In their path-finding 1973 paper Black and Scholes presented two separate derivations of their famous option pricing partial differential equation (pde). The second derivation was from the standpoint that was Black’s original motivation, namely, the capital asset pricing model (CAPM). We show here, in contrast, that the option valuation is not uniquely determined; in particular, strategies based on the delta-hedge and CAPM provide different valuations of an option although both hedges are instantaneouly riskfree. Second, we show explicitly that CAPM is not, as economists claim, an equilibrium theory.Capital asset pricing model (CAPM); nonequilibrium; financial markets; Black-Scholes; option pricing strategies;

Gunaratne, Gemunu H.

McCauley, Joseph L.

English

Research Papers in Economics

 
On CAPM and Black-Scholes 
Differing risk-return strategies 
 
Joseph L. McCauley and Gemunu H. Gunaratne 
Department  of  Physics 
University of Houston 
Houston, Texas 77204 
 
PACs numbers: 
89.65.Gh    Economics, business, and financial markets 
89.75.Fb    Structures and organization in complex systems 
 
 
Abstract 
 
In their path-finding 1973 paper Black and Scholes presented two 
separate  derivations  of  their  famous  option  pricing  partial 
differential  equation  (pde).  The  second  derivation  was  from  the 
standpoint  that  was  Black’s  original  motivation,  namely,  the 
capital asset pricing model (CAPM).  We show here, in contrast, 
that the option valuation is not uniquely determined; in particular, 
strategies based on the delta-hedge and CAPM provide different 
valuations of an option although both hedges are instantaneouly 
riskfree.  Second,  we  show  explicitly  that  CAPM  is  not,  as 
economists claim,  an  equilibrium theory. 
 
 
1. The CAPM portfolio selection strategy 
 
The  Capital  Asset  Pricing  Model  (CAPM)  is  very  general:  it 
assumes no particular distribution of returns and is consistent with 
any distribution with finite first and second moments. Therefore, 
in this section, we generally  assume the empirical distribution of 
returns  but  also  will  apply  the  model  to  Gaussian  returns 
(lognormal prices) in part 2 below.  The CAPM is not, as is often 
claimed, an equilibrium model because the distribution of returns 
is  not  an  equilibrium  distribution.  We  will  exhibit  the  time-
dependence of some of the parameters in the model in the familiar 
lognormal price approximation. Economists and finance theorists 
(including Sharpe [1| and  Black [2];  see also Bodie and Merton 
[3])) have adopted and propagated the strange notion that random motion of returns defines ‘equilibrium’, which disagrees with the 
requirement that in equilibrium no averages of any moment of the 
distribution can change with time. Random motion in the market is 
due  to  trading  and  the  excess  demand  of  unfilled  limit  orders 
prevents equilibrium at all or almost all times.  Apparently, what 
many economists mean by ‘equilibrium’ is more akin to assuming 
the EMH (efficient market hypothesis), which has nothing to do 
with  vanishing  excess  demand  in  the  market.    The  only 
dynamically  consistent  definition  of  equilibrium  is  vanishing 
excess  demand:  if  p  denotes  the  price  of  an  asset  then  excess 
demand  ε(p,t)    is  defined  by  dp/dt= ε(p,t)  including  the  case 
where  the  right-hand  side  is  drift  plus  noise,  as  in  stochastic 
dynamical models of the market. These issues have been discussed 
in detail in a previous paper [4]. Bodie and Merton [3] claim that 
vanishing excess demand is necessary for the CAPM, but one sees 
in part 2 below that no such assumption comes into play during 
the derivation and would even cause all returns to vanish in the 
model! 
 
