When are Options Overpriced? The Black–Scholes Model and Alternative Characterisations of the...

European Finance Review3: 79–102, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

79

When are Options Overpriced? The Black–ScholesModel and Alternative Characterisations of thePricing Kernel

GÜNTER FRANKE1, RICHARD C. STAPLETON2 andMARTI G. SUBRAHMANYAM 3

1Fakultät für Wirtschaftswissenschaften und Statistik, University of KonstanzE-mail: [email protected] of StrathclydeE-mail: [email protected] School of Business, New York UniversityE-mail: [email protected]

Abstract. An important determinant of option prices is the elasticity of the pricing kernel usedto price all claims in the economy. In this paper, we first show that for a given forward price ofthe underlying asset, option prices are higher when the elasticity of the pricing kernel is decliningthan when it is constant. We then investigate the implications of the elasticity of the pricing kernelfor the stochastic process followed by the underlying asset. Given that the underlying informationprocess follows a geometric Brownian motion, we demonstrate that constant elasticity of the pricingkernel is equivalent to a Brownian motion for the forward price of the underlying asset, so that theBlack–Scholes formula correctly prices options on the asset. In contrast, declining elasticity impliesthat the forward price process is no longer a Brownian motion: it has higher volatility and exhibitsautocorrelation. In this case, the Black–Scholes formula underprices all options.

1. Introduction

Following Black and Scholes (1973), the traditional approach to the pricing ofEuropean-style options on an underlying asset assumes that the asset price followsa given, exogenous process and prices the options using an arbitrage-free hedgingargument. An alternative equilibrium approach, followed by Rubinstein (1976) andBrennan (1979), assumes that the asset price and the value of the market portfolioat the end of a single period have a given joint probability distribution and thata representative investor exists, with a given utility function for end of periodwealth. It has been shown that both these approaches can lead to the same risk-neutral valuation relationship for the option price. A third approach, followingHarrison and Kreps (1979), assumes a no-arbitrage economy which in turn impliesthe existence of a pricing kernel. This pricing kernel variable has the importantproperty that the option forward price equals the expected value of the product of

80 GÜNTER FRANKE ET AL.

the option payoff and the pricing kernel. This third approach is consistent with theequilibrium approach, since the Brennan–Rubinstein assumptions imply a pricingkernel which equals the relative marginal utility of the representative investor.

In this paper, we adopt the more general pricing kernel framework. Assumingthat the asset-specific pricing kernel exhibits constant elasticity, yields the Black–Scholes assumption of a geometric Brownian motion of the asset price. Assuminga representative investor exists with constant relative risk aversion, implies theBrennan–Rubinstein world. However, the general framework permits the pricingof options under less restrictive assumptions. In particular, it turns out that thecurvature of the pricing kernel is critical for the pricing of options. Alternativecharacterizations of the elasticity of the pricing kernel with respect to the price ofthe underlying asset lead to different option prices. In order to investigate the effectof alternative pricing kernels, we start with an assumption regarding the price ofthe underlying asset. Option pricing models typically take as given either the priceof the underlying asset and the risk-free rate of interest, or alternatively, the assetforward price. In this paper, we assume throughout that the current forward price,for delivery at a fixed terminal date, is given. Thus, when we compare the effect onoption prices of different characterizations of the pricing kernel, we do so assumingthat the different pricing kernels lead to the same current forward price of the asset.

We first investigate the relative pricing of options in a general setting, wherethe forward price of the asset is given and the asset-specific pricing kernel exhibitseither constant or declining elasticity. We find that the prices of all options arehigher in the economy with declining elasticity than in the economy with constantelasticity. These higher prices are the result of the increased convexity of the pricingkernel. In the special case where asset prices on the terminal date are lognormal, alloption prices exceed the Black-Scholes prices, if the pricing kernel has decliningelasticity.

How can it be, then, that the Black–Scholes model underprices all options, whenwe know that if the asset forward price follows a geometric Brownian motion,no-arbitrage arguments can be used to establish the Black–Scholes prices? Weinvestigate the answer to this puzzle and find that, in a declining elasticity pricingkernel economy, the asset forward price doesnot follow a geometric Brownianmotion, even though the information process does. We first establish the conditionsunder which the asset forward price follows a geometric Brownian motion. We theninvestigate the effect on the price process of the alternative assumptions regardingthe elasticity of the pricing kernel.

The organization of this paper is as follows. In the following section we reviewprevious related work. Then, in Section 3, we establish our principal result: alloptions have higher prices in the declining elasticity economy than in the constantelasticity economy. In Section 4, we consider a Black–Scholes world in whichthe terminal asset price is lognormal, and we establish the equivalence of twoalternative assumptions: constant elasticity of the pricing kernel, and a geometricBrownian motion of the asset forward price. Section 5 then investigates the effect

WHEN ARE OPTIONS OVERPRICED? 81

on the stochastic process of the asset forward price, of the alternative assumptionof declining elasticity. In Section 6, we assume a more traditional, representativeagent economy, and establish sufficient conditions for declining elasticity of thepricing kernel in an economy in which the asset price and aggregate consumptionare related by a log-linear regression. Section 7 summarizes the main conclusionsof our analysis.

2. Recent Literature on the Mispricing of Options by the Black–ScholesModel

Empirical research in the last few years has suggested that options are underpricedby the Black–Scholes model, i.e., the implied volatility of options typically exceedsthe historical volatility of the price of the underlying asset (see, for example Caninaand Figlewski (1993)). This evidence is corroborated by studies that estimate theexpected value of the implied pricing kernel and the parameters of the risk-neutraldistribution, using index options data (for example, see Longstaff (1995), Brennerand Eom (1998), and Buraschi and Jackwerth (1998)). Although many alternativeexplanations have been proposed for these findings, ranging from jumps in theprice process to the existence of ‘fat tails’ in the return distribution of the under-lying asset, most of the explanations relate one way or another to the stochasticprocess followed by the price of the underlying asset. We suggest an alternativeexplanation. We derive a model in which all options are underpriced by the Black–Scholes model, even though the underlying asset price has a lognormal distributionon the terminal date. Also, the price process exhibits excess volatility, even thoughthe information process for the underlying asset follows a geometric Brownianmotion. In our model, it is the characteristics of the pricing kernel, i.e., of the riskadjustment, that produces the excess-pricing of the options.

