Algorithmic pricing theoretical framework

The determination of prices for goods, services and tradable securities is an indispensable task for the proper functioning of any economic system. There has been always the belief that there should be an optimum price as to maximize the revenues of any sell-side agent in the economy (producers, intermediaries, final sellers) and that fairness in price should be interpreted as setting the prices as lowest as possible as to cover all costs due to commercialization process plus a revenue²⁸⁵.

281 Melvin, Michael and Sultan, Jahangir. “South African Political Unrest, Oil Prices, and the Time-Varying Risk Premium in the Gold Futures Market”. The Journal of Futures Market, Vol. 10. 1990.

282 Source: Investopedia.

283 Clark, Jeff. ‘There's no reason to own a gold ETF’. Business Insider. April 9, 2016. Available at:

http://www.businessinsider.com/theres-no-reason-to-own-a-gold-etf-2016-4?r=US&IR=T&IR=T

284 Krige, D.G. and Austin, J. D. “Theshold rates of return on new mining projects as indicated by market investment results”. Journal of the South African Institute of Mining and Metallurgy. October, 1980.

285 A theoretical price-setting agent more often than not will be interested “fairness” of her prices for a number of reasons, for example:

 Buyers must be convinced the prices they pay are fair if they are to become loyal, repeat customers.

 Sales personnel must be convinced the prices they charge customers are fair if they are to be effective salespersons for the business.

 Regulatory agencies must be convinced prices charged are fair or they may be motivated to increase their oversight and control of an industry.

The concept of “fairness” I am referring here to should not be confused with the technical notion of fair price as the equilibrium price for futures contracts. It is also called the theoretical futures price, and equals the spot price continuously compounded at the cost of carry rate for some time interval.

In the context of corporate governance, fair-price provisions limit the range of prices a bidder can pay in two-tier offers (offers to purchase a sufficient number of stockholders' shares so as to gain effective control of a firm at a certain price per share, followed by a lower offer at a later date for the remaining shares).

Sources:

Rouse, Margaret. ‘Fair and reasonable price’. At:

http://searchitchannel.techtarget.com/definition/fair-and-reasonable-price. July, 2007.

Hayes, David and Miller, Allisha. ‘What Is A Fair Price? - And Who Gets to Decide?’. At http://www.hospitalitynet.org/news/4049159.html, November 17, 2010.

94 However, there are not many contemporary studies regarding the theory of prices, as if the matter didn’t deserved a thoroughly and systematic approach from the economic point of view. Price theory was for a long time set in the context of models of price-taking agents existing in a partial equilibrium:

supply and demand models where the clearance (operational capabilities subject to a set of restrictions) on the market of some specific goods/services is obtained independently from prices and quantities in other markets. In other words, the prices of all substitutes and complements, as well as income levels of consumers, are exogenous to these models and therefore taken as given.²⁸⁶ This approach relies in simple models which very often resemble a game-theoretic problem posing of the relevant concepts, and almost always places the main thrust of the effort in the measurement side (the resulting output quantity price) rather than in the theory that lie underneath.

Computational price theory

First, a definition for economic price theory is needed. The only good definition to be found in the literature is that from Weyl (2017): <<Abstractly, I define price theory as neoclassical microeconomic analysis that attempts to simplify a rich (high-dimensional heterogeneity, many agent, dynamics, etc.) and often incompletely specified model for the purposes of answering a simple (scalar or low-dimensional) allocative question.>>²⁸⁷ A definition for computational price theory should take this price theory definition as a blueprint. This is an essential requirement to fulfill if the theory is to be of practical use. After all, all claims concerning the functionality and performance of an algorithm assume a particular computational model.

The first aspect to be considered is the simplification of the rich, high-dimensional heterogeneity, (many agents, dynamics, etc.). This idea matches an important characteristic of the definition of a pricing algorithms, that is, Ω, the space of market scenarios is n-dimensional, while the price p is 1-dimensional, i.e. a scalar, except for one thing, the heterogeneity. In our case, we need the high-dimensional space to be homogeneous. This is achieved by considering that the space made up of financial variable only, so basically an n-dimensional space is a space consisting in n variables.