The CAPM [5] can be stated in the following way: Let Ro denote 
the risk-free interest rate,  
 
    
   (1) 
 
is the fluctuating return on asset k where pk(t) is the price of the 
kth asset at time t. The total return x on the portfolio of n assets 
relative to the risk free rate is given by 
 
 
 
 
(2) 
 
where fk is the fraction of the total budget that is bet on asset k. The 
CAPM minimizes the mean square fluctuation 
 
  (3) 
xk = ln(pk(t +  t)/ pk(t))
x   Ro = fi(
i=0
n
  xi   Ro)
 
2 = fi fj
i,j   (xi   Ro)(xj   Ro) = fi fj
i, j    ij 
subject to the constraints of fixed expected return R, 
 
(4) 
 
and fixed normalization 
 
 
(5) 
 
where σij is the correlation matrix 
 
(6) 
 
Following Varian, we solve 
 
 
(7) 
 
for the f’s where ΔRe = Re–Ro and Re is the expected return of the 
‘efficient portfolio’, the portfolio constructed from f’s that satisfy 
the condition (7). The expected return on asset k can be written as 
 
 
(8) 
 
where σee is the mean square fluctuation of the efficient portfolio, 
σke is the correlation matrix element between the kth asset and the 
efficient portfolio, and βΔRe is the risk premium for asset k. 
 
For  many  assets  n  in  a  well-diversified  portfolio,  studying  the 
largest eigenvalue of the correlation matrix σ seems to show that 
fi
i=0
n
  =1
 ij = (xi   Ro)(x j   Ro)
 ki
i   fi =   ke =   ee(Rk   Ro)/ Re
R   Ro = (x   Ro) = fi
i   (xi   Ro) = fi(Ri   Ro)
i  
 Rk =
  ke
  ee
 Re =  k Rethat eigenvalue represents the market as a whole, and that clusters 
of eigenvalues represent sectors of the market like transportation, 
paper, etc. [6]. However, in formulating and deriving the CAPM 
above, nothing is assumed either about diversification or how to 
choose  a  winning  portfolio  (the  strategies  of  agents  like  Buffet, 
Soros and Lynch have not been mathematized and apparently do 
not  depend  on  the  CAPM  notion  of  diversification  and  risk 
minimization), only how to try to minimize the fluctuations in any 
arbitrarily-chosen portfolio of n assets, which portfolio may or may 
not be well-diversified relative to the market as a whole, and which 
may well consist of a basket of losers.  Negative f represents a short 
position, positive f a long position. Large beta implies both greater 
risk and larger expected return. Without larger expected return a 
trader will not likely place a bet to take on more risk. Negative 
returns R can and do occur systematically in market downturns, 
and in other bad bets.  
 
We define a liquid market as one where an agent can reverse his 
trade  over  a  very  short  time  interval  Δt  with  only  very  small 
transaction  costs  and net losses, as in the stock market on the scale 
of seconds during normal trading. A market crash is  by definition 
a  liquidity  drought  where  limit  orders  placed  for  selling 
overwhelmingly  dominate  limit  orders  placed  for  buying.  Large 
deviations  in  the  theory  of  Gaussian  returns  (lognormal  price 
distribution) are by far too unlikely to match the empirical data on 
crashes and bubbles. 
 
In what follows  we consider a portfolio of 2 assets, e.g. a bond 
(asset #1) and the corresponding European call option (asset # 2). 
For two assets the solution for the CAPM portfolio can be written 
in the form needed in part 2 below, 
 
 
 (9) 
 
Actually there are 3 assets in this model because a fraction xo can 
be invested in a risk free asset, or may be borrowed in which case  
xo < 0.  
 
So far we have used the notation of the CAPM. In all that follows 
we will write x=ln(p(t)/p(0)) and Δx=ln(p(t+Δt)/p(t)). 
 
f1 / f2 = ( 12 R2    22 R1)/( 12 R1    11 R2) 
2. Black-Scholes theory of option pricing 
 
Let p denote the price of asset #1, a bond or stock, e.g., and w(p,t) 
the price of a corresponding European call option. In this section, 
in  order  to  discuss  the  original  Black-Scholes  derivation  [7],  we 
follow Osborne [8] and assume that  asset returns are distributed 
normally, with stochastic differential equation 
 
 
(10) 
 
where  ΔB(t)=B(t+Δt)-B(t)  is  an  identically  and  independently 
distributed  Gaussian  random  variable  (B(t)  is  a  Wiener  process 
with<ΔB>=0, <ΔB
2>=dt), and σ1 is assumed constant. Throughout 
this article we use Doob’s notation [9] for stochastic calculus [10]. 
For very small returns over very small time intervals Δt we can 
approximate (10) for small returns as
1 
 
