Continuous-time trading and the emergence of volatility

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.


Introduction
This note is part of the recent revival of interest in game-theoretic probability (see, e.g., [7,8,4,2,3]). It concentrates on the study of the " √ dt effect", the fact that a typical change in the value of a non-degenerate diffusion process over short time period dt has order of magnitude √ dt. Within the "standard" (not using non-standard analysis) framework of game-theoretic probability, this study was initiated in [9]. In our definitions, however, we will be following [11], which also establishes some other randomness-type properties of continuous price processes. The words such as "positive", "negative", "before", and "after" will be understood in the wide sense of ≥ or ≤, respectively; when necessary, we will add the qualifier "strictly".

Null and almost sure events
We consider a perfect-information game between two players, Reality (a financial market) and Sceptic (a speculator), acting over the time interval [0, T ], where T is a positive constant fixed throughout. First Sceptic chooses his trading strategy and then Reality chooses a continuous function ω : [0, T ] → R (the price process of a security).
Let Ω be the set of all continuous functions ω : [0, T ] → R. For each t ∈ [0, T ], F t is defined to be the smallest σ-algebra that makes all functions ω → ω(s), s ∈ [0, t], measurable. A process S is a family of functions S t : Ω → [−∞, ∞], t ∈ [0, T ], each S t being F t -measurable (we drop the adjective "adapted"). An event is an element of the σ-algebra F T . Stopping times τ : Ω → [0, T ] ∪ {∞} w.r. to the filtration (F t ) and the corresponding σ-algebras F τ are defined as usual; ω(τ (ω)) and S τ (ω) (ω) will be simplified to ω(τ ) and S τ (ω), respectively (occasionally, the argument ω will be omitted in other cases as well). The class of allowed strategies for Sceptic is defined in two steps. An elementary trading strategy G consists of an increasing sequence of stopping times τ 1 ≤ τ 2 ≤ · · · and, for each n = 1, 2, . . ., a bounded F τn -measurable function h n . It is required that, for any ω ∈ Ω, only finitely many of τ n (ω) should be finite. To such G and an initial capital c ∈ R corresponds the elementary capital process (with the zero terms in the sum ignored); the value h n (ω) will be called the portfolio chosen at time τ n , and K G,c t (ω) will sometimes be referred to as Sceptic's capital at time t. A positive capital process is any process S that can be represented in the form where the elementary capital processes K Gn,cn t (ω) are required to be positive, for all t and ω, and the positive series ∞ n=1 c n is required to converge. The sum (1) is always positive but allowed to take value ∞. Since K Gn,cn 0 (ω) = c n does not depend on ω, S 0 (ω) also does not depend on ω and will sometimes be abbreviated to S 0 . The upper probability of a set E ⊆ Ω is defined as where S ranges over the positive capital processes and I E stands for the indicator of E. Notice that P is not a probability measure, even if restricted to F T : for example, the events E (c) := {ω ∈ Ω | ω(t) = c, ∀t ∈ [0, T ]} are disjoint for different c, but P(E (c) ) = 1 for all c; see [11] for less trivial examples. We say that E ⊆ Ω is null if P(E) = 0. A property of ω ∈ Ω will be said to hold almost surely (a.s.), or for almost all ω, if the set of ω where it fails is null. Upper probability is countably (and finitely) subadditive: In particular, a countable union of null sets is null.

