Drift and the Risk-Free Rate
It is proven, under a set of assumptions differing from the usual ones in the unboundedness of the time interval, that, in an economy in equilibrium consisting of a risk-free cash account and an equity whose price process is a geometric Brownian motion on , the drift rate must be close to the risk-free rate; if the drift rate and the risk-free rate are constants, then and the price process is the same under both empirical and risk neutral measures. Contributing in some degree perhaps to interest in this mathematical curiosity is the fact, based on empirical data taken at various times over an assortment of equities and relatively short durations, that no tests of the hypothesis of equality are rejected.
In the Black-Scholes model of a market with a single equity, its price is a geometric Brownian motion (GBM) satisfying for time the stochastic differential equation where the volatility , the drift rate , and the rate for the risk-free security are all constants. The stochastic process , is a standard Brownian motion. In the formulation of Harrison and Kreps  the process is on , , and is defined on the probability space , where the filtration , , is that generated by . They show in this case that under equilibrium pricing for their securities market model, allowing only simple trading strategies, there is a measure equivalent to and prices can be expressed as expectations with respect to . Furthermore, under , is a martingale on with respect to . There are three free parameters in the model, , , and . It is shown here that, if the equity's prices are given by (1.1) on and again only simple trading strategies are allowed on finite but arbitrary sets of nonrandom times, then there is an equivalent martingale measure and pricing with respect to it in the same manner represents a viable pricing system in the sense of Kreps  if and only if . In this case, there are really only two free parameters.
Besides the results found in Lemma 4.5 relating to the Black-Scholes model, results of a somewhat more general nature in which , , and depend upon time deterministically, can be found in Lemma 4.1.
The arguments given here are for an economy consisting of a single equity and a cash account. To the extent, therefore, that such models are pertinent to actual equities prices, an empirical investigation of for real market data is of interest. Assuming that the model is true for our empirical data consisting of daily closes of some selected equities, the hypothesis that is tested and in no case is the hypothesis of equality rejected by these optimal tests.
The organization of the paper is as follows. First the terms, definitions, and basic results of Harrison and Kreps  are recalled in the context of our model on . Then connections are made between a presumed empirical GBM process with drift and the martingale arising from viability. It is not assumed but shown that the price process must also be a GBM under the equivalent measure; its drift is and its volatility agrees with that of the empirical GBM. It is shown also that the arguments used in  to obtain this result on cannot generally be used here. Next, the main result on is presented; namely, that under equilibrium the drift of the empirical GBM must be the risk-free rate. If the price process is a GBM under the empirical measure, then a consequence of viability is that it is also a GBM under an equivalent (risk-neutral) measure. Finally, the development and results of our hypothesis tests appear in Tables 1 and 2.
Proofs of technical details most pertinent to the main ideas appear in the body of the paper; proofs of more tangential ones have been placed in the appendix.
2. Viability, the Extension Property, and Equivalent Martingale Measures
The notation and assumptions are those of  except that here there is an infinite rather than a finite horizon. Thus, there is a linear space of functions which are random variables defined on a probability space . The points represent states of the world; the points represent bundles of goods in some abstract economy. A subspace represents the space of bundles that can be constructed out of marketed bundles of goods. There is a bounded linear functional defined on , with representing the market price of and a collection of agents represented by complete transitive binary relations on the space . The pair is viable (as a model of economic equilibrium) if there is an order of the above specifications and an such that and for all such that . Letting denote the collection of positive bounded linear functionals on , Kreps  proves the following lemma.
Lemma 2.1. The price system is viable with respect to if and only if there is a such that .
Here the securities market model of Harrison and Kreps  is extended to and involves, as there, a risk-free security with rate 0 and a security whose price at time under the state of the world is , where the second-order stochastic process is measurable with respect to a filtration . Only simple trades are allowed. Simple trading strategies are denoted by and implicit in 's description is a finite set of nonrandom trading times . The 2-vector of functions has elements which indicate the units held in the risk-free asset and the equity so that the value of the portfolio at time is , the ordinary inner product of the vector with . The function is -adapted, and, for each is constant on . As in  simple trades involve finite arbitrary collections of nonrandom points of time at which trades occur but, in contrast, here there is no fixed “consumption time.” Instead, if the trading strategy has its last trading time at as above, then at the last time all is placed in the risk-free cash account so that, at times . The subspace is the linear span of the random variables , where is a simple trading strategy. It is implicit that for each , an assumption made throughout. A simple trading strategy is self-financing if for each . If a security market model is viable and if is self-financing, then for . The existence of an equivalent martingale measure is asserted in Lemma 2.2 and its proof can be carried out as by Harrison and Kreps.
