4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY 6-439 Email: maxmasuda@maxmasuda.com hp://www.maxmasuda.com/ December 4 his paper presens everyhing you need o know abou Black-choles model which is ruly single mos imporan revoluionary work in he hisory of quaniaive finance. Alhough B model has is flaws such as he normally disribued (i.e. zero skewness and zero excess kurosis) log reurn densiy and he assumpion of consan volailiy across srike prices and he ime o mauriy, i ouperforms more (so-called) advanced models in numerous cases. 4 azuhisa Masuda All righs reserved.
4 azuhisa Masuda All righs reserved. [] andard Brownian Moion: Building Block of B Model A sandard Brownian moion filered probabiliy space ( [, ) B ) ( Ω, F, P) [, ) is a real valued sochasic process defined on a saisfying: () Is incremens are independen. In oher words, for < <... < n <: P ( B... ) B B B B B B n n = P ( B ) ( ) ( )... ( ) P B B P B B B B P n. n () Is incremens are saionary (ime homogeneous): i.e. for h, B+ h B has he same disribuion as B h. In oher words, he disribuion of incremens does no depend on. (3) P( B = ) =. he process sars from almos surely (wih probabiliy ). (4) B Normal(, ). Is incremens follow a Gaussian disribuion wih he mean and he variance. I urns ou ha a sandard Brownian moion ( [, ) B ) saisfies he following condiions: () he process is sochasically coninuous: ε >, lim P ( X X ε ) =. () Is sample pah (rajecory) is coninuous in (i.e. coninuous rcll) almos surely. For more deails abou Brmwoan moion, consul Masuda (5) Inroducion o Brownian Moion. h [] Black-choles Disribuional Assumpions on a ock Price In radiional finance lieraure almos every financial asse price (socks, currencies, ineres raes) is assumed o follow some variaions of Brownian moion wih drif process. B (Black-choles) models a sock price incremen process in an infiniesimal ime inerval d as a log-normal random walk process: d µ d σ db + h = +, () where he drif is µ which is a consan expeced reurn on a sock µ proporional o a sock price and he volailiy is σ which is a consan sock price volailiy σ proporional o a sock price. he reason why he process () is called a log-normal random walk process will be explained very soon. Alernaively, we can sae ha B models a percenage change in a sock price process in an infiniesimal ime inerval d as a Brownian moion wih drif process:
4 azuhisa Masuda All righs reserved. d = µ d + σdb, () d ( d / µ d) P ( ) = exp[ ]. πσ d σ d Le be a random variable whose dynamics is given by an Io process: d = a(, ) d + b(, ) db, and V be a funcion dependen on a random variable V(, ) is given by an Io formula: and ime. he dynamics of V V V dv = d + d + b d, (3) or in erms of a sandard Brownian moion process B : V V V dv = d + ( ad + bdb) + b d, V V V V dv = + a + b d b db +. (4) Dynamics of a log sock price process ln can be obained by applying (4) o () as: ln ln ln ln dln = + µ + σ d σ + db. ubsiuing ln =, ln =, and ln = yields: ln = µ σ + σ d d db, (5) or: ln ln = µ σ ( ) + σ( B B) ln = ln + µ σ + σb. (6) 3
4 azuhisa Masuda All righs reserved. he equaion (6) means ha B models a log sock price l n as a Brownian moion wih drif process whose probabiliy densiy is given by a normal densiy: ln ln + ( µ σ ) (ln ) exp[ P = ]. (7) πσ σ Alernaively, he equaion (6) means ha B models a log reurn ln ( / ) as a Brownian moion wih drif process whose probabiliy densiy is given by a normal densiy: ln ( / ) = µ σ + σb, ln ( / ) µ σ P ( ln ( / )) = exp[ ]. (8) πσ σ An example of B normal log reurn ln ( / ) densiy of (8) is illusraed in Figure. Of course, B log reurn densiy is symmeric (i.e. zero skewness) and have zero excess kurosis because i is a normal densiy. Densiy.5.5.5 - -.5.5 log reurn lnh ê L Figure : An Example of B normal log reurn ln( / ) Densiy. Parameers and variables fixed are µ =., σ =., and =.5. Le y be a random variable. If he log of y is normally disribued wih mean variance b such ha ln y N( a, b ), hen y is a log-normal random variable whose densiy is a wo parameer family ( ab), : a and a+ b a+ b b (, ( y Lognormal e e e )), 4
