The Capital Asset Pricing Model as a corollary of the Black Scholes model

he Capital Asset Pricing Model as a corollary of the Black Scholes model Vladimir Vovk he Game-heoretic Probability and Finance Project Working Paper #39 September 6, 011 Project web site: http://www.probabilityandfinance.com

Abstract We consider a financial market in which two securities are traded: a stock and an index. heir prices are assumed to satisfy the Black Scholes model. Besides assuming that the index is a tradable security, we also assume that it is efficient, in the following sense: we do not expect a prespecified self-financing trading strategy whose wealth is almost surely nonnegative at all times to outperform the index greatly. We show that, for a long investment horizon, the appreciation rate of the stock has to be close to the interest rate assumed constant) plus the covariance between the volatility vectors of the stock and the index. his contains both a version of the Capital Asset Pricing Model and our earlier result that the equity premium is close to the squared volatility of the index. Contents 1 Introduction 1 heoretical performance deficit 3 Capital Asset Pricing Model 5 4 A more direct derivation of the Sharpe Lintner CAPM 8 5 Conclusion 10 References 11

For me, the strongest evidence suggesting that markets are generally quite efficient is that professional investors do not beat the market. Burton G. Malkiel [3] 1 Introduction his article continues study of the efficient index hypothesis EIH ), introduced in [4] under a different name) and later studied in [8] and [6]. he EIH is a hypothesis about a specific index I t, such as FSE 100. Let Σ be any trading strategy that is prudent, in the sense of its wealth process being nonnegative almost surely at all times. We consider only self-financing trading strategies in this article.) rading occurs over the time period [0, ], where the investment horizon > 0 is fixed throughout the article, and we assume that I 0 > 0. he EIH says that, as long as Σ is chosen in advance and its initial wealth K 0 is positive, K 0 > 0, we do not expect K /K 0, where K is its final wealth, to be much larger than I /I 0. he EIH is similar to the Efficient Market Hypothesis EMH; see [1] and [3] for surveys) and in some form is considered to be evidence in favour of the EMH see the epigraph above). But it is also an interesting hypothesis in its own right. For example, in this article we will see that in the framework of the Black Scholes model it implies a version of the Capital Asset Pricing Model CAPM), whereas the EMH is almost impossible to disentangle from the CAPM or similar asset pricing models see, e.g., [1], III.A.6). Several remarks about the EIH are in order following [6]): Our mathematical results do not depend on the EIH, which is only used in their interpretation. hey are always of the form: either some interesting relation holds or a given prudent trading strategy outperforms the index greatly almost surely or with a high probability). Even when using the EIH in the interpretation of our results, we do not need the full EIH: we apply it only to very basic trading strategies. Our prudent trading strategies can still lose all their initial wealth they are only prudent in the sense of not losing more than the initial wealth). A really prudent investor would invest only part of her capital in such strategies. We start the rest of the article by proving a result about the theoretical performance deficit in the terminology of [8]) of a stock S t as compared with the index I t, Namely, in Section we show that, for a long investment horizon and assuming the EIH, ln S /S 0 σ S σ I, 1.1) I /I 0 1

where I 0 is assumed positive and σ S and σ I are the volatility vectors formally defined in Section ) for the stock and the index. We can call σ S σ I / the theoretical performance deficit as it can be attributed to insufficient diversification of S t as compared to I t. Section 3 deduces a version of the CAPM from 1.1); this version is similar to the one obtained in [8] but our interpretation and methods are very different. Section 4 gives a more direct derivation of the CAPM, which improves some constants. Section 5 concludes. heoretical performance deficit he value of the index at time t is denoted I t and the value of the stock is denoted S t. We assume that these two securities satisfy the multi-dimensional Black Scholes model { dit I t = µ I dt + σ I,1 dwt 1 + + σ I,d dwt d ds t S t = µ S dt + σ S,1 dwt 1 + + σ S,d dwt d,.1) where W 1,..., W d are independent standard Brownian motions. For simplicity, we also assume, without loss of generality, that I 0 = 1 and S 0 = 1. he parameters of the model are the appreciation rates µ I, µ S R and the volatility vectors σ I := σ I,1,..., σ I,d ) and σ S := σ S,1,..., σ S,d ). We assume σ I σ S, σ I 0, and σ S 0. he number of sources of randomness W 1,..., W d in our market is d. he interest rate r is constant. We interpret e rt as the price of a zero-coupon bond at time t. Let us say that a prudent trading strategy beats the index by a factor of c if its wealth process K t satisfies K 0 > 0 and K /K 0 = ci. Let N 0,1 be the standard Gaussian distribution on R and z p, p > 0, be its upper p-quantile, defined by the requirement Pξ z p ) = p, ξ N 0,1, when p 0, 1), and defined as when p 1. We start from the following proposition. Proposition.1. Let δ > 0. here is a prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ln S + σ S σ I I < z δ/ σ S σ I..) We assumed σ S 0, but Proposition.1 remains true when applied to the bond B t := e rt in place of the stock S t. In this case.) reduces to ln I e r σ I < z δ/ σ I..3) Informally,.3) says that the index outperforms the bond approximately by a factor of e σ I /. For a proof of this statement which is similar to, but simpler than, the proof of Proposition.1 given later in this section), see [6], Proposition.1.

