Drift and the Risk-Free Rate

Abstract

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 [0,∞), the drift rate must be close to the risk-free rate; if the drift rate 𝜇 and the risk-free rate 𝑟 are constants, then 𝑟=𝜇 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.

1. Introduction

In the Black-Scholes model of a market with a single equity, its price 𝑆_𝑡 is a geometric Brownian motion (GBM) satisfying for time 𝑡≥0 the stochastic differential equation𝑑𝑆_𝑡=𝜇𝑆_𝑡𝑑𝑡+𝜎𝑆_𝑡𝑑𝐵_𝑡, (1.1) where the volatility 𝜎, the drift rate 𝜇, and the rate 𝑟 for the risk-free security are all constants. The stochastic process 𝐵_𝑡, 0≤𝑡 is a standard Brownian motion. In the formulation of Harrison and Kreps [1] the process is on 𝑡∈[0,𝑇], 𝑇<∞, and is defined on the probability space (Ω,ℱ_𝑇,𝑃_𝑇), where the filtration ℱ_𝑡=𝜎(𝐵_𝑠,0≤𝑠≤𝑡), 𝑡≥0, is that generated by 𝐵_𝑡. They show in this case that under equilibrium pricing for their securities market model, allowing only simple trading strategies, there is a measure 𝑃^∗_𝑇 equivalent to 𝑃_𝑇 and prices can be expressed as expectations with respect to 𝑃^∗_𝑇. Furthermore, under 𝑃^∗_𝑇, 𝑒^−𝑟𝑡𝑆_𝑡 is a martingale on [0,𝑇] with respect to ℱ_𝑡. There are three free parameters in the model, 𝜇, 𝜎, and 𝑟. It is shown here that, if the equity's prices are given by (1.1) on [0,∞) 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 [2] if and only if 𝑟=𝜇. 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 𝑟, 𝜇, and 𝜎 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 𝜇=𝑟 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 𝑟=𝜇 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 [1] are recalled in the context of our model on [0,∞). Then connections are made between a presumed empirical GBM process with drift 𝜇 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 𝑟 and its volatility agrees with that of the empirical GBM. It is shown also that the arguments used in [1] to obtain this result on [0,𝑇] cannot generally be used here. Next, the main result on 𝜇=𝑟 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.

Table 1 

𝑃 values for testing 𝑟=𝜇. None are significant at 0.05.

Table 2 

𝑃 values for testing 𝑟=𝜇, March 1, 2007, to August 30, 2007.

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 [1] except that here there is an infinite rather than a finite horizon. Thus, there is a linear space 𝑋 of functions 𝑥∶Ω→𝑅 which are random variables defined on a probability space (Ω,ℱ,𝑃). The points 𝜔∈Ω represent states of the world; the points 𝑥∈𝑋 represent bundles of goods in some abstract economy. A subspace 𝑀⊂𝑋 represents the space of bundles that can be constructed out of marketed bundles of goods. There is a bounded linear functional 𝜋 defined on 𝑀, with 𝜋(𝑚) representing the market price of 𝑚∈𝑀 and a collection 𝐴 of agents represented by complete transitive binary relations ≽ on the space 𝑋. The pair (𝑀,𝜋) is viable (as a model of economic equilibrium) if there is an order ≽∈𝐴 of the above specifications and an 𝑚^∗∈𝑀 such that 𝜋(𝑚^∗)≤0 and 𝑚^∗≽𝑚 for all 𝑚∈𝑀 such that 𝜋(𝑚)≤0. Letting Ψ denote the collection of positive bounded linear functionals on 𝑋, Kreps [2] proves the following lemma.

Lemma 2.1. The price system (𝑀,𝜋) is viable with respect to 𝑋 if and only if there is a 𝜓∈Ψ such that 𝜓∣𝑀=𝜋.

Here the securities market model of Harrison and Kreps [1] is extended to [0,∞) and involves, as there, a risk-free security with rate 0 and a security whose price at time 𝑡 under the state of the world 𝜔∈Ω is 𝑍_𝑡(𝜔), where the second-order stochastic process 𝑍_𝑡 is measurable with respect to a filtration ℱ_𝑡⊂ℱ_∞=𝜎(∪_𝑠≥0ℱ_𝑠)=ℱ. Only simple trades are allowed. Simple trading strategies are denoted by 𝜃 and implicit in 𝜃's description is a finite set of nonrandom trading times 0≤𝑡₁<t₂<⋯<𝑡_𝑘. The 2-vector of functions 𝜃(𝑡)=(𝜃_𝑅(𝑡),𝜃_𝐸(𝑡)) has elements which indicate the units held in the risk-free asset and the equity so that the value of the portfolio at time 𝑡 is 𝜃(𝑡)⋅𝑉(𝑡), the ordinary inner product of the vector 𝜃 with 𝑉(𝑡,𝜔)=(1,𝑍_𝑡(𝜔)). The function 𝜃 is ℱ_𝑡-adapted, and, for each 𝜔,𝜃(𝑡) is constant on 𝑡_𝑖−1≤𝑡<𝑡_𝑖. As in [1] 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 𝑡_𝑘 as above, then at the last time all is placed in the risk-free cash account so that, at times 𝑡≥𝑡_𝑘,𝜃(𝑡)=(𝜃(𝑡_𝑘)⋅𝑉(𝑡_𝑘),0). The subspace 𝑀 is the linear span of the random variables 𝜃⋅𝑉, where 𝜃 is a simple trading strategy. It is implicit that 𝐸_𝑃[‖𝜃(𝑡_𝑗)‖²]<∞ for each 𝑗=1,2,…,𝑘, an assumption made throughout. A simple trading strategy 𝜃 is self-financing if 𝜃(𝑡_𝑗−1)⋅𝑉(𝑡_𝑗)=𝜃(𝑡_𝑗)⋅𝑉(𝑡_𝑗) for each 𝑗=1,2,…,𝑘. If a security market model (𝑀,𝜋) is viable and if 𝜃 is self-financing, then 𝜃(0)⋅𝑉(0)=𝜋(𝑚) for 𝑚=lim_𝑡→∞𝜃(𝑡)⋅𝑉(𝑡)=𝜃(𝑡_𝑘)⋅𝑉(𝑡_𝑘). 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 (𝑀,𝜋) is viable with respect to 𝑋=𝐿₂(Ω,ℱ,𝑃), then there is an equivalent measure 𝑃^∗ with 𝑑𝑃^∗/𝑑𝑃∈𝑋 and under this measure 𝑍_𝑡 is a martingale with respect to ℱ_𝑡.

The risk-free security of concern here has instantaneous rate 𝑟(𝑡) at time 𝑡, the price process 𝑆_𝑡 solves (3.1), and the corresponding trades relative to the process of real interest here, 𝑉_𝑡=(𝑒^{∫𝑡0𝑟(𝑠)𝑑s},𝑆_𝑡), can be obtained by taking 𝑍_𝑡=𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡 and 𝑉_𝑡=𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑉_𝑡 (see [1, Section 7]). Under a viable pricing system, it follows that 𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡 is a martingale with respect to ℱ_𝑡. Thus, pricing for a final transaction time 𝑇 is given by 𝜋(𝑚)=𝐸_𝑃^∗[𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}𝑚].

