A possible interpretation of financial markets affected by dark volatility

1.   Introduction

Einstein warped product manifolds have been investigated both in mathematics and physics. Several papers in literature are dedicated to the study of warped products as solutions of Einstein's field equations, (to name a few, among others, see ^[1,2,3]. They are mathematical objects which are also considered in current research. Here, we consider $M$ an Einstein warped product manifold of special kind, the so-called PNDP-manifold, introduced by A. Pigazzini et al. in ^[4,5], and we try to use this new approach to "virtual" dimensions and "emerging spaces" (see also ^[6]), in order to describe the behavior of the financial market conditioned by a ghost field. To present these applications, we take as basic knowledge what is stated in ^[4,5,6], recalling the definition of PNDP-manifold.

In recent decades, physicists have increasingly contributed to the modeling of "complex systems" using tools and methodologies developed in statistical mechanics and theoretical physics, introducing what today can be called Econophysics (see ^[7]), and geometric theories, such as the Geometric Arbitrage Theory have also been introduced (^[8]). Recently, works have been published (see ^[9,10,11]) in which the authors introduce the concepts of Ghost Field, hidden space and extra dimensions in the economic field, also considering spaces with negative dimensions.

In ^[10] we want to emphasize that the authors used an original idea in the Minkowski spacetime embedded in Kolmogorov space in time series data with the behavior of traders, for GARCH $(1, 1)$ process; to learn more about the GARCH model see also ^[12]. The result of the work ^[10] is equivalent to the dark volatility or the hidden risk fear field induced by the interaction of the behavior of the trader in the financial market panic when the market crashes. In particular the authors induce a dark volatility equivalent to dark matter in theoretical physics discovered by Wolfgang Pauli. They consider $g_{ij} \in G$ be a cycle (group $G$) and $g^{ij} \in G^*$ be a cocycle, a group action acting on the fibre bundle of manifold, $g_{ij} = \sqrt{g_{ij}} \sqrt{g_{ij}}$ and $x_t$ is a time series in the section of the Riemannian manifold, $x_t \in T_{X_0}X = p^{-1}(X)$. Thus there is a differential 2-form of time series arising from the scalar production of the group transformation of the vector bundle, the so-called principal bundle of the time series, $\sigma^2_t = \langle x_t, x_t \rangle = g_{ij}||x_t||^2$ with $g_{ij} = 1$, where $\sigma^2_t$ indicates the volatility, and the authors are interested in the case of dark volatility, i. e. $\sigma^2_t < 0$, which happens when we change the Euclidean metric to the Minkowski metric $g_{ij} = -1$ which allows the negative volatility, $\sigma^2_t < 0$. Because $\sigma^2_t = \langle x_t, x_t \rangle = g_{ij}||x_t||^2 = -||x_t||^2$; the metric used for dark volatility corresponds to the metric used for $F$ in ^[4].

Our work can also be understood as an extension of the geometric approach described in these papers ^{[13,14,15,16]}. For instance in ^[13], objects are represented by open string with $2$-endpoints and $D2$-brane, which are continuous enhancement of $1$-endpoint open string model. We can, in fact, reinterpret strings and branes as PNDP-strings and PNDP-brane showing the geometric aspects of the "hidden" spaces that our approach introduces, but also as an extension to more "classic" works as ^[17,18].

Later, in Section 4, we have demonstrated possible empirical nature of our approach. We have tried to show a particular case, when the forward price represented in its exponential form coincides with the warping function. We have outlined the possible links between the solutions provided by the systems that determine a PNDP-manifold and some classic stochastic models. We have tried to find answer how our concept of dark volatility could be able to stabilize the market crisis.

On the other hand, it would be useful to mention that there is increasing knowledge about the nonstationary, far-from-equilibrium dynamics. The proper understanding such dynamics, e. g., the violation of the ergodic assumption ^[19], in the systems which econophysics deals, can bring a new perspective on the investigated phenomena ^[20]. Regarding the performed analysis with volatilities, a number of interesting generalizations of geometric Brownian motion (GBM) models emerged booth in economic and physical literature on stochastic processes. In the future it would be very useful to look at the obtained results from the point of view of the computation with time averaged mean squared displacement for scaled GMB ^[21,22] with volatility varying as a power law in time.

