We introduce in this section the novel notion of
additive functional generation of trading strategies, and study its properties. To simplify notation, and when it is clear from the context, we write from now on
\(V^{\vartheta}\) (respectively,
\(Q^{\vartheta}\)), to denote the value process
\(V^{\vartheta}(\cdot; \mu)\) given in (
2.5) (respectively, the defect of self-financibility process
\(Q^{\vartheta}(\cdot;\mu)\) of (
2.6)) for
\(X = \mu \). Proposition
2.2 allows us then to interpret
\(V^{\vartheta}=V^{\vartheta}(\cdot; \mu) = V^{\vartheta}(\cdot; S) / \varSigma \) as the “relative value” of the trading strategy
\(\vartheta \in \mathscr {T} (S)\) with respect to the market portfolio.
4.1 Additive generation
For any given function
\(G : \mathrm{supp\,} ( \mu) \rightarrow \mathbb {R}\) which is regular for the vector process
\(\mu \) of market weights as in Definition
3.1, we consider the vector
\({ \vartheta} ={( { \vartheta}_{1} , \dots, { \vartheta}_{d} )}^{\prime}\) of processes
\({ \vartheta}_{i} := D_{i} G ( \mu )\) as in (
3.1), and the trading strategy
\({ \varphi} ={( { \varphi}_{1} , \dots, { \varphi}_{d} )}^{\prime}\) with components
$$ { \varphi}_{i} (t) := { \vartheta}_{i} (t) - Q^{ { \vartheta} } (t)- { C} (0) , \qquad i=1, \dots, d, t \ge0, $$
(4.1)
in the manner of (
2.7) and (
2.6), and with the real constant
$$ { C}(0) := \sum_{j=1}^{d} \mu_{j} (0) D_{j} G \big( \mu(0) \big)- G \big( \mu(0) \big). $$
(4.2)
In other words, we adjust each
\({ \vartheta}_{i} (t) \) both for the “defect of self-financibility”
\(Q^{ { \vartheta} } (t) = Q^{ { \vartheta} } (t; \mu) \) at time
\(t\ge0\), and for the “defect of balance”
\({ C}(0)\) at time
\(t=0\).
If the function \(G\) is nonnegative and concave, the following result guarantees that the strategy it generates holds a nonnegative amount of each asset, even if \(D_{i} G(\mu(t))\) is negative for some \(i = 1, \dots , d\).
The proof of Proposition
4.5 requires some convex analysis and is presented in Sect.
7.1 below.
4.2 Multiplicative generation
Let us study now, in the generality of the present paper, the class of functionally generated portfolios introduced in [
7‐
9]. Suppose the function
\(G : \mathrm{supp\,} (\mu) \rightarrow[0, \infty) \) is regular for the vector process
\(\mu \) of market weights in (
2.2), and that
\(1/G(\mu)\) is locally bounded. This holds if
\(G\) is bounded away from zero, or if (
2.4) is satisfied and
\(G\) is strictly positive on
\(\varDelta ^{d}_{+}\). We introduce the predictable portfolio weights
$$ { \varPi}_{i} := \mu_{i} \bigg( 1 + \frac{1 }{ G ( \mu ) } \Big( D_{i} G ( \mu ) - \sum_{j=1}^{d} D_{j} G ( \mu ) \, \mu_{j} \Big) \bigg), \qquad i=1, \dots, d. $$
(4.8)
These processes satisfy
\(\sum_{i=1}^{d} { \varPi}_{i} \equiv1\) rather trivially; and it is shown as in Proposition
4.5 that they are nonnegative if one of the three conditions in Theorem
3.7 holds.
In order to relate these portfolio weights to a trading strategy, let us consider the vector process
\(\eta = ( \eta_{1} , \dots, \eta_{d} )' \) given in the notation of (
3.1) by
$$\begin{aligned} \eta_{i} :={ \vartheta}_{i} \exp\left( \int_{0}^{\cdot} \frac{\mathrm{d} \varGamma^{G} (t) }{ G ( \mu(t)) }\right) = D_{i} G (\mu ) \exp\left( \int _{0}^{\cdot} \frac{\mathrm{d} \varGamma^{G} (t) }{ G ( \mu(t)) }\right) \end{aligned}$$
(4.9)
for
\(i=1, \dots, d \). We note that the integral is well defined, as
\(1/G(\mu)\) is locally bounded by assumption. We have moreover
\(\eta \in\mathscr {L} (\mu)\), since
\({ \vartheta} \in\mathscr {L} (\mu)\) and the exponential process is locally bounded.
As before, we turn the predictable process
\({ \eta} \) into a (self-financed) trading strategy
\({ \psi} = ( { \psi}_{1} , \dots, { \psi}_{d} )' \) by setting
$$\begin{aligned} { \psi}_{i} : = \eta_{i} - Q^{ \eta} - { C} (0) , \qquad i=1, \dots, d, \end{aligned}$$
(4.10)
in the manner of (
2.7) and (
2.6), and with the real constant
\(C(0)\) given by (
4.2).
Proposition
4.3 has the following counterpart.