3. Price Process under 𝑃 and 𝑃^∗

It is assumed that under 𝑃 the price process 𝑆_𝑡(𝜔), 𝑡≥0 solves the SDE𝑑𝑆_𝑡=𝜇(𝑡)𝑆_𝑡𝑑𝑡+𝜎(𝑡)𝑆_𝑡𝑑𝐵_𝑡 (3.1) analogous to (1.1) but with deterministically varying 𝑟(⋅), 𝜇(⋅), and 𝜎(⋅) subject to the following assumptions: (A1)the functions 𝜇(⋅) and 𝑟(⋅) are continuous on [0,∞) and 𝜎(⋅) is absolutely continuous with a derivative bounded on compact intervals, (A2)for some 0<𝜎_𝐿<𝜎_𝑈<∞ and all 𝑠≥0, 𝜎_𝐿≤𝜎(𝑠)≤𝜎_𝑈 holds, (A3)the risk premium 𝜌(𝑠)=𝜎⁻¹(𝑠)(𝜇(𝑠)−𝑟(𝑠)) is uniformly bounded: for some 𝜌_𝑈<∞ and all 𝑠>0, |𝜌(𝑠)|≤𝜌_𝑈,(A4)the risk free rate is uniformly bounded: for some 0<𝑟_𝑈<∞ and all 𝑠≥0, |𝑟(𝑠)|≤𝑟_𝑈.

The solution 𝑆 to (3.1) at time 𝑇 given its value at time 0≤𝑡<𝑇 can be written explicitly as𝑆_𝑇=𝑆_𝑡exp^𝑇_𝑡𝜇(𝑠)1𝑑𝑠−2^𝑇_𝑡𝜎²(𝑠)𝑑𝑠+^𝑇_𝑡𝜎(𝑠)𝑑𝐵_𝑠. (3.2)

Under the measure 𝑃^∗ the continuous discounted value process 𝑋_𝑡=𝑆_𝑡exp−^𝑡₀𝑟(𝑠)𝑑𝑠 (3.3) is a martingale, so by Theorems 4.2 (Chapter 3) of Karatzas and Shreve [3] and IX.5.3 of Doob [4], for example, provided the quadratic variation process ⟨𝑋⟩_𝑡(𝜔) is an absolutely continuous function of 𝑡 for 𝑃^∗-almost every 𝜔 with a nonzero derivative, there exists 𝑊_𝑡 a Wiener process under 𝑃^∗ and a measurable ℱ_𝑡-adapted process Φ(𝑡,𝜔) such that 𝑑(𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡(𝜔))=Φ(𝑡,𝜔)𝑑𝑊_𝑡(𝜔). Moreover ⟨𝑋⟩_𝑡(𝜔)=∫^𝑡₀Φ²(𝑠,𝜔)𝑑𝑠. It also follows by Lemma 3.1 that under the equivalence of 𝑃 and 𝑃^∗ one has Φ²(𝑠,𝜔)=𝑋²_𝑠(𝜔)𝜎²(𝑠).

Lemma 3.1. Let conditions (A) be satisfied, and let 𝑆_𝑡 satisfy (3.1), where 𝐵_𝑡 is a standard Brownian motion under 𝑃. Suppose that 𝑋_𝑡=𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡 is a martingale under the equivalent measure 𝑃^∗. Then, under the measure 𝑃^∗, ⟨𝑋⟩_𝑡(𝜔)=∫^𝑡₀𝑋²_𝑠(𝜔)𝜎²(𝑠)𝑑𝑠. See proof in the Appendix.

By Ito’s formula, 𝑑(𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡)=−𝑟(𝑡)𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡𝑑𝑡+𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑑𝑆_𝑡, so𝑑𝑆_𝑡=𝑟(𝑡)𝑆_𝑡𝑑𝑡+𝑒^{∫𝑡0𝑟(𝑠)𝑑𝑠}Φ𝑡,𝑆_𝑡𝑑𝑊_𝑡. (3.4) It follows from Lemma 3.1 that, under the equivalence of 𝑃 and 𝑃^∗, one has Φ(𝑡,𝜔)=𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡(𝜔)𝜎(𝑡) (3.5) and hence that𝑑𝑆_𝑡=𝑟(𝑡)𝑆_𝑡𝑑𝑡+𝜎(𝑡)𝑆_𝑡𝑑𝑊_𝑡. (3.6) Thus, if 𝑇>𝑡, then under 𝑃^∗𝑆_𝑇=𝑆_𝑡exp^𝑇_𝑡𝑟(𝑠)1𝑑𝑠−2^𝑇_𝑡𝜎²(𝑠)𝑑𝑠+^𝑇_𝑡𝜎(𝑠)𝑑𝑊_𝑠. (3.7)

It has been shown that under the equivalent martingale measure 𝑃^∗ the price process satisfies on [0,∞) an SDE with a standard Brownian motion 𝑊_𝑡, 𝑡≥0, the same volatility as the empirical one, and a drift coincident with the risk-free rate. How does this result on [0,∞) relate to the results of [1] on [0,𝑇], 0<𝑇<∞?

Harrison and Kreps show that there is a probability measure 𝑃^∗_𝑇 equivalent to 𝑃_𝑇 under which 𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡, 𝑡≤𝑇 is a martingale, where 𝑃_𝑇 is 𝑃 restricted to ℱ_𝑇. Karatzas and Shreve [3, Section 3.5A] (see also [5, Section 1.7, Proposition 7.4]), show that there is a probability measure 𝑃^∗ on (Ω,ℱ_∞) with the property that for every 𝑇>0 the measure restricted to ℱ_𝑇 is 𝑃^∗_𝑇. They point out, however, that generally 𝑃^∗ need not be equivalent to 𝑃. That is, in fact the case if ∫^∞₀𝜌²(𝑠)𝑑𝑠=∞ as can be seen from Proposition 3.2 (see also the remark preceding Example 7.6, Chapter 1, of [5]). In that case, not only is our measure 𝑃^∗ 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 𝑃^∗ satisfies 𝜁=𝑑𝑃^∗/𝑑𝑃∈𝐿₂(Ω,ℱ,𝑃) and, for each 𝑡>0, 𝑃^∗|_ℱ𝑡=𝑃^∗_𝑡 then 𝑃^∗⟂𝑃^∗ if ∫^∞₀𝜌²(𝑠)𝑑𝑠=∞. See proof in the Appendix.