The paper is organized as follows: in Sections 2 and 3 we introduce basic properties of PNDP-manifolds with some Ricci-flat examples. In Section 4 we show the relationship between PNDP-manifolds approach and exponential models and provide some empirical analysis on real financial data. In Section 5 we summarize the result of presented work and discuss future ideas.

2.   PNDP-manifolds

Before proceeding with the definitions, it is useful to remember the following: if $A\rightarrow M$ and $B \rightarrow M$ are transversal sub-manifolds of co-dimensions $a$ and $b$, respectively, then their intersection $C$, is a new sub-manifold of co-dimensions $a+b$. By removing the requirement for transversality, derived geometry explains how to interpret a non-transversal intersection $C$ as a derived smooth manifold of co-dimension $a+b$. Specifically, $dim(C) = dim(M)-a-b$, and the latter number can be negative. Therefore, we are dealing with "virtual dimensions", which are not linked to the usual geometrical conception of "dimension". In fact, in differential geometry, the transversality condition is a condition that establishes how two submanifolds interact with each other in a larger space. More precisely, two submanifolds are transversal if they intersect in a "non-degenerate" way, i. e., if the sum of their dimensions is equal to the dimension of the largest manifold containing them. The removal of the transversality condition implies that the submanifolds no longer satisfy this condition and therefore their intersection is no longer non-degenerate. This means that their interaction can no longer be described simply as a sum of their dimensions, but requires more advanced techniques than derived geometry, to define what is called "virtual" dimensions.

In this paper, we apply derived geometry and the "virtual" dimension, by utilizing the concept of Kuranishi neighborhood $(R^d, E, s)$, where, by definition, it is the intersection of zero section of the obstruction bundle with itself, therefore, we define the following virtual dimension: $dim(R^d)-rank(E)$, see ^[23] for more details.

Definition 1: A warped product manifold $(M, \bar{g}) = (B, g)\times_f(F, \ddot{g})$, with metric tensor $\bar{g} = g+f^2 \ddot{g}$, is Einstein if only if

where $\lambda$ and $\mu$ are constants, $d$ is the dimension of $F$, $\nabla^2f$, $\Delta f$ and $\nabla f$ are, respectively, the Hessian, the Laplacian and the gradient of $f$ for $g$, with a smooth positive function $f:(B) \rightarrow {R}^+$.

Definition 2: We call PNDP-manifold a warped product manifold $(M, \bar{g}) = (B, g) \times_f (F, \ddot{g})$ satisfying (1), where the base-manifold $(B, g)$ is a Riemannian (or pseudo-Riemannian) product-manifold $B = B_1 \times B_2$ with $g = \Sigma g_i$, where $B_2$ is an Einstein manifold (i. e., $Ric_2 = \lambda g_2$ where $\lambda$ is the same for (1) and $g_2$ is the metric for $B_2$), with $dim(B_1) = n_1$, $dim(B_2) = n_2$, so $dim(B) = n = n_1+n_2$ (Figure 1). The warping function $f:B \rightarrow {R}^+$ is $f(x, y) = f_1(x)+ f_2(y)$ (where each is a function on its individual manifold, i. e., $f_1:B1\rightarrow {R}^+$ and $f_2:B_2 \rightarrow {R}^+$) and can also be a constant function. The fiber-manifold $(F, \ddot{g})$ is a derived Riemann-flat manifold with negative "virtual" integer dimensions $m$, where with derived smooth manifold is considered a smooth Riemannian flat manifolds by adding a vector bundle of obstructions. In particular for $F$ we consider the space form $R^d$ as a underlying manifold, with orthogonal Cartesian coordinates such that $g_{ij} = -\delta_{ij}$, and as vector bundle of obstructions, $E \rightarrow {R}^d$, a bundle with $rank(E) = 2d$, i. e., twice the dimension of the space form $R^d$. In this way the dimension of $F$ will always be $m = d - rank(E) = - d$. In fact, in this circumstance, if we consider a Kuranishi neighborhood $({R}^d, E, S)$, with manifold ${R}^d$, obstruction bundle $E \rightarrow {R}^d$, and section $S: {R}^d \rightarrow E$, then the dimension of the derived smooth manifold $F$ is $dim({R}^d) - rank(E)$. Moreover in the case $n-d > 0$ (i.e., $M$ with positive "virtual" dimension) we consider it as PNDP-manifold only the case $n_1 = d = -m$ (so the "virtual" dimension of $M$, $dim(M)_{\mathcal{V}}$, must coincides with $dim(B_2)$). In the specal case where $n-d > 0$ in which also $B_1$ is an Einstein-manifold, then we consider it as PNDP-manifold only the case in which $B_1: = B_2$.

