Model-free portfolio theory and its functional master formula arxiv:66.335v [q-fin.pm] 7 Jun 6 Alexander Schied, Leo Speiser, and Iryna Voloshchenko Department of Mathematics University of Mannheim 683 Mannheim, Germany First version: June, 6 This version: June 7, 6 Abstract We use functional pathwise Itô calculus to prove a strictly pathwise version of the master formula in Fernholz stochastic portfolio theory. Moreover, the portfolio-generating function may depend on the entire history of the asset trajectories and on an additional continuous trajectory of bounded variation. Our results are illustrated by several examples and shown to work on empirical market data. Keywords: Pathwise Itô calculus; Föllmer integral; functional Itô formula; portfolio analysis; market portfolio; portfolio generating functionals; functional master formula on path space; entropy weighting Introduction The purpose of this paper is twofold. On the one hand, it deals with an extension of the master formula in stochastic portfolio theory to path-dependent portfolio generating functions. On the other hand, it yields a new case study in which continuous-time trading strategies can be constructed in a probability-free manner by means of pathwise Itô calculus. Stochastic portfolio theory (SPT was introduced by Fernholz [8, 9, ]; see also Karatzas and Fernholz [3] for an overview. Its goal is to construct investment strategies that outperform a certain reference portfolio such as the market portfolio µ(t; see, e.g., [, 6, 7]. Here, our focus is mainly on functionally generated portfolios, which in standard SPT are generated from functions G(t, µ(t of the current state, µ(t, of the market portfolio. The performance of the functionally generated portfolio relative to the market portfolio can be described in a very convenient way by the so-called master formula of SPT. See Strong [5] for an extension of the The authors gratefully acknowledge support by Deutsche Forschungsgemeinschaft through the Research Training Group RTG 953.
master formula to the case in which G may additionally depend on the current state, A(t, of a continuous trajectory A of bounded variation. In practice, portfolios are often constructed not just from current market prices or capitalizations but also from past data, such as econometric estimates, moving or rolling averages, running maxima, realized covariance, Bollinger bands, etc. It is therefore a natural question whether it is possible to develop a master formula for portfolios that are generated by functionals of the entire past evolution, µ t := (µ(s s t, of the market portfolio and maybe also other factors. The first contribution of this paper is an affirmative answer to this question. Our main result, Theorem.9, contains a master formula for portfolios that are generated by sufficiently smooth functionals of the form G(t, µ t, A t µ, where A t µ = (A µ (s s t is an additional m-dimensional continuous trajectory of bounded variation that may depend on µ in an adaptive manner. We then turn to analyzing several concrete examples for portfolios that are generated by functions of mixtures of current asset prices and their moving averages. Our analysis is carried out both on a mathematical level and on empirical market data from Reuters Datastream. The second contribution of this paper concerns the basis for the modeling framework of SPT. While price processes for SPT are usually modeled as Itô processes, it has often been remarked that both sides of the (standard master formula can be understood in a strictly pathwise manner. So the question arises to what extent a stochastic model is actually needed in setting up SPT. Do price processes really need to be modeled as Itô processes driven by Brownian motion or is it possible to relax this condition and consider more general processes, perhaps even beyond the class of semimartingales? Can one get rid of the nullsets that are inherent in stochastic models, or can one prove the master formula strictly path by path? Our approach gives affirmative answers also to the questions raised in the preceding paragraph. To this end, we show that SPT can be formulated within the pathwise Itô calculus introduced by Föllmer [4] and further developed to path-dependent functionals by Dupire [7] and Cont and Fournié [, 3]. Here we will use the slightly extended formalism from [4] and heavily rely on the associativity rule developed there. Thus, the only assumption on the trajectories of the price evolution is that they are continuous and admit quadratic variations and covariations in the sense of [4]. This assumption is satisfied by all typical sample paths of a continuous semimartingale but also by non-semimartingales, such as fractional Brownian motion with Hurst index H / and many deterministic fractal curves [8, 3]. In this sense, our paper is also a contribution to robust finance, which aims to reduce the reliance on a probabilistic model and, thus, to model uncertainty; see, e.g., [, 5, 6,, ] for similar analyses on other financial problems. Robustness results for discrete-time SPT were previously obtained also by Pal and Wong [9], where the relative performance of portfolios with respect to a certain benchmark is analyzed using the discrete-time energy-entropy framework [, 8, 9]. The paper is organized as follows. In Section, we introduce our market model. Our main result on the pathwise functional extension of the master equation is stated in Theorem.9; the corresponding proof is given in Section 3 and essentially relies on the associativity of the pathwise Itô integral in the functional setting, which is derived in [4]. A discussion of examples and simulation results follows Theorem.9. Notation and key results of functional pathwise Itô calculus are recalled in Appendix A.