In a closely related recent paper, Mathur and Ritchken (1995) consider theprice of options on the market portfolio, in a single-period, representative agentmodel. Restricting their analysis to agents with declining absolute risk aversion,they conclude that the price of an option given constant proportional risk aversion(CPRA), is the minimum option price. The implication is that declining propor-tional risk aversion will produce higher option prices. In the special case of alognormal market portfolio payoff, the Black–Scholes price, resulting from CPRA,is the minimum option price. Our results, cast in terms of the characteristics of theasset-specific pricing kernel rather than risk attitudes, generalize and explain thisconclusion in several ways. First, we consider options on assets in a multi-asseteconomy. In the special case where we consider options on the market portfo-lio, our results are consistent with those of Mathur and Ritchken. The secondgeneralization is that we do not assume a representative agent economy. In con-trast, we assume, in Section 3, that the pricing kernel has declining elasticity. Thisis consistent with, but does not require, declining proportional risk aversion of


the representative agent.1 Thirdly, our conclusions hold in a general, multi-periodeconomy rather than only in the single-period economy. Our conclusions in Section5, regarding the effect of declining elasticity on the stochastic process followed byasset forward prices also help to explain Mathur and Ritchken’s results. Althoughthey assume a single period economy, the question arises as to how their resultsare consistent with the stochastic process followed by prices between the twodates. The answer in our model is that underpricing by the Black–Scholes modelis consistent with a process for the forward price that exhibits excess volatility.

Benninga and Mayshar (1997) analyze a model in which heterogeneous in-vestors with different levels of CPRA act like a representative investor with declin-ing proportional risk aversion. They also find that certain options are underpricedby the Black–Scholes model. Our paper is also closely related to the prior work ofBick ((1987) and (1990)), Franke (1984) and Stapleton and Subrahmanyam (1990).These authors investigated the consistency of various asset price processes in arepresentative investor economy, Bick in a continuous-time setting and Franke andStapleton and Subrahmanyam in a discrete-time setting.2 Our analysis, in Section4, on the equivalence of constant elasticity of the pricing kernel and a randomwalk in the asset forward price, parallels that of Bick. Again, our analysis here issomewhat more general, relying on the existence of a pricing kernel, rather than arepresentative investor who is limited to purchasing claims on the market portfolio.

3. Contingent Claims Prices Given Declining Elasticity of the Pricing Kernel

In this section, we analyze the prices of contingent claims in a perfect capitalmarket, where arbitrage possibilities do not exist. We do so by examining theproperties of the pricing kernel, a variable which can be used to price any claimin this economy.

Consider a datet in the interval[0, T ] where 0 is the current date andT is someterminal date. LetST,j be the price of the assetj at timeT . The forward price atdatet , for delivery of the asset at dateT is denotedFt,T ,j .

Based on the absence of arbitrage there exists a pricing kernel,ψt,t+1, such thatfor any asset or claim on an assetj ,

Ft,T ,j = Et [Ft+1,T ,jψt,t+1] (1)

whereEt is the expectation operator conditional on the information set at timet .ψt,t+1 is a positive random variable. Since a risk-free claim on a dollar to be paid

1 Another set of conditions in which CPRA investors act as if they have declining proportional riskaversion is provided by Franke et al. (1998). They show that if investors face non-hedgeable back-ground risks, they act like investors with declining proportional risk aversion and demand options tohedge the marketable risks that they face.

2 Franke (1984) and Stapleton and Subrahmanyam (1990) use a somewhat different approach tocharacterize the preferences that support a geometric random walk. They start with a process for thecash flows, the fundamental exogenous variable, and derive the restrictions required for the processfor cash flows to be transformed into a geometric random walk for returns.


at dateT always has a forward price of one dollar, it follows from the no-arbitragecondition in Equation (1) that the pricing kernel has an expectation of unity, i.e.,Et(ψt,t+1) = 1. Now, defining the pricing kernel over the interval fromt to T as

ψt,T = ψt,t+1ψt+1,t+2...ψT−1,T

it follows by successive substitution and using the unbiased expectations propertyof conditional expectations, that the asset forward price is

Ft,T ,j = Et [ST,jψt,T ], (2)

sinceFT,T ,j = ST,j . Also, it follows thatEt(ψt,T ) = 1.The pricing kernel,ψt,T , prices any dateT claim. If we now consider claims

contingent on a single asset,j , with price ST,j , we can define and use a pricingkernel unique to assetj . Defining

φt,T ,j = Et [ψt,T |ST,j ]and using the property of conditional expectations we can rewrite Equation (2) as

Ft,T ,j = Et [ST,jφt,T ,j ] (3)

where the expectation is over states ofST,j and φt,T ,j is a timeT measurablerandom variable, unique to assetj . Clearly,φt,T ,j is a function ofST,j . 3

Since we are concerned here with the pricing of contingent claims on (any)single asset, we drop the subscriptj in Equation (3) and write the basic pricingequation simply as

Ft,T = Et [ST φt,T ] (4)

We assume thatφt,T is twice differentiable inST . Having described the basiceconomy, we can now proceed to price contingent claims.

3.1. PRICING OF EUROPEAN OPTIONS: THE GENERAL CASE

A similar no-arbitrage pricing argument can be used to evaluate the forward priceof a European-style contingent claim on the risky asset. If the payoff on the con-tingent claim at timeT is g(ST ), then the forward price of the contingent claim attime t , for delivery atT , denotedCt,T , is given by

Ct,T = Et [g(ST )φt,T ] (5)

3 The pricing kernel,φt,T , can also be derived using the first order condition for the optimalportfolio choice of the investor in a representative agent economy. This is discussed in more detail inSection 6 below.


In option pricing, we generally take the price of the underlying asset as given,and consider only therelative pricing of the option. We take a similar approachhere,Ft,T is assumed to be at a given level,F ∗t,T :

Ft,T = F ∗t,T = Et [ST φt,T ]. (6)

We then ask the following question. How does the forward price of the optionCt,Tdepend on the pricing kernel,φt,T , given thatFt,T = F ∗t,T ? Clearly, assuming onlythatFt,T = F ∗t,T leaves room for several alternative shapes of the pricing kernelφt,T , since there is an infinite number of possible pricing kernels that satisfy theconstraint in Equation (6). We now establish a result which characterizes theφt,Tfunctions which satisfy Equation (6).

