.8) akes he form α r α r (ds+ Sδd ) (ds+ Sδd ) + rd = d. υs υs υ υ υ υ Therefore, o preclude arbirage, we mus have α r α r =. (3) υ υ Many auhors use he erm marke price of risk for he raio ( α r)/ υ. The Sharpe raio is he same as he marke price of risk if he denominaor, υ, is posiive. Two imporan applicaions of he no-arbirage condiion (3) are α r αopion r =, (4) σ sgn( Ω) σ opion 7 3/9/04.
which is a correced version of (.), and α( r (),, T) r () α( r (),, T) r () =, qr ( ( ), T, ) qr ( ( ), T, ) which is equivalen o (5.4). Here, sgn(ω) denoes he sign of Ω, he opion s elasiciy. In a Black-Scholes model, he marke price of risk for every raded asse is he sock s Sharpe raio, (α r)/σ, which does no depend on ime or sock price. In he ineres rae model in Secion 5., he marke price of risk a ime for every bond is φ(r, ), where r is he value of he shor-erm ineres rae a ha ime. Besides (.), formulas (.7), (.8) and (.9) also conain ypographical errors. The expecaion operaor E in (.7) and (.8) should be wrien as E, because he expecaion is condiional on informaion up o ime. The wo erms on he lef-hand side of (.8) should be swiched. In (.9), sgn(ω) is missing. The correced (.8) and (.9) are: dv E (d V ) SVS dz V V = V σ (.8) = sgn( Ω) σ opiondz (.9) Also, he unnumbered equaion beween (.9) and (.30) on page 637 should be E (d S) = S ( r δ )d. I follows from (4) or from (.) (or from (.0)) ha αopion r = Ω (α r). ( T ) For a pu opion, dela a ime is e δ N( d ), which is negaive. Hence, sgn(ω) is negaive. If α > r (which seems o be a naural assumpion), we have αpu opion < r. This means ha if one wans o value a pu opion using he acual probabiliy measure, hen for discouning, one needs o use an ineres rae ha is lower han he risk-free ineres rae or even a negaive ineres rae. This may seem srange o some acuaries. 7. Risk-neural valuaion I is a raher amazing resul ha, under he assumpion of no arbirage (and he securiies marke being fricionless), he price of each derivaive securiy is he expeced presen value of is payoffs. Of course, acuaries have been calculaing expeced presen values for over wo cenuries. However, he expeced presen value in finance is differen in wo respecs: (i) he discouning is calculaed using he risk-free ineres rae; (ii) he expecaion is aken wih respec o a so-called risk-neural probabiliy measure. For an opion or a claim ha pays V( ST ( ), T ) a ime T, is price a ime, T, is rt ( ) E [ e V( ST ( ), T)]. (5) Wih = 0, (5) is he righ-hand side of (9.) and (0.3). The aserisk in (5) signifies ha he expecaion is aken wih respec o he risk-neural probabiliy measure, and he subscrip means ha he expecaion is aken condiional on informaion up o ime, in paricular, he sock price a ime, S(). Noe ha he righ-hand side of he firs displayed equaion on page 639 is he inegral form of (5) for he ime-t payoff 8 3/9/04
V( ST ( ), T) = max[0, ST ( ) K]. If he risk-free ineres rae is no consan, hen he discoun facor T rt ( ) e is rs ( )ds replaced by e ; see formula (5.9). For example, consider a zero-coupon bond mauring for a ime T. Then, is price a ime, T, is T rs ( )ds e E [ ], which is he righ-hand side of (5.0). (Noe ha boh E on page 759 should be changed o E.) The noion of prepaid forward prices does no seem o be in oher finance exbooks. In any case, (5) gives you a formula for deermining he prepaid forward price. Tha is, he ime- prepaid forward price for delivery of V( ST ( ), T ) a ime T is (5). As an applicaion, le us derive he relaionship of pu-call pariy. I follows