Statement of results We work in a strictly pathwise setting, based on the pathwise Itô calculus developed by Föllmer [4], Dupire [7], and Cont, Fournié []. In this framework, we derive a functional, pathdependent master formula for the Stochastic Portfolio Theory (SPT [8, 9,, 3], and thus can overcome uncertainty issues in specifying a probabilistic model. In the following, we consider a financial market model consisting of d risky assets and a locally riskless bond. The price of the bond is given by ( B(t = exp r(s ds, where r : [, R is a measurable short rate function satisfying T r(s ds < for all T >. The prices of the risky assets are described by a d-dimensional continuous path S with values in an open subset U R d +. For the remainder of this paper we fix a refining sequence of partitions (T n of [,. That is, T n = {t, t,... } is such that = t < t <... and t k as k, we have T T, and the mesh of T n tends to zero on each compact interval as n. Moreover, it will be convenient to denote the successor of t T n by t. That is, t = min{u T n u > t}. Writing S+ d for the class of symmetric nonnegative definite d d matrices, we assume that S admits the continuous S+-valued d quadratic variation [S] along (T n in the sense of [4] (notation: S QV d, i.e., for any i, j d and all t the sequence (S i (s S i (s (S j (s S j (s (. s Tn s t converges to a finite limit, denoted [S i, S j ](t, such that t [S i, S j ](t is continuous. In order to define admissible integrands for S in our functional setting, we need a functional version of the Itô formula of [, Theorem 3] and its slight extension in the form of [4, Theorem.], which, for the convenience of the reader, is recalled in Theorem A.4. Note that although we need a functional Itô formula for continuous paths X, we still have to consider functionals defined on the space of càdlàg paths, since the definition of functional derivatives requires us to apply discontinuous shocks. Our notation follows closely the one in [], with the slight extensions from [4]. More precisely, for T > and D R n, a D-valued càdlàg function is a rightcontinuous function f : [, T ] D with left limits such that for each t (, T ], f(t D. We write f(t := f(t f(t for the jump of f at time t. We write f t = (f(u t u T for the stopped path at t. We denote by D T = D([, T ], D the space of D-valued càdlàg functions on [, T ], and by D t D T the space of D-valued càdlàg paths stopped at t. Analogously, D t I D I is the set of D-valued càdlàg paths on a subinterval I [, T ], stopped at time t. By C(I, D we denote the set of continuous functions on I with values in D. In the following we let U R d + be an open subset of R d + and W R m a Borel subset of R m. An interesting class of functionals on path space is formed by non-anticipative functionals 3
F : [, T ] U T W T R, which only depend on the past values, but not on the future values, i.e., (t, X, A [, T ] U T W T, F (t, X, A = F (t, X t, A t. It will be convenient for us to require that the second argument A has components of bounded variation. If I is a time interval, the class of right-continuous functions A : I R m whose components are of bounded variation will be denoted by BV m (I. The subset of continuos functions in BV m (I will be denoted by CBV m (I. We will also use the notation DI,BV t := DI t BV m (I and DI,CBV t := Dt I CBV m (I. With this at hand, the definition of an admissible integrand ξ for S is given as follows, so as to guarantee the existence of the pathwise Itô integral T ξ(s ds(s, T >, as the limit of non-anticipative Riemann sums in the sense of (A.4. Definition.. A function ξ : [, R d is called a locally admissible integrand for S if for every T > there exists a partition T = {t,..., t N } of [, T ] such that k =,..., N, there are constants m k N, functions A k W [tk,t k ],CBV, and non-anticipative functionals F k as in Theorem A.4 such that N ξ(t = X F k (t, Sk, t A t k t [tk,t k. k= Here, X F k is the vertical derivative of F k with respect to X in the sense of Definition A., S k := S [tk,t k ] is the restriction of X to [t k, t k ], and we require F k (t k, S t k k, A t k k = F k+ (t k, S t k k, A t k k for k =,..., N. Suppose that ξ is a locally admissible integrand for S and η is a real-valued measurable function on [, such that T η(s d B (s < for all T >. Then the pair (ξ, η is called a trading strategy, where ξ i (t corresponds to the number of shares held in the i-th stock at time t, and η(t is the number of shares held in the riskless