Since option prices are dependent on the joint relationship of the pricing kernel,φt,T , and the price of the asset on the terminal date, we can analyse option pricesby investigating theelasticityof the pricing kernel,φt,T , with respect to the assetprice on the terminal date. The elasticity is defined in the conventional manner as

η(ST ) = −∂φt,Tφt,T

/∂ST

ST(7)

We define the elasticity of two different pricing kernels, both of which satisfyEquation (6) as follows. The first pricing kernelφt,T ,1, written henceforth asφ1,has constant elasticityη1, i.e.,η′1 = 0. The second pricing kernelφt,T ,2, written asφ2, has declining elasticityη2, whereη′2 is negative for all values ofST . We firstestablish the following result about the properties of the two pricing kernels.

LEMMA 1 (Intersections of Pricing Kernels with Different Elasticities). Considertwo pricing kernels,φ1 andφ2, each of which yields the same forward asset priceF ∗t,T . Suppose that forφ1, the elasticity is constant, i.e.,η′1 = 0, and forφ2, theelasticity is declining, i.e.,η′2 < 0,∀ST , then the pricing kernelsφ1 andφ2 intersecttwice atSAT andSBT , whereSAT < SBT . Also, forST < SAT andST > SBT , φ2 > φ1.

Proof.Consider the two pricing kernelsφ1 andφ2 with corresponding elastici-tiesη1 andη2 for whichη′1 = 0 andη′2 < 0. This implies that

∂

∂ST

[η2

η1

]< 0. (8)

Suppose that both pricing kernels satisfy Equation (6). First, it is necessary thatthe two pricing kernels intersect at least once. Otherwise, it would be impossiblefor them to have the propertyE(φ1) = E(φ2) = 1. Second, the two pricing kernelsmust intersect more than once, since otherwise the forward price of the risky asset,F ∗t,T , cannot be the same under both pricing kernels. To see this, suppose that the

two pricing kernels intersect only once atST = ST . Suppose thatφ1 > [<]φ2 forST < [>]ST . Then, consider a claim paying(ST − ST ) at dateT . Then,E[(ST −


ST )φ2] > E[(ST − ST )φ1] follows since(ST − ST )(φ2−φ1) ≥ 0,∀ST . AsE[(ST −ST )φ] = E[ST φ] − ST , the forward price of the risky asset would be higher underpricing kernelφ2 than underφ1. Hence, the forward price can be the same only ifthe pricing kernels intersect at least twice. Finally, we show in Appendix A thatmore than two intersections contradicts the assumption in Equation (8). �

Now consider the two intersections atSAT andSBT whereSAT < SBT . For prices below

SAT , φ2 > φ1. This implies that for contingent claims that pay off only in the regionST < SAT , contingent claim prices will be higher underφ2 than underφ1. Also, forprices aboveSBT , we haveφ2 > φ1. Again, for contingent claims that pay off onlyin the regionST > SBT , contingent claim prices will be higher underφ2 than underφ1. In particular, put options with strike prices at or belowSAT and call optionswith strike prices at or aboveSBT have higher prices under the declining elasticitypricing kernel. However, the following Theorem establishes thatall options havehigher prices.

THEOREM 1 (The Pricing of European-Style Options). Consider two pricing ker-nels,φ1 andφ2, both of which yield the same forward price of the risky asset.Suppose that for pricing kernelφ1, the elasticity is constant and for pricing ker-nel φ2, the elasticity is declining. Then, the price of any European-style option isgreater under pricing kernelφ2 than underφ1.

Proof.We show in Appendix A that the two pricing kernelsφ1 andφ2 intersecttwice, at points which we denote asSAT andSBT . That is

φ2 > φ1 for ST < SAT ,

φ2 < φ1 for SAT < ST < SBT , (9)

φ2 > φ1 for SBT < ST .

Now letLk(ST ) = ak + bkST , whereak andbk are chosen so that

Lk(ST ) = (ST − k)+, for ST = SAT , andST = SBT . (10)

The forward price of a call option with strike pricek is

Ck,j = E[(ST − k)+φj ], j = 1,2 (11)

which can be written

Ck,j = E[((ST − k)+ − Lk(ST ))φj ] + E[Lk(ST )φj ], j = 1,2 (12)

Since the forward price of a linear payoff is the same under both pricing kernels,i.e.,

E[Lk(ST )φ1] = E[Lk(ST )φ2], (13)


it follows that

Ck,2− Ck,1 = E[((ST − k)+ − Lk(ST ))(φ2− φ1)]. (14)

It follows from the definition ofLk(ST ) that(ST −k)+−Lk(ST ) ≥ [=][≤]0, whenφ2− φ1 > [=][<]0, and henceCk,2 > Ck,1.

Also, by put-call parity, all puts must have higher forward prices underφ2 thanunderφ1. �

Theorem 1 shows that given the same forward price for the underlying asset, alloptions, both puts and calls atany strike price are more highly priced by thedeclining elasticity pricing kernel,φ2, compared to the constant elasticity pricingkernel,φ1.4 The intuitive reason for this ‘mispricing’ is that the declining elasticitypricing kernel is more convex than the one with constant elasticity. This convexityimplies that convex claims, such as options, are valued more highly by the decliningelasticity pricing kernel, all else being the same. In other words, extreme payoffson either side of the mean are priced more highly by the declining elasticity pricingkernel. However, linear claims such as the forward contract on the asset are pricedthe same, by assumption. Although the payoffs close to the mean are priced lowerby the declining pricing kernel, this is not sufficient to outweigh the higher pricingof the more extreme payoffs.

Theorem 1 is a general result for the pricing of European-style options: it holdsfor any probability distribution ofST . An important implication of the result is thatoption pricing models that implicitly assume a constant elasticity for the pricingkernel yield lower option prices than those that assume declining elasticity. Ifthe true pricing kernel has declining elasticity, the use of such models leads tomispricing.