from he ideniy max[x, 0] max[ x, 0] = x ha max[s(t) K, 0] max[k S(T), 0] = S(T) K. Discouning each erm wih he risk-free ineres rae and aking expecaion wih respec o he risk-neural probabiliy measure, we obain he ideniy C(K, T) (K, T) = F0, ( S ) V0,T(K), T which is (3.). Secion 0.6, eniled Risk-neural valuaion, is an imporan secion, bu i has several ypographical errors. In (0.9), Z () should be Z(). In (0.30), Z() should be Z (). Noe ha (0.9) and (0.30) are he same equaion: d S () = ( α δ )d+σ d Z ( ) S () α r = ( r δ )d+σ d+ d Z ( ) σ = ( r δ )d+σd Z ( ), because r Z α () = + Z () σ. (6) Under he risk-neural probabiliy measure, he sochasic process { Z ( )} is a sandard Brownian moion. Noe ha he fracion in (6) is he marke price of risk. (As poined ou in he las secion, he marke price of risk in Secion 5. is φ(r, ). Analogous o (6), { Z ( )} in Secion 5. is given by d Z () = φ ( r (),)d + d Z (). You can check his by comparing (5.6) wih (5.3).) 9 3/9/04
Every Z on page 69 and 60 should be changed o Z. The expecaion operaor E in he middle of page 69 should be wrien as E, or more precisely, as E 0. The word expeced in he hird senence of he second paragraph on page 69 should be deleed. To obain he firs displayed, bu un-numbered, formula on page 69 (wih Z(T) changed o ZT ( )), we can sar wih (0.). By applying (6) o (0.), we have ( r δ ½ σ ) +σz () S ( ) = S(0) e. Then, replace by T, and raise boh sides o power a. Here is an ineresing resul ha follows from (0.3) in roposiion 0.3. If a is a number such ha r + a(r δ) + ½a(a )σ = 0, (7) hen a a FT, ( S ) = [ S ( )]. (8) Equaion (7) is a quadraic equaion in a. Thus, here are wo values of a for which (8) holds. They are given in Exercise. wih γ = 0. (They are also given by h and h on page 373 and page 695. h and h have appeared in Quesion 6 of he May 007 MFE examinaion, bu Secion.6 is no on he MFE syllabus anymore.) 8. Furher references This secion is NOT par of he MFE syllabus. A popular exbook ha reas opics similar o hose in McDonald is John Hull, Opions, Fuures, and Oher Derivaives, renice-hall. On he Inerne, you can obain an early ediion of his book raher cheaply. For a deeper undersanding of he maerial in he MFE syllabus, here are wo of many fine books: Marin Baxer and Andrew Rennie (996) Financial Calculus: An Inroducion o Derivaive ricing, Cambridge Universiy ress. Tomas Börk (009) Arbirage Theory in Coninuous Time, 3 rd ed., Oxford Universiy ress. The following survey paper can be downloaded via JSTOR G. Malliaris (983) Io's calculus in financial decision making, SIAM Review 5 no. 4, 48-496. There are now numerous lecure noes and videos available on he Inerne. The Universiy of Chicago has a Maser of Science in Financial Mahemaics program. Inroducory lecures can be found in hp://www-finmah.uchicago.edu/sudens/sep_review.shml In paricular, I recommend viewing rofessor Rober Fefferman s lecures on probabiliy. Also, you may wan o read he following -page aricle abou rofessor Kyosi Iô, whose lemma is he heme of Secion 0.4. hp://www.ams.org/noices/00706/x070600744p.pdf Noe he senence: Indeed, one can argue ha mos of applied mahemaics radiionally comes down o changes of variables and Taylor-ype expansions. Finally, here is a ime-honored ool in acuarial science called he Esscher ransform. I urns ou o be a very efficien mehod for pricing financial opions, especially hose involving muliple asses. hp://www.soa.org/library/research/ransacions-of-sociey-of-acuaries/990-95/994/anuary/sa94v467.aspx 0 3/9/04