4 azuhisa Masuda All righs reserved. { ln y a} P ( y) = exp[ ]. y π b b From he equaion (6), we can sae ha B models a sock price disribued random variable whose densiy is given by: as a log-normally P ( ) Is annualized momens are calculaed as: ln ln + ( µ σ ) exp[ = ]. (9) πσ σ Mean[ ] = e µ, ( ) σ ( ) Variance[ ] e e σ µ =, kewness = e + e, σ [ ] 3 σ σ 4σ Excess urosis[ ] 6 3e e e = + + +. An example of B log-normal sock price densiy of (9) is illusraed in Figure. Noice ha B log-normal sock price densiy is posiively skewed..5.4 Densiy.3.. 3 4 5 6 7 8 ock price Figure : An Example of B Log-Normal Densiy of a ock Price. Parameers and variables fixed are = 5, µ =., σ =., and =.5. able Annualized Momens of B Log-Normal Densiy of A ock Price in Figure Mean andard Deviaion kewness Excess urosis 55.585.63.6495.678366 5
4 azuhisa Masuda All righs reserved. From he equaion (6), we can obain an inegral version equivalen of (): exp[ln ] = exp[ln + µ σ + σb ] = exp[ln ]exp µ σ σb + = exp µ σ + σb. () Equaion () means ha B models a sock price dynamics as a geomeric (i.e. exponenial) process wih he growh rae given by a Brownian moion wih drif process: + B. µ σ σ [3] radiional Black-choles Opion Pricing: PDE Approach by Hedging Consider a porfolio P of he one long opion posiion V(, ) on he underlying sock wrien a ime and a shor posiion of he underlying sock in quaniy o derive opion pricing funcion. P = V(, ). () Porfolio value changes in a very shor period of ime d by: dp = dv (, ) d. () ock price dynamics is given by a log-normal random walk process of he equaion (7.): d µ d σ db = +. (3) Opion price dynamics is given by applying Io formula of he equaion (3): dv = d + d + d. (4) V V V σ Now he change in he porfolio value can be expressed as by subsiuing (3) and (4) ino (): 6
4 azuhisa Masuda All righs reserved. V V V σ dp = d + d + d d. (5) eing = V / (i.e. dela hedging) makes he porfolio compleely risk-free (i.e. he randomness d has been eliminaed) and he porfolio value dynamics of he equaion (5) simplifies o: V V dp = + σ d. (6) ince his porfolio is perfecly risk-free, assuming he absence of arbirage opporuniies he porfolio is expeced o grow a he risk-free ineres rae r : EdP [ ] = rpd. (7) Afer subsiuion of () and (6) ino (7) by seing = V /, we obain: V V V + σ d r V = Afer rearrangemen, Black-choles PDE is obained: d. V V V + + r rv(, ) =. (8) (, ) (, ) (, ) σ B PDE is caegorized as a linear second-order parabolic PDE. he equaion (8) is a linear PDE because coefficiens of he parial derivaives of V(, ) (i.e. σ / and r ) are no funcions of V(, ) iself. he equaion (8) is a second-order PDE because i involves he second-order parial derivaive V(, )/. Generally speaking, a PDE of he form: is said o be a parabolic ype if: V V V V V a+ b + c + d + e + g = g 4de=. (9) he equaion (8) is a parabolic PDE because i has g = and e = which saisfies he condiion (9)., 7