In the next section we will need the following one-sided version of Proposition.1. Proposition.. Let δ > 0. here is a prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ln S + σ S σ I < z δ σ S σ I..4) I here is another prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ln S + σ S σ I > z δ σ S σ I. I In the rest of this section we will prove Proposition.1 Proposition. can be proved analogously). Without loss of generality suppose δ 0, 1). We let W t stand for the d-dimensional Brownian motion W t := Wt 1,..., Wt d ). he market.1) is incomplete when d >, as it has too many sources of randomness, so we start from removing superfluous sources of randomness. he standard solution to.1) is { I t = e µ I σ I /)t+σ I W t S t = e µ S σ S /)t+σ S W t.5). Choose two vectors e 1, e R d that form an orthonormal basis in the - dimensional subspace of R d spanned by σ I and σ S. Set W 1 t := e 1 W t and W t := e W t ; these are standard independent Brownian motions. Let the decompositions of σ I and σ S in the basis e 1, e ) be σ I = σ I,1 e 1 + σ I, e and σ S = σ S,1 e 1 + σ S, e. Define σ I := σ I,1, σ I, ) R and σ S := σ S,1, σ S, ) R, and define W t as the -dimensional Brownian motion W t := W 1 t, W t ). We can now rewrite.5) as { I t = e µ I σ I /)t+ σ I W t S t = e µ S σ S /)t+ σ S W t. In terms of our new parameters and Brownian motions,.1) can be rewritten as { dit I t = µ I dt + σ I,1 d W t 1 + σ I, d W t ds t S t = µ S dt + σ S,1 d W t 1 + σ S, d W t.6). he risk-neutral version of.6) is { dit I t = rdt + σ I,1 d W t 1 + σ I, d W t ds t S t = rdt + σ S,1 d W t 1 + σ S, d W t, whose solution is { I t = e r σ I /)t+ σ I W t S t = e r σ S /)t+ σ S W t. 3

Let b R and let 1{...} be defined as 1 if the condition in the curly braces is satisfied and as 0 otherwise. he Black Scholes price at time 0 of the European contingent claim paying I 1{S /I b} at time is e r E = e σ I / E e r σ I /) + σ I ξ 1 e σi ξ 1 { e r σ S /) + σ S ξ e r σ I /) + σ I ξ b }) { }) σs σ I ) ξ ln b + σ S σ I,.7) where ξ N 0,1. o continue our calculations, we will need the following lemma. Lemma.3. Let u, v R, v 0, c R, and ξ N0,1. hen E e u ξ 1{v ξ c} ) ) u v c = e u / F, v where F is the distribution function of N 0,1. Proof. his follows from E e u ξ 1{v ξ c} ) = 1 π = 1 π e u / = 1 π e u / e u z 1{v z c} e z R / dz 1{v z c} e z u R / dz 1 {v w c u v} e w R v = e u / P v ξ c u v ) v ) u v c = e u / F. v / dw Now we can rewrite.7) as F σ I σ S σ I ) ln b σ S σ I σ S σ I ) Let us define b by the requirement = F σ S σ I + ln b σ S σ I = z δ/, σ S σ I + ln b σ S σ I ). i.e., ln b = σ S σ I + z δ/ σ S σ I..8) 4