4. Constant Elasticity of The Pricing Kernel: The Black–Scholes Economy

We have shown above that if the pricing kernel exhibits declining elasticity, thenEuropean options are underpriced by any model that assumes, either explicitly orimplicitly, that the pricing kernel has constant elasticity. Hence, the question arisesas to what pricing kernel property would yield the same option prices as the Black–Scholes model. Since the Black–Scholes model follows from the assumption thatthe forward price of the underlying asset follows a geometric Brownian motion,we need to investigate the relationship between the properties of the pricing kerneland the asset price process.

In this section, we first examine the relationship between the two assumptions:the elasticity of the pricing kernel is constant, and the asset forward price follows

4 We exclude cases where there is a zero probability of finishing out-of-the-money. For example,a call option at a strike price of zero always finishes in-the-money. By definition, its forward price isthe same as the forward price of the underlying asset, and hence equal under the two pricing kernels.


a geometric Brownian motion. We then illustrate the case of constant elasticityusing an example, where the forward price follows a stationary geometric binomialprocess. In the following section, we relax the assumption of constant elasticity andinvestigate the effects on the price process.

4.1. THE GENERAL CASE

We assume here that the conditional expectation of the underlying asset price attime T , ST , evolves as a geometric Brownian motion. We show in the followingtheorem that two properties: (A) the pricing kernel has constant, non-state depen-dent elasticity, and (B) the forward price of the asset follows a geometric Brownianmotion, are equivalent.5 In the following section we then proceed to derive theimplications of declining elasticity for the forward price process.

First, letBτ be a Brownian motion on the probability space(�,F ,P). We definethe information process for the priceST as the conditional expectation process ofST , Iτ = Eτ(ST ), τε(t , T ). We assume that the behaviour ofIτ is governed by thestochastic differential equation:

dIτIτ= α dτ + σdBτ (15)

whereσ is a constant andα, the mean of the process, is zero, simply because itis an information process. It follows thatST is lognormally distributed. We nowinvestigate conditions under which the forward priceFτ,T follows a geometricBrownian motion process of the form

dFτ,T = Fτ,T µτ dτ + Fτ,T σdBτ , t ≤ τ ≤ T , (16)

where the drift,µτ is non-stochastic, but possibly time dependent. It is knownthat, if the forward price is governed by (16), then the Black–Scholes prices forEuropean-style options must obtain. Hence, we are also looking at conditions forthe Black–Scholes theorem to hold. We establish:

THEOREM 2 (Constant Elasticity of the Pricing Kernel). Given that the informa-tion process for the underlying asset is

dIτIτ= α dτ + σ dBτ

with α = 0, then the following statements are equivalent:

5 Note that in the multi-period world (A) includes the condition of non-state dependency of theelasticity of the pricing kernel. In principle, it is possible for the pricing kernel elasticity to be statedependent, i.e., for the elasticity ofφt,T to depend on the state att , for t < T .


(A) The pricing kernel,φt,T has constant elasticity,

ηt,T =∫ Ttητ dτ

(T − t) =∫ Ttµτ dτ

σ 2(T − t)in each state and at each date, whereητ = µτ/σ 2.

(B) The asset forward price,Fτ,T follows a geometric Brownian motion, withdrift µτ and standard deviationσ .

Also, if A) or B) holds, then the Black–Scholes formula for the price of aEuropean-style option onST holds, at each date and in each state.

Proof.B ⇒ A Assume that the forward price follows the geometric Brownianmotion

dFτ,T = Fτ,T µτ dτ + Fτ,T σ dBτ , t ≤ τ < T .For notational convenience, sinceT is fixed, we write this as

dFτ = Fτµτ dτ + Fτσ dBτ . (17)

We now consider the process for the conditional expectation of the pricing kernel,φt,T . φt,T is a timeT measurable random variable, and its conditional expectationisEτ (φt,T ). For simplicity, we denote

Eτ(φt,T ) ≡ θτ = θτ (Fτ , τ )where, by assumption,θτ is a twice continuously differentiable function of theforward priceFτ and of timeτ . By Ito’s lemma,

dθτ =(∂θτ

∂τ+ 1

2

∂2θτ

∂F 2τ

F 2τ σ

2

)dτ + ∂θτ

∂FτdFτ

Sinceθτ is the conditional expectation of the pricing kernel, it is aP martingale. Itfollows that the terms in dτ must add to zero. Hence, we have

dθτ = ∂θτ

∂FτFτσ dBτ . (18)

In Appendix B, we show that it follows from the definition of the forward pricethatEτ(dFτ θτ+dτ ) = 0. Sinceθτ+dτ = θτ + dθτ , we have, using the expressionsfor dθτ and dFτ ,

µτdτθτ + σ 2Fτ∂θτ

∂Fτdτ = 0

which implies that, for the elasticityητ ,

ητ ≡ − ∂θτ∂Fτ

Fτ

θτ= µτ

σ 2, ∀τ. (19)


From (18) and (19) it follows that

dθτ = −θτητσ dBτ , ∀τ.θτ follows a geometric Brownian motion. Hence,θT = φt,T is lognormal. Sinceφt,T = φt,T (ST ), whereST is also lognormal, thenφt,T has constant elasticity withrespect toST . From (19),µτ = ητσ 2 so that∫ T

t

µτ dτ = σ 2∫ T

t

ητ dτ ≡ σ 2(T − t)ηt,T

This establishes that the pricing kernel has constant, non-state-dependent elas-ticity, ηt,T , with respect to the terminal spot price, when the forward price followsa geometric Brownian motion. �

A⇒ B Assume that the pricing kernelφt,T has constant, non-state-dependentelasticity, ηt,T . Constant elasticity with respect toST implies that we can writethe pricing kernel as

φt,T = λt,T S−ηt,TT .

Hence, from the conditionEt(φt,T ) = 1,

λ−1t,T F

ηt,Tt = Et

[(ST

Ft

)−ηt,T ].

Also, fromFt = Et(ST φt,T )

λ−1t,T F

ηt,Tt = Et

[(ST

Ft

)−ηt,T+1].

