A Simple Approach to CAPM, Option Pricing and Asset Valuation

Transcription

1 A imple Approach o CAPM, Opion Pricing and Asse Valuaion Riccardo Cesari (*) Universià di Bologna, Dip. Maemaes, viale Filopani, Bologna, Ialy rcesari@economia.unibo.i Carlo D Adda Universià di Bologna, Dip. cienze economiche, srada Maggiore, Bologna, Ialy dadda@spbo.unibo.i Absrac In his paper we propose a simple, inuiive approach o asse valuaion in erms of marginal conribuions o he characerisics (momens) of he marke porfolio. Considering only he firs wo momens, mean and variance, he valuaion equaion is shown o correspond o harpe s CAPM. A risk-neural pricing formula is easily derived, showing he equivalence beween CAPM and he Black and choles model. Exensions o higher momens like skewness and kurosis are sraighforward, providing a generalized valuaion equaion. Finally, he generalized equaion is derived in a differen, more rigorous way, as a resul of a classical ineremporal general equilibrium model. We would like o hank Emilio Barone, Umbero Cherubini and Fabio Panea for encouragemen and helpful commens o a previous version. The auhors bear any responsibiliy. (*) Corresponding auhor

2 . Moivaion The Capial asse pricing model and he opion pricing heory are wo of he bes known and mos imporan developmens in he subjec of Finance. The firs model is provided by William harpe (964) even if Tobin (958), Treynor (965), Linner (965) and Mossin (966) reached similar resuls during he same period and all of hem are indebed o he Markowiz (95, 959) porfolio model. The Opion pricing heory, on he oher hand, derives from he seminal paper of Black and choles (973), in which an arbirage argumen is developed o solve he old problem of pricing opion conracs in a compleely new way. Nowadays, he wo models have become he cornersones of any financial curriculum sudiorum. In paricular, a suden of Economics learns abou CAPM during his second year courses and he golden formula he finds in his handbook is: R = R + β ( R R ) () j F jm M RF where R j E (R j ) is he expeced (a ime ) rae of oal reurn of sock j, R RF is he risk-free rae, R M is he marke rae of reurn and β jm is he bea coefficien, measuring he risk of he sock and defined by he covariance beween R j and R M divided by he variance of R M. If our suden is clever enough, he will undersand ha CAPM, as a capial asse pricing model, is an equilibrium model o price financial asses of any kind, even if sandard implemenaion is usually limied o common socks. In fac, if () is he price a ime of asse j and M() is he price (index) of he marke, using he definiion of rae of reurn beween curren ime and a fuure dae T (excluding dividends for simpliciy): R j = T ( ) ( ) () () and, subsiuing ino Equaion, he can obain he CAPM formula in price erms: () = E ( ( T)) E( M( T) M( )( + RRF)) Cov + R ( + R ) Var ( M( T)) RF RF ( ( T), M( T)) (3) The inerpreaion of he price formula is sraighforward: he curren price of he asse is he fuure price expeced oday E ((T)) and discouned a he risk-free rae minus a risk adjusmen ha depends on he covariance beween he asse and he marke.

3 In a more compac form: ( ) = P E ( T ( )) + P Cov ( T ( ), MT ( )) (4) RF M where P RF is he curren price of a zero coupon bond, giving one uni of money a ime T, and P M is he (negaive) price of one uni of risk (i.e. covariance). The following year, during his Finance classes, our suden learns abou opion pricing using a compleely differen se-up and obaining a compleely differen resul for he price of derivaive asses. In he simples case of a European call opion, which gives he righ o buy a specified asse (underlying) a a given dae T, paying a given amoun K (srike price), he celebraed Black and choles (973) model provides he price of he call: r( T ) C () = () Φ( d ) Ke Φ( d ) where d ln(( ) / K) + ( r+ / )( T ) = T d = d T (5) r is he consan, coninuously compounded risk-free rae and Φ(x) is he probabiliy of a number less han or equal o x according o he sandard normal disribuion funcion. Even if he recognises ha coninuous and discree compounding are equivalen, in he sense ha e r( T ) is he same as, he wo + R RF approaches will sill appear o be quie differen. Can hey be compaible? Presened in differen conexs, by differen eachers, in differen academic years, he wo models seem o belong o differen secions of Finance and he link beween hem, if i exis, appears compleely lacking 3. Moreover, CAPM is very general, concerning socks, bonds and derivaive asses, including pus and calls, bu opion pricing is less special han i appears if you bear in mind ha common socks are call opions wrien on he asses of he firm and corporae bonds are equivalen o defaul-free bonds plus a shor posiion in pus. Are hey herefore wo compeing models 4 of asse pricing? In he following secions we shall show ha he wo models are special cases of a more general valuaion equaion. In paricular, we shall presen a simple and inuiive pricing model, which includes CAPM (secion ), riskneural pricing and herefore he opion approach (secion 3), a direc roue o generalizaions wih explici expressions for prices and esable 3