bond at time t. Definition.. Let ξ be a locally admissible integrand for S and η a real-valued measurable function on [, such that T η(s d B (s < for all T >. The trading strategy (ξ, η is said to be self-financing if the associated wealth V (t := ξ(t S(t + η(tb(t satisfies the identity V (t = V ( + ξ(s ds(s + η(s db(s, t. (. Denote the discounted prices by S(t := ( S (t B(t,..., S d(t, Ṽ (t := V (t B(t B(t = ξ(t S(t + η(t, and B(t. Lemma.3. It holds that S QV d if and only if S QV d. The following proposition is standard in stochastic calculus. In our pathwise setting, however, its proof needs the associativity rule from [4], which, for the convenience of the reader, is recalled in Theorem A.5. 4
Proposition.4. If ξ is a locally admissible integrand for S, then ξ is also a locally admissible integrand for S, and a trading strategy (ξ, η is self-financing if and only if its associated discounted wealth satisfies Ṽ (t = Ṽ ( + In this case, the riskless component η is given by η(t = V ( + t ξ(s d S(s, t. (.3 ξ(s d S(s ξ(t S(t. In particular, if ξ is a locally admissible integrand for S and w R, then there exist a real-valued function η such that the pair (ξ, η is a self-financing trading strategy with V ( = w. Often it is convenient to describe such self-financing trading strategies by the vector π(t = (π (t,..., π d (t, where π i (t denotes the proportion of the current wealth V π (t that is invested into the i-th asset at time t, i.e., ξ i (t = π i(tv π (t, i =,..., d, and η(t = S i (t ( d π i(t V π (t B(t. (.4 Taking this point of view, the trading strategy (.4 will be self-financing if and only if the associated wealth, V π, satisfies the Itô differential equation ( dv π (t = V π (t π(t d S(t ds(t + V π π i(t (t db(t, (.5 B(t where π(t S(t := ( π (t S (t,..., π d(t S d (t. By [4, Theorem 4.], it follows that V π (t = V ( exp π(s S(s ds(s i,j= π i (sπ j (s t S i (ss j (s d[s i, S j ](s + ( π i (sr(s ds for any π such that π/s is a locally admissible integrand for S and the second and third integrals in (.6 exist as Riemann-Stieltjes integrals. Definition.5. An R d -valued measurable function π is called a portfolio if π/s is a locally admissible integrand for S and the second and third integrals in (.6 exist as Riemann-Stieltjes integrals, and if π (t + + π d (t =, t. (.7 A portfolio is called long-only if π i (t for all t. (.6 5
In the following, we will use the notation V w,π for the wealth associated to a portfolio π with initial investment w. As in [3, Section ], we normalize the market, i.e., we suppose that at any time t each stock has only one share outstanding. Then, the stock prices S i (t represent the capitalizations of the individual companies, and the quantities S Total (t := S (t + + S d (t and µ i (t := S i(t, i =,..., d, S Total (t correspond to the total capitalization of the market and the relative capitalizations of the individual firms, which are called the respective market weights. The portfolio µ is called the market portfolio. In our model-free version of portfolio theory, we do not wish to make assumptions on the structure of the covariations [S i, S j ] apart from (.. In particular, we do not assume that [S i, S j ](t is absolutely continuous in t. Growth rates and covariances, which are functions in [3], therefore need to be modeled as measures in our strictly pathwise setting. Definition.6. The covariance of the stocks in the market is described by the positive semidefinite matrix-valued Radon measure a = (a ij i,j d defined as a ij (dt := d[log S i, log S j ](t = S i (ts j (t d[s i, S j ](t, i, j =,..., d. The excess growth rate of the portfolio π is defined as the signed Radon measure ( γπ(dt := π i (ta ii (dt π i (tπ j (ta ij (dt. For any portfolio π, we define the covariances of the individual stocks relative to the portfolio π as follows for i, j =,..., d, i,j= τ π ij(dt : = (π(t e i a(dt(π(t e j. (.8 It follows from Lemma 3. below that τ π = ( τ π ij is always a positive semidefinite matrixvalued Radon measure. Moreover, if π is long-only, Lemma 3. gives that the excess growth rate is a positive Radon measure. Note that τij(dt π = a ij (dt π j (ta ij (dt j= π i (ta ij (dt + π i (tπ j (ta ij (dt. (.9 i,j= Thus, the relative covariances of the individual stocks from (.8 satisfy the elementary property π j (tτij(dt π =, i =,..., d. (. j= Lemma.7. The return from a one-unit investment according to the portfolio π is given by d log V π (t = π(t d log S(t + γπ(dt, (. where log S(t denotes the vector of the log-prices and V π (t := V,π (t. 6