Equating these expressions, defining[µ(Ft )−σ 2/2](T − t) as the mean of the log-arithm ofST /Ft , given the forward priceFt , and using the properties of lognormalvariables, yields6

µ(Ft ) = σ 2ηt,T

But, by assumptionηt,T and henceµ(Fτ ) is state independent. Hence, sinceηt,T =∫ Ttµτ dτ/σ 2(T − t) andST /Ft is lognormal, for alltε[0, T ),

Et(ln ST )− lnFt =∫ T

t

µτ dτ − σ2

2(T − t). (20)

6 If X is lognormally distributed withE(lnX) = µ − σ2/2, thenE(Xa) = exp[a(µ + (a −1)σ2/2)].


Now, consider the information processIt . It has zero drift, i.e.,

dItIt= σ dBt.

This implies thatST is lognormal with

Et(ln ST ) = lnEt(ST )− σ2

2(T − t).

From this equation and (20) it follows that

lnFt = lnEt(ST )−∫ T

t

µτ dτ

and since lnEt(ST ) is a Brownian motion, so is lnFt .Finally, it is well known that condition B above implies that the Black–Scholes

model holds. The proof is similar to the original Black–Scholes proof, with theforward price process substituted for the spot price process. �

Theorem 2 shows that the assumption of the Black–Scholes model, that theasset (forward) price follows a Brownian motion, is equivalent to constant elasticityof the pricing kernel. It follows, using Theorem 1 that the Black–Scholes modelunderprices options in a declining elasticity economy. We have the following:

COROLLARY 1 (Declining Elasticity and Black–Scholes Underpricing). Supposethat the information process ofST follows a standard geometric Brownian motionand that the forward priceFt,T is given. Then, if the pricing kernel has the propertyof declining elasticity, all options onST will have higher forward prices at datetthan those given by the Black–Scholes model.

Proof. First, from Theorem 2, the Black–Scholes formula holds if the pricingkernel has constant, non-state-dependent elasticity. Further, from Theorem 1 weknow that, if the pricing kernel has declining elasticity, all options have higherprices than in the case of constant elasticity. Hence, the forward prices of optionsin the case of declining elasticity exceed the Black–Scholes prices. �

In Theorem 2 we show that the assumption of either a Brownian motion or a pricingkernel with constant, non-state-dependent elasticity is sufficient for the Black–Scholes model to hold. The prior work of Brennan (1979), who showed that, ina representative agent single-period economy, constant relative risk aversion is anecessary condition for Black–Scholes to price options on the market portfolio,suggests that these conditions may also be necessary. However this is not the case.The Black–Scholes model does not require a pricing kernel with constant, state-independent elasticity, or a Brownian motion in the forward price. If, however, weadd a mild restriction on the pricing kernel in an intertemporal setting, to the effect


that the pricing kernel is path independent, we can show necessity of the Brownianmotion. First, we define path-independence of the pricing kernel.

DEFINITION (Path-independence of the pricing kernel). A pricing kernel is path-independent if for any two outcomes ofST : ST,1, ST,2, the ratio

φt,T (ST ,1)

φt,T (ST ,2)

does not depend on the stateIt , ∀t < T . We now establish

COROLLARY 2 (Necessity of a Brownian Motion in the Forward Price for Black–Scholes Pricing). Assume the same information process as in Theorem 2 and pathindependence of the pricing kernel. Then the Black–Scholes formula correctlyprices European-style options on an asset with priceST at time T , only if theunderlying asset has a forward price which follows a Brownian motion.

Proof. If the Black–Scholes model holds at datet , the risk-adjusted density ofST must be lognormal. This density equals the true density multiplied byφt,T (ST ).Since the true density is lognormal, by assumption, it follows thatφt,T (ST ) hasconstant elasticity,ηt,T , which may, however, depend onIt . Hence

φt,T (ST ,1)

φt,T (ST ,2)=(ST,1

ST,2

)−ηt,T (It ),

so that the pricing kernel is path-dependent. But this path dependency is ruled outby assumption. Hence, by the equivalence of (A) and (B) in Theorem 2 it followsthat lnFτ is a Brownian motion. �

Corollary 2 shows that a geometric Brownian motion information process and pathindependence of the pricing kernel imply a Brownian motion of the asset for-ward price, if the Black–Scholes model is to hold. Many financial models assumetime additive utility of a representative investor, an assumption which guaran-tees path independence of the pricing kernel. Hence, the Black–Scholes world isonly slightly more general than a world where the asset forward price follows aBrownian motion.

4.2. CONSTANT ELASTICITY: AN EXAMPLE IN THE CASE OF A BINOMIAL

PROCESS

In order to clarify the restrictions implied by constant elasticity of the pricing ker-nel, we now look at an example where the asset forward price follows a binomialprocess. The example allows us to specify the process followed by the conditionalexpectation ofφt,T . In order to be consistent, in the limit, with geometric Brownianmotion, we assume that the information process ofST follows a multiplicativebinomial process.


Given a forward asset priceFt,T , we now assume ann-stage, stationary mul-tiplicative binomial process for the forward priceFτ,T , over the period fromt toT . Specifically, letu andd be the proportionate up and down movements of thebinomial process over each sub-interval, then

Fτ+1,T

Fτ,T={u, q

d, 1− q},∀τ (21)

whereq is the probability of an up-movement in the forward price over any sub-interval. Whenn is large, the process in (21) converges to a geometric Brownianmotion process. We now show, consistent with Theorem 2, that the pricing kernelhas constant elasticity.

First, we need to specify the pricing kernel process. DefiningFt+1t,T = Ft,T +1Ft and noting thatE[1Ftθt+1] = 0] from the results in Appendix B, it followsthat

Ft,T = Et [Ft+1,T θt+1],whereθt+1 is the conditional expectation, at timet + 1, of the pricing kernelφt,T .In the binomial case, there are only two states at timet + 1, so we can write

Ft,T = qFt+1,T ,uθt+1,u + (1− q)Ft+1,T ,dθt+1,d (22)

whereθt+1,u andθt+1,d are the values of the conditional expectationEt+1(φt,T ), inthe up-state and down-state respectively.