4 resricions on risk premia (secion 4) and a derivaion of he same pricing funcion hrough a more rigorous ineremporal general equilibrium model (secion 5). A final secion concludes he paper.. Inuiion In (micro)economics, goods are priced a he margin (uiliy of a marginal quaniy). We could, herefore, ry o use his principle in finance, o price financial asses. In order o do his, we need a simple, basic assumpion according o which a financial good (or asse) is jus a bundle of characerisics. Jus as consumer goods are physical objecs wih physical characerisics (see Lancaser (966)), financial asses are random variables (random processes) wih momens as characerisics: mean, variance, skewness ec. Each asse is priced a he margin, in erms of is marginal conribuions o he measures of he characerisics of he global marke porfolio, he price of he asse being he sum of price, P i, imes marginal quaniy, Mch i, of each characerisics: ( ) = P ( ) Mch ( ) + P ( ) Mch ( ) +... (6) If his is he inuiion, le us give he simples conceivable example. Example. uppose ha he expeced fuure (ime T) value (he mean or firs momen) is he only relevan characerisic. The issue of g unis of asse in a perfecly compeiive marke has a differenial effec on he characerisic of he marke porfolio: ch () = E (M(T)+g(T))-E (M(T)) = ge ((T)) and he marginal characerisic is jus he limi of he raio ch () as he g quaniy g goes o zero: Mch ch () () lim g 0 g Therefore he price of he asse is he marginal effec in he characerisic muliplied by he price of he characerisic: () = P () Mch () = P () E (( T)) (7) Wha abou P ()? By definiion, i is he compeiive price of one uni of 4

5 he characerisic (he mean). If a defaul-free zero-coupon bond exiss mauring a T and paying a ha ime one uni of money, is price P RF ()=e -r(t-) mus saify Equaion 7: P RF ()=P ()E ()=P ()= e -r(t-) so ha he price of he firs characerisic can be idenified wih he discoun funcion, and Equaion 7 becomes: () = e -r(t-) E ((T)) (8) If he mean is he only relevan characerisic, prices reflec he risk-neural valuaion principle: he price of an asse is he expeced fuure value discouned a he risk-free rae. Example. uppose, as a second sep, ha mean and variance (he firs wo momens) are he relevan characerisics. The issue of g unis of asse implies he following differenial effecs: ch ()= E (M(T)+g(T))-E (M(T)) = ge ((T)) ch () = Var (M(T)+g(T))-Var (M(T)) = g Var ((T))+gCov ((T),M(T)) (9) Therefore, aking he limis: Mch i ch i () () lim i =, g 0 g () = P () Mch () + P () Mch () = P ( ) E (( T)) + P ( ) Cov (( T), M( T)) (0) Once again, we have o idenify P () and P (). The zero-coupon bond and he marke porfolio can be used o inver he price formula, obaining P and P : P RF () = P ()E ()+P ()Cov (,M(T)) = P () M() = P RF ()E (M(T))+P ()Var (M(T)) so ha P () is obained in erms of observable variables: P () = M () P () E( MT ( )) RF Var ( M( T)) 5

6 ubsiuing, we have harpe s CAPM of Equaion 3: () = P () E(( T)) + RF M () PRF () E ( MT ( )) Cov Var ( M( T)) (( T), M( T)) Noe ha he asse price is made by a risk-neural componen plus a riskadjusmen componen. 3. Equivalen pricing funcions Le us wrie Equaion 0 in an equivalen form, collecing P RF and he expecaion operaor: r( T ) P () () = e E T ( ) + MT ( )(( T) E (( T))) PRF () () In his way, he asse price appears as he discouned (naural) expecaion of a risk-adjused argumen, he expression in square brackes. Now define an expecaion operaor $E such ha: P $ () E(( T)) E ( T) + MT ( )(( T) E (( T))) PRF () Clearly, he risk-adjusmen in $E has been made hrough he probabiliy disribuion, no he argumen. We can, herefore, wrie Equaion as: r( T ) () e E$ (( T)) () and, comparing i wih Equaion 8 of Example, i should be no surprise ha $E is called he risk-neural expecaion operaor. Bu how can we obain $E from an operaional poin of view? Le µ be he compounded average rae of reurn of asse. We have, by definiion: µ T r T () E ( e ( ) T ( )) E$ ( e ( ) T ( )) so ha he risk-neural expecaion $E is he naural expecaion E wih he average rae µ subsiued by he risk-free rae r. Given ha µ in he heory of probabiliy is called drif coefficien, he previous resul is an applicaion 6