As the Black Scholes price of the European contingent claim I 1{S /I b} is δ/, there is a prudent trading strategy Σ 1 with initial wealth δ/ that almost surely beats the index by a factor of /δ if S /I b. Now let a R and consider the European contingent claim paying I 1{S /I a}. Replacing b by a and ln b by ln a in.7) and defining a to satisfy ln a = σ S σ I z δ/ σ S σ I in place of.8), we obtain a prudent trading strategy Σ that starts from δ/ and almost surely beats the index by a factor of /δ if S /I a. he sum Σ := Σ 1 + Σ will beat the index by a factor of 1/δ if S /I / a, b). his completes the proof of Proposition.1. 3 Capital Asset Pricing Model In this section we will derive a version of the CAPM from the results of the previous section. Our argument will be similar to that of Section 3 of [6]. Proposition 3.1. For each δ > 0 there exists a prudent trading strategy Σ = Σσ I, σ S, r,, δ) that satisfies the following condition. For each ɛ > 0, either µ S µ I + σ I σ S σ I < z δ/ + z ɛ ) σ S σ I 3.1) or Σ beats the index by a factor of at least 1/δ with probability at least 1 ɛ. Proof. Suppose 3.1) is violated; we are required to prove that some prudent trading strategy independent of ɛ) beats the index by a factor of at least 1/δ with probability at least 1 ɛ. We have either µ S µ I + σ I σ S σ I z δ/ + z ɛ ) σ S σ I 3.) or µ S µ I + σ I σ S σ I z δ/ + z ɛ ) σ S σ I. 3.3) he two cases are analogous, and we will assume, for concreteness, that 3.) holds. As.5) solves.1), we have ln S = µ S µ I ) + σ I σ S + σ S σ I ) ξ, 3.4) I where ξ N d 0,1. In combination with 3.) this gives ln S I σ I + σ S σ I + z δ/ + z ɛ ) σ S σ I ) 5

+ σ I σ S + σ S σ I ) ξ = σ S σ I + z δ/ + z ɛ ) σ S σ I + σ S σ I ) ξ. 3.5) Let Σ be a prudent trading strategy that, almost surely, beats the index by a factor of 1/δ unless.) holds. It is sufficient to prove that the probability of.) is at most ɛ. In combination with 3.5),.) implies i.e., z δ/ σ S σ I > z δ/ + z ɛ ) σ S σ I + σ S σ I ) ξ, 3.6) he probability of the last event is ɛ. σ S σ I σ S σ I ξ < z ɛ. 3.7) Allowing the strategy Σ to depend, additionally, on µ I, µ S, and ɛ, we can improve 3.1) replacing δ/ by δ. Proposition 3.. Let δ > 0 and ɛ > 0. Unless µ S µ I + σ I z δ + z ɛ ) σ S σ I σ S σ I <, 3.8) there exists a prudent trading strategy Σ = Σµ I, µ S, σ I, σ S, r,, δ, ɛ) that beats the index by a factor of at least 1/δ with probability at least 1 ɛ. Proof. We modify slightly the proof of Proposition 3.1: assuming 3.) with δ/ replaced by δ) we now take as Σ a prudent trading strategy that, almost surely, beats the index by a factor of 1/δ unless.4) holds. Combining 3.5) with δ/ replaced by δ) and.4), we get 3.6) with δ/ replaced by δ), and we still have 3.7). Notice that Σ now depends on which of the two cases, 3.) or 3.3) with δ/ replaced by δ), holds. Propositions 3.1 and 3. are similar to Black s version of the CAPM, and we will derive corollaries of Proposition 3. similar to the Sharpe Lintner CAPM we do not state the analogous easy corollaries of Proposition 3.1). But before stating and proving these corollaries, we will discuss them informally, to give us a sense of direction. Assuming δ 1, ɛ 1, and 1, we can interpret 3.8) as saying that µ S µ I σ I + σ S σ I. 3.9) his approximate equality is applicable to the bond as well as the stock by results of [6]), which gives µ I r + σ I. 3.10) Combining 3.9) and 3.10) we obtain µ S r + σ S σ I. 3.11) 6