However, since the forward price moves fromt to t+1 as a two-state branchingprocess we have a dynamically complete market economy. It follows that thereexists a unique ‘risk neutral’ probability measure under which the forward priceof the asset is a martingale. Also the probability of an up movement under thismeasure over any sub-period is a constant:

p = 1− du− d , 0≤ p ≤ 1

The forward price of the risky asset at any point of timet must also therefore begiven by the equation:

Ft,T = pFt+1,T ,u + (1− p)Ft+1,T ,d

or

Ft,T = qFt+1,T ,u

(p

q

)+ (1− q)Ft+1,T ,d

(1− p1− q

). (23)

Equating (23) and (22) for the conditional expectation of the pricing kernel,Et+1(φt,T ),

θt+1,u = p

q, θt+1,d = 1− p

1− q


in the up-states and down-states. Also, ifj is the number of up movements of theasset price over then sub-periods fromt to τ ,

θτ,j = (θt+1,u)j (θt+1,d)

n−j

We show now that lnFτ,T and lnθτ are perfectly correlated. First, the forward price,afterj up-moves, is given by

Fτ,T ,j = Ft,T ujdn−j

Hence, taking the logarithm of the pricing kernel expectation and of the forwardprice, yields

ln θτ,j = j ln θt+1,u + (n− j) ln θt+1,d

and

lnFτ,T ,j = lnFt,T + j ln u+ (n− j) ln d

Thus, lnθτ and lnFτ,T are linear inj . It follows that we can write in general

ln θτ = ατ + β lnFτ,T

for appropriateατ andβ, and in particular:

lnφt,T = αT + β lnST (24)

Equation (24) establishes the perfect correlation of lnST and lnφt,T .We can now investigate the elasticity of the pricing kernel. Equation (24) is the

key to understanding the restrictions imposed on the pricing kernel by the assump-tion of the lognormal process for the asset price. It implies that the pricing kernelhas the same stochastic properties as the asset price itself. In particular, in the limitasn → ∞, the unconditional pricing kernel and the asset price are lognormallydistributed, as in Rubinstein (1976) and Brennan (1979).

Although for a finite binomial process withn sub-periods, there exists onlya finite number ofST values, we can think of a largen so that, approximately,ST may be considered a variable which is continuous on the range(0,∞). Thendifferentiating Equation (24) with respect to lnST yields the elasticity of the pricingkernel,

∂ lnφt,T∂ lnST

= −ηt,T = β (25)

Hence, a stationary multiplicative binomial process ofFτ,T implies a constant andstate-independent elasticity of the pricing kernel. This binomial example illustratesthe result in Theorem 2, where a geometric Brownian motion for the asset forward


price was shown to imply a constant, non-state-dependent elasticity of the pricingkernel. Here, starting with a multiplicative binomial distribution for the forwardprice, we have also shown that the pricing kernel is perfectly correlated with theasset price and has constant, non-state-dependent elasticity.7 In the limit, both theasset price and the pricing kernel are lognormal and the Black–Scholes model holdsfor European-style claims on the asset.

5. Declining Elasticity and Excess Volatility

So far, we have shown, in Section 3, that options have higher prices when thepricing kernel has declining elasticity than when it has constant elasticity. We havethen shown, assuming that the information process follows a geometric Brownianmotion, that the asset forward price follows a geometric Brownian motion if andonly if the pricing kernel has constant elasticity. It remains to be shown exactlyhow declining elasticity affects the forward price process. We now derive the im-plications for the forward price process, of relaxing the assumption of constantelasticity of the pricing kernel. We show in the case of declining elasticity, that thevariance of the forward price, vart (Fτ,T ) increases relative to the constant elasticitycase, and also that returns exhibit negative autocorrelation.8

THEOREM 3. Consider an economy for datesτ ∈ [t , T ]. Assume that the infor-mation process for the asset price at dateT follows a geometric Brownian motion.LetFτ,T ,1 andFτ,T ,2 be the forward prices of the asset, at timeτ , under the constantand declining elasticity pricing kernels respectively. Then,

(a) across states, the ratio of the two pricesFτ,T ,2/Fτ,T ,1 increases monotoni-cally in Fτ,T ,1, ∀τ ∈ (t , T ),

(b) there exists aF ∗τ,T ,1, such that

Fτ,T ,2 < [=] [>] Fτ,T ,1 if

Fτ,T ,1 < [=] [>] F ∗τ,T ,1 ∀τ ∈ (t, T ),(c) the variance of the forward price is higher under the declining elasticity

pricing kernel,

vart (Fτ,T ,2) > vart (Fτ,T ,1) ∀τ ∈ (t, T ).(d) For datesτ = t1, t2, . . . , tj , . . . , T , the price relatives(Ftj ,T ,2/Ftj−1,T ,2)

exhibit negative autocorrelation.

7 The results here relate closely to those in Stapleton and Subrahmanyam (1984b). They showedthat, if the forward price is multiplicative binomial, a risk-neutral valuation relationship holds for thevaluation of options on the asset, if the utility function of the representative agent is a power function.The results here are analogous to those, but in a multi-period setting.

8 In the case of increasing elasticity of the pricing kernel, the variance declines relative to theconstant elasticity case, although the returns exhibit negative autocorrelation in this case also.


Proof. (a) Constant elasticity and the lognormality ofST imply that the expectedreturnEτ(ST /Fτ,T ,1) is independent of the state at timeτ . In the case of decliningelasticity, the rate of return in the high states (highFτ,T ,1), is relatively low and therate of return in the low states, is relatively high, compared to the constant elasticitycase. Therefore, the forward price at timeτ , Fτ,T ,2, for the declining elasticity caseis relatively higher thanFτ,T ,1 in the high states and relatively lower in the lowstates. Since elasticity is monotonically declining, it follows thatFτ,T ,2/Fτ,T ,1 ismonotonically increasing inFτ,T ,1, ∀τ ∈ (t , T ).

(b) Given the same initial priceF ∗t,T , it must be that the forward prices underthe two pricing kernels do not dominate each other. Hence, given thatFτ,T ,1/ Fτ,T ,2increases monotonically inFτ,T ,1, there can be only one value ofFτ,T ,1 whereFτ,T ,2 = Fτ,T ,1. In other words, there is aF ∗τ,T ,1, such thatFτ,T ,2 = Fτ,T ,1 = F ∗τ,T ,1,and the result (b) follows.