Important Note: Since $F: = ({R}^d, E, S)$, and on $E$ (obstruction bundle) the (pseudo-)Riemannian geometry does not work, each (pseudo-)Riemannian geometry operation is performed and defined only on the underlying ${R}^d$, but is considered performed and defined also on $F$ (e. g., we will say that the Ricci curvature of $F$ is zero because the Ricci curvature of ${R}^d$ is zero). The usual (pseudo-)Riemannian geometry works for the underlying smooth manifold (because it is an ordinary manifolds). From now on we will work with the (pseudo-)Riemannian geometry on the derived fibers-manifold $F$. We will define all (pseudo-)Riemannian geometry operations not directly on $F$, but on ${R}^d$, and considering them made on $F$, paying attention only to the dimension. Obviously for what has been said, the tangent space and the vector fields are those of ${R}^d$. The scalar product with two arbitrary vector fields $\ddot g\langle V, W \rangle$ is define on $F$ as: $g_{ij}v^iw^j = -\delta_{ij} v^i w^j = -(v^iw^i)$.

The analysis does not differ from the usual Einstein sequential warped product manifold analysis, $(M_1 \times_h M_2) \times_{\bar{h}}M_3$, (see ^[24,25]), where $h = 1$, $M_2$ is an Einstein-manifold and $M_3$ is a derived-smooth-manifold with negative "virtual" dimensions. The Riemannian curvature tensor and the Ricci curvature tensor of the product Riemannian manifold can be written respectively as the sum of the Riemannian curvature tensor and the Ricci curvature tensor of each Riemannian manifold (see ^[26]).

Proposition: If we write the B-product as $B = B_1\times B_2$, where:

$({\rm{i}})$ $Ric_i$ is the Ricci tensor of $B_i$ referred to $g_i$, where $i = 1, 2$,

$({\rm{ii}})$ $f(x, y) = f_1(x)+ f_2(y)$, is the smooth warping function, where $f_i:B_i\rightarrow {R}^+$,

$({\rm{iii}})$ $Hess(f) = \sum_i {\tau_i^*Hess_i(f_i)}$ is the Hessian referred on its individual metric, where $\tau_i^*$ are the respective pullbacks, (and $\tau_2^* Hess_2(f_2) = 0$ since $B_2$ is Einstein),

$({\rm{iv}})$ $\nabla f$ is the gradient (then $|\nabla f|^2 = \sum_i{|\nabla_if_i|^2}$), and

$({\rm{v}})$ $\Delta f = \sum_i{\Delta_if_i}$ is the Laplacian, (from (ⅲ) then also $\Delta_2 f_2 = 0$),

then the Ricci curvature tensor will be

where $f^* = \frac{\Delta_1 f_1}{f}+(d-1) \frac{|\nabla f|^2}{ f^2}$, and $X_i$, $X_j$, $Y_i$, $Y_j$, $U_i$, $U_j$ are vector fields on $B_1$, $B_2$ and $F$, respectively.

A warped product manifold with derived differential fiber-manifold $F: = ({R}^d, E, S)$, and $dim(F)$ a negative integer, is a PNDP-manifold, as defined in Definition 2, if and only if

(since $Ric$ is the Ricci curvature of $B$, then $Ric = Ric_1+ Ric_2 = \lambda(g_1+g_2)+ \frac{d}{f}\tau_1^*\nabla_1^2 f_1$). Therefore the system (2.3), for $n-d = 0$ and $n-d < 0$, become

where $n_1$, $n_2$, $R_1$ and $R_2$ are the dimension and the scalar curvature of $B_1$ and $B_2$ respectively. While for $n-d > 0$, as by Definition 2, we must set $d = n_1$, so we have