4 azuhisa Masuda All righs reserved. B solves PDE of (8) wih boundary condiions: ( ( max,) for a plain vanilla call, ) max, for a plain vanilla pu, and obains closed-form soluions of call and pu pricing funcions. Exac derivaion of closed-form soluions by solving B PDE is omied here (i.e. he original B approach). Insead we will provide he exac derivaion by a maringale asse pricing approach (his is much simpler) in he nex secion. [4] radiional Black-choles Opion Pricing: Maringale Pricing Approach Le { B ; } be a sandard Brownian moion process on a space ( Ω, F, P). Under acual probabiliy measure P, he dynamics of B sock price process is given by equaion (9) in he inegral form (i.e. which is a geomeric Brownian moion process): exp. () = µ σ + σb B model is an example of a complee model because here is only one equivalen maringale risk-neural measure Q P under which he discouned asse price process r { e ; } becomes a maringale. B finds he equivalen maringale risk-neural measure Q P B by changing he drif of he Brownian moion process while keeping he volailiy parameer σ unchanged: exp QB = r σ + σb. () B Noe ha B Q is a sandard Brownian moion process on ( Ω, F, Q B ) and he r discouned sock price process { e ; } is a maringale under and wih respec o he filraion { F ; }. hen, a plain vanilla call opion price C (, ) which has a erminal payoff funcion max, is calculaed as: ( ) ( ) C e E r ( ) QB (, ) = max, Q B F. () Le Q( ) (drop he subscrip B for simpliciy) be a probabiliy densiy funcion of in a risk-neural world. From he equaion (9), a erminal sock price is a log-normal random variable wih is densiy of he form: 8
4 azuhisa Masuda All righs reserved. Q ( ) ln ln + ( r σ ) τ exp[ = ]. (3) στ πσ τ Using (3), he expecaion erm in () can be rewrien as: max (,) = ( ) Q( ) + ( ) ( ) Q F max (,) F = ( ) Q( F) d. Q E F F d d Q E Using his, we can rewrie () by seing τ as: ( ) ( ) C(, τ ) = e Q F d. (4) ince is a log-normal random variable wih is densiy given by he equaion (3): ln F Normal m ln + ( r σ ) τστ,. (5) For he noaional simpliciy, le ln from a log-normal random variable wih: From (6): Z F ln. We use a change of variable echnique + r σ τ σ τ ln ln ( ) o a sandard normal random variable ln m N (,) ( Z ) σ τ Z exp[ ] Z =. π Z hrough: ormal, (6) = exp( Zσ τ + m). (7) We can rewrie (4) as: ( ( ) ) ( ) C ( τ, ) e r = τ exp Zσ τ + m Z Z dz, (ln m) / σ τ 9
4 azuhisa Masuda All righs reserved. and we express his wih more compac form as: C(, ) C C τ =, (8) where = exp( σ τ + ) ( ) C e Z m Z dz Consider C : Z and ( ) (ln m) / σ τ ( σ τ ) exp( ) ( ) C e = exp Z m Z Z dz (ln m) / σ τ C = e Z Z dz. (ln m) / σ τ C = exp r exp ln + ( r ) exp Z Z ( τ) σ τ ( σ τ ) ( ) Z dz (ln m) / σ τ ( ) ( ) Z dz Z C = exp ln στ exp( Z ) exp[ ] σ τ dz (ln m)/ σ τ π C = exp ln στ exp Zσ τ Z (ln m) / σ τ Z Zσ τ C = exp ln στ exp[ ] dz (ln m) / σ τ π ( ) Z σ τ σ τ C = exp ln στ exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp ln στexp στ exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp( ln ) exp[ ] dz (ln m) / σ τ π ( Z σ τ ) C = exp[ ] dz. (9) (ln m) / σ τ π Use he following relaionship: Equaion (9) can be rewrien as: b a ( ) Z c b c Z exp[ ] dz = exp[ ] dz a c π π. Z exp[ ] (ln m)/ σ τ σ τ π C = dz. (3)
4 azuhisa Masuda All righs reserved. Le N ( ) be he sandard normal cumulaive densiy funcion. Using he symmery of a normal densiy, (3) can be rewrien as: (ln m)/ σ τ + σ τ Z exp[ ] C = dz π ln + m C = N + σ τ σ τ. (3) From (5), subsiue for m. he equaion (3) becomes: ln + ln + ( r σ ) τ C = N + σ τ σ τ ln ( r ) ln ( r ) σ τ σ τ σ τ + + + + C = N = N σ τ σ τ (3) Nex, consider C in (8): (ln m)/ σ τ ( ) ( ) C e r Z dz e τ Z dz = Z = (ln m)/ σ τ Z r ln + m C = e τ N. (33) σ τ From (5), subsiue for m. he equaion (33) becomes: ln + ln + ( r σ ) τ ln ( r ) rτ + σ τ C = e N = e N. (34) σ τ σ τ ubsiue (3) and (34) ino (8) and we obain B plain vanilla call opion pricing formula: ( ) ( ) C(, τ ) = N d e N d, (35) where d ln + ( r + σ ) τ = and σ τ d ln ( ) + r σ τ = = d σ τ. σ τ
