ORIGINAL ARTICLE
How Can We Engineer Stock Markets like NSEIL with BSE SENSEX
Index Data Using String Theory
Soumitra Kumar Mallick QC
1 The Earl Fitzpatrick of York, Sir Asutosh Mukherjee Chair Professor, Indian Institute of Social Welfare & Business Management, Management House, College Square West, Kolkata 700 073, India
Abstract: This paper considers the problem of sustainably developing stock exchanges like NSEIL and measurable index systems like BSE SENSEX by developing strategies to achieve sales volume which achieve optimal growth rates. Experimental data on BSE SENSEX and Companies over time are used which closely mimics now active NSEIL. This requires consideration of consumer choice in intertemporal markets with endogenous stock market products. A sequence of five steps is derived to characterize the venturing technology which will achieve such desired stock market sales volume with fixed prices and hence the optimal growth rate. It is derived how String Theory is sufficient to build such stock markets in value and volumes by using depositories. Keywords: aengineering stock markets, Indian stock markets, factors, indices, string theory, technology venturing algorithm
1. Introduction
Mallick, Sarkar, Roy, Duttachaudhuri, Chakrabarty (2007) have estimated a dynamic asset pricing model for Indian Stock Markets using disaggregated SENSEX data from the BSEIL and NSEIL without controlling for participation or sales volumes and have found good fit with fundamental and monetary policy variables over time and space of companies and information field (Mallick (2014b)). As has been argued in Mallick (2009) this model can be used as a model for studying the welfare properties of the Indian population with stock markets if there is ”enough” participation in various kind of stocks, so that market for each individual stock is not imperfect (Mallick (2007)). Which may not be the case in practice as there is a significant deviation in sales volumes across stocks thus the stock market values are not conformal with first best efficiency which includes physical as well as allocative efficiency (see for e.g. Mallick et al.(2007)).
Thus, a well developed ”desired” marketing technology (which may include stock exchanges, stock depositories, stock price indices, financial laws) needs to be characterized for such markets and the implementation problems need to be addressed. This paper solves some of the steps in such stock market engineering problem. In a recent paper regarding marketing in stock market products, Chan, Dahan, Kim, Lo & Poggio (2002) have considered the problems associated with marketing of stock market products where the information contained in such products about goods derives the demand for the products, in this paper with optimal growth sales of stock market products is generated by the necessity to save and invest.
Copyright © 2018 Soumitra Kumar Mallick QC. doi: 10.18686/fm.v3i1.916
This is an open-access article distributed under the terms of the Creative Commons Attribution Unported License
(http://creativecommons.org/licenses/by-nc/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Coincidentally, the optimal growth model is a representative agent planning model, for a context with marketing of savings and investment products. However the exact decentralized properties of such a model with stock market products like stocks and bonds is yet relatively unknown (see Mallick (2007)). Experimental data on BSE SENSEX and Companies over time are used which closely mimics now active NSEIL and NSDL.
From a different angle the purpose of this paper is also to analyze steps in constructing stock markets using algoritmic steps by using the physical methods of String Theory. We find that Strings can be constructed from simple to advanced within the stock market architecture. The heterotic process engineers modern stock markets comprised of the NSEIL and NSDL in India. However, since M Strings are not possible as discussed later hence relativistic strings need to be developed separately and tickwise price adjustment processes cannot completely be constructed in the absence of Relativistic or M-Branes resolution. In the absence of market crashes this would create genetic algorithms.
Mazumdar and Mitra (1994) introduced wealth effects in the optimal growth model through utility and discover chaotic paths. Nyarko & Ohlson (1994) studied the role of uncertainty in an optimal growth problem where utility depends on consumption and resource stock and find that the first best growth path is "stationary" under certain stochastic assumptions on theprocess of growth. Khan & Mitra (1986) in contrast to Radner (1961) turnpike model proved the existence of a stationary path of capital market sales (which is optimal), where there are multiple sectors in the economy, without any exogenous imposition of path of expansion. Ohlson & Roy (1996) in contrast to Mallick (1993) decentralized and single budget constraint framework have analyzed the growth properties of an optimal growth model with utility depending on both consumption and resource stock and derive conditions under which the dynamically optimal resource utilization does not lead to extinction of the resource in the limit. In all of the above papers, which are variants of the optimal growth set up, aggregate wealth is seen to affect the characteristics of the optimal solution. However, it is not clear that the interiority assumptions made on the optimal path of sales of savings and investment products would necessarily remain valid if the capital stock affected utility at some fundamental stage of the model. Thaler (1985) has considered the importance of discretion on consumer choice in marketing of goods. This paper considers a model with consumer choice but analyzes the nature of technology venturing “desired” to achieve “golden rule” sales volume with stock market products.The new String Theory technology creates new analytical as well as genetic engineering framework.