In particular, equation (. yields the following dynamics for the market weights Lemma.8. Equivalently, (. can be written as d log µ i (t = ( e i µ(t d log S(t γ µ(dt. (. dµ i (t µ i (t = ( e i µ(t d log S(t γµ( dt + τ µ ii ( dt, i =,..., d. We can now introduce the notion of portfolio generating functionals. These are smooth functionals that may depend on the entire past evolution of the trajectories µ,..., µ d. We use the concepts and notation of functional pathwise Itô calculus from [7, ] in the version of [4]; see also the Appendix. Denote by d the simplex in R d and introduce the notation d + := {(π,..., π d d π >... π d > }. We suppose that we are given a nonanticipative functional G : [, T ] V T WBV T (,, where V d + open, and W R m is a Borel subset of R m. We assume further that G is regular enough in the sense that it is of class C, ([, T ] and satisfies the regularity conditions from Theorem A.4. Then the portfolio π with weights [ ] π i (t = i log G(t, µ t, A t µ + µ j (t j log G(t, µ t, A t µ µ i (t, i d, (.3 j= is called the portfolio generated by G. Here, A is a CBV m -functional, i.e., the map A : C([, T ], V X A X WCBV T is such that A X(t, t [, T ], is a function of t and (X(s s t. An example would be the running maximum, A X (t = max s t X(s. The following questions arise: What is the relation between the wealth of the market portfolio and the functionally generated portfolio (.3? Is it possible to establish descriptive conditions on the market structure that lead to portfolios that outperform the market? The next theorem provides the required tools for answering these questions. Theorem.9 (Functional Master equation. The relative wealth of the portfolio π generated by G, with respect to the market, is given by the following functional master equation ( ( V π (T G(T, µ T, A T µ log = log + g([, T ] + h([, T ], T <, (.4 V µ (T G(, µ, A µ where g(dt := G(t, µ t, A t µ is the second-order drift term, and i,j= ijg(t, µ t, A t µµ i (tµ j (tτ µ ij (dt (.5 h(dt := D log G(t, µ t, A t µν(dt = G(t, µ t, A t µ DG(t, µt, A t µν(dt (.6 is the horizontal drift term. Here, ν is the measure associated to the function A µ CBV m ([, T ]. 7
Note that we do not have to know or estimate the volatility of the model in order to compute the second-order drift term: The master equation does it for us, in terms of quantities that are completely observable, ( V π (T G(µ( g([, T ] = log h([, T ]. V µ (T G(µ(T Next we discuss simulation results in order to get some basic idea about the behavior of portfolio generating functionals and their associated portfolios. In the following examples, we will work with convex combinations X(t := αx(t + ( αϑ (t, where α (,, and ϑ is the (modified moving average defined by for δ >. ϑ δ i (t := δ t δ X i (s ds δ t δ X i ( ds, t [, δ, X i (s ds, t [δ, T ], j= i =,..., d, Example. (Geometric mean. Consider the functional G(t, X t := ( d k= Xk (t, which generates the portfolio with weights [ ] α π i (t = d µ i (t + αµ j (t µ i (t. (.7 d µ j (t The master equation (.4 implies that the relative performance of this portfolio with respect to the market is given by where log ( ( d V π (T k= = log ( µ i(t T T d V µ (T d k= (µ + g( dt + h( dt, k( d ( g( dt = α µ i (t d ( µ i (t τ µ ii (dt d j= µ i (tµ j (t µ i (t µ j (t τ µ ij (dt d and h( dt = α dδ { dt µ i (t µ i (t µ i (, t < δ, µ i (t µ i (t δ, < δ t. The following two figures display the results of a simulation of such a geometrically weighted portfolio with the parameters δ = 6 days and α =, 7. We used the stock data base from Reuters Datastream; our data included years of daily values for the DAX index. In Figure we see the relative performance of the portfolio (.7 with respect to the DAX index. In Figure we see the decomposition of the curve(s in the left-hand panel according to the master 8