Lemma 2.2. If is viable with respect to , then there is an equivalent measure with and under this measure is a martingale with respect to .
The risk-free security of concern here has instantaneous rate at time , the price process solves (3.1), and the corresponding trades relative to the process of real interest here, , can be obtained by taking and (see [1, Section 7]). Under a viable pricing system, it follows that is a martingale with respect to . Thus, pricing for a final transaction time is given by .
3. Price Process under and
It is assumed that under the price process , solves the SDE analogous to (1.1) but with deterministically varying , , and subject to the following assumptions: (A1)the functions and are continuous on and is absolutely continuous with a derivative bounded on compact intervals, (A2)for some and all , holds, (A3)the risk premium is uniformly bounded: for some and all , ,(A4)the risk free rate is uniformly bounded: for some and all , .
The solution to (3.1) at time given its value at time can be written explicitly as
Under the measure the continuous discounted value process is a martingale, so by Theorems 4.2 (Chapter 3) of Karatzas and Shreve  and IX.5.3 of Doob , for example, provided the quadratic variation process is an absolutely continuous function of for -almost every with a nonzero derivative, there exists a Wiener process under and a measurable -adapted process such that . Moreover . It also follows by Lemma 3.1 that under the equivalence of and one has .
Lemma 3.1. Let conditions (A) be satisfied, and let satisfy (3.1), where is a standard Brownian motion under . Suppose that is a martingale under the equivalent measure . Then, under the measure , . See proof in the Appendix.
By Ito’s formula, , so It follows from Lemma 3.1 that, under the equivalence of and , one has and hence that Thus, if , then under
It has been shown that under the equivalent martingale measure the price process satisfies on an SDE with a standard Brownian motion , , the same volatility as the empirical one, and a drift coincident with the risk-free rate. How does this result on relate to the results of  on , ?
Harrison and Kreps show that there is a probability measure equivalent to under which , is a martingale, where is restricted to . Karatzas and Shreve [3, Section 3.5A] (see also [5, Section 1.7, Proposition 7.4]), show that there is a probability measure on with the property that for every the measure restricted to is . They point out, however, that generally need not be equivalent to . That is, in fact the case if as can be seen from Proposition 3.2 (see also the remark preceding Example 7.6, Chapter 1, of ). In that case, not only is our measure not obtained from the arguments of Harrison and Kreps, it is singular with respect to one that is. The scope of our main result below would be more limited if the finiteness of the integral were assumed.
Proposition 3.2. If the measure satisfies and, for each , then if . See proof in the Appendix.
4. Relationship between and
Let be arbitrary. Suppose that there is an essentially disjoint collection of subintervals of the real line satisfying Under the assumption that the functions and are continuous observe that, unless for all , for sufficiently small, the collections are nonempty. Denote the Lebesgue measure of an interval by and fix for which is nonempty. The equities price process under the actual probability measure is given in (3.2) and if the pricing system is viable then under the same process is given by (3.7). Lemmas 4.2, 4.3, and 4.4 are used in the proof of Lemma 4.1, the key to our main result.
Lemma 4.1. Under the assumptions (A) and for some , if is a viable pricing system for , the class of marketable claims under simple trading strategies, then for every , the set of indices for which is at most finite.
Proof. Suppose that is viable and the claim is not true. Then, for some , there is an infinite collection of such intervals , with . Writing
assume without loss of generality that thereon . Then, also without loss of generality, one can assume that there are points , where , and , , where .