(c) From (a) and (b), it follows that

Fτ,T ,2 = Fτ,T ,1+ E[Fτ,T ,2− Fτ,T ,1] + ε (26)

whereE(ε) = 0 and cov(ε, Fτ,T ,1) > 0 sinceFτ,T ,2 gets larger relative toFτ,T ,1 asFτ,T ,1 increases. Hence,

var(Fτ,T ,2) = var(Fτ,T ,1)+ var(ε)+ 2cov(ε, Fτ,T ,1) > var(Fτ,T ,1). (27)

(d) For the constant elasticity pricing kernel, the autocorrelation of returns iszero, since the forward price process is generated by a geometric Brownian motion.Now, assume non-constant elasticity of the pricing kernel. Consider datest , t1, Tand the price relativesFt1,T /Ft,T andST /Ft1,T . If the price relative in the period[t , t1] is lower [higher] under non-constant elasticity, then the conditional expectedprice relative in the period[t1, T ] must be higher [lower] implying negative au-tocorrelation. Second, we split the period[t1, T ] into subperiods[t1, t2] and [t2,T ]. By the same argument as before, given some state att1, the price relativesFt2,T /Ft1,T and ST /Ft2,T must be negatively autocorrelated under non-constantelasticity. Similarly, the period[t2, T ] can be split sequentially into arbitrarily manysubperiods so that, by induction, negative autocorrelation of the price relatives isobtained for any number of subperiods. �

Theorem 3 shows that a geometric random walk for the forward price is ruledout by declining elasticity. Moreover, the forward price at any intermediate dateis more volatile under the declining elasticity than under the constant elasticitypricing kernel.


6. Option Pricing and the Elasticity of the Pricing Kernel in aRepresentative Agent Economy

The analysis of option prices using the pricing kernel approach in a no-arbitragesetting is quite general. However, it is useful to relate the analysis to an equilibriumsetting in order to interpret the pricing kernel in economic terms. For example,what kind of equilibrium would lead to pricing kernels with constant or decliningelasticity? What restrictions on preferences would lead to such pricing kernels?In order to answer these questions, we now make the more traditional assumptionof a representative investor economy, where the agent has utility for end of periodconsumption. The analysis below provides a set of restrictive, sufficient conditions,under which the pricing kernel for an asset has the characteristics assumed inprevious sections of the paper.

We now assume that aggregate end-of-period consumption,CT , and the spotprice of the asset on the terminal dateT , ST , have a constant elasticity with respectto each other, but with an error.9 In other words, the two variables are log-linearlyrelated with an independent error term as follows.10

lnCT = a + b ln ST + ε, (28)

whereε is independent ofST . A special case is analysed by Rubinstein (1976), andBrennan (1979), who show that the Black–Scholes model holds in a single-periodeconomy where a representative investor exists with a utility that exhibits constantrelative risk aversion, and where aggregate wealth is lognormally distributed.11 TheRubinstein–Brennan assumptions imply a pricing kernel with constant elasticity.Now denoting the utility function of the representative investor asu(CT ) we canestablish:

THEOREM 4 (Elasticity of the Pricing Kernel in a Representative Investor Econ-omy). Consider an economy in which consumption takes place at timeT . Assumethat an asset with priceST and aggregate consumptionCT are log-linearly relatedas in Equation (28) above and that a representative investor exists with relative riskaversionR(CT ). Then, at any datet , the pricing kernel,φt,T (ST ) for the asset haselasticity

ηt (ST ) = bRt (ST )9 CT can be literally interpreted as aggregate consumption or as aggregate wealth in a single

period setting. More generally, it can be thought of as a state variable which is the argument in thepricing kernel function.

10 We do not assume here that either the asset price or aggregate consumption is lognormallydistributed. Joint lognormality of the variables is sufficient, but not necessary for the log-linearrelationship to hold.

11 Following up on a result in Merton (1973), Rubinstein (1976) and Brennan (1979) showed thatthe Black–Scholes model holds under these assumptions. Brennan shows that the constant relativerisk aversion assumption is also a necessary condition. Stapleton and Subrahmanyam ((1984a) and(1984b)) and Heston (1993) have extended this work in various directions.


where

Rt (ST ) = Et[R(CT )

u′(CT )Et [u′(CT ) | ST ] | ST

]Proof. In a representative investor economy the pricing kernel is

ψt,T = u′(CT )Et [u′(CT )] .

Sincet andT are fixed, we denote the pricing kernel asψ , whereψ = ψ(CT ),CT > 0. The asset-specific pricing kernel is

φ = Et [ψ | ST ]where we can writeφ = φ(ST ).

The elasticity of the asset specific pricing kernel,φ, is η = −∂ lnφ/∂ ln ST .Using (28), and the fact that the partial derivative,∂ψ/∂ST = 0, we have

η = −E[u′′(CT )CT b | ST

]E [u′(CT ) | ST ]

,

where for notational convenience we writeEt(·) asE(·). Hence, we can write

η = bE[R(CT )u

′(CT ) | ST]

E [u′(CT ) | ST ]= bR(ST )

where

R(ST ) = E[R(CT )

u′(CT )E[u′(CT ) | ST ] | ST

]is the representative agent’s asset specific relative risk aversion. �

COROLLARY 3 (Declining Elasticity). Suppose that relative risk aversion of therepresentative investor,R(CT ), is declining inCT , then if b 6= 0, the elasticity ofthe asset specific pricing kernel,φt,T (ST ), declines inST .

Proof. See Appendix C.

The significance of Theorem 4 and Corollary 3 is as follows. In a representa-tive investor economy, the elasticity of the pricing kernel is closely related to therelative risk aversion of the investor. However, for a specific asset, the elasticitydepends on a ‘risk adjusted’ relative risk aversion, which accounts for the risk ofaggregate consumption, given the asset price. In Corollary 3, we find that this riskadjusted relative risk aversion declines with the asset price, if the actual relativerisk aversion declines with aggregate consumption.


The results in Theorem 4 and Corollary 3 allow us to generalize the conclusionsof Brennan (1979) and Rubinstein (1976). They showed that the Black–Scholesformula priced European-style options if the asset price and aggregate consump-tion are joint-lognormally distributed and if a representative investor exists withCPRA utility. In our Theorem 4, we first show that lognormality of aggregate con-sumption is not required. In fact, if we have CPRA and the log-linear relationshipbetween the asset price and consumption in Equation (28), then the pricing kernelwill have constant elasticity, and from Theorem 2, the Black–Scholes model willhold. Furthermore, if the representative investor has declining proportional riskaversion (DPRA), then this will translate into a declining elasticity pricing kernel.The result, in that case, is that all options on the asset have higher prices than thosegiven by the Black–Scholes model.