equation. The blue curve is the change in the generating functional, while the red and the green ones are the respective drift terms. Each curve shows the cumulative value of the daily changes induced in the corresponding quantities by capital gains and losses. As can be seen, the cumulative second-order drift term was the dominant part over the period, with a total contribution of about 5 percentage points to the relative return. The second-order drift term was quite stable over the considered period, with an exception of the period around the financial crisis of 8. Figure : LHS vs. RHS of the master formula (.4 for geometric weighting Figure : Componentwise representation of the RHS of (.4 for geometric weighting Example. (Functional diversity-weighting. Consider the functional ( ( p p G(t, X t := Xk (t, p (,, k= which generates the portfolio with weights [ α ( µ i (t p π i (t = d k= ( µ k(t + p j= αµ j (t ( µ j (t p d k= ( µ k(t p ] µ i (t. (.8 The relative performance of this portfolio with respect to the market is given by ( ( d V π (T k= ( µ k(t p p T T log = log V µ (T ( d k= (µ k( p + g( dt + h( dt, p where g( dt = α ( p and h( dt = α δ µ i (t ( µ i (t p d k= ( µ k(t p τ µ ii ( dt j= µ i (tµ j (t ( µ i (t p ( µ j (t p ( d k= ( µ k(t p τ µ ij ( dt { ( µ i (t p d k= ( µ k(t dt µ i (t µ i (, t < δ, p µ i (t µ i (t δ, < δ t. 9
To simulate such a diversity weighted portfolio with actual stocks, we used again Reuters Datastream to obtain our data base, containing now the monthly average prices for the period from 973 to 5 of the S&P 5 index. We filtered the data so as to consider only those stocks for which the prices are known at each time point of the considered time period. The results of a simulation of the portfolio (.8 using the parameters δ = months, α =, 6, and p =, are presented below: Figure 3 shows the relative performance of this portfolio with respect to the filtered index, and Figure 4 shows its decomposition in the three components according to the master equation. Each curve represents the cumulative value of the monthly changes induced in the corresponding quantities by capital gains and losses, but contrarily to above, it is now the cumulative change in the generating functional that was the dominant part over the period, with a total contribution of about 7 percentage points to the relative return. The second-order drift term was quite stable over the period with a total contribution of about 3 percentage points, whereas the horizontal drift term can be viewed as the price we have to pay for more generality (see the discussion following the next example. Figure 3: LHS vs. RHS of the master formula (.4 for diversity weighting Figure 4: Componentwise representation of the RHS of (.4 for diversity weighting Example. (Functional entropy-weighting. Consider the functional G(t, X t := k= ( X k (t log Xk (t, which generates the portfolio with weights [ α log ( µ i (t π i (t = d k= µ k(t log ( µ k (t + and associated drift rates g( dt = α G(t, µ t j= ] αµ j (t log ( µ j (t d k= µ µ i (t, (.9 k(t log ( µ k (t µ i (t µ i (t τ µ ii (dt
and h( dt = α δg(t, µ t (log ( µ i (t + dt { µ i (t µ i (, t < δ, µ i (t µ i (t δ, < δ t. Using the same data set as in the previous example, we ran a simulation of the portfolio (.9 with respect to the filtered index taking the parameters δ = 6 months and α =, 9, which is presented in Figure 5 and Figure 6, respectively. Setting γ µ( dt := µ i (t µ i (t τ µ ii (dt, (. the master equation (.4 reads for the portfolio (.9: log ( V π (T = log V µ (T log ( d k= µ k(t log ( µ k (T d k= µ + k( log (µ k ( ( d k= µ k(t log ( µ k (T d k= µ k( log (µ k ( T α γ µ( dt + h([, T ] (. G( µ(t + h([, T ] + α γ µ([, T ]. (. log d Note that the positive Radon measure γ µ from (. describes the market s intrinsic volatility, in some extended sense, since it is a weighted average of the variances of the individual stocks relative to the market. If we now assume that γ µ is a strictly positive measure, (. yields V π (T > V µ (T, under the additional condition that the horizontal drift term is significantly outperformed, i.e., log ( d k= µ k(t log ( µ k (T d k= µ k( log (µ k ( + α γ µ([, T ] log d + h([, T ] >. In this sense, availability of intrinsic volatility can be regarded as a property that admits strong relative arbitrage with respect to the market, provided that the horizontal drift term behaves nicely. Moreover, we see that the horizontal drift term h([, T ] can be considered as a trade-off term, appearing due to our functional extension. The above argument is supported by real market data, as can be seen in Figure 5 and Figure 6: Indeed, since the cumulative second-order drift term is continually increasing, we infer that the (extended excess growth rate of the market γ µ is a strictly positive measure. Moreover, the horizontal drift term does not seem to have a large influence on the relative performance of the entropy-weighted portfolio, with a total contribution of less than percentage point. Thus, entropy-weighting should significantly outperform the market on the considered time interval, which is confirmed in Figure 5.