Consider a sequence of trading strategies . Under the simple strategy , buy shares of the equity at time to spend units. At this time also sell of the risk free security to spend at time . Cash flow at time is then 0. At time , sell the equity to spend and redeem the bond to spend . “Cash” on hand at this stage is Invest it in the risk-free security. At times , repeat this, buying at time , shares of the equity and selling of the risk-free security to spend a total of 0. At time , sell the shares to obtain and redeem the bond to spend and invest the “cash” in the risk-free security. At time , there will be an amount The price of this will be Under the geometric Brownian motion model (3.1), the term inside the expectation is , where an average of independent, but not identically distributed, random variables. By Lemma 4.2, there is a subsequence and an such that and, under , Since a sequence converging in probability has a subsequence which converges almost surely, there is a subsequence such that . By equivalence of and , that convergence is also almost surely . By Lemma 4.4, the sequence is uniformly integrable under , so expectations converge (see [6, Theorem 5.4]) and one concludes that . On the other hand, by Lemma 4.3, for every . It follows that , a contradiction.
Lemma 4.2. Under conditions (A), there is a subsequence and such that (see proof in the Appendix).
Lemma 4.3. (see proof in the Appendix).
Lemma 4.4. Under conditions (A) the sequence is uniformly integrable under (see proof in the Appendix).
Lemma 4.5. If and are constant and if the model is viable, then .
Proof. Suppose that . Then, for , and any , essentially disjoint intervals in can be chosen in such a way that there is an infinite collection satisfying . By Lemma 4.1 this violates the viability of the model.
The next most interesting case is when where , the possibly varying risk premium, is assumed constant.
Lemma 4.6. Under conditions (A), if the pricing system is viable and if (4.10) holds for all , then .
Proof. Assume that , and let . Then, for any essentially disjoint intervals in can be chosen in such a way that there is an infinite collection, contradicting Lemma 4.1 unless .
Returning to the market expressed in terms of , define the functional on , where is the empirical measure under which satisfies (3.1), by . Our interest in the following theorem centers on the pricing system defined on by , and is the measure corresponding to the process solving (3.6).
Theorem 4.7. For the Black-Scholes model on (, , and are constants), the pricing system is viable with respect to if and only if .
Proof. By Lemma 4.5 it is known that, if the system is viable then . It suffices to show that, if , then the pricing system given by is viable. According to Lemma 2.1, it suffices to show, as it plainly does here, that extends .
5. Empirical Considerations
For these considerations to have relevance to real data, equities prices should be adequately modeled as solutions to SDE (3.1). For roughly a century, models in agreement with (3.1) [7–10] have appeared in the literature, and we will assume this model here. Statistical tests of , assuming the Black-Scholes model over a suitably brief time span, are then employed on a small set of data. The results found in Tables 1 and 2 are consistent with the truth of the hypothesis of equality.
5.1. UMPU Test
Assuming the model (3.1) with and constant, some tests of are developed here based on readily available daily log return data, , where are i.i.d. , and applied to different underlying equities at various historical times.
Let . Setting and , the hypothesis pair versus becomes versus , which will be tested based on observing i.i.d. . Writing one has that are joint sufficient statistics for . It can be seen, based upon the theory of tests of a single parameter from an exponential family (see ), that a uniformly most powerful unbiased (UMPU) test of size exists for testing the hypothesis of interest here, Furthermore, denoting by the power function of the test , one has the following result.
Lemma 5.1. The test which rejects if , where satisfies for all in See proof in the Appendix.
The results of testing hypothesis (5.2) for various equities at varying times are found in Table 1 and for a fixed set of times in Table 2. The risk-free rates were determined from the US treasury for the corresponding time spans at each initial time and, in the latter case, reveal that none of the drift rates differ significantly (at ) from , the daily rate for 26-week treasury bills during that stretch. In the tables, the entry Effect Size is , a rough estimate of in terms of the volatility.
It is perhaps surprising that there were no significances especially in the case of C in Table 1 and AAPL in Table 2, the former because of the long time span and the approximation assuming a fixed risk-free rate in a world in which it is constantly changing, and the latter because on June 29, 2007 Apple introduced its first iPhone a major milestone in the company's rising fortunes. The former is in line with the sample size and observed effect size while the latter is consistent with a fundamental change.
5.2. Likelihood Ratio Tests
Likelihood ratio tests present an alternative possibility. They are known to be optimal (see, e.g., ) in large samples. According to Wilks’ theorem, the null hypothesis should be rejected when is too large. Setting one has the following.
Lemma 5.2. The likelihood ratio test of (5.2) of size rejects if the test statistic exceeds . See proof in the Appendix.
values are found in Tables 1 and 2 both for the UMPU test and for the LRT and one observes that they are quite close and, again, there are no significances at .