4. Relationship between 𝜇 and 𝑟

Let 𝛼>0 be arbitrary. Suppose that there is an essentially disjoint collection 𝒞_𝛼={𝐼_𝑗}_𝑗≥1 of subintervals of the real line satisfying _𝑗≥1𝐼_𝑗⊂𝑠≥0∶𝜇(𝑠)−𝑟(𝑠)≥𝛼. (4.1) Under the assumption that the functions 𝜇 and 𝑟 are continuous observe that, unless 𝜇(𝑠)=𝑟(𝑠) for all 𝑠>0, for 𝛼>0 sufficiently small, the collections 𝒞_𝛼 are nonempty. Denote the Lebesgue measure of an interval 𝐼 by 𝑚(𝐼) and fix 𝛼>0 for which 𝒞_𝛼 is nonempty. The equities price process under the actual probability measure 𝑃 is given in (3.2) and if the pricing system is viable then under 𝑃^∗ 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 𝑃[𝑆₀=𝑠₀]=1 for some 𝑠₀>0, if (𝑀,𝜋) is a viable pricing system for 𝑀, the class of marketable claims under simple trading strategies, then for every 𝛾>0, the set of indices 𝑗 for which 𝑚(𝐼_𝑗)≥𝛾 is at most finite.

Proof. Suppose that (𝑀,𝜋) is viable and the claim is not true. Then, for some 𝛾>0, there is an infinite collection of such intervals [𝑎_𝑗,𝑏_𝑗]⊂𝐼_𝑖𝑗, 𝑗=1,2,… with 𝑏_𝑗−𝑎_𝑗≥𝛾. Writing 𝜇(𝑠)−𝑟(𝑠)=𝜇(𝑠)−𝑟(𝑠)₊−𝜇(𝑠)−𝑟(𝑠)₋, (4.2) assume without loss of generality that thereon |𝜇(𝑠)−𝑟(𝑠)|=(𝜇(𝑠)−𝑟(𝑠))₊≥𝛼. Then, also without loss of generality, one can assume that there are points 𝑑_𝑗<𝑢_𝑗, where 𝑎_𝑖𝑗≤𝑑_𝑗<𝑢_𝑗≤𝑏_𝑖𝑗, and ∫^𝑢𝑗_𝑑𝑗(𝜇(𝑠)−𝑟(s))𝑑𝑠=𝛽/2=𝑐, 𝑗=1,2,…, where 𝛽=𝛾𝛼.
Consider a sequence of trading strategies 𝜃_𝑛. Under the simple strategy 𝜃_𝑛, buy (𝑛𝑆_𝑑1)⁻¹𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠} shares of the equity at time 𝑑₁ to spend 𝑛⁻¹𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠} units. At this time also sell 𝑛⁻¹𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠} of the risk free security to spend −𝑛⁻¹𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠} at time 𝑑₁. Cash flow at time 𝑑₁ is then 0. At time 𝑡=𝑢₁, sell the equity to spend −𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠}𝑆_𝑢1/𝑛𝑆_𝑑1 and redeem the bond to spend 𝑛⁻¹𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠}𝑒^{∫𝑢1𝑑1𝑟(𝑠)𝑑𝑠}. “Cash” on hand at this stage is 𝑒^{∫𝑢10𝑟(𝑠)𝑑𝑠}𝑛𝑆_𝑢1𝑆_𝑑1−𝑒^{∫𝑢1𝑑1𝑟(𝑠)𝑑𝑠}. (4.3) Invest it in the risk-free security. At times 𝑑₂,𝑑₃,…,𝑑_𝑛, repeat this, buying at time 𝑑_𝑗, (𝑛𝑆_𝑑𝑗)⁻¹𝑒^{∫𝑢𝑗0𝑟(𝑠)𝑑𝑠} shares of the equity and selling 𝑛⁻¹𝑒^{∫𝑢𝑗0𝑟(𝑠)𝑑𝑠} of the risk-free security to spend a total of 0. At time 𝑢_𝑗, sell the shares to obtain −𝑒^{∫𝑢𝑗0𝑟(𝑠)𝑑𝑠}𝑆_𝑢𝑗/𝑛𝑆_𝑑𝑗 and redeem the bond to spend 𝑛⁻¹𝑒^{∫𝑢𝑗0𝑟(𝑠)𝑑𝑠}𝑒^{∫𝑢1𝑑𝑗𝑟(𝑠)𝑑𝑠} and invest the “cash” in the risk-free security. At time 𝑢_𝑛, there will be an amount 𝐶_𝑛=𝑛^𝑛−1_𝑗=1𝑒^{∫𝑢𝑛𝑢𝑗𝑟(𝑠)𝑑𝑠}×𝑒^{∫𝑢𝑗0𝑟(𝑠)𝑑𝑠}𝑆_𝑢𝑗𝑆_𝑑𝑗−𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}. (4.4) The price of this will be 𝜋𝐶_𝑛=𝐸_𝑃^∗𝑒^{−∫𝑢𝑛0𝑟(𝑠)𝑑𝑠}𝐶_𝑛=𝐸_𝑃^∗1𝑛^𝑛_𝑗=1𝑆_𝑢𝑗𝑆_𝑑𝑗−𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}. (4.5) Under the geometric Brownian motion model (3.1), the term inside the expectation is 𝑛⁻¹∑^𝑛_𝑗=1𝑌_𝑗, where 𝑌_𝑗=𝑒^{∫𝑢𝑗𝑑𝑗(𝜇(𝑠)−𝜎2(𝑠)/2)𝑑𝑠+∫𝑢𝑗𝑑𝑗𝜎(𝑠)𝑑𝐵𝑠}−𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}, (4.6) an average of independent, but not identically distributed, random variables. By Lemma 4.2, there is a subsequence 𝑛_𝑘 and an 𝐿>0 such that 1𝑛_𝑘^𝑛𝑘_𝑗=1𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}⟶𝐿 (4.7) and, under 𝑃, 𝐽_𝑘=𝑉_𝑛𝑘=1𝑛_𝑘^𝑛𝑘_𝑖=1𝑌⟶_𝑖^𝑝𝐿𝑒^𝑐−1. (4.8) Since a sequence converging in probability has a subsequence which converges almost surely, there is a subsequence 𝑘^′ such that 𝐽−−−−→_𝑘′^a.s.𝑃𝐿(𝑒^𝑐−1). By equivalence of 𝑃 and 𝑃^∗, that convergence is also almost surely 𝑃^∗. By Lemma 4.4, the sequence 𝐽_𝑘 is uniformly integrable under 𝑃^∗, so expectations converge (see [6, Theorem 5.4]) and one concludes that 𝐸_𝑃^∗[𝐽_𝑘′]→𝐿(𝑒^𝑐−1). On the other hand, by Lemma 4.3, 𝐸_𝑃^∗[(𝑛_𝑘)⁻¹∑^𝑛𝑘_𝑖=1𝑌_𝑖]=0 for every 𝑘. It follows that 𝑐=0, a contradiction.

Lemma 4.2. Under conditions (A), there is a subsequence 𝑛_𝑘 and 𝐿>0 such that 1𝑛_𝑘^𝑛𝑘_𝑗=1𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}1⟶𝐿,𝑛_𝑘^𝑛𝑘_𝑗=1𝑌⟶_𝑗^𝑝𝐿𝑒^𝑐−1 (4.9) (see proof in the Appendix).