And combining 3.11) and 3.10) we obtain µ S r + σ S σ I σ I µ I r). 3.1) Equation 3.1) is a continuous-time version of the Sharpe Lintner CAPM. he standard Sharpe Lintner CAPM see, e.g., [], pp. 8 9) can be written in the form ER S ) = r + covr S, R I ) σ ER I ) r), 3.13) R I ) where R S and R I are the returns of a risky asset and the market portfolio, respectively. he correspondence between 3.1) and 3.13) is obvious. Equation 3.9) can be regarded as an analogue of Black s version of the CAPM, not involving the interest rate. Now we state formal counterparts of 3.10) 3.1). he following proposition, which would have been a corollary of Proposition 3. had we allowed σ S = 0 or of heorem 4.3 below had we allowed σ S = σ I ), is proved in [6], Proposition 3.. Proposition 3.3. Let δ > 0 and ɛ > 0. Unless µ I r σ I < z δ + z ɛ ) σ I, 3.14) there exists a prudent trading strategy Σ = Σµ I, σ I, r,, δ, ɛ) that beats the index by a factor of at least 1/δ with probability at least 1 ɛ. he following two corollaries of Propositions 3. and 3.3 assert existence of trading strategies that depend on everything, namely, on µ I, µ S, σ I, σ S, r,, δ, and ɛ. he first corollary formalizes 3.11). Corollary 3.4. Let δ > 0 and ɛ > 0. Unless µ S r σ S σ I < z δ + z ɛ ) σ I + σ S σ I, 3.15) there exists a prudent trading strategy that beats the index by a factor of at least with probability at least 1 ɛ. 1 δ Proof. Let Σ 1 be a prudent trading strategy satisfying the condition of Proposition 3., and let Σ be a prudent trading strategy satisfying the condition of Proposition 3.3. Without loss of generality suppose that the initial wealth of both strategies is 1. hen Σ 1 + Σ will beat the index by a factor of at least 1 δ with probability at least 1 ɛ unless both 3.8) and 3.14) hold. he conjunction of 3.8) and 3.14) implies 3.15). Finally, we have a corollary formalizing the Sharpe Lintner CAPM 3.1). 7

Corollary 3.5. Let δ > 0 and ɛ > 0. Unless µ S r σ S σ I σ I µ I r) z δ + z ɛ ) σ I + σ S + σ S σ I, there exists a prudent trading strategy that beats the index by a factor of at least with probability at least 1 ɛ. 1 3δ Proof. Let Σ 1 be a prudent trading strategy satisfying the condition of Proposition 3.3 and Σ be a prudent trading strategy satisfying the condition of Corollary 3.4. Without loss of generality suppose that the initial wealth of Σ 1 is 1 and the initial wealth of Σ is. hen Σ 1 + Σ will beat the index by a factor of at least 1 3δ with probability at least 1 ɛ unless both 3.14) and 3.15) hold. he conjunction of 3.14) and 3.15) implies µ S r σ S σ I σ I µ I r) µ S r σ S σ I σ I σ I + σ S σ I σ I z δ + z ɛ ) σ I + σ S σ I + σ S σ I σ I z δ + z ɛ ) σ I + σ S + σ S σ I. z δ + z ɛ ) σ I z δ + z ɛ ) 4 A more direct derivation of the Sharpe Lintner CAPM In the previous section we deduced the Sharpe Lintner CAPM Corollary 3.5) from our result about the theoretical performance deficit. In this section we will derive it in a more direct manner, which will allow us to improve some constants in Corollaries 3.4 and 3.5. We start from modifying Propositions.1 and.: whereas Propositions.1 and. measure the performance of the stock in terms of the index, our new propositions will measure it in terms of the bond. Proposition 4.1. Let δ > 0. here is a prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ) ln S e r + σ S σ S σ I < z δ/ σ S. 4.1) Proposition 4.. Let δ > 0. here is a prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ) ln S e r + σ S σ S σ I < z δ σ S. 4.) 8

here is another prudent trading strategy Σ = Σσ I, σ S, r,, δ) that, almost surely, beats the index by a factor of 1/δ unless ) ln S e r + σ S σ S σ I > z δ σ S. he proofs of Proposition 4.1 and the two parts of Proposition 4. are very similar, and we will only prove 4.), again assuming δ 0, 1). Proof of Proposition 4. part 4.)). Let b R; we will be using the notation σ I and σ S introduced in Section. he Black Scholes price at time 0 of the European contingent claim paying I 1{S / e r b} at time is e r E e r σ I /) + σ I ξ 1 = e σ I / E { e r σ S /) + σ S ξ e σi ξ 1 e r b }) { }) σs ξ ln b + σ S, where ξ N 0,1 cf..7)). By Lemma.3 this can be rewritten as F ) ln b + σs σ I σ S. σs It remains to define b by the requirement ln b + σ S σ I σ S σs = z δ and remember that σ S = σ S and σ S σ I = σ S σ I. he following result strengthens Corollary 3.4; its proof is similar to that of Proposition 3.. heorem 4.3. Let δ > 0 and ɛ > 0. Unless µ S r σ S σ I < z δ + z ɛ ) σ S, 4.3) there exists a prudent trading strategy Σ = Σµ S, σ I, σ S, r,, δ, ɛ) that beats the index by a factor of at least 1/δ with probability at least 1 ɛ. Proof. Suppose 4.3) is violated. For concreteness, let µ S r σ S σ I z δ + z ɛ ) σ S. 4.4) 9