7. Conclusions and Extensions

We have derived the main implications, for the asset price process and for optionprices, of declining elasticity of the pricing kernel. Firstly, under declining elastic-ity, options have higher prices than under the more familiar assumption of constantelasticity. Secondly, in the special case where the information process of the assetprice follows a geometric Brownian motion, the Black–Scholes model underpricesEuropean-style options. Also, given the terminal probability distribution of theasset price, the stochastic process of the asset forward price has higher volatilityand exhibits negative autocorrelation under declining elasticity. Thirdly, declin-ing elasticity is consistent, in a representative investor economy, with decliningproportional risk aversion of the representative investor.

The model in which the asset (forward) price follows a geometric Brownianmotion is one of the standard work-horses of finance. It has been useful in de-riving many empirically testable propositions, but its characteristics and valuationimplications are not always in line with the empirical evidence. Examples of suchempirical anomalies include the high volatility of stock returns, their autocorrela-tion and the underpricing of contingent claims. The question, therefore, is whetherthe implicit assumption of constant elasticity of the pricing kernel can be modifiedfor the resultant models to better fit the data. An alternative proposed and analyzedin this paper is to assume a pricing kernel that exhibits declining elasticity withrespect to the payoff on the asset. This model could help explain a number ofempirical anomalies relating to the return generating process and the pricing ofcontingent claims.

Several other directions of research can be pursued, based on the research re-ported in this paper. First, the properties of the pricing kernel that lead to a broaderclass of stochastic processes for returns than the standard geometric Brownianmotion could be explored. These properties could be tested directly to assess theirempirical validity as has been proposed in the literature on the term structure ofinterest rates. Second, the further implications of declining elasticity of the pricing


kernel for option pricing, such as for the ‘smile’ effect, that relates implied volatil-ities to strike prices, could be explored further. This could, in turn, provide a bettertheoretical justification for recent work on fitting binomial trees using observedoption prices.

Appendix A: Proof that More than Two Intersections of the Pricing Kernelcannot Exist

Suppose there are three or more intersections of the two pricing kernels. Considerthe first three intersections at forward pricesSAT , SBT andSCT respectively. Supposethat atSAT , φ2 intersectsφ1 from above, i.e.,

−∂φ1(SAT )

∂ST< −∂φ2(S

AT )

∂ST.

Since, at the first intersection,

φ1(SAT ) = φ2(S

AT )

it follows that

η1(SAT ) = −

∂φ1(SAT )

∂ST· SAT

φ1(SAT )

< η2(SAT ) = −

∂φ2(SAT )

∂ST· SAT

φ2(SAT )

(29)

Similarly atSBT , φ2 intersectsφ1 from below, it follows that

η1(SBT ) > η2(S

BT ) (30)

Again, atSCT , sinceφ2 intersectsφ1 from above, we must have

η1(SCT ) < η2(S

CT ) (31)

However, this would contradict inequality (8). Thus, three or more intersections ofthe two pricing kernels are not possible. In conclusion, the two pricing kernels mustintersect twice and, in order to satisfyη′2 ≤ 0, φ2 must intersectφ1 from above atthe first intersection,SAT , and from below at the second intersection,SBT �


Appendix B: Proof that Eτ(dFτθτ+dτ ) = 0

To establish this result, we will first consider the discrete quantityEτ(1Fτ θτ+1τ ),whereFτ+1τ = Fτ +1Fτ , and then take the limit. Consider the quantity

Eτ(Fτ+1τ θτ+1τ ) = Eτ [Fτ+1τEτ+1τ (φt,T )]= Eτ [Eτ+1τ (ST φτ+1τ,T )Eτ+1τ (φt,T )]= Eτ [Eτ+1τ (ST φτ+1τ,T )φt,τφτ,τ+1τ ]= Eτ [ST φτ+1τ,T φt,τφτ,τ+1τ ]= Eτ [ST φτ,T ]φt,τ= Fτφt,τ . (32)

By definition

Fτ+1τ = Fτ +1Fτ ,hence

Eτ [Fτ+1τ θτ+1τ ] = Eτ [Fτ θτ+1τ ] + Eτ [1Fτθτ+1τ ]= Eτ [FτEτ+1τ (φt,T )] + Eτ [1Fτθτ+1τ ]= FτEτ (φt,T )+ Eτ [1Fτθτ+1τ ]= Fτφt,τ + Eτ [1Fτθτ+1τ ] (33)

Combining (32) and (33), it follows that

Eτ [1Fτθτ+1τ ] = 0.

Hence, taking limits,

lim1τ→0

Eτ [1Fτθτ+1τ ] = Eτ(dFτ θτ+dτ ) = 0.

�

Appendix C: Proof of Corollary 3

From Theorem 4, the elasticity of the pricing kernel is

η = bE[R(CT )

u′(CT )E [u′(CT ) | ST ]

| ST].

Hence, forb 6= 0,


1

b2

∂η

∂ ln ST= E

[R′(CT )CT

u′(CT )E [u′(CT ) | ST ]

| ST]

−E{R(CT )2u′(CT )E[u′(CT ) | ST ]}E[u′(CT ) | ST ]2

+E[R(CT )u′(CT ) | ST ]2

E[u′(CT ) | ST ]2

and therefore

1

b2

∂η

∂ ln ST= E

[R′(CT )CT

u′(CT )E[u′(CT ) | ST ] | ST

]−E[R(CT )u

′(CT ){R(CT )− R(ST )} | ST ]E[u′(CT ) | ST ] . (34)

AsE[u′(CT ){R(CT )− R(ST )} | ST ] = 0, we can expand the second term to

−E[{R(CT )− R(ST )}u′(CT ){R(CT )− R(ST )} | ST ]

E[u′(CT ) | ST ] < 0. (35)

R′(CT ) ≤ 0 means that the first term is negative. We have shown that the secondterm in (34) is also negative. Therefore,η declines in lnST . �

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