Lemma 4.3. 𝐸_𝑃^∗[𝑌_𝑗]=0 (see proof in the Appendix).

Lemma 4.4. Under conditions (A) the sequence 𝑛⁻¹∑^𝑛_𝑗=1𝑌_𝑗 is uniformly integrable under 𝑃^∗ (see proof in the Appendix).

Lemma 4.5. If 𝜇(⋅) and 𝑟(⋅) are constant and if the model is viable, then 𝑟=𝜇.

Proof. Suppose that |𝑟−𝜇|=𝑑>0. Then, for 0<𝛼<𝑑, and any 𝛾>0, essentially disjoint intervals 𝐼 in 𝒞_𝛼 can be chosen in such a way that there is an infinite collection satisfying 𝑚(𝐼)≥𝛾. By Lemma 4.1 this violates the viability of the model.
The next most interesting case is when 𝜇(𝑠)−𝑟(𝑠)=𝜌𝜎(𝑠), (4.10) where 𝜌, the possibly varying risk premium, is assumed constant.

Lemma 4.6. Under conditions (A), if the pricing system (𝑀,𝜋) is viable and if (4.10) holds for all 𝑠≥0, then 𝜌=0.

Proof. Assume that 𝜌≠0, and let 0<𝛼<|𝜌|𝜎_𝐿. Then, for any 𝛾>0 essentially disjoint intervals in 𝒞_𝛼 can be chosen in such a way that there is an infinite collection, contradicting Lemma 4.1 unless 𝜌=0.
Returning to the market expressed in terms of 𝑉(𝑡), define the functional 𝜓 on 𝑋=𝐿₂(Ω,ℱ,𝑃), where 𝑃=𝑃_𝜇 is the empirical measure under which 𝑆_𝑡 satisfies (3.1), by 𝜓(𝑥)=∫𝑥(𝜔)𝑑𝑃(𝜔). Our interest in the following theorem centers on the pricing system defined on 𝑀 by 𝜋(𝑚)=∫𝑚(𝜔)𝑑𝑃_𝑟(𝜔), and 𝑃_𝑟 is the measure corresponding to the process solving (3.6).

Theorem 4.7. For the Black-Scholes model on [0,∞) (𝑟=0, 𝜇, and 𝜎 are constants), the pricing system (𝑀,𝜋) is viable with respect to 𝑋 if and only if 𝜇=0.

Proof. By Lemma 4.5 it is known that, if the system (𝑀,𝜋) is viable then 𝑟=𝜇. It suffices to show that, if 𝑟=𝜇, then the pricing system given by 𝜋 is viable. According to Lemma 2.1, it suffices to show, as it plainly does here, that 𝜓 extends 𝜋.

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 𝑟=𝜇, 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 𝜇 and 𝑟 constant, some tests of 𝜇=𝑟 are developed here based on readily available daily log return data, 𝑅_𝑡=ln(𝑆_𝑡+1/𝑆_𝑡), where 𝑅_𝑡 are i.i.d. 𝑁(𝜇−𝜉²/2,𝜉²), and applied to different underlying equities at various historical times.

Let 𝑋_𝑖=𝑅_𝑖−𝑟. Setting 𝜂=𝐸[𝑋_𝑖] and 𝜉²=Var(𝑋_𝑖), the hypothesis pair 𝐻^′₀∶𝜇=𝑟 versus 𝐻^′_𝑎∶𝜇≠𝑟 becomes 𝐻₀∶𝜂+𝜉²/2=0 versus 𝐻_𝑎∶𝜂+𝜉²/2≠0, which will be tested based on observing 𝑋₁,…,𝑋_𝑛 i.i.d. 𝑁(𝜂,𝜉²). Writing 𝑓𝐱∣𝜂,𝜉²=12𝜋𝜉^2𝑛/2−1exp2𝜉^2𝑛_𝑖=1𝑥_𝑖−𝜂², (5.1) one has that (∑𝑋²_𝑖,∑𝑋_𝑖) are joint sufficient statistics for (−1/2𝜉²,𝜂/𝜉²)=(𝜃₁,𝜃₂). It can be seen, based upon the theory of tests of a single parameter from an exponential family (see [11]), that a uniformly most powerful unbiased (UMPU) test 𝜙_𝛼 of size 𝛼 exists for testing the hypothesis of interest here,𝐻₀∶𝜃₂1=−2versus𝐻_𝑎∶𝜃₂1≠−2. (5.2) Furthermore, denoting by 𝑃_𝛼(𝜃) the power function of the test 𝜙_𝛼, one has the following result.

Lemma 5.1. The test which rejects 𝐻₀ if |𝜏_𝑛|>𝑧_𝛼/2, where 𝜏_𝑛=√𝑛𝑋_𝑛/𝑆²_𝑛+1/21/𝑆²_𝑛+3/2+𝑆²_𝑛/4, (5.3) satisfies for all 𝜃 in 𝐻₀lim_𝑛→∞𝑃𝜏_𝑛>𝑧_𝛼/2∣𝜃=𝑃_𝛼(𝜃). (5.4) 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 𝛼=0.05) from 𝑟=2.0113×10⁻⁴, the daily rate for 26-week treasury bills during that stretch. In the tables, the entry Effect Size is 𝜂/𝜎+𝜎/2, a rough estimate of 𝜇−𝑟 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., [12]) in large samples. According to Wilks’ theorem, the null hypothesis should be rejected when −2ln𝜆(𝑥) is too large. Setting𝑤^∗=2∑1+^𝑛𝑗=1𝑥²_𝑗𝑛−1, (5.5) one has the following.

Lemma 5.2. The likelihood ratio test of (5.2) of size 𝛼 rejects 𝐻₀ if the test statistic 𝜏=𝑛ln𝑤^∗+1−1𝑤^∗𝑛_𝑗=1𝑥_𝑗+𝑤^∗2²∑−𝑛ln^𝑛𝑗=1𝑥_𝑗−𝑥²𝑛 (5.6) exceeds 𝜒²_𝛼. See proof in the Appendix.

𝑃 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 𝛼=0.05.