Remarks: In the particular case where $d = 1$ the systems (2.4) and (2.5) should be modified, in fact for $d = 1$, from the system (2.3), we get

from which the system (2.4) becomes

and the system (2.5) becomes

Finally with regard to the projections/desuspensions we remember that

● if $dim(M) > 0$ (i. e., system solutions (2.5)) we have the projection

so $M$ is "virtually" $B_2$ (see Figure 2),

● if $dim(M) = 0$, (i. e., system solutions (2.4)), we have the projection

where with $P$ we mean a point-like manifold (zero dimension), so $M$ is "virtually" $P$, and

● if $dim(M) < 0$, (i. e., system solutions (2.4)), we have the projection

with $\Sigma^{dim(M) < 0}(p)$, we mean the $(||dim(M)||)$-th desuspension of point. For example, if $dim(M) = -4$ the projection $\pi_{-4}$ will "virtually" project $M$ into an object which will be given by the fourth desuspension of a point.

Definition 3: We say that dimension of $X$ is $-1$ if suspension of $X$, $\Sigma X$ is diffeomorphic to a point. By induction and suspension operation we can define all negative dimensions. Since if $X$ and $Y$ are diffeomorphic then their suspensions $\Sigma X$ and $\Sigma Y$ are diffeomorphic. Also a metric on $X$ is compatible with $\Sigma X$. It is also well-known that differentials and suspensions are compatible on manifolds. This means that the differential on the suspension space $\Sigma X$ is induced from the differential on $X$.

3.   PNDP-manifolds in Financial Market

The aim of this work is to interpret the "dark volatility" ghost field, using the fiber-manifold ($F$) of the PNDP-manifolds. In fact, as per the speculative interpretation of negative dimensions provided earlier, we consider dark volatility as a negative dimension that influences the market (base-manifold ($B$)), making it emerge as a new manifold.

Let's consider the global market ($B$) and the hidden dark volatility ($F$) that influences it.

In the case of the PNDP-manifolds, it is possible to combine both of them in a single Einstein-manifold ($M$) in which $F$ will have a negative "virtual" dimension, $M = B \times_f F$, where $B = B_1 \times B_2$ with Einstein-$B_2$.

Let us proceed with the following examples to show in advance the types of scenarios that may arise. By utilizing PNDP-manifolds, we can analyze the situation that is related to the Global Market to see how it can possibly react differently if stimulated by a ghost field such as dark volatility, when it is in equilibrium, i.e. when the Ricci curvature is zero, that is Ricci-flat scenario, (arbitrage opportunity disappears and the system of the time series data contains no curvature).

Example 1: Ricci-flat with $\dim(M) < 0$.

Let $M$ be a PNDP-manifolds with $dim(M) < 0$, where $B = B_2 \times B_2$ represents the global market manifold and $F$ the dark volatility.

In this example we show a type of Ricci-flat PNDP-manifold, i. e., with $\lambda = 0$. If we set $B_2 = R^2$, $B_1$ a 1-dimensional manifold, and $F = (R^4+E)$, then (2.4) becomes

The solution is $f_1 = c$, where $c$ is a constant and $\pi_{dim(M) = -1}:(R^2 \times B_1) \times_c(R^4+E) \rightarrow \Sigma^{-1}p$, in which the global market $B = (R^2 \times B_1)$ under the influence of dark volatility $F$, changes dimension "virtually", from $3$-dimensional manifold to "virtual" $-1$-dimensional manifold, i. e., an object with negative dimension, so even if its Ricci-curvature remains preserved, we cannot consider it as an "equilibrium" situation because it "virtually" degenerates into something like the desuspension of a point. In this case, the scenario points to an extremely negative effect, where the market fails to maintain stability and crashes.

Example 2: Ricci-flat with $\dim(M) = 0$, point-like.

For this purpose, we consider fiber-manifold $F = (R^4+E)$ (with $rank(E) = 8$), 2-dimensional Ricci-flat $B_1$-manifold and $\lambda = 0$. In this case we still have the situation of (2.4). If for example we choose $B_1 = B_2 = R^2$, then our PNDP-manifold will be $M = (R^2 \times R^2)\times_f(R^4+E)$, with the following metric: $ds^2 = dr^2+dx^2+dy^2+dz^2 + f^2(du^2+dv^2+dw^2)_{(-4)}$, where $f$ is a constant function.