(11) 
 
In the corresponding Langevin equation for price p(t), 
 
 (12) 
 
with  η  white  noise,  the  right  hand  side  represents  the  excess 
demand ε(p,t) for the asset,  dp/dt=ε(p,t), as is emphasized in [4]. 
The excess demand does not vanish, either in the market or in the 
stochastic model, nor (due to limit orders)  does the total excess 
demand of the market vanish. There is no equilibrium, either in the 
market or in the model.  
 
                                                 
1 We note that the corresponding equation (10.7) in Hull {11} cannot be correct for all values of Δp/p, only 
for very small returns. 
 p/ p   R t + 1 B
  
dp = R1pdt +  1pdB(t)
  
dp
dt
= R1p + 1p (t)The  stochastic  differential  equation  for  the  price  change  of  the 
option is then (by Ito’s lemma [10]) 
 
 (13) 
 
In equations (12) and (13) the initial data  p(t), w(p,t), w’ and w’’ 
are deterministic  at the first instant (p,t) while the changes dp and 
dw as well as p(t+dt) and w(p+dp,t+dt) are random due to noise 
dB. In deriving a deterministic diffusive equation of motion for the 
option price w(p,t), two separate methods were presented in the 
original  Black-Scholes  paper  [7].  The  claim  there  is  that  both 
methods yield the same option pricing pde but we will show that 
this is not so. 
 
 
2.1. The Delta Hedge Strategy 
 
The standard idea is to construct a riskfree hedge [7,11]. The delta 
hedge, defined as a portfolio with value 
 
 
(14) 
  
does  this  because  the  portfolio  is  instantaneosly  riskfree:  the 
variance of the return rate (ΔΠ/ΠΔτ−R)  vanishes to O(dt), 
 
 
 
(15) 
 
 
 for any expected rate of return R. Setting the portfolio return equal 
to a constant yields the Black- Scholes  pde 
 
  =  w+    w p
dw = ( ˙  w +    w Rp +
1
2
      w   1
2p
2)dt +    w  1pdB
   R ow = ˙  w + Rop    w  +  1p
2       w  /2
  
  t
  R
  
  
  
  
  
  
2
= 0  
 (16) 
 
for  the  option  price  w(p,t)  if  in  addition  we  assume  the  no-
arbitrage coondition R=Ro where Ro is the riskfree rate of return.  
 
 Note that the ratio invested is given by 
 
  
(17) 
 
We  will  need  this  result  below  for  comparison  with  the 
corresponding CAPM strategy of option pricing, and will see, in 
contrast with the claim of the original Black-Scholes  paper [7], that 
these two strategies do not and cannot agree with each other, even 
in the limit where Δt goes to zero.  
 
 
2.2. The CAPM option pricing strategy 
 
 
From (13) the  fluctuating option price change over a finite time 
interval Δt is given by 
 
 
(13b) 
where the dot in the last term denotes the usual Ito product. In 
what that follows we assume sufficiently small time intervals Δt to 
make the small returns approximation whereby ln(w(t+Δt)/w(t)≈ 
Δw/w  and  ln(p(t+Δt)/p(t)≈Δp/p.  In  the  CAPM  strategy  of 
portfolio construction the expected return on the option is given by 
 
(17) 
where  from  the  small  returns  approximation  (local  solution  of 
(13b)) 
 
R2 = Ro +  2 Re
f1 / f 2=   pw' /w
  
 w = ( ˙  w +    w R 1p+
1
2
      w    1
2p
2)dt
t
t+ t
  +  1(    w p)• B
  
 w   ( ˙  w +    w R 1p+
1
2
      w   1
2p
2) t +  1(    w p) B 
(13c) 
we get 
 
  
(18) 
 
The expected return on the stock is given from CAPM by 
 
(19) 
 
According to Black  and Scholes [7], we should be able to prove 
that 
 
 
 (20) 
 
Were this the case then, combining (17), (18) and (19), we would 
get a cancellation of the two beta terms in (21) below: 
 
 
  
(21) 
 
leaving us with the riskfree rate of return and the original option 
pricing pde (16).  
 