Appendix

Proof of Lemma 3.1. Fix 𝑡>0 and 0<𝑡₁<⋯<𝑡_𝑛=𝑡 an equidistant partition of the interval [0,𝑡]. To ease notation let us denote 𝑋_𝑗∶=𝑋_𝑡𝑗, 𝐴_𝑡∶=∫^𝑡₀(𝜇(𝑠)−𝑟(𝑠)−(1/2)𝜎²(𝑠))𝑑𝑠, 𝑀_𝑡∶=∫^𝑡₀𝜎(𝑠)𝑑𝐵_𝑠, and correspondingly 𝐴_𝑗∶=𝐴_𝑡𝑗, 𝑀_𝑗∶=𝑀_𝑡𝑗. Also, denote Δ𝐴_𝑗∶=𝐴_𝑗−𝐴_𝑗−1, Δ𝑀_𝑗∶=𝑀_𝑗−𝑀_𝑗−1.
Notice that, under 𝑃, 𝑀_𝑡 is a normal variable with mean zero and variance ∫^𝑡₀𝜎²(𝑠)𝑑𝑠. Let us evaluate ^𝑛_𝑗=1𝑋_𝑗−𝑋_𝑗−1²−^𝑡₀𝑋²_𝑠𝜎²(𝑠)𝑑𝑠=^𝑛_𝑗=1𝑋²_𝑗−1𝑒^{Δ𝐴𝑗+Δ𝑀𝑗}−1²−^𝑡𝑗_𝑡𝑗−1𝑋²_𝑠𝜎²(𝑠)≤𝑑𝑠_𝑛𝑗=1𝑋²_𝑗−1𝑒^{Δ𝐴𝑗+Δ𝑀𝑗}−1²−𝑋²_𝑗−1Δ𝐴_𝑗+Δ𝑀_𝑗²+^𝑛_𝑗=1𝑋²_𝑗−1Δ𝐴_𝑗+Δ𝑀_𝑗²−^𝑡𝑗_𝑡𝑗−1𝑋²_𝑠𝜎²(𝑠)≤𝑑𝑠_𝑛𝑗=1𝑋²_𝑗−1𝑒^{Δ𝐴𝑗+Δ𝑀𝑗}−1²−𝑋²_𝑗−1Δ𝐴_𝑗+Δ𝑀_𝑗²+^𝑛_𝑗=1𝑋²_𝑗−1Δ𝐴_𝑗²+^𝑛_𝑗=12𝑋²_𝑗−1Δ𝐴_𝑗⋅Δ𝑀_𝑗+^𝑛_𝑗=1𝑋²_𝑗−1Δ𝑀_𝑗²−^𝑡𝑗_𝑡𝑗−1𝑋²_𝑠𝜎²(𝑠)𝑑𝑠∶=𝑇_1,𝑛+𝑇_2,𝑛+𝑇_3,𝑛+𝑇_4,𝑛. (A.1) Since lim_𝑥→0𝑥⁻³[(𝑒^𝑥−1)²−𝑥²]=1, for 𝑥 sufficiently close to zero |(𝑒^𝑥−1)²−𝑥²|≤2|𝑥³|. Moreover, max_𝑗≤𝑛|Δ𝐴_𝑗|≤(𝑡/𝑛)(|𝜇_𝑈|+|𝑟_𝑈|+(1/2)𝜎²_𝑈)∶=(𝑡/𝑛)𝐶(𝜇,𝑟,𝜎)→0, and since max_𝑗≤𝑛𝐸_𝑃(Δ𝑀_𝑗)²=max_𝑗≤𝑛∫^𝑡𝑗_𝑡𝑗−1𝜎²(𝑠)𝑑𝑠≤(𝑡/𝑛)𝜎²_𝑈→0, there exists a subsequence 𝑛_𝑘 on which convergence holds almost surely. In order to simplify notation, we assume that 𝑛_𝑘=𝑛. Then, for 𝑛 large enough and for almost all 𝜔 we have 𝑇_1,𝑛≤2^𝑛_𝑗=1𝑋²_𝑗−1Δ𝐴_𝑗+Δ𝑀_𝑗³≤2^𝑛_𝑗=1𝑋²_𝑗−1Δ𝐴_𝑗³+3Δ𝐴_𝑗²Δ𝑀_𝑗+3Δ𝐴_𝑗Δ𝑀_𝑗²+Δ𝑀_𝑗³≤2^𝑛_𝑗=1𝑋²_𝑗−1𝐶𝜇,𝑟,𝜎³𝑡³/𝑛³+3𝐶𝜇,𝑟,𝜎²𝑡²/𝑛²Δ𝑀_𝑗+3𝐶𝜇,𝑟,𝜎(𝑡/𝑛)Δ𝑀_𝑗²+Δ𝑀_𝑗³. (A.2) We have 𝐸_𝑃Δ𝑀_𝑗=2𝜋^𝑡𝑗_𝑡𝑗−1𝜎²(𝑠)𝑑𝑠≤2𝜋𝜎_𝑈𝑡^1/2𝑛^1/2, (A.3)𝐸_𝑃(|Δ𝑀_𝑗|²)≤(𝑡/𝑛)𝜎²_𝑈, and 𝐸_𝑃(|Δ𝑀_𝑗|³)=(2√2/√𝜋)(∫^𝑡𝑗_𝑡𝑗−1𝜎²(𝑠)𝑑𝑠)^3/2≤(𝑡^3/2/𝑛^3/2)𝜎²_𝑈. From conditions (A) it follows that there exists 𝐾>0 such that 𝐸_𝑃𝑋²_𝑗−1=𝑆²₀𝑒^{2𝐴𝑗−1}𝑒^{2∫𝑡0𝑗−1𝜎2(𝑠)𝑑𝑠}≤𝐾. (A.4) Moreover, using that 𝑋_𝑗−1 is independent of Δ𝑀_𝑗, we have 𝐸_𝑃𝑇_1,𝑛≤2𝐾𝐶𝜇,𝑟,𝜎³𝑡³𝑛²+𝐶₁𝑡𝜇,𝑟,𝜎^5/2𝑛^3/2+𝐶₂𝑡𝜇,𝑟,𝜎²𝑛+𝐶₃𝑡𝜇,𝑟,𝜎^3/2𝑛^1/2⟶0, (A.5) and it follows that 𝑇_1,𝑛 has a subsequence which converges 𝑃-almost surely to zero.
Let us now evaluate 𝐸_𝑃𝑇_2,𝑛≤^𝑛_𝑗=1𝐾𝐶²𝑡𝜇,𝑟,𝜎²𝑛²=𝐾𝐶²𝑡𝜇,𝑟,𝜎²𝑛⟶0, (A.6) therefore 𝑇_2,𝑛 has a subsequence that converges 𝑃-almost surely to zero. 𝐸_𝑃𝑇_3,𝑛≤^𝑛_𝑗=1𝐶₄𝑡𝜇,𝑟,𝜎^3/2𝑛^3/2=𝐶₄𝑡𝜇,𝑟,𝜎^3/2𝑛^1/2⟶0, (A.7) and therefore 𝑇_3,𝑛 has a subsequence convergent to zero 𝑃-almost surely
For the last term 𝑇₄, we write 𝑇_4,𝑛≤^𝑛_𝑗=1^𝑡𝑗_𝑡𝑗−1𝑋_𝑗−1𝑑𝑀_𝑠²−^𝑡𝑗_𝑡𝑗−1𝑋_𝑠𝑑𝑀_𝑠²+^𝑛_𝑗=1^𝑡𝑗_𝑡𝑗−1𝑋_𝑠𝑑𝑀_𝑠²−^𝑡𝑗_𝑡𝑗−1𝑋²_𝑠𝜎²(𝑠)𝑑𝑠∶=𝑇_5,𝑛+𝑇_6,𝑛. (A.8) Since ∫^𝑡₀𝑋_𝑠𝑑𝑀_𝑠 is a continuous 𝑃 square integrable martingale, by Theorem 5.8 of [3, Chapter 1] it follows that ∑^𝑛_𝑗=1(∫^𝑡𝑗_𝑡𝑗−1𝑋_𝑠𝑑𝑀_𝑠)²→⟨∫^𝑡₀𝑋_𝑠𝑑𝑀_𝑠⟩ in probability 𝑃. By [13, Proposition 2.3, Chapter 2], ⟨∫^𝑡₀𝑋_𝑠𝑑𝑀_𝑠⟩=∫^𝑡₀𝑋²_𝑠𝜎²(𝑠)𝑑𝑠; therefore, 𝑇_6,𝑛 converges to zero 𝑃-almost surely on a subsequence.
As for 𝑇_5,𝑛, from Hölder’s inequality we have 𝐸𝑇_5,𝑛≤^𝑛_𝑗=1𝐸^𝑡𝑗_𝑡𝑗−1𝑋_𝑗−1−𝑋_𝑠𝑑𝑀_𝑠^21/2𝐸^𝑡𝑗_𝑡𝑗−1𝑋_𝑗−1+𝑋_𝑠𝑑𝑀_𝑠^21/2. (A.9) By Itô’s isometry, 𝐸^𝑡𝑗_𝑡𝑗−1𝑋_𝑗−1−𝑋_𝑠𝑑𝑀_𝑠²=𝐸^𝑡𝑗_𝑡𝑗−1𝑋_𝑗−1−𝑋_𝑠²𝜎²(𝑠)=𝑑𝑠_{𝑡𝑗𝑡𝑗−1}𝐸𝑋_𝑗−1−𝑋_𝑠²𝜎²(𝑠)=𝑑𝑠_{𝑡𝑗𝑡𝑗−1}𝐸𝑋_𝑗−1²𝐸𝑋_𝑠𝑋_𝑗−1−1²𝜎²(𝑠)𝑑𝑠=𝑆²₀^𝑠_𝑡𝑗−1𝑒^{2𝐴𝑗−1+2∫𝑡𝑗0𝜎2(𝑢)𝑑𝑢}𝑒^{2𝐴𝑠−2𝐴𝑗−1+2∫𝑡𝑗𝑡𝑗−1𝜎2(𝑢)𝑑𝑢}−2𝑒^{𝐴𝑠−𝐴𝑗−1+(1/2)∫𝑠𝑡𝑗−1𝜎2(𝑢)𝑑𝑢}+1𝑑𝑠≤𝐾^𝑡𝑗_𝑡𝑗−1𝑒^{𝐴𝑠−𝐴𝑗−1+(1/2)∫𝑠𝑡𝑗−1𝜎2(𝑢)𝑑𝑢}−1²𝑑𝑠. (A.10) For 𝑛 large enough 𝑒^{𝐴𝑠−𝐴𝑗−1+(1/2)∫𝑠𝑡𝑗−1𝜎2(𝑢)𝑑𝑢}−1²∼𝐴_𝑠−𝐴_𝑗−1+12^𝑠_𝑡𝑗−1𝜎²(𝑢)𝑑𝑢²≤𝐶²𝜇,𝑟,𝜎𝑠−𝑡_𝑗−1². (A.11) By a similar argument, [𝐸(∫^𝑡𝑗_𝑡𝑗−1(𝑋_𝑗−1+𝑋_𝑠)𝑑𝑀_𝑠)²]^1/2≤𝐶 for some constant depending on 𝜇, 𝑟, 𝜎, and 𝑡. Then, 𝐸𝑇_5,𝑛≤𝐶^𝑛_𝑗=1𝑡_𝑗−𝑡_𝑗−1^3/2⟶0, (A.12) and therefore 𝑇_5,𝑛 has a subsequence converging to zero 𝑃-almost surely It has been shown that ^𝑛_𝑗=1𝑋_𝑗−𝑋_𝑗−1²−^𝑡₀𝑋²_𝑠𝜎²(𝑠)𝑑𝑠⟶0 (A.13) almost surely 𝑃 on a subsequence, and, since 𝑃∼𝑃^∗, convergence holds almost surely in 𝑃^∗ as well. Thus, |∑^𝑛_𝑗=1|𝑋_𝑗−𝑋_𝑗−1|²−⟨𝑋⟩(𝑡)|→0 in 𝑃^∗, and it follows that ⟨𝑋⟩_𝑡(𝜔)=∫^𝑡₀𝑋²_𝑠(𝜔)𝜎²(𝑠)𝑑𝑠.