7 of Girsanov s heorem of drif change (Duffie, 99 p. 37). Example 3. Consider he case of a European call opion, giving he righ, a mauriy T, o he payoff C(T)=max(0, (T)-K). According o he wo-momen pricing of Example, we have he call price: C () = P () E( CT ( )) + P() Cov( CT ( ), MT ( )) = P ()$ E( CT ( )) RF RF The firs expression is he naural pricing funcion, which requires he calculaion of naural expecaions, he second one is he risk-neural pricing funcion, requiring o subsiue he average rae µ wih r. For example, if (T) and M(T) are joinly normally disribued 5 we obain he naural pricing formula: ( K ( ) e ) µ τ e K C () = PRF ()[ exp( ) + (( e ) K) ( () Φ )] π e K P () Cov (( T), M( T)) ( () + Φ ) µ τ ( e ) τ T µ µ τ µ τ and he equivalen risk-neural pricing formula: µ τ rτ r rτ $ ( K ( ) e ) rτ e C () = e exp( ) + (( ) Ke ) Φ( () π $ $ wih $ (exp( rτ) ) r τ T τ K ) If (T) and M(T) are joinly lognormally disribued 6 we obain he naural wo-momen pricing formula: C() = P E (C(T)) + P () Cov (C(T),M(T)) where: 7

8 µ + log K µ K E C T K log ( ( )) = exp( µ + ) Φ( ) Φ( ) Cov ( C( T), M( T)) = M µ + + ρmm log K exp( µ + + µ M + + ρmm ) Φ( ) M µ + log K exp( µ + + µ M + ) Φ( ) M µ + ρmm log K µ K exp( µ M + ) Φ( ) Φ( µ log(( )) + ( µ / ) τ τ τ T log K ) and he risk-neural formula is given by he Black and choles model in Equaion Exensions Lognormaliy and oher non normal disribuions sugges ha no only mean and variance bu also higher momens like skewness and kurosis should be included in he pricing funcion 7. Along he line of secion we calculae he differenial effec on he marke porfolio of g unis of asse in erms of hird (skewness) and fourh (kurosis) cenral momens, and le g go o zero in order o ge a marginal effec. The differences in he marke momens before and afer he issue of g unis of asse are, respecively: ch 3 () = E[[(M(T)+g(T))-E(M(T)+g(T))] 3 ] -E[(M(T)-E(M(T))) 3 ] = E[g 3 ((T)-E((T))) 3 ]+ 3E[g ((T)-E((T))) (M(T)-E(M(T)))]+ (3) 3E[g((T)-E((T)))(M(T)-E(M(T))) ] ch 4 () = E[[(M(T)+g(T))-E(M(T)+g(T))] 4 ] -E[(M(T)-E(M(T))) 4 ] = E[g 4 ((T)-E((T))) 4 ]+ 4E[g 3 ((T)-E((T))) 3 (M(T)-E(M(T)))]+ (4) 8

9 6E[g ((T)-E((T))) (M(T)-E(M(T))) ]+ 4E[g((T)-E((T)))(M(T)-E(M(T))) 3 ] so ha, a he margin, we obain he following equaion: () = P ()E ((T)) + P ()Cov ((T),M(T))+ P 3 ()3Cosk ((T),M(T)) + P 4 ()4Coku ((T),M(T)) (5) where: Cosk ((T),M(T)) E[((T)-E((T)))(M(T)-E(M(T))) ] can be defined as co-skewness beween he asse and he marke and Coku ((T),M(T)) E[((T)-E((T)))(M(T)-E(M(T))) 3 ] can be defined analogously as co-kurosis 8. In his case, four differen (observable) asses are required o subsiue he (unknown) prices of he characerisics, P,P,P 3,P 4, bu if a risk free zerocoupon bond exiss is price is always P RF =P. Noe ha, in general, higher momens can be expressed in erms of covariances: Cosk ((T),M(T)) = Cov ((T),M (T))-E (M(T))Cov ((T),M(T)) Coku ((T),M(T)) = Cov ((T),M 3 (T))-3E (M(T))Cov ((T),M (T))+ 3E (M(T))Cov ((T),M(T)). so ha, subsiuing ino he price funcion, we obain: () = PRF () E ( T ( )) + P ~ () Cov ( T ( ), MT ( )) P ~ () Cov ( ( T ), M ( T )) P ~ (6) () Cov ( ( T ), M ( T )) 3 As in he sandard wo-momen CAPM, i is possible o ranslae he valuaion equaion in reurn erms. From Equaion 6 we obain: E( R j) = RRF + π ( ) Cov( R j, RM) (7) 3 + π () Cov( R, R ) + π () Cov( R, R ) 3 j M 4 j M which is clearly an exension of he harpe s model: he covariances measure he risk facors and he π s represen explici forms of he marke prices of risks. 4 9