Figure 5: LHS vs. RHS of the master formula (.4 for entropy weighting Figure 6: Componentwise representation of the RHS of (.4 for entropy weighting Note also that the above example is valid in a very general context, since we have not imposed in the above discussion any assumption on the volatility structure of the market model beyond the absolutely minimal condition that the stock price vector S should admit continuous quadratic variation. 3 Proofs Proof of Lemma.3. Without loss of generality assume d =. First, let S QV. Föllmer s pathwise Itô formula [4] applied to the function f(s, a = s a and to the paths S and /B yields that S(t = S( + t B(s ds(s B(s S(s db(s, (3. in conjunction with the fact that [S, /B] vanishes, due to [, Remark 8]. Since is a Riemann-Stieltjes integral, its quadratic variation along (T n vanishes. Since S(s db(s B(s ds(s is B(s a pathwise Itô integral, it belongs to QV, by [4, Proposition.]. Moreover, both admit (vanishing covariation along (T n, by polarization. It thus follows that S QV. On the other hand, if S QV, an application of Föllmer s pathwise Itô formula [4] to the function f(s, a = s a with the paths S and B and a completely analogous argument yield that S QV. Proof of Proposition.4. Again, without loss of generality assume d =. It is easy to see that ξ is locally admissible for S. To see the equivalence of (. and (.3, we first assume that
(ξ, η is self-financing in the sense of (.. Then Ṽ (t = Ṽ ( + t t dv (s B(s (ξ(ss(s + η(sb(s db(s = B (s t (s db(s = B(sṼ t ξ(s ds(s + B(s ξ(s ds(s B(s η(s db(s B(s B(s ξ(s S(s db(s. Here, we have used Föllmer s pathwise Itô formula [4] applied to the function f(v, a = v a and to the paths V and /B in the first step. In the second step, we have used (. and the associativity rule in functional pathwise Itô calculus, Theorem A.5, and we have inserted the definition of Ṽ. Using (3. and once again Theorem A.5, we infer that t ξ(s ds(s B(s t B(s ξ(s S(s db(s = (3. ξ(s d S(s, (3.3 which is (.3. To show the converse direction, we assume that (.3 holds, apply (3.3, and reverse the steps in (3. so as to obtain (.. Proof of Lemma.7. On the one hand, Föllmer s pathwise Itô formula [4], applied to f(x = log x and the paths S i individually, yields d log S i (t = S i (t ds i(t (S i (t d[s i](t. On the other hand, (.6 implies, in conjunction with the portfolio condition (.7, d log V π (t = π(t S(t ds(t π i (tπ j (t S i (ts j (t d[s i, S j ](t. Combining the above two equations we infer, using the associativity of the Stieltjes integral from [7, Theorem I.6 b] and the associativity of the pathwise functional Itô integral, Theorem A.5, d log V π (t = π(t d log S(t + π i (t d[log S i ](t π i (tπ j (t d[log S i, log S j ](t, which implies the assertion via the definition of γ π. Proof of Lemma.8. Let i {,..., d} be given. In the first step, observe that (. implies [ d[log µ i ](t = d = i,j= ( ei µ(s d log S(s] (t i,j= ( (ei k µ k (s ( (e i l µ l (s d[log S k, log S l ](t k,l= = ( a( ( µ(t e i dt µ(t ei = τ µ ii ( dt. (3.4 3
Analogously (see also Lemma 3., it can be inferred that τ µ ij ( dt = d[log µ i, log µ j ](t = Thus, (. becomes ( µ i (t = exp ei µ(s d log S(s µ i (tµ j (t d[µ i, µ j ](t, i, j =,..., d. (3.5 τ µ ii ( ds exp τ µ ii ( ds t γµ(ds. Denoting I(t = ( ei µ(s d log S(s, we infer with (3.4 that [I](t = τ µ ii ( ds. Thus, applying Föllmer s pathwise Itô formula [4] to the function f(k, l = e k+l and to the paths K(t := I(t [I](t, L(t := τ µ ii ( ds t γ µ(ds yields that µ i (t = µ i ( + = µ i ( + µ i (s dk(s + µ i (s d[k](s + µ i (s dl(s µ i (s ( e i µ(s d log S(s µ i (sγµ(ds + µ i (sτ µ ii ( ds, where we have used the associativity rule for pathwise functional Itô calculus in the form of Theorem A.5, with η = µ i and ξ = e i µ, the associativity of the Stieltjes integral from [7, Theorem I.6 b], and the fact that [K](t = [I](t, as can be seen from [, Remark 8]. But this is what we had to show. In order to prove Theorem.9, let us first collect some useful properties of the relative covariances τij π of the individual stocks from (.8. The proofs of the following lemmas are left to the reader, since they follow straightforwardly by adapting the respective proofs from [3, Section 3] to our functional pathwise setting. For every stock i and every portfolio π, we denote by ( Ri π Si (t (t := log, t <, V (t w=si w,π ( the relative return of the i-th stock with respect to the portfolio π. Lemma 3.. For every portfolio π, for all i, j d, we have τ π ij(dt = d[r π i, R π j ](t; in particular, τ π ii(dt = d[r π i ](t, and τ π = (τ π ij i,j d is a positive semidefinite matrix-valued Radon measure. 4