Proof of Lemma 4.1. The random variables 𝜁_𝑡=𝐸[𝜁∣ℱ_𝑡] constitute a martingale and 𝜁_𝑡 is the Radon-Nikodym (RN) derivative of the probability measure 𝑃^∗|_ℱ𝑡 with respect to 𝑃|_ℱ𝑡. By Theorem 4.1 of [4, page 319] since 𝜁_𝑡 are nonnegative and 𝐸[𝜁_𝑡]=1 for all 𝑡, one has that lim_𝑡→∞𝜁_𝑡=𝜁 with 𝑃-probability 1. One has from [1, Theorem 3], that for each 𝑡>0, 𝜁_𝑡=𝑒^{∫𝑡0𝜌(𝑠)𝑑𝐵𝑠−(1/2)∫𝑡0𝜌2𝑠𝑑𝑠}. Then since ∫^𝑡₀𝜌(𝑠)𝑑𝐵_𝑠∼𝑁(0,∫^𝑡₀𝜌²_𝑠𝑑𝑠), one has for any 𝑎 such that 𝑒^𝑎 is a continuity point of the distribution of the random variable 𝜁 that 𝑃𝜁_𝑡≥𝑒^𝑎=𝑃^𝑡₀𝜌(𝑠)𝑑𝐵_𝑠≥12^𝑡₀𝜌²_𝑠1𝑑𝑠+𝑎=𝑃𝑍≥2^𝑡₀𝜌²_𝑠𝑎𝑑𝑠+∫^𝑡₀𝜌²_𝑠.𝑑𝑠 (A.14) So, unless ∫^∞₀𝜌²_𝑠𝑑𝑠<∞, one has 𝑃[𝜁≥𝑒^𝑎]=0. But then 𝑃[𝜁<𝑒^𝑎]=1 so that choosing 𝑎<0 shows that 𝜁 cannot be the RN derivative of a probability measure absolutely continuous with respect to 𝑃.