In this case, the Global Market $B = (R^2\times R^2)$, under the influence of dark volatility $F$, "virtually" changes dimension from $4$-dimensional to "virtual" $0$-dimensional manifold, $\pi_{dim(M) = 0}:(R^2 \times R^2)_f(R^4+E) \rightarrow$ "virtual" point-like manifold ("virtual" zero-dimension), so we cannot consider it as an "equilibrium" situation because it "virtually" degenerates into a point-like manifold. It is a highly unstable situation, as the market may immediately crash.

Example 3a: Ricci-flat with $\dim(M) > 0$.

Let $M$ be a PNDP-manifold with $dim(M) > 0$, where $B = B_1 \times B_2$ represents the global market manifold and $F$ is the dark volatility. Let $B_1 = B_2$, both Ricci-flat, i. e., $R_1 = R_2 = 0$, and both 2-dimensional, i. e., dimension $n_1 = n_2 = 2$. Then we can consider $B_1$ and $B_2$ to be both $R^2$, with metric: $ds^2 = (dx^2+dy^2)$. Now let $F = (R^2+E)$ have "virtual" negative dimension $m = -2$.

In this case the system (2.5) is satisfied for $f = c$, where $c$ is constant ($f_1$ and $f_2$ are both constant). Therefore the PNDP-manifold will be $M = :(R^2 \times R^2) \times_c (R^2+E)$, with metric $ds^2 = dt^2+dx^2+dy^2+dz^2-c^2(du^2+dv^2)_{(-2)}$ and "virtual" dimension $dim(M) = 2$ (see Figure 3).

It follows from our projection that $\pi_{dim(M) = 2}:(R^2 \times R^2) \times_c (R^2+E) \rightarrow R^2$, which the global market $B = (R^2 \times R^2)$ under the influence of dark volatility $F$ changes from $4$-dimensional flat-manifold to its $2$-dimensional flat-submanifold, but in this case where the curvature remains preserved and the market maintains a positive dimension $(dim(M) > 0$), we consider that the market also preserves its equilibrium situation.

Example 3b: Ricci-flat with $\dim(M) > 0$.

Let $M$ be a PNDP-manifolds with $dim(M) > 0$, where $B = B_1 \times B_2$ represents the global market manifold and $F$ is the dark volatility. Let $B_1 = B_2$, both Ricci-flat, i. e., $R_1 = R_2 = 0$, and both 2-dimensional, i. e., dimension $n_1 = n_2 = 2$, then we can consider $B_1$ and $B_2$ as $S^1 \times S^1$, i. e. $B_1$ and $B_2$ are both $T^2$ torus, and let $F = (R^2+E)$ with "virtual" negative dimension $m = -2$. Also in this case the system (2.3) is satisfied for $f = c$, where $c$ is constant ($f_1$ and $f_2$ are both constant), so the PNDP-manifold will be $M = :(S^1 \times S^1 \times S^1 \times S^1) \times_c (R^2+E)$, with "virtual" dimension $dim(M) = 2$.

This implies that $\pi_{dim(M) = 2}:(S^1 \times S^1 \times S^1 \times S^1) \times_c (R^2+E) \rightarrow (S^1 \times S^1)$, which the global market $B = (S^1 \times S^1 \times S^1 \times S^1)$ under the influence of dark volatility $F$ changes dimension, from $4$-torus Ricci-flat to its submanifold, a $2$-torus Ricci-flat, but, as in the previous example, it is still in equilibrium.

Time series of financial asset returns often demonstrates volatility clustering (Figure 4). In a time series of stock prices, for instance, it is observed that the variance of returns or log-prices is high for extended periods and then low for extended periods.

What this means in practice and in the world of investing is that as markets respond to new information with large price movements (volatility), these high-volatility environments tend to endure for a while after the first shock. In other words, when a market suffers a volatile shock, more volatility should be expected.

In our scenario, we assume that the more $\dim F$ is negative, the more shocks the market ($B$) will suffer.