Section 2 discusses the Stock Market development model with consumer choice and savings and investment products, section 3 characterizes the dynamic sales volume equation in stock market products, section 4 concludes with a set of “strategic” engineering steps to implement the sustainable stock exchanges. This technology will be ventured into because it can implement the desired sustainable sales volume and value.
2. The Model
There is one representative saver and investor in the economy, who receives an exogenous wealth - wt in each period t. The buyer or seller in stock market products makes three decisions the current consumption (ct), the planning horizon (θt) - the period over which to roll over the present wealth in consumption and the time stationary consumption in each period in the future (ct+1)∀t + n,n ∈ I+.The stock of stock market products which results at the end of each period is
assumed to be invested entirely in the stock market to finance consumption in the future.
The felicity function is the Riemann summation over time of a stationary, concave per period component given by u : R+− > R+,u(0) ≤ 0,u′(.) > 0,u″(.) < 0
The Riemann summation is specified for identifying the welfare aggregation procedure in this representative agent framework, as discussions on optimality would depend on this specification crucially.
max u(ct) + θtu(ct+1) θt θt+2−t (ct,(ct+1)t+1,θt)∈R+
s.t.ct + T(θt)ct+1 = wt (1)
T(θt) is the multiplicative investment function with the assumption T(θt) > 0 for θt > 0 and
T′ (θ) > 0 (2)
Its some other properties which are endogenous to the planner’s problem in this paper will be derived later..
3. Dynamic Stock Market Sales Equation
Stock Market Sales is given by : kt+1=θt∗c
t+1
∗ +ωt+1 (3)
at the end of t where solves (1). Thus, kt+1 measures the volume of sales in stock market products (assumed to have fixed unit monetary price) at the end of t to finance consumption in the future. The amount invested serves as an instrument for the volume. An indirect measure of the ”sales volume” is the incremental ”market capitalization” realized in each period (however the assumption of unit price in such a case may not hold). However, both these measures have fundamental similarities and dissimilarities with ”sales volume” in goods market. Thus we can generate the genetic algorithm for implementing the market system “optimal sales volume” in finite steps with stationarity with potential for stochastic complementary production and utility as golden rule is market efficient.
The budget balance constraint in (1) is different from an additive formulation as is usually done. This is done for ease of getting explicit solutions. However, consider the constraint :
ct + θtct+1 + γ(θt) = wt (4) The following theorem establishes the qualitative equivalence.
Theorem 1:
If T(θt) is differentiable, then for every T(θt) ∃ a γ(θt) such that the solution space of (1) is identical with the
additive balance constraint replaced in (1). Proof:
Consider the decomposition ct + T(θt)ct+1=ct + (γ θt
ct+1∗ +θt)ct+1 and The
Kuhn-Tucker Saddle function of (1) is taken with the alternative resource constraints obtained by the above decomposition. By the concavity properties of u(.) and the linearity of the constraints an interior solution exists to the choice of ct, ct+1 in either case for a given T(θt) or γ(θt), holding fixed in the second case at the latter solution (on the Kuhn-Tucker theorem see for e.g. Rockafeller (1970) theorem 28.4).
Since u is differentiable and positive the solutions are positive and locally unique by the same theorem. Hence, substitute the values of ct∗＆c
t+1
∗ in the two cases and from the fact that differentiability of T(θt) ensures invertibility one can solve the above decomposition as an equation to obtain the solution to γ(.).
Since the objective function is linear in θt and T(θt) is differentiable a solution to the optimum θt will also exist by the Kuhn-Tucker theorem for some T(.). The invertibility argument works using the Inverse function theorem Kolmogorov & Fomin (1970) as before. Hence the theorem.
This result helps in decomposing the impacts of stock market sales volume and the consumption streams which result over time, on the welfare levels of participants as well as optimal growth of the market.
Stock market Sales Volume is ”golden rule” iff dkt+1
dkt =
u'(ct)
u'(ct+1) (5)
4. Stock Market Sales Volume & Technology Venturing
Technology is ventured into in the stock market to buy and sell stocks with the objective of saving and investing. Villanueva,Yoo & Hanssens (2003) considers the problem of optimal allocation of budgets over alternative marketing channels which generate different customers with lifetime values. This paper considers a similar problem over marketing technology for stock markets where the technology determines the ”lifetime of the consumer” in terms of planning span with respect to demand and supply of different intertemporal stock market products for savings and investments, in a representative agent model. However, this paper characterizes the ventured technology exante. Also, it integrates, in a rudimentary framework, the marketing technology problem of consumer and stock markets in an intertemporal setting.