Proof of Lemma 4.2. Existence of the subsequence and nonzero limit 𝐿 is obvious from assumptions (A) and the sequential compactness of the real line. Since under 𝑃^𝑢𝑗_𝑑𝑗𝜎(𝑠)𝑑𝐵_𝑠∼𝑁0,^𝑢𝑗_𝑑𝑗𝜎²(𝑠)𝑑𝑠 (A.15) and 𝐸_𝑃[𝑒^{∫𝑢𝑗𝑑𝑗𝜎(𝑠)𝑑𝐵𝑠}] is just the mgf 𝜙(𝑢) of this random variable evaluated at 𝑢=1, it follows that 𝑋_𝑗=𝑒^{∫𝑢𝑗𝑑𝑗𝜇(𝑠)𝑑𝑠}𝑒^{∫𝑢𝑗𝑑𝑗𝜎(𝑠)𝑑𝐵𝑠−∫𝑢𝑗𝑑𝑗𝜎2(𝑠)/2𝑑𝑠}−1 (A.16) is zero mean under 𝑃 and that, defining 𝑋_𝑛,𝑗=(1/𝑛)𝑋_𝑗, one has (1/𝑛)∑^𝑛_𝑗=1(𝑌_𝑗−𝐸_𝑃[𝑌_𝑗])=∑^𝑛_𝑗=1𝑋_𝑛,𝑗, where 𝑌_𝑗 are given in (4.6). Under 𝑃, 𝑋_𝑛,𝑗 are independent across 𝑗 for each 𝑛. Let 𝑉_𝑛=(1/𝑛)∑^𝑛_𝑗=1𝑌_𝑗, 𝑊_𝑛=(1/𝑛)∑^𝑛_𝑗=1𝐸_𝑃[𝑌_𝑗], and 𝑊₀=𝐿(𝑒^𝑐−1). Then, 𝑊_𝑛 are constants converging to 𝑊₀ along a subsequence and 𝐸_𝑃[(𝑉_𝑛−𝑊_𝑛)²]=𝑏²_𝑛, where 𝑏²_𝑛=∑^𝑛_𝑗=1𝜎²_𝑛,𝑗 and 𝜎²_𝑛,𝑗=Var(𝑋_𝑛,𝑗). Setting 𝜂_𝑗=∫^𝑢𝑗_𝑑𝑗𝜎²(𝑠)𝑑𝑠, 𝐸_𝑃𝑒^{𝑘∫𝑢𝑗𝑑𝑗𝜎(𝑠)𝑑𝐵𝑠−𝑘∫𝑢𝑗𝑑𝑗𝜎2(𝑠)/2𝑑𝑠}=𝑒^{((𝑘2−𝑘)/2)𝜂𝑗}, (A.17) so one has 𝜎²_𝑛,𝑗=𝑒^{2∫𝑢𝑗𝑑𝑗𝜇(𝑠)𝑑𝑠}𝑒^𝜂𝑗−1𝑛². (A.18) In case assumptions (A) hold, then since 𝑐=^𝑢𝑗_𝑑𝑗𝜌(𝑠)𝜎(𝑠)𝑑𝑠≥𝛼𝑢_𝑗−𝑑_𝑗 (A.19) so that 𝜂_𝑗≤𝜎²_𝑈(𝑢_𝑗−𝑑_𝑗)≤𝜎²_𝑈𝑐/𝛼<∞ for all 𝑗, it follows that 𝑏²_𝑛 is 𝑂(𝑛⁻¹). Therefore 𝑉_𝑛−𝑊₀²_𝑃=𝑏²_𝑛+2⋅0⋅𝑊_𝑛−𝑊₀+𝑊_𝑛−𝑊₀²⟶0 (A.20) on the subsequence for which 𝑊_𝑛→𝑊₀. This convergence implies convergence in probability.

Proof of Lemma 4.3. Under 𝑃^∗, 𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡 is a martingale so that for 0≤𝑡<𝑇 one has 𝑒^{−∫𝑡0𝑟(𝑠)𝑑𝑠}𝑆_𝑡=𝐸_𝑃^∗𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}𝑆_𝑇∣𝑆_𝑡=𝑒^{−∫𝑇0𝑟(𝑠)𝑑𝑠}𝑆_𝑡𝑒^{∫𝑇𝑡𝜇(𝑠)𝑑𝑠−∫𝑇𝑡𝜎2(𝑠)/2𝑑𝑠}𝐸_𝑃^∗𝑒^{∫𝑇𝑡𝜎(𝑠)𝑑𝐵𝑠}. (A.21) Therefore, 𝐸_𝑃^∗𝑒^{∫𝑇𝑡𝜎(𝑠)𝑑𝐵𝑠}=𝑒^{∫𝑇𝑡(𝑟(𝑠)−𝜇(𝑠))𝑑𝑠}𝑒^{∫𝑇𝑡𝜎2(𝑠)/2𝑑𝑠}. (A.22) Thus, 𝐸_𝑃^∗𝑌_𝑗=exp^𝑢𝑗_𝑑𝑗𝜇(𝑠)𝑑𝑠−^𝑢𝑗_𝑑𝑗𝜎²(𝑠)2𝑑𝑠+^𝑢𝑗_𝑑𝑗𝑟(𝑠)−𝜇(𝑠)𝑑𝑠+^𝑢𝑗_𝑑𝑗𝜎²(𝑠)2𝑑𝑠−exp^𝑢𝑗_𝑑𝑗𝑟(𝑠)𝑑𝑠 (A.23) and the claim follows.

Proof of Lemma 4.4. First consider the distribution of the random variable 𝑉_𝜎=∫^𝑇_𝑡𝜎(𝑠)𝑑𝐵_𝑠 under 𝑃^∗. Under 𝑃^∗, ^𝑇_𝑡𝜎(𝑠)𝑑𝐵_𝑠∼𝑁−^𝑇_𝑡𝜌(𝑠)𝜎(𝑠)𝑑𝑠,^𝑇_𝑡𝜎²(𝑠)𝑑𝑠. (A.24) To see this, compute the MGF of 𝑉_𝜎 as 𝜑_𝜎(𝑢)=𝐸[𝑒^𝑢𝑉𝜎]. Since 𝑢𝑉_𝜎=𝑉_𝑢𝜎, by (A.22) one has 𝜑_𝜎(𝑢)=𝜑_𝑢𝜎(1)=𝑒^{∫𝑇𝑡(𝑢𝑟(𝑠)−𝑢𝜇(𝑠))𝑑𝑠}𝑒^{∫𝑇𝑡(𝑢𝜎(𝑠))2/2𝑑𝑠} (A.25) and the claim has been established. Next the claim is that 𝐸_𝑃^∗𝑌²_𝑗=𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}𝑒^𝜂𝑗−1. (A.26) To see this, set 𝑉_𝜎,𝑗=∫^𝑢𝑗_𝑑𝑗𝜎(𝑠)𝑑𝐵_𝑠 and recall that ∫^𝑢𝑗_𝑑𝑗𝜌(𝑠)𝜎(𝑠)𝑑𝑠=𝑐 and ∫^𝑢𝑗_𝑑𝑗𝜎²(𝑠)𝑑𝑠=𝜂_𝑗, so that 𝑌²_𝑗=𝑒^{∫𝑢𝑗𝑑𝑗𝜇(𝑠)𝑑𝑠+𝑉𝜎,𝑗−𝜂𝑗/2}−𝑒^{∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠2}. (A.27) Therefore, one has 𝐸_𝑃^∗𝑌²_𝑗=𝑒^{2∫𝑢𝑗𝑑𝑗𝜇(𝑠)𝑑𝑠−𝜂𝑗}𝑒^{−2𝑐+2𝜂𝑗}−2𝑒^{∫𝑢𝑗𝑑𝑗𝜇(𝑠)𝑑𝑠+∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠−𝜂𝑗/2}𝑒^{−𝑐+𝜂𝑗/2}+𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}=𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠+2𝑐+𝜂𝑗−2𝑐}−2𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}+𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}=𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}𝑒^𝜂𝑗−1. (A.28) We do not know that 𝐵_𝑢𝑗−𝐵_𝑑𝑗 are independent under 𝑃^∗ but by Jensen's inequality 1𝑛^𝑛_𝑗=1𝑤_𝑗²≤1𝑛^𝑛_𝑗=1𝑤²_𝑗 (A.29) so that 𝐸_𝑃^∗1𝑛^𝑛_𝑗=1𝑌_𝑗²≤1𝑛^𝑛_𝑗=1𝑒^𝜂𝑗−1max_{1≤𝑗≤𝑛}𝑒^{2∫𝑢𝑗𝑑𝑗𝑟(𝑠)𝑑𝑠}, (A.30) and under (A) this is uniformly bounded in 𝑛. Therefore, the sequence (1/𝑛)∑^𝑛_𝑗=1𝑌_𝑗 is uniformly integrable under 𝑃^∗.