For example, if we consider Example 2 and Example 3a, we see that the same Market $B$ (which is represented by $R^2 \times R^2$, i. e., $R^4$) suffers a greater shock when dark volatility ($F$) increases to the point that everything degenerates into a point-like manifold (Example 2), compared to when it settles into $R^2$-manifold (Example 3a).

Therefore, we should expect, in our approach, that the clustering of volatility manifests itself, following a shock, as the tendency of a market to a certain change in dimension (sometimes also in curvature) which can be represented by the manifold on which the shock leads it to self projection.

In summary, it is worth noticing that we are dealing with dark volatility, and not volatility, in the sense that $B$ already takes volatility into account, so $F$ (dark volatility) influences B (which already takes into account volatility) by transforming $B$ into another "object" through our projection.

If, for example, we start from market with "virtual" dimension $\dim(M) > 0$ and we have a trend, therefore Dark Volatility Clustering, for example negative, then F will gradually increase its negative dimension, so remaining in $\dim(M) > 0$, i. e., the market is projected in manifolds with positive dimension. However, if the trend turns negative even further, then the market will gradually change into manifolds with ever smaller dimensions and in the end we will reach $\dim(M) = 0$. Here there is a transition point, i. e., beyond this, we have the Crash, identified with $\dim(M) < 0$ (desuspensions of point). If the trend improves then we go back to $\dim(M) > 0$.

Maintaining these trends over a period of time, even a short one, make it possible to determine $F$, and therefore obtaining a projection on an "object" that will represent the market in that period.

The situation is different if we have a large fluctuation of dark volatility ($F$), i. e., in every instant the dark volatility continues uninterruptedly to go from very high to very low (Figure 5), which contributes to creating the "Market Panic" (Figure 6). Thus, according to our approach, the big jump in dark volatility corresponds to a change in dimension of $F$ and this means that $F$ continues to change in dimension every instant without a trend takes shape over a period of time, and then the "virtual" dimension of $M$ ($dim (M)$) keeps changing instantaneously by increasing and decreasing, being able to pass from positive to negative and vice versa instantly.

So our projection continues to vary instantly without identifying the market in a specific "object". It is as if the possible states overlap, until a trend takes shape, at which point our projection returns to project the market into one specific "object", since the dark volativity re-identifies with an $F$, and thus a precise dimension, for a period of time.

4.   Relationship between PNDP-manifolds approach and exponential models

In this section we want to relate the PNDP manifolds with the classical exponential models and to do this we want to highlight a particular case in which the warping function $f$, defined on the base-manifold $B$ of a PNDP-manifold, can coincide with the forward price represented in its exponential form. The objective of this paragraph is in fact to stipulate the possible links between the solutions provided by the systems that determine a PNDP-manifold and some classic stochastic models, to show the possible applicative and empirical nature of this new approach to all effects. Furthermore we demonstrate how it is possible to apply the concept of dark volatility to the well-known classical models.

4.1.   Exponential models

Let's start by recalling that the classic Black '76 model ^[27] is a generalization of the Black-Scholes ^[28] model in which a log-normal trend of the underlying is assumed. It also assumes that the forward price of an underlying has the following exponential trend

where $\sigma$ is the constant log-normal volatility, $W_t$ is the Brownian motion and $F_s(t)$ is the forward price of an underlying with expiry in $t$ and valued in $s \leq t$.

The above model can be seen as a particular case of the exponential Levy models. These models provide for the use of Levy processes (see ^[29]) for the description of the forward price of the underlying. In general, the forward price is represented in its exponential form as

where $f_t$ is a Levy process.

It is worth noticing that (from Definition 4 in ^[4]) an Einstein warped product manifold is a PNDP-manifold, if and only if, the statements in system (2.3) are satisfied.

If, for example, we consider the $dim(B_1) = n_1 = 1$ (in case with positive dimension), we have $n_1 = d = 1$, then system (2.5) becomes

At this point if we set $n_2 = 2$, $\lambda = -1$, $R_1 = 0$ and $f_2 = 1$, the system (4.3) becomes

Since the last equation of (4.4) is an identity, the system is reduced to find the solution of a simple linear second order ODE: $\Delta_1 f_1 = f_1 +1$, by noticing that $f = f_1+ f_2 = f_1 + 1$. The solution will be: $f_1 = C_1e^t+C_2e^{-t}-1$.