+1 = ∗+1 + +1 +1
∗ = +1 +1 ∗
Plugging this into (1) :
u(kt+1 ωt+1
θt∗ )= u'(ct)T(θt)[ ωt ct T
'θt
T2θt ] (6) u'(c
t+1)θ1tdkt+1= u'(ct)T(θt)T
'(θt)
T2(θt)dωt dkt+1
dωt =
u'(ct)
u'(ct+1)T '(θt)
T(θt)θt (7)
Now, if T'(θt)θt
T(θt) = 1, then
dkt+1
dkt =
dkt+1
dωt =
u'(ct)
u'(ct+1)in (7)
which becomes the same as the ”golden rule” (see definition). Solving the differential equation we arrive at the particular solution
T(θt) = Aθt
This leads to the following theorem: Theorem 2:
Even when θ < ∞(∞ = lifetime of the agent) ∃ marketing technology T(θ) s.t. the stock market sales volume is ”golden rule”.
Comparison of the golden rule definition in this paper with the usual definition of golden rule which implies finding the steady state stock of capital such that the steady state consumption is maximized with respect to the stationary capital is given below for the two forms of the budget constraint presented.
Corollary:
Under the budget constraint presented in equation 1 and the stock sales (capital accumulation rule) given by (3), golden rule (usual) implies θ∗ =∞ and a solution to our problem does not exist.
Proof:
From the budget constraint in equation 1 and substituting the stock sales (capital accumulation) rule given in equation 3 and setting all values at their stationary values treating wt=kt we get :
dc∗ /dk∗ = 1/[1 + T 0(θ) + T(θ)]
Therefore at golden rule (usual) dc∗ /dk∗ = 0 => θ∗ = ∞, which is not a solution to our problem. Q.E.D.
Corollary 2:
Under the form of the budget constraint given by Theorem 1 and the stock sales (capital accumulation) rule given in (3), golden rule (usual) implies θ∗ = ∞ and a solution to our problem does not exist.
Proof:
Under the linear form of the budget constraint derived in Theorem 1 and using the stock sales (capital accumulation) rule in (3) and treating wt=kt under steady state values we get:
dc∗ dk∗ = 1 [1+θ+γ'(θ)1 c∗]
Therefore at the golden rule (usual) dc∗ /dk∗ = 0 => θ∗ = ∞ , which is not a solution to our problem. Q.E.D.
Dbranes String Theory as we will show works in Riemann Algebra by the "jump quantum" transitivity conductance provided by equation (5).This also solves for the Nyarko et. al. stochasticity and complementarity problem. See Mallick, Hamburger, Mallick (2016) for proof.
5. A Note on the String Theory of the National Stock Exchange and the
Existence of Measurable Stock Depositories-Quantum Scaling Law of
Sustainable Stock Markets i.e. Sustainable Stock Depositories
Theorem 3: There exists Probability Distributions of [wt] say for t ranging from 0 to some finite T, such that a D-Branes string solution exists for the representative planning problem for stocks given by program (1) and the Dynamic Stock Sales Rule (1) which defines a ”stock depository”.
Proof:
This proves that the Econophysics Stock Sales (”Stock Depository”) equation defined by the Definition , is capable of giving rise to a D-Branes String solution if the exogenous stock depository variable [wt] of this model follows a stochastic path given by some piecewise differentiable distribution, say, [P(w(t))] which makes the r.h.s. of the equation ”Dirichlet” differentiable.
One such distribution generation function could be the measurable function [P(wt|kt∗|kt∗ϵ definition 1)]. This is a ”D-Branes string function” which it can be easily proved gives rise to the distribution of[P ωt tT|P ω
0 ]
In any case existence of such D-Branes String ensures existence of measurable sets with the r.h.s. distribution of [P(w0)] in such cases giving rise to a ”stock depository”. Suitable properties of such stock depositories (field equation) in terms of the stock sales measures can give rise to the ”String Theory of National Stock Depository from National Stock Exchange” and will be useful in the development of depositories when a networked (national) stock exchange already exists, say with respect to the ratio of its volume and value.
In the above example the sales sequence will be given by [E ωt ω0 ]0T [E c∗t kt∗ ]0Twhere E stands for Expectation with respect to the P measure, which are all Lebeque measures, hence by applying the Karush-Kuhn-Tucker theorem as before, we get the Dirichlet differentiable set for all possible values of k0, it can be easily verified Q.E.D. Notice that the development problem of stock markets into (a) stock depositories, which essentially consists of the dynamic dual solution to the sustainable stock sales development and (b) stock sales realisation, which unlike S-duality concepts exhibits unique solutions under certain conditions as demonstrated by Theorem 3 and are not gauge invariant. Thus we derive the ”Quantum Scaling Law for sustainable stock markets” which is a D-Branes String solution to Differentiable Probability Fields over stock sales prospects. A M-Branes String solution is not possible at this point in the absence of good analytical Strings over Research & Development data (Mallick, Krichel & Novarese (2010)), although it is clearly possible to derive such Dirichlet Differentiable sets of data which are relativistically dual in the
sense of gauge invariance.The potential for stochasticity and complementarity with implicit String condensation and therefore Dbranes String differentiability results in the systems genetic algorithms with functions.