Proof of Lemma 3.1. Fix and an equidistant partition of the interval . To ease notation let us denote , , , and correspondingly , . Also, denote , .
Notice that, under , is a normal variable with mean zero and variance . Let us evaluate Since , for sufficiently close to zero . Moreover, , and since , there exists a subsequence on which convergence holds almost surely. In order to simplify notation, we assume that . Then, for large enough and for almost all we have We have , and . From conditions (A) it follows that there exists such that Moreover, using that is independent of , we have and it follows that has a subsequence which converges -almost surely to zero.
Let us now evaluate therefore has a subsequence that converges -almost surely to zero. and therefore has a subsequence convergent to zero -almost surely
For the last term , we write Since is a continuous square integrable martingale, by Theorem 5.8 of [3, Chapter 1] it follows that in probability . By [13, Proposition 2.3, Chapter 2], ; therefore, converges to zero -almost surely on a subsequence.
As for , from Hölder’s inequality we have By Itô’s isometry, For large enough By a similar argument, for some constant depending on , , , and . Then, and therefore has a subsequence converging to zero -almost surely It has been shown that almost surely on a subsequence, and, since , convergence holds almost surely in as well. Thus, in , and it follows that .
Proof of Lemma 4.1. The random variables constitute a martingale and is the Radon-Nikodym (RN) derivative of the probability measure with respect to . By Theorem 4.1 of [4, page 319] since are nonnegative and for all , one has that with -probability 1. One has from [1, Theorem 3], that for each , . Then since , one has for any such that is a continuity point of the distribution of the random variable that So, unless , one has . But then so that choosing shows that cannot be the RN derivative of a probability measure absolutely continuous with respect to .
Proof of Lemma 4.2. Existence of the subsequence and nonzero limit is obvious from assumptions (A) and the sequential compactness of the real line. Since under and is just the mgf of this random variable evaluated at , it follows that is zero mean under and that, defining , one has , where are given in (4.6). Under , are independent across for each . Let , , and . Then, are constants converging to along a subsequence and , where and . Setting , so one has In case assumptions (A) hold, then since so that for all , it follows that is . Therefore on the subsequence for which . This convergence implies convergence in probability.
Proof of Lemma 4.3. Under , is a martingale so that for one has Therefore, Thus, and the claim follows.
Proof of Lemma 4.4. First consider the distribution of the random variable under . Under , To see this, compute the MGF of as . Since , by (A.22) one has and the claim has been established. Next the claim is that To see this, set and recall that and , so that Therefore, one has We do not know that are independent under but by Jensen's inequality so that and under (A) this is uniformly bounded in . Therefore, the sequence is uniformly integrable under .
Proof of Lemma 5.1. As it is well known, the test will satisfy the derivative condition for exponential class densities
and the size condition
where the expectations refer to the conditional distribution of given . By the theory of Lehmann , the UMPU test can be based upon the statistic
since it is a monotonic function of given . The latter follows simply from
There is no closed-form expression for the conditional distribution of given under the condition but a large sample approximation can be made as follows.
It is shown below that under the hypothesis , as the sample size , where . Therefore, by Slutsky’s theorems, a test suggests itself; namely, reject if exceeds in absolute value the upper cutoff of the standard normal. That test, which is more complicated than the one given in (5.3), would have asymptotically the same power function as the UMPU test, but it is sufficient here to operate under the null hypothesis. Under the null hypothesis one has so a suitable test statistic is given by (5.3). The claimed asymptotic distribution in (A.35) is verified. It is well known that with one has Here and . With , and the latter, under the null hypothesis, is 0. Hence, by the -method and under the null hypothesis where Upon simplification, the claim has been verified.
Proof of Lemma 5.2. Write the maximum of the density under as , denote the denominator which is the unrestricted maximum similarly, and find Since for and it is the case that attained at , one has For the numerator is sought as is the location of the maximum. The location is therefore where is attained. More simply, , where maximizes Thus, and the derivative is zero at so that taking account of the proper sign and that the derivative is zero at the maximum, (5.5), the LRT rejects if exceeds the upper cutoff of a chi square with 1 degree of freedom. Simplifying, the test statistic is that in (5.6).
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