10 In general, wih respec o he classical, linear CAPM, a nonlinear relaion holds beween asse reurns and he marke porfolio, induced by higher momen preferences. Empirical analysis is required o assess he relevan risk facors beyond he linear relaion wih marke reurns General equilibrium approach Le us consider he classical ineremporal consumpion-invesmen model (e.g. Meron, 98) of a represenaive agen wih addiive, concave uiliy in consumpion, U(C,), N financial asses wih prices i () and oal reurns R i () and wealh W a he beginning of period, before he choice of he opimal consumpion C and porfolio allocaions x i () of residual N wealh, wih xi () =. i= Following he Bellman approach o sochasic dynamic programming 0 we have he consrained problem in erms of uiliy value funcion J: JW (, ) = max( UC (, ) + E( JW ( +, + ))) N W+ = ( W C)( + xi( ) Ri( + )) i= N xi () = i= giving, by derivaion, he envelope condiion, J W (W,)=U C (C,) and he sochasic Euler equaion: E J W W( +, + ) ( + Ri ( + )) JW( W, ) = or E J W W(, + ) i() = ( i( + ) + Di( + )) (8) JW( W, ) + In he case of a one-period defaul-free zero coupon bond we have: P E J W W(, + ) RF () = ( ) J ( W, ) W + (9) so ha, ignoring dividends and using he propery ha 0

11 E(XY)=E(X)E(Y)+Cov(X,Y), he valuaion Equaion (8) becomes: JW( W+, + ) i( ) = PRF( ) E( i( + )) + Cov i( + ), J ( W, ) W (0) Noing ha, from a Taylor expansion, he marginal uiliy can be wrien as: JW( W+, + ) = JW( W, + ) + JWW( W, + )( W+ W) + JWWW ( W, + )( W + W ) +... he price equaion (0) becomes: i() = PRF() E(( i + )) + ~ P () Cov( i ( + ), W ( + ) ) + ~ () P ( ) Cov ( + ), W ( + ) ( i ) where W is he aggregae wealh and he global marke porfolio. We have herefore obained he same valuaion funcion in Equaion 6 of secion 4 following an ineremporal general equilibrum approach. 6. Conclusions In recen years, he proliferaion of financial asses of many ypes has been enormous. This paper ries o explore wheher he apparen mulipliciy of righs and obligaions may be ackled hrough one simple valuaion approach, in which any asse is evaluaed hrough is marginal conribuion o he relevan characerisics (momens) of he marke porfolio. For example, in a Gaussian world, asse prices are obained hrough he marginal conribuions o wo basic characerisics, expeced value and variance. The equivalen risk-neural pricing funcion is easily obained, so ha he valuaion formula agrees boh wih he CAPM and he Black and choles no-arbirage pricing of opions. Exensions o hree or more characerisics are simply obained by considering he marginal conribuions of each asse o firs, second and higher-order marke momens, providing a generalized valuaion equaion. Finally, he generalized equaion is derived in a differen, more rigorous way, as a resul of a classical ineremporal general equilibrium model of opimal consumpion and invesmen decisions.