Equation  (20)  is  in  fact  impossible  to  derive  without  making  a 
serious error. Within the context of CAPM it is impossible to use 
(20) in (21). Let us now calculate correctly and show this.  
 
R1 = Ro +  1 Re
 2 =
p    w 
w
 1
R2 = Ro +  2 Re =
˙  w 
w
+
p    w 
w
R1 +
1
2
 1
2 p
2       w 
w
=
˙  w 
w
+
p    w 
w
Ro +
p    w 
w
 1 Re +
1
2
 1
2p
2       w 
w
  
R 2  
 w
w t
 
˙  w 
w
+
p    w 
w
R 1 +
1
2
 1
2p
2       w 
wWe can easily calculate the fluctuating option return x2≈Δw/wΔt at 
short  times.  With  x1≈Δp/pΔt  denoting  the  short  time 
approximation to the asset return, we obtain  
 
 
  (22) 
 
Taking the average would yield (20) if we were to assume that (16) 
holds,  but  we  are  trying  to  derive  (16),  not  assume  it.  Therefore, 
taking the average yields  
 
  
(23) 
 
which is true but does not reduce to (20), in contrast with the claim 
made by  Black and  Scholes [7] in their otherwise very beautiful 
paper. 
 
To see that assuming (16) in order to get (20) from (23) is wrong, 
we  go  further  and  calculate  the  ratio  invested  f2/f1  by  our 
hypothetical  CAPM  risk-minimizing  agent.  Here,  we  need  the 
correlation matrix for Gaussian returns only to leading order in Δt: 
 
 
 (24) 
 
 
   (25) 
 
and 
 
 11   1
2 / t
 12  
p    w 
w
 11
  
x 2  R o  
1
w
( ˙  w +
 1
2p
2       w 
2
+ R op    w    Row)+
p    w 
w
(x1  R 1)
  
 2  
1
w R 2
( ˙  w +
 1
2p
2       w 
2
+R op    w    Row) +
p    w 
w
 1 
 (26) 
 
The variance of the portfolio vanishes to lowest order, but it is also 
easy to show, also to leading order in Δt, that 
 
 
(27) 
 
and 
 
 
 (28) 
 
so that  it is impossible that (20) could be satisfied! Note that the 
ratio f1/f2 is exactly the same as for the delta-hedge. It is also easy 
to show that the CAPM portfolio is instantaneously riskfree, like 
the delta hedge. 
 
That CAPM is not an equilibrium model is exhibited explicitly by 
the time dependence of the terms in (24-26). It would be a serious 
mistake to try to think of an arbitrary time t dynamically as a point 
of equilibrium: the self-contradiction in the economists’ notion of 
‘temporary price equilibria’  [12] has been exhibited elsewhere [4].  
 
 The CAPM simply does not predict either the same option pricing 
equation as does the delta-hedge. Furthermore, if traders actually 
would use the delta-hedge in option pricing then this means that 
agents  do  not  trade  in  a  way  that  minimizes  the  mean  square 
fluctuation ala CAPM. The CAPM and the delta-hedge do not try 
to reduce risk in exactly the same way. In the delta-hedge the main 
fluctuating terms are removed directly from the portfolio return, 
thereby lowering the expected return, whereas in CAPM nothing is 
subtracted from the return in forming the portfolio and the idea 
there is not only diversification but also increased expected return 
through increased risk. This is illustrated explicitly  by the fact that 
the  expected  return  on  the  CAPM  portfolio  is  not  the  risk-free 
return, but is instead proportional to the factor set equal to zero by 
 22   (
p    w 
w
)
2 11
f1  ( 1p    w / w   2)p    w /w
f2   ( 2    1p    w  /w)Black  and  Scholes,  shown  above  as  equation  (20).  With 
Rcapm=Ro+ΔRcapm we have 
 
 
 
(29) 
 