As we can see, in this case the market is no longer in equilibrium. In fact we no longer have zero scalar curvature, but we obtain that the warped function $f = f_1 + f_2$ is

Just to make a parallel with the Black '76 model (4.1), we can set $C_1 = 0$, $C_2 = F_0$ (where $F_0$ is the forward price, with expiry in $t$, and constant value in $s \leq t$) and $f_t = -t$. We can also indicate that our solution is considering a Brownian motion in a non-Newtonian fluid. Also our warped function $f:(B_1 \times B_2) \rightarrow R^+$ is equivalent to

that is, the description of the forward price of the underlying using exponential type model.

This particular case analyzed here suggests that our approach can be traced back to a Levy-type exponential model, only in a market that is not in equilibrium, and highlight that the warping function (solution of the systems that determine the PNDPs and defined on the $B$-manifold with real positive values), can be linked to exponential models. In particular, in this case, it shows us that the presence of dark volatility is linked to a manifold ($F$) with only one "virtual" negative dimension and that the forward price of an underlying, again in this particular market situation, will be destined to drop to an asymptotic zero; in other words, it predicts an upper hand of dark volatility and therefore an exponential collapse of the price.

This will lead to a consequent increase in dark volatility corresponding to a market crash, but continuing with the analysis we find that the collapse of the market will continue. In effect, if we consider the increase in dark volatility, subordinated to the collapse of the price, and therefore no longer $F$ with a "virtual" negative dimension, but $F$ with three "virtual" negative dimensions, then the system of Eqs. (4.4) becomes

We obtain that the first equation has a solution: $f_1 = C_1{\rm{e}}^{t/\sqrt{3}}+C_2{\rm{e}}^{-t/\sqrt{3}}-1$, while from the fifth equation we get: $f_1 = {\rm{e}}^{\frac{\sqrt{3}t+C_3}{3}}-1$ and $f_1 = {\rm{e}}^{\frac{-\sqrt{3}t-C_3}{3}}-1$. At this point, it is easy to see that, always referring to (4.1) as in the previous case, for $C_1 = C_3 = 0$ and $C_2 = 1$, the solutions of the first equation will coincide with the solution of the fifth, that is, we get

which confirms the ongoing crash.

4.2.   Empirical analysis for traded options

To demonstrate the behavior of our approach, we performed the empirical analysis of the forward price represented in its exponential form. Further we use the standard formula for the call and put prices ^[28]

where $S$ is an underlying asset price, $K$ is a strike price, $t$ is time to expiration in years, $r$ is a risk-free interest rate (for simplification $q$ as foreign risk-free interest rate was set to $q = r$) and $N(x)$ is a cumulative distribution function for a Gaussian distribution with standard deviation $\sigma$. Forward price was then set as difference of $F = p_c-p_p$.

As the practical implementation of above formulae for the calculations of Black-Scholes option prices we used $\texttt{py_vollib}$ python library ^[30]. For the purpose of analysis we used the online available historical data Yahoo! Finance ^[31] for selected traded options, namely, Alphabet Inc. (GOOGL), Tesla, Inc. (TSLA), SAP SE (SAP), Diageo plc (DEO), Anheuser-Busch InBev SA/NV (BUD), which provided us with values of parameters $S$ and $K$. The choosen time period was between 2021/03/24 and 2022/03/24 and the option calls were set for the day 2022/04/22 (time $t$ one month in the future). The risk-free interest rate $r$ was set to $0.0235$.

The visualization of comparative difference between the classical Black-Scholes model ^[28], denoted (A), and our modified model based on the Eq. (4.8) with incorporated dark volatility (B) is presented on Figures 7–8. For both figures, the first column of plots, the histograms for the estimation error, shows how correctly are predicted the market values of the options. In the case of the model (A), for GOOGL options the estimation error is on average underpricing the options, for other options (SAP, DEO, BUD and TSLA) the mean of the estimation error is near the zero. For the model (B), the mean is moving to the positive numbers, the estimation has slightly overpricing effect. The middle column of plots shows the influence of the considering dark volatility on the call and put options values, which is the largest for BUD options. The last column of plots show the difference of forward price values for different strike expectations. The difference has increasing trend for all indexes for both models (A) and (B), but for GOOGL and TSLA contracts is rather stable in the central region.