12 References Black F. and choles M The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, vol. 8, no. 3 (May/June): Cooner, P. H. (ed.) 964. The Random Characer of ock Marke Prices, Cambridge, Ma: Mi Press Cox, J.C. and M. Rubinsein Opion Markes. Englewood Cliffs, NJ: Prenice Hall. Dimar R.F. 00. Nonlinear Pricing Kernels, Kurosis Preference, and Evidence from he Cross ecion of Equiy Reurns. Journal of Finance, vol. 57, no. (February): Duffie D. 99, Dynamic Asse Pricing Theory, Princeon, NJ: Princeon Universiy Press Fang H. and T. Lai Co-Kurosis and Capial Asse Pricing. Financial Review, vol. 3, no. (May): Kendall, M. and A. uar The Advanced Theory of aisics. Volume. Disribuion Theory, 4h ediion. London: Charles Griffin & Co. Lancaser, K.J A New Approach o Consumer Theory Journal of Poliical Economy, vol. 74, April: 3-57 Leland, H. E Beyond Mean-Variance: Performance Measuremen in a Nonsymmerical World Financial Analyss Journal, vol. 55, no. : 7-36 Linner, J The Valuaion of Risk Asses and he elecion of Risky Invesmens in ock Porfolios and Capial Budges., Review of Economics and aisics, vol. 47, 3-37 Markowiz, H. 95. Porfolio elecion. Journal of Finance, vol. 7, 77-9 Markowiz, H Porfolio elecion: Efficien Diversificaion of Invesmens, New Haven: Yale Universiy Press Meron, R.C. 98. On he Microeconimic Theory of Invesmen under Uncerainy, in Arrow, K.J. and Inrilligaor, M.D. 98. Handbook of Mahemaical Economics, Vol.II, Amserdam: Norh-Holland, ch. 3 Mossin, J Equilibrium in a Capial Asse Marke. Economerica,

13 vol. 34: Rendleman, R.J. Jr Opion Invesing from a Risk-Reurn Perspecive. Journal of Porfolio Managemen, vol. 5, May: 09- Rubinsein, M The Valuaion of Uncerain Income reams and he Pricing of Opions. The Bell Journal of Economics and Managemen cience, vol. 7: harpe, W. F Capial Asse Prices: a Theory of Marke Equilibrium under Condiions of Risk. Journal of Finance, vol. 9: harpe, W.F., G.J. Alexander and J.V. Bailey Invesmens. 5h ediion. Englewood Cliffs, NJ: Prenice Hall. Tobin, J Liquidiy Preference as Behavior owards Risk. Review of Economic udies, vol. 5: Treynor, J How o Rae Managemen of Invesmen Funds. Harvard Business Review, vol. 43: VV.AA. 99. From Black-choles o Black Holes. New Froniers in Opions. London: Risk Magazine Ld 3

14 Noes Early models can be found in Cooner (ed.) (964). More recen developmens are colleced in VV.AA. (99). For example harpe, Alexander and Bailey (995), chaper 0. 3 For example, in he handbook of harpe, Alexander and Bailey (995), opions are analysed en chapers and 400 pages laer han CAPM. 4 During he las hiry years only a few auhors have addressed he quesion of he relaion beween CAPM and opion pricing. Black and choles (973), in heir famous paper, derived he link using he absrac approach of sochasic calculus and a coninuous-ime version of CAPM. In discree ime, Rubinsein (976) analysed he relaion under he special assumpion of lognormaliy and Cox and Rubinsein (985, p.85) and Rendleman (999) under he binomial model of price dynamics. 5 In paricular assume ha he asse price () is a diffusion process wih drif µ() and diffusion coefficien so ha he condiional disribuion of (T) given () is normal wih mean ()exp(µ(t-)) and variance /(exp(µ(t-))-)/µ. 6 In paricular assume ha he asse price () is a diffusion process wih drif µ() and diffusion coefficien () so ha he condiional disribuion of log((t)) given () is normal wih mean log(())+(µ- /)(T-) and variance (T-). 7 kewness is a momen used o measure he asymmery of he probabiliy disribuion around he mean. A symmerical disribuion has a skewness equal o zero bu i mus be reckoned ha some special disribuions exis having hird (and all odd-order momens) equal o zero bu ha are no symmerical. ee Kendall and uar (977), p.87. On he oher hand, kurosis measures he faness of he ails of he disrubuion, i.e. he probabiliy of exreme evens. For normal disribuions, kurosis is 3 imes he square of he variance. 8 In he case of normaliy, he firs produc-momen (coskewness) is zero and he second (cokurosis) is 3Cov(X,M)Var(M). 9 For recen empirical ess of a four-momen CAPM see Fang and Lai (997) and Dimar (00). kewness from lognormal reurns is considered in Leland (999). 0 According o he Bellman principle, he opimal consumpion-invesmen pah over he agen s ime horizon mus be such ha a any poin in ime i mus be opimal for he remaining period. 4