Note also that the expected return ΔRcapm in excess of the risk-free 
rate depends on time, not only through the term pw’/w, but also 
through the terms of higher order neglected in (26-28), even if the 
β’s were t-independent (but we know that they are not).  Note also 
from (29) that beta for the CAPM hedge is given by 
 
(30) 
 
The notion of increased expected return via increased risk is not 
present in the delta-hedge strategy, which tries to eliminate risk 
and to minimize return. We see now that the way that options are 
priced  (even  in  the  riskfree  Gaussian  returns  case)  is  strategy-
dependent,  which  may  be  closer  to  the  notion  that  psychology 
plays  a  role  in  trading.    The  CAPM  option  pricing  equation 
depends on the expected returns for both stock and option, 
 
 
(31) 
 
and so differs from the original Black-Scholes  equation (16) of the 
delta-hedge strategy. There is no such thing as a universal option 
pricing equation independent of the chosen strategy, even if that 
strategy is reflected in this era by the market. Economics is not like 
physics  (non-thinking  nature),  but  depends  on  human  behavior 
and expectations [13]. 
 
In  a  paper  to  follow  we  will  show  how  to  use  the  empirical 
distribution of returns, which is far from Gaussian, to construct a 
stochastic differential equation and corresponding Fokker-Planck 
equation that not only reproduces the empirical distribution but 
also prices options correctly without the use of ‘implied volatility’. 
 Rcapm =
 1p    w  /w    2
p    w / w  1
 Re
 CAPM =
 1p    w /w    2
p    w  /w  1
R2w = ˙  w + p    w R1 +
1
2
 1
2p
2       w Instead  of  implied  volatility  we  use  the  returns  distribution  to 
deduce  a  correct  local  volatility,  namely,  an  (x,t)-dependent 
diffusion coefficient. 
 
    
Acknowledgement 
 
One of us (GHG) is grateful to collegues at the TradeLink Corporation 
during 1990.  The other (JMC) is grateful to Chairman Larry Pinsky 
and    the  Econophysics  Program  at  the  University  of  Houston 
Department of Physics for encouragement including financial support, 
and to Kevin Bassler and George Reiter for very helpful discussions 
and comments during the Fall Econphysics Seminar at UH. We are 
extremely grateful to Harry Thomas for email correspondence, helpful 
criticism, and for sharing with us his notes on Ito calculus which led us 
to correct errors in stochastic calculus in the first two versions of this 
manuscript as preprint. 
 
 
 
References 
 
1.  W.F. Sharpe, J. Finance XIX (1964) 425. 
 
2. F. Black, J. Portfolio Management 4 (1989)1. 
 
3. Z. Bodie, Z., R.C. Merton, Finance, Prentice-Hall,  Saddle River, N.J., 
1998. 
 
4. J. L. McCauley, Physica A285 (2000) 506. 
 
5. H.R. Varian, Microeconomics, Norton, N.Y., 1992. 
 
6. P. Gopikrishnan, B. Rosenow, V. Plerou, H.E. Stanley, Phys. Rev. 
E64 (2001); B. Rosenow, V. Plerou, P. Gopikrishnan, P., H.E. Stanley, 
Portfolio Optimization and the Random Magnet Problem, (preprint, 
2001) 
 
7. F. Black, M. Scholes, J. Political Economy 81 (1973) 637. 
 
8. M.F.M. Osborne, in P. Cootner, (Ed.), The Random  Character of 
Stock Market Prices, MIT, Cambridge, 1964. 
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10. M. Baxter, A. Rennie, Financial Calculus, Cambridge, Cambridge, 
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11. J. Hull, Options, Futures, and  Other  Derivatives, Prentice-Hall, 
Saddle river, N.J., 1997. 
 
12.  H.  Föllmer  in  S.  Howison,  P.R.  Kelly,  P.  Wilmott,  (editors) 
Mathematical Models in Finance, Chapman & Hall, London, 1995. 
 
13.  J.  L.  McCauley,  Physica  A237  (1997)  387;  Discrete  Dynamical 
Systems in Nature and Society 1 (1997) 17. 
 

On CAPM and Black-Scholes, differing risk-return strategies

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