Lemma 3.. For any pair of portfolios π and ρ we have the following numéraire invariance property ( γπ(dt = π i (tτ ρ ii (dt π i (tπ j (tτ ρ ij (dt. In particular, the excess growth rate of a portfolio π can be represented as a weighted average of the variances τ π ii of the individual stocks relative to the portfolio π, i.e., j= γ π(dt = π i (tτii(dt. π (3.6 Furthermore, for any long-only portfolio π we get γ π(dt. Remark 3.3. We infer with the dynamics (. that ( V π (t d log = (π(t µ(t d log S(t + (γ V µ π γ (t µ(dt. (3.7 On the other hand, using Lemma.8 and the associativity of the Stieltjes integral from [7, Theorem I.6 b] together with the associativity of the pathwise functional Itô integral, Theorem A.5, yields that π(t µ(t dµ(t = ( π(t µ(t d log S(t γ µ( dt + π i (tτ µ ii ( dt, thanks to the fact that the portfolio weights sum up to one. numéraire invariance property from Lemma 3. gives us Furthermore, applying the π(t µ(t dµ(t = ( π(t µ(t d log S(t γ µ( dt + π i (tτ µ ii ( dt + γ π( dt. Thus, we receive for any portfolio π the relative return formula ( ( V π (t d log = π(t V µ (t µ(t dµ(t π i (tπ j (tτ µ ij (dt. (3.8 We are now ready to proceed toward the proof of the master equation. Proof of Theorem.9. We introduce the notation j= g i (t := i log G(t, µ t, A t µ and N(t := µ j (tg j (t. j= 5
Then definition (.3 becomes π i (t = (g i (t + N(tµ i (t, i =,..., d. Moreover, with g(t := (g (t,..., g d (t, we obtain ( π(t dµ(t = g(t dµ(t + N(t d µ i (t = g(t dµ(t µ(t and π i (tπ j (tτ µ ij ( dt = j= g i (tg j (tµ i (tµ j (tτ µ ij ( dt, in conjunction with the elementary property (.. Thus, formula (3.8 from the preceding remark gives d log ( V π (t = g(tdµ(t V µ (t j= j= g i (tg j (tµ i (tµ j (tτ µ ij (dt. (3.9 On the other hand, the chain rule for vertical derivatives (see, e.g., [4, Lemma 3.] yields ij log G(t, µ t, A t µ = ijg(t, µ t, A t µ G(t, µ t, A t µ Thus, the change of variables formula, Theorem A.4, implies d log G(t, µ t, A t µ = D log G(t, µ t, A t µν(dt + + X log G(t, µ t, A t µdµ(t = g(tdµ(t + ( ij G(t, µ t, A t µ G(t, µ t, A t µ i,j= + D log G(t, µ t, A t µν(dt, i log G(t, µ t, A t µ j log G(t, µ t, A t µ. ij log G(t, µ t, A t µ d[µ] ij (t i,j= g i (tg j (t µ i (tµ j (tτ µ ij (dt in conjunction with (3.5. Using (3.9 we infer that this last expression equals d log ( V π (t + V µ (t G(t, µ t, A t µ ijg(t, µ t, A t µµ i (tµ j (tτ µ ij (dt j= + D log G(t, µ t, A t µν(dt = d log whence the assertion follows with log ( V π ( =. V µ ( ( V π (t V µ (t g( dt h( dt, 6
A Appendix: Functional pathwise Itô calculus In the following, we first give a short overview on the definitions and notation of the pathwise Itô calculus for non-anticipative functionals that depend on an additional argument A that corresponds to a general path of bounded variation [4]. In the second step, we present its slightly extended functional change of variables formula for a continuous path X, which then allows us to establish an associativity rule in the pathwise functional It calculus. Let us note again that, as the definition of functional derivatives requires us to apply discontinuous shocks even in case X is continuous, we still have to consider functionals defined on the space of càdlàg paths. Our framework is based on [, 3, 7], our notation is as close as possible to the one in [] with the slight extensions from [4, Section.], as described at the beginning of Section. Consider U R d open and W R m Borel. Let X D([, T ], U and h R d be small enough, then the vertical perturbation X t,h of the stopped path X t is defined as the càdlàg path obtained by shifting the value at t by the quantity h, that is, X t,h (u = X t (u + h [t,t ] (u. Moreover, since we work with stopped instead of restricted paths, we can use the standard supremum norm on path space: d ((X, A, (X, A = sup X(u X (u + sup A(u A (u u [,T ] u [,T ] (A. for (X, A, (X, A U T W T. We will need the following regularity properties: A non-anticipative functional F is said to be left-continuous (notation: F F l if for all t [, T ] and all positive ɛ >, for all stopped paths (X t, A t there exists η > such that for all (X, A stopped at t h we have the following implication d ( (X t, A t, (X, A + h < η F (t, X t, A t F ((t h, X, A < ɛ. (A. A non-anticipative functional F is said to be F is