5.   Conclusions and remarks

We have shown how the interpretation and the differential geometric approach expressed by the authors in ^[10], can be re-expressed and reinterpreted in the light of PNDP-manifolds.

Analyzing the above Examples, we can say that if the hidden "virtual" dimension of the dark volatility is greater than the positive dimension of the Global Market itself, the result will be an "emergent" Global Market with negative dimensions, something like the desuspension/s of a point, which we could interpret as the boundary of a point and so on. This occurs regardless of which manifold represents the starting Global Market. We consider this situation as a "Market Crash".

When the hidden "virtual" dimension of the dark volatility is equal to the positive dimension of the Global Market itself, will "emerge" a Global Market, which changes into a point-like object, and also in this case, regardless of which manifold represents the starting Global Market. We see the point-like market as a transitional state, in which it can suddenly collapse and crash.

We also want to analyze that when we are in presence of a Global Market expressed by a manifold with zero Ricci curvature, if the hidden "virtual" dimension of the dark volatility is less than the positive dimension of the Global Market itself, the dark volatility can intervene without changing the curvature, but changing only the dimension. We will obtain a manifold with smaller positive dimensions, but in this case, it could preserve its Ricci-flat curvature.

In the case of Ricci-flat scenario, i. e., market in equilibrium, or when arbitrage opportunity disappears and physiology of the time series data contains no curvature, the interpretation with PNDP-model with positive "virtual" dimension, shows that the dark volatility may have no influence on it. It will decrease its dimension, but the emerging market may remain in equilibrium. So in the Examples 3a and 3b we have shown how it would be possible to construct and interpret a type of differential geometric model of markets, with $dim(M) > 0$, where dark volatility does not affect its curvature.

In Section 4, we have instead shown a possible correlation to classic exponential stochastic models, analyzing a specific case in which for a market, not in equilibrium, the approach suggests a strong influence of dark volatility, predicting an exponential collapse of the price. Thus highlighting the effective negative influence that dark volatility brings also by applying it to the well-known classical models. This could be used to describe the influence of dark volatility on the market crisis that occurred in the United States in 1929, or in Argentina in 2001, but also on the financial crisis of 2007–2009, created by the United States housing bubble. Numerical simulations in this regard will be the subject of our next works.

In fact, here we have only shown a new possible interpretative geometrical approach considering this new manifold, but the main goal of our future work is to investigate this further from the empirical point of view, by analyzing in more detail the Black-Scholes equation. Black-Scholes is a widely used pricing model, also with differential geometric techniques (see ^[32,33,34]). We intend to study the Black-Scholes option pricing on Riemannian manifold influenced by dark volatility, this considering Brownian motion on Riemannian manifold as $B$-market, to determine such pricing on a B-market that changes under the influence of dark volatility $F$. We will compare the results as $F$ varies, enabling us to see how the change of manifold influences the price trend, and how the dark volatility acts. Since the concept of "virtual" dimensions, from speculative point of view, can create a new concept of "hidden" dimensions, the PNDP-manifolds can be considered in different types of applications, a possible one has already been shown in ^[6]. Our work can also be understood as an extension of the geometric approach described in these papers ^{[13,14,15,16]}. For instance, objects are represented by open string with $2$-endpoints and $D2$-brane, which are continuous enhancement of $1$-endpoint open string model (^[13]). We can, in fact, reinterpret strings and branes as PNDP-strings and PNDP-brane showing the geometric aspects of the "hidden" spaces that our approach introduces, but also as an extension to more "classic" works as ^[17,18].

In this paper we are using PNDP-manifolds to describe some dark field related aspects in the financial market such as the new concept of dark volatility, in which the latter is considered as the dark matter in the cosmological model. The study considers classical exponential models as possible solutions to our new approach, to show how dark volatility, combined with specific conditions, leads to the collapse of a forward price.

Acknowledgments

The support of the Scientific Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic and Slovak Academy of Sciences (VEGA) under Grant No. 2/0076/23 is acknowledged.

Conflict of interest

The authors declare there is no conflict of interest.