Proof of Lemma 5.1. As it is well known, the test 𝜙=𝜙_𝛼 will satisfy the derivative condition for exponential class densities 𝐸_𝜃2=−1/2𝑆₂𝜙𝑆₂,𝑠∣𝑆₁=𝑠=𝛼𝐸_𝜃2=−1/2𝑆₂∣𝑆₁=𝑠 (A.31) and the size condition 𝐸_𝜃2=−1/2𝜙𝑆₂,𝑠∣𝑆₁=𝑠=𝛼, (A.32) where the expectations refer to the conditional distribution of 𝑆₂=∑𝑋_𝑖 given 𝑆₁=∑𝑋²_𝑖. By the theory of Lehmann [11], the UMPU test can be based upon the statistic 𝑋_𝑛𝑆²_𝑛+12 (A.33) since it is a monotonic function of ∑𝑋_𝑖 given ∑𝑋²_𝑖. The latter follows simply from 𝜕𝑣𝜕𝑣𝑛𝑡−𝑣²=𝑛𝑡+𝑣²𝑛𝑡−𝑣²²>0. (A.34) There is no closed-form expression for the conditional distribution of ∑𝑋_𝑖 given ∑𝑋²_𝑖 under the condition 𝜃₂=−1/2 but a large sample approximation can be made as follows.
It is shown below that under the hypothesis 𝐻₀∶𝜂/𝜉²=−1/2, as the sample size 𝑛→∞, √𝑛𝑋_𝑛𝑆²_𝑛+12^𝐿⟶𝑁(0,𝑣), (A.35) where 𝑣=(𝜉⁴+6𝜂²𝜉²+4𝜂⁴)/𝜉⁶. Therefore, by Slutsky’s theorems, a test suggests itself; namely, reject 𝐻₀ if √𝑛𝑋_𝑛/𝑆²_𝑛+1/21/𝑆²_𝑛+6𝑋²/𝑆⁴_𝑛+4𝑋⁴/𝑆⁶_𝑛 (A.36) exceeds in absolute value the upper 𝛼/2 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 𝑣=1/𝜉²+3/2+𝜉²/4 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 𝜉Σ=²002𝜉⁴ (A.37) one has √𝑛𝑚_1,𝑛−𝜂₁𝑚_2,𝑛−𝜂₂^𝐿⟶𝑁₂(0,Σ). (A.38) Here 𝑚_𝑖,𝑛=(1/𝑛)∑^𝑛_𝑗=1𝑋^𝑖_𝑗 and 𝜂_𝑖=𝐸[𝑋^𝑖]. With 𝑓(𝑢,𝑣)=𝑢/(𝑣−𝑢²)+1/2, 𝑓(𝜂₁,𝜂₂)=(𝜂/𝜉²)+(1/2) and the latter, under the null hypothesis, is 0. Hence, by the 𝛿-method and under the null hypothesis √𝑛𝑓𝑚_1,𝑛,𝑚_2,𝑛−0⟶^𝐿𝑁0,∇𝑓^′Σ∇𝑓, (A.39) where ∇𝑓^′=𝑓_𝑢𝜂₁,𝜂₂,𝑓_𝑣𝜂₁,𝜂₂=𝜂₂+𝜂²₁𝜉⁴𝜂,−¹𝜉⁴. (A.40) Upon simplification, the claim has been verified.

Proof of Lemma 5.2. Write the maximum of the density under 𝐻₀ as 𝑁(𝑢,𝑣), denote the denominator 𝐷 which is the unrestricted maximum similarly, and find 𝜆(𝑥)=sup_{𝑢+𝑤/2=0}𝑓(𝑥∣𝑢,𝑤)sup_𝑢,𝑤>0𝑓()=𝑁𝑥∣𝑢,𝑤𝐷. (A.41) Since for 𝑏>0 and 𝑣>0 it is the case that max_𝑡>0𝑡^−𝑏𝑒^−𝑣/𝑡=(𝑣/𝑏)^−𝑏𝑒^−𝑏 attained at 𝑡=𝑣/𝑏, one has ∑𝐷=𝑒2𝜋^𝑛𝑗=1𝑥_𝑗−𝑥²𝑛^−𝑛/2. (A.42) For the numerator max_{𝑢+𝑤/2=0}ln𝑓(𝑥∣𝑢,𝑤) is sought as is the location of the maximum. The location (𝑢^∗,𝑤^∗) is therefore where max_{𝑢+𝑤/2=0}−𝑛21ln𝑤−2𝑤^𝑛_𝑗=1𝑥_𝑗−𝑢² (A.43) is attained. More simply, 𝑢^∗=−𝑤^∗/2, where 𝑤^∗ maximizes ℎ(𝑤)𝑛=−21ln𝑤−2𝑤^𝑛_𝑗=1𝑥_𝑗+𝑤2². (A.44) Thus, 𝜕ℎ=𝜕𝑤−𝑛+∑2𝑤^𝑛𝑗=1𝑥²_𝑗2𝑤²−𝑛8 (A.45) and the derivative is zero at 𝑤^∗=𝑛±𝑛²+𝑛∑^𝑛_𝑗=1𝑥²_𝑗−𝑛/2 (A.46) so that taking account of the proper sign and that the derivative is zero at the maximum, (5.5), the LRT rejects 𝐻₀ if −𝑛−22ln𝑤^∗−12𝑤^∗𝑛_𝑗=1𝑥_𝑗+𝑤^∗2²+𝑛2∑ln𝑒+ln^𝑛𝑗=1𝑥_𝑗−𝑥²𝑛 (A.47) exceeds the upper 𝛼 cutoff of a chi square with 1 degree of freedom. Simplifying, the test statistic is that in (5.6).