6. Stock Depositories as “Community of Stacks” as opposed to
“Community of Beehives”
It is possible to derive the unique optimal ratio sequence by the above theorem for a given T. A similar path of reaching the minima has been used in the “Simple String Theory Method” in Shepperd, Terrell & Henkleman (2008) which consists in considering paths of lower energy to reach a particular potential difference. In our method the Probability Distribution and the Expectation function (defined as the Lebesque integral over the String function) gives rise to a similar solution over the “stock depository accumulation” definition in the Definition. This gives a “Community of Stacks” as opposed to ”Community of Beehives” architecture (Mallick (2014a, 2014b, 2014c)).
7. “Strategic” Venturing Technology
The above theorems characterize some features of the marketing technology which may be ventured into and may include the system of education with respect to stock sales, the network of terminals, the distribution of trained agents of stock brokers and so on, and which determines sales volume on any given date in the stock market, with respect to different stocks. Each of these types of ventured technology will have features of its own, which will be different from the more developed banking and non-banking-financial technologies like postoffice savings bank and insurance company networks in emerging markets like India, requiring further indepth modeling.
The steps to ENGINEER STOCK MARKETS therefore are:
venturing stock marketing technology of the NSEIL ⇒ golden rule stock market participation (Theorem 1)⇒
golden rule stock market sales volume (Mallick (2011) examples)⇒golden rule realized consumer welfare (Theorem 2) ⇒optimal growth of the economy with sustainable stock market depositories like the NSDL⇒optimal growth of the
economy with D-Branes String endogenous stock market products (Theorem 3). Given the stochastic and functional generality these results can be applied to a wide panel of e-bourses.
8. Experimental Results
This algorithmic procedure has been demonstrated experimentally in Tables 2.4.1.3.1, 2.4.1.3.2 and 2.4.1.3.3 in http://planningcommission.nic.in/reports/sereport/ser/scoiwel.pdf. The above String has been experimentally and mathematically-statistically derived with 0.53 R-squared in Indian Manufacturing Sector Stock Markets in Mallick (2009) and in Banking Sector Stock Market Results in Mallick, Sarkar, Roy, Duttachaudhuri & Chakraborty (2010).These results are generalisable to e-bourses with their clientele in developed as well asdeveloping countries subject to satisfying the String Theoretic conditions.
The origin of the idea of strings in stock market research with optimal (i.e. dynamic social welfare optimising) growth in consumption materials is due to Mallick (2014a) as it is shown to be dependant (in an interactive way i.e. vice versa, starting from a particular dataset similar to the “interaction picture” in Quantum Mechanics). Mallick, Krichel and Novarese (2010) have demonstrated experimentally that this phenomenon is always observable in digital information systems like research libraries and R&D. The idea that string theoretic econophysics models can be converted into interactive algorithms with D-Branes String Field data is also due to Mallick (2014a, 2014b, 2014c).The stochasticity and complementarity gives it the quantum mechanical industrial direction with the fundamental Dbranes String differentiability in a wide class of mathematical-statistical systems.
endogenous marketing of stock market (i.e. world volume and sheet value, because of fixed price resolution) products to achieve intertemporal golden rule welfare levels hence physical and financial flow efficiency. What we have not achieved is how to relate this representative agent framework to disaggregated modeling for individual stocks and types of markets with and without variable prices. Dbranes String Theory works in Lie algebras, however our analysis has been in Riemann Algebra with equation (5) allowing for non-commutative “jump quantum” transitivity (Mallick (2016) et. al.).
Acknowledge
The author acknowledges helpful discussions with Jess Benhabib, Yaw Nyarko, Jota Ishikawa, Fumio Hayashi, Ed Prescot, B. Friedman, M. Ramachandran & Sandra Dawson, Egil Tzaland, T. Johnsen, R. N. Childs, L. Goldberg, S. Raychaudhury, A. Sen, for some free time to complete this work. This paper is derived from papers presented at the Far Eastern meetings of the Econometric Society 2001, Kobe, the European Economic Association Congress, Stockholm, 2002, South & South East Asia Econometric Society Meeting, Chennai, 2006, IIM Lucknow 2006, ICFAI Conference Hyderabad 2010, & Strings 2015 Bengaluru. The author thanks the organizers of the meetings in Chennai and IISWBM faculty research fund, for financial support. The usual disclaimer applies.
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