Proof
The more difficult part of this proof (vex(ω) ≤ 2 a.s.) will be modelled on the proof in [1], which is surprisingly game-theoretic in character. The proof of the easier part is modelled on [10]. (Notice, however, that our framework is very different from those of [1] and [10], which creates additional difficulties.) Without loss of generality we impose the restriction ω(0) = 0.
We need to show that the event vex(ω) < 2 & nc(ω) is null, where nc(ω) stands for "ω is not constant". By Lemma 1 it suffices to show that vex(ω) < p & nc(ω) is null for each p ∈ (0, 2). Fix such a p. It suffices to show that var p (ω) < ∞ & nc(ω) is null and, therefore, it suffices to show that the event var p (ω) < C & nc(ω) is null for each C ∈ (0, ∞). Fix such a C. Finally, it suffices to show that the event The elementary capital process corresponding to the elementary gambling strategy G := (τ n , h n ) ∞ n=1 and initial capital c := δ 2−p C will satisfy provided τ n+1 (ω) ≤ T ∧ T A (ω) and n + 1 < C/δ p , and so satisfy provided τ N (ω) ≤ T ∧ T A (ω) and N < C/δ p . On the event E p,C,A we have T A (ω) < T and N < C/δ p for the N defined by τ N = T A . Therefore, on this event We can see that K G,c t (ω) increases from δ 2−p C, which can be made arbitrarily small by making δ small, to A 2 over [0, T ]; this shows that the event E p,C,A is null. The only remaining gap in our argument is that K G,c t may become strictly negative strictly between some τ n < T ∧T A and τ n+1 with n+1 < C/δ p (it will be positive at all τ N ∈ [0, T ∧T A ] with N < C/δ p , as can be seen from (3)). We can, however, bound K G,c t for τ n < t < τ n+1 as follows: , and so we can make the elementary capital process positive by adding the negligible amount 2Aδ to Sceptic's initial capital.
We need to show that the event vex(ω) > 2 is null, i.e., that vex(ω) > p is null for each p > 2. Fix such a p. It suffices to show that var p (ω) = ∞ is null, and therefore, it suffices to show that the event The strong p-variation var p (f, [0, t]) of f ∈ Ω over an interval [0, t], t ≤ T , is defined as where n ranges over all positive integers and κ over all subdivisions 0 = t 0 < t 1 < · · · < t n = t of the interval [0, t] (so that var p (f ) = var p (f, [0, T ])). The following key lemma is proved in [1] (Lemma 1; in fact, this lemma only requires p > 1).

Lemma 2.
For all f ∈ Ω, t > 0, and q ∈ [1, p), Another key ingredient of the proof is the following game-theoretic version of Doob's upcrossings inequality: Lemma 3. Let c < a < b be real numbers. For each elementary capital process S ≥ c there exists a positive elementary capital process S * that starts from S * 0 = a − c and satisfies, for all t ∈ [0, T ] and ω ∈ Ω, (a, b)), where S(ω) stands for the sample path t → S t (ω).
Proof. The following standard argument is easy to formalize. Let G be an elementary gambling strategy leading to S (when started with initial capital S 0 ). An elementary gambling strategy G * leading to S * (with initial capital a − c) can be defined as follows. When S first hits a, G * starts mimicking G until S hits b, at which point G * chooses portfolio 0; after S hits a, G * mimics G until S hits b, at which point G * chooses portfolio 0; etc. Since S ≥ c, S * will be positive. Now we are ready to finish the proof of the theorem. Let T A := inf{t | |ω(t)| = A} be the hitting time for {−A, A} (with T A := T if {−A, A} is not hit). By Lemma 3, for each k ∈ N and each i ∈ {−2 k + 1, . . . , 2 k } there exists a positive elementary capital process S k,i that starts from A + (i − 1)A2 −k and satisfies Summing 2 −kq S k,i /A2 −k over i ∈ {−2 k + 1, . . . , 2 k }, we obtain a positive elementary capital process S k such that A + (i − 1)A2 −k A2 −k ≤ 2 −kq 2 2k+1 and S k TA ≥ 2 −kq M TA (ω, A2 −k ).
On the event E p,A we have T A = T and so, by Lemma 2, c q,A,TA (ω) = ∞. This shows that S T = ∞ on E p,A and completes the proof.
The situation for p = 2 remains unclear. It would be very interesting to find the upper probability of the event {var 2 (ω) < ∞ and ω is not constant}. (Lévy's [6] result shows that this event is null when ω is the sample path of Brownian motion, while Lepingle [5] shows this for continuous, and some other, semimartingales.)