It is easy to see how the portfolio process
\({ \varPi} \) in (
4.8) is obtained from (
4.12) in the same manner as (
4.6), since
\(V^{ \psi} \) is strictly positive. The representation in (
4.11) is a
generalized master equation in the spirit of Theorem 3.1.5 in [
7]; both it and its additive version (
4.3) have the remarkable property that they do not involve any stochastic integration at all.
4.3 Comparison of additive and multiplicative functional generation
It is instructive at this point to compare additive and multiplicative functional generation. On a purely formal level, the multiplicative generation of Definition
4.7 requires a regular function
\(G\) with the property that
\(1/G(\mu )\) is locally bounded. On the other hand, additive functional generation requires only the regularity of the function
\(G\).
At time
\(t=0\), the additively generated strategy agrees with the multiplicatively generated one; that is, we have
\(\varphi(0) = \psi (0)\) in the notation of (
4.5) and (
4.12). However, at any time
\(t>0\) with
\(\varGamma^{G}(t) \neq0\), these two strategies usually differ; this is seen most easily by looking at their corresponding portfolios (
4.7) and (
4.8). More precisely, the two strategies differ in the way they allocate the proportion of their wealth captured by the finite-variation “cumulative earnings” process
\(\varGamma^{G} \). The additively generated strategy tries to allocate this proportion uniformly across all assets in the market, whereas the multiplicatively generated strategy tends to correct for this amount by proportionally adjusting the asset holdings.
To see this, consider again (
4.12) and assume for concreteness that the regular function
\(G\) is also balanced for the vector process
\(\mu \) of market weights; see, for instance, the geometric mean function of (
4.13) right below. We have then from (
4.12) the representation
$$\begin{aligned} { \psi}_{i} (t) = D_{i} G \big( \mu(t) \big) \, \exp\bigg( \int_{0}^{\,t} \frac{\mathrm{d} \varGamma^{G} (s) }{ G ( \mu(s)) }\bigg) , \qquad i=1, \dots, d; \end{aligned}$$
thus, in this situation, the multiplicatively generated
\({\psi}(t)\) does not invest in assets with
\(D_{i} G(\mu(t)) = 0\), for any
\(t \geq0\), but instead adjusts the holdings proportionally. By contrast, the additively generated trading strategy
\({\varphi}(\cdot)\) of (
4.5) buys
$$\begin{aligned} { \varphi}_{i} ( t)= D_{i} G \big( \mu( t) \big) + \varGamma^{G} (t) , \qquad i=1, \dots, d, \end{aligned}$$
shares of the different assets at time
\(t\), and does
not shun stocks with
\(D_{i} G(\mu(t)) = 0\).
Ramifications: The above difference in the two strategies leads to two observations.
First, if one is interested in a trading strategy that invests through time only in a subset of the market, such as, for example, the set of “small-capitalization stocks”, then strategies generated multiplicatively by functions
\(G\) that satisfy the “balance” property
\(\sum_{j=1}^{d} x_{j} D_{j} G ( x) = G(x)\) for all
\(x \in \varDelta ^{d}\) are appropriate. If, on the other hand, one wants to invest the trading strategy’s earnings in a proportion of the whole market, additive generation is better suited. This is illustrated further by Examples
6.2 and
6.3.
Secondly, the trading strategy which holds equal weights across all assets can be generated multiplicatively, as long as (
2.4) holds, by the “geometric mean” function
$$ \varDelta ^{d} \ni x \mapsto G(x) = \bigg(\prod^{d}_{i=1} x_{i} \bigg)^{1/d} \in(0,1) ; $$
(4.13)
indeed, the portfolio weights in (
4.8) become now
\({ \varPi}_{i} = 1/d\) for all
\(i=1, \dots, d\) (the so-called “equal-weighted” portfolio). However, such a trading strategy cannot be additively generated; for instance, the portfolio in (
4.7), namely
$${ \pi}_{i} (t) = \frac{\, 1\, }{1 + R^{G} (t)} \,\frac{1}{\,d\,} + \frac{\, R^{G} (t)\, }{1 + R^{G} (t)} \, \mu_{i} (t) , \quad\text{with}\quad R^{G} (t) := \frac{\, \varGamma^{G} (t)\,}{\, G (\mu(t))\,} $$
for all
\(i=1, \dots, d, t \ge0\), that corresponds to the strategy generated additively by this geometric mean function
\(G \), distributes the cumulative earnings captured by
\(\varGamma^{G} \) uniformly across stocks, and this destroys equal weighting.
Comparison of portfolios: Let us compare the two portfolios in (
4.7) and (
4.8) more closely. These portfolios differ only in the denominators inside the brackets on their right-hand sides.
Computing the quantities of (
4.8) needs, at any given time
\(t \geq0\), knowledge of the configuration of market weights
\(\mu_{1} (t), \dots, \mu_{d} (t)\) prevalent at that time –
and nothing else. By contrast, the quantities of (
4.7) need, in addition to the current market weights
\(\mu_{1} (t), \dots, \mu_{d} (t)\), the current value
\(V^{{ \varphi}} (t)\) of the wealth generated by the portfolio. One computes this value from the entire history of the market weights during the interval
\([0,t]\), via the Lebesgue–Stieltjes integrals in, say, (
3.5). This is also the case when these portfolios in (
4.7), (
4.8) are expressed as trading strategies, as in (
4.5), (
4.12).