boundedness-preserving (notation: F B if for any compact subset K U there exists a constant C K such that t [, T ], (X, A K t S t BV, F (t, X, A < C K. (A.3 Such functionals are, in particular, locally bounded in the neighborhood of any given path, i.e., for all (X, A there exist constants C, η > such that for all paths (X, A stopped at t we have the following implication d ( (X t, A t, (X, A < η t [, T ], F (t, X, A C. (A.4 We next introduce our notion of horizontal derivative (with respect to some measure, which is motivated by the desire to lessen smoothness assumptions on those functionals, for which a change of variables formula can be derived (see [4, Definition.4]. Definition A. (Horizontal derivative. Let F be a non-anticipative functional and (X, A U t WBV t. Since A has components of bounded variation, which correspond to finite measures 7
ν k, k =,..., m, on [, T ], we can write ν( ds := ( ds, A ( ds,..., A m ( ds. Then, the horizontal derivative of F at (t, X, A (with respect to ν is given by the vector where DF (t, X t, A t = ( D F (t, X t, A t, D F (t, X t, A t,..., D m F (t, X t, A t, D F (t, X t, A t F (t, X t h, A t h F ((t h, X t h, A t h := lim h + h (A.5 D k F (t, X t, A t F (t, X t h, A t h,..., A t k := lim m F (t, X t h, A t h, h + ν k ((t h, t] (A.6 for k =,..., m, if the corresponding limits exist. In addition it will be convenient to set DF (t, X t, A t = for t =. If (A.5 and (A.6 are well-defined for all (X, A, then the map DF : [, T ] U T WBV T R m+ (t, X, A DF (t, X t, A t defines a non-anticipative vector-valued functional DF : [, T ] U T WBV T extended horizontal derivative of F. (A.7 Rm+, the Note that the definition (A.5 is based on a left-hand limit, and thus only the past evolution of the underlying path is relevant while no assumptions on the possible future values are imposed (this is inspired by [4, Definition.9]. Definition A. (Vertical derivative. Denote (e i, i =,..., d the canonical basis in R d. A non-anticipative functional F is said to be vertically differentiable with respect to X at (X t, A t U t W t BV if the map Rd e F (t, X t,e, A t is differentiable at. Its gradient at X F (t, X t, A t = ( i F (t, X t, A t, i =,..., d, where i F (t, X t, A t F (t, X t,hei, A t F (t, X t, A t = lim, (A.8 h h is called the vertical derivative of F at (t, X t, A t, with respect to X. If (A.8 is well-defined for all (X, A, the map X F : [, T ] U T W T BV R d (t, X, A X F (t, X t, A t (A.9 defines a non-anticipative functional X F with values in R d, which we call the vertical derivative of F with respect to X. If the functional F admits horizontal and vertical derivatives DF and X F, we may iterate the above operations in order to define higher order horizontal and vertical derivatives. Definition A.3. Let I [, T ] be a subinterval of [, T ] with nonempty interior, I. We denote by C j,k (I the set of all non-anticipative functionals F on t I U I t Wt I,BV such that: 8
F is continuous at fixed times, locally uniformly. That is, t [, T ], ɛ >, (X, A U t I W t I,BV, η > such that for all t [, T ] and (X, A U t I W t I,BV, d ((X, A, (X, A + t t < η F (t, X, A F (t, X, A < ɛ. (A. F admits j horizontal derivatives and k vertical derivatives with respect to X at all (X, A U t I Wt I,BV, t I. D l F, l j, n XF, n k, are left-continuous on I. The following result is a slight extension of the functional change of variables formula from [, 7] in the form of [4, Theorem.]. Theorem A.4. ([7,, 4]. Let (X, A QV d U T W T CBV and denote X n (t := s T n X(s [s,s (t + X(T I {T } (t, t T, (A. A n (t := s T n A(s [s,s (t + A(T I {T } (t, t T, (A. h n s := s s, s, s T n. (A.3 Suppose moreover that F is a left-continuous non-anticipative functional of class C, ([, T ] such that DF, X F, X F B. Denote Xn,s the n-th approximation of X stopped at time s. Then the pathwise Itô integral along (T n, defined as T exists and X F (s, X s, A s dx(s := lim X F (s, X n,s, A n,s (X(s X(s, n s T n F (T, X T, A T F (, X, A = T + DF (s, X s, A s µ( ds + T X F (s, X s, A s dx(s. T i,j= (A.4 ijf (s, X s, A s d[x] ij (s (A.5 Theorem A.5 ([4, Theorem 3.]. Let X QV d U T and ξ (,..., ξ (ν be locally admissible integrands for X, and define Y (l (t := ξ (l (s dx(s, l =,..., ν. (A.6 Moreover, let η = (η,... η ν be a locally admissible integrand for Y. Then ν l= η lξ (l is a locally admissible integrand for X and T η(s dy (s = T ν η l (sξ (l (s dx(s. (A.7 l= 9
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