From.5), ln S e r = µ S r) σ S + σ S ξ, 4.5) where ξ N d 0,1. In conjunction with 4.4) this implies ln S e r σ S σ I + z δ + z ɛ ) σ S σ S + σ S ξ. 4.6) Let Σ be a prudent trading strategy that, almost surely, beats the index by a factor of 1/δ unless 4.) holds. o see that the probability of 4.) is at most ɛ, notice that the conjunction of 4.6) and 4.) implies i.e., z δ σ S > z δ + z ɛ ) σ S + σ S ξ, σ S σ S ξ < z ɛ. he strategy Σ in heorem 4.3 depends on µ S but does not depend on µ I. We can make Σ independent of µ S if we replace δ in 4.3) by δ/: take as Σ a prudent trading strategy that, almost surely, beats the index by a factor of 1/δ unless 4.1) holds. Cf. Propositions 3.1 and 3..) Using heorem 4.3 in place of Corollary 3.4, we can strengthen Corollary 3.5 as follows. Corollary 4.4. Let δ > 0 and ɛ > 0. Unless µ S r σ S σ I σ I µ I r) z δ + z ɛ ) σ S, there exists a prudent trading strategy that beats the index by a factor of at least with probability at least 1 ɛ. 1 δ 5 Conclusion Let us summarize our results at the informal level of approximate equalities such as 3.9) 3.1). At this level, our only two results are the CAPM 3.1) and the equity premium relation 3.10) established earlier in [6]); the rest follows. Indeed, 3.1) and 3.10) imply 3.11), and 3.11) and 3.10) imply 3.9). he crude form 1.1) of.) also follows from 3.1) and 3.10): just combine the crude form ln S µ S µ I ) + σ I σ S I of 3.4) with 3.9). Finally, the crude form ) ln S e r σ S σ I σ S 10

of 4.1) follows from 3.1) and 3.10) by combining the crude form ln S e r µ S r) σ S of 4.5) with 3.11). An alternative, simpler, summary of our results at the informal level is given by the approximate equality 3.11) in which we allow S = I. We can allow S = I even in heorem 4.3: when S = I, it reduces to Proposition 3.3. he approximate equality 3.11) implies both 3.10) it is a special case for S := I) and 3.1) combine 3.11) and 3.10)). herefore, at the informal level, heorem 4.3 or its weaker version Corollary 3.4) is the core result of this article. One interesting direction of further research is to derive probability-free and continuous-time versions of our results e.g., in the framework of [5]). he results of [8] are probability-free and very similar to the results of this article, but the discrete-time framework of [8] makes them mathematically unattractive. he results of [7] are probability-free, very similar to the results of this article, and are stated and proved in a continuous-time framework; they, however, use nonstandard analysis. Acknowledgments his research has been supported in part by NWO Rubicon grant 680-50-1010. References [1] Eugene F. Fama. Efficient capital markets: A review of theory and empirical work. Journal of Finance, 5:383 417, 1970. [] Eugene F. Fama and Kenneth R. French. he Capital Asset Pricing Model: heory and evidence. Journal of Economic Perspectives, 18:5 46, 004. [3] Burton G. Malkiel. Reflections on the Efficient Market Hypothesis: 30 years later. Financial Review, 40:1 9, 005. [4] Glenn Shafer and Vladimir Vovk. Probability and Finance: It s Only a Game! Wiley, New York, 001. [5] Vladimir Vovk. Continuous-time trading and the emergence of probability. he Game-heoretic Probability and Finance project, Working Paper 8, http://probabilityandfinance.com, July 011. he journal version is to appear in Finance and Stochastics. Older versions: http://arxiv.org/ abs/0904.4364. [6] Vladimir Vovk. he efficient index hypothesis and its implications in the BSM model. he Game-heoretic Probability and Finance project, http://probabilityandfinance.com, Working Paper 38, September 011. 11

[7] Vladimir Vovk and Glenn Shafer. Game-theoretic capital asset pricing in continuous time. he Game-heoretic Probability and Finance project, http://probabilityandfinance.com, Working Paper, December 001. [8] Vladimir Vovk and Glenn Shafer. he Game-heoretic Capital Asset Pricing Model. he Game-heoretic Probability and Finance project, http://prob abilityandfinance.com, Working Paper 1, November 001. Published in International Journal of Approximate Reasoning, 49:175 197, 008. 1