4 azuhisa Masuda All righs reserved. Following he similar mehod, B plain vanilla pu opion pricing formula can be obained as: ( ) ( ) P(, τ ) = e N d N d. (36) We conclude ha boh PDE approach and maringale approach give he same resul. his is because in boh approaches we move from a hisorical probabiliy measure P o a riskneural probabiliy measure Q. his is very obvious for maringale mehod. Bu in PDE approach because he source of randomness can be compleely eliminaed by forming a porfolio of opions and underlying socks, his porfolio grows a a rae equal o he riskfree ineres rae. hus, we swich o a measure Q. For more deails, we recommend Nefci () pages 8-8 and 358-365. [5] Alernaive Inerpreaion of Black-choles Formula: A ingle Inegraion Problem Under an equivalen maringale measure Q P under which he discouned asse price r process { e ; } becomes a maringale, a plain vanilla call and pu opion price which has a erminal payoff funcion as: ( ) and max (,) max, ( ) ( ) C e E r ( ) Q (, ) = max, P e E r ( ) Q (, ) = max, are calculaed F, (37) F. (38) Noe ha an expecaion operaor E [ ] is under a probabiliy measure Q and wih respec o he filraion F. Le Q( F ) be a condiional probabiliy densiy funcion of a erminal sock price. For he noaional simpliciy we use Q ( F ) Q( ) and τ. he expeced erminal payoffs in he equaions (37) and (38) can be rewrien as: ( ) F = ( ) Q( ) Q E max, ( ) F = ( ) Q( ) Q E max, Using hese, we can rewrie (37) and (38) as: d, d. ( ) Q( ) C(, τ ) = e d, (39)
4 azuhisa Masuda All righs reserved. ( ) ( ) P(, τ ) = e Q d. (4) B assumes ha a erminal sock price densiy of he form: is a log-normal random variable wih is ln ln + ( r σ ) τ Q ( ) = exp[ ]. στ πσ τ herefore, B opion pricing formula comes down o a very simple single inegraion problem: ln ln + ( r σ ) τ = ( ) στ πσ τ C( τ, ) e exp[ ] d ln ln + ( r σ ) τ = ( ) στ πσ τ P( τ, ) e exp[ ] d, (4). (4) his implies ha as far as a risk-neural condiional densiy of he erminal sock price Q F ) is known, plain vanilla opion pricing reduces o a simple inegraion problem. ( [6] Black-choles Model as an Exponenial Lévy Model he equaion () ells us ha B models a sock price process as an exponenial Brownian moion wih drif process: which means: = +, exp µ σ σb =, e L where he sock price process { : } is modeled as an exponenial of a Lévy process { L ; }. Black and choles choice of he Lévy process is a Brownian moion wih drif (coninuous diffusion process): 3
4 azuhisa Masuda All righs reserved. + L µ σ σb. (43) he fac ha an sock price is modeled as an exponenial of Lévy process L means ha is log-reurn ln( ) is modeled as a Lévy process such ha: ln( ) = L = + µ σ σb. B model can be caegorized as he only coninuous exponenial Lévy model apparenly because a Brownian moion wih drif process is he only coninuous (i.e. no jumps) Lévy process. his indicas ha he Lévy measure of a Brownian moion wih drif process is zero: ( dx ) =, and obviously is arrival rae of jumps is zero: ( dx ) =. References Black, F. and choles, M., 973, he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy 3. Hull, J. C.,, Opions, Fuures, and Oher Derivaives (5h Ediion), Prenice Hall. Masuda,., 5, Inroducion o Brownian Moion. Nefci,. N.,, An Inroducion o he Mahemaics of Financial Derivaives, Academic Press. Wilmo, P., 998, Derivaives, John Wiley & ons. 4