Second aspect refers to the “often incompletely specified models” part of the definition. For computational finance purposes, the model has to be well-specified, and in our case the model is the pricing algorithm itself. Recall from Section 4.3 that the particular selection of the probability measure ℙ was up to some degree, a subjective exercise. But once ℙ was selected, the rest of the calculation procedure was objective. The third aspect focuses on the purpose of the analysis:

answering a simple (scalar or low-dimensional) allocative question. I agree with this part and appreciate the brilliance of recognizing that pricing is, indeed, performed with the goal of allocating resources among market participants. The question here, is the price. Prices are the solution of the allocative problem we are interested in.

So my own definition of computational price theory is: Computationally, price theory is the analysis of the reduction of an n-dimensional range (n being the number of financial variables involved), by means of the application of pricing algorithms, into prices that provide approximate solutions to resource a specific allocation problem.

How efficient the reduction might be depends on the degree of correlation between the variable. The lower the correlation, the higher the efficiency of the theoretical algorithm.

So, computational price theory relies on pricing algorithms for its practical implementation. Normally, a computational model is assumed to permit certain operations with associated costs. In the case of

“Front-End Loaded Tender Offers: The Application of Federal and State Law to an Innovative Corporate Acquisition Technique” (Editorial). University of Pennsylvania Law Review, Vol 131, pp. 389-422, 1982. Available at: http://scholarship.law.upenn.edu/penn_law_review/vol131/iss2/3/ .

286 Partial equilibrium. Wikipedia. At https://en.wikipedia.org/wiki/Partial_equilibrium.

287 Weyl, E. Glen. “Price Theory”. Journal of Economic Literature. April, 2014.

Take a look also to Weyl, E. Glen. ‘What Is “Price Theory”? Marginal Revolution. July 29, 2015. Available at: http://marginalrevolution.com/marginalrevolution/2015/07/what-is-price-theory.html

95 pricing algorithms, as we have already made clear, that would be very difficult. However, a binary relation connecting pricing algorithms and expressing the fact that any two algorithms yield the same results if fed with the same given inputs is necessary. So it is to specify a way to derive those algorithms from a model description. Following is the definition of the equivalence of pricing algorithms, and in the next subsection, the derivation of these artefacts is discussed.

Let W be the set of all possible states of a financial market, and let R the set of financial assets in that market at time T0. Let G be a pricing algorithm that computes a price vector V1 for an asset r ∈ R under conditions C1, C2, C3,…, Cn. If within that particular state w ∈ W, there is another algorithm H that also computes a price vector V2 for asset r, and if V2 – V1 ≤ ε and it holds that H ⊆ G ∩ (C1 ∩ C2 ∩ C3,…, ∩ Cn) then both algorithms are said to be equivalent. The intuition of the conditions is as follows. First the elements of the price vectors does not need to be equal but similar, and the measure of the required similarity is set by the constant ε. Second, the set of conditions under which G works is established, but that is not the case of H, all we know from the latter is that it can compute the target price. So, it can be the case that there are scenarios in which H will work but G not.

Derivation of a pricing algorithm

To derive a pricing algorithm is to specify its structure using an asset pricing kernel model. A pricing kernel or a state price density is a vector, M = {m1, ..., mS}, such that for any asset with payoff vector X = {x1, ..., xS}, the value is given by:

S

𝓟 (X) = E[ MX ] = ∑ πs ms xs , where π is state price (“Arrow-Dubreu” price)²⁸⁸

s= 1

The asset pricing kernel (APK), or stochastic discount factor, is an important concept in mathematical finance and financial economics: it is a crucial link between economics and finance and plays a pivotal role in assessing the risk aversion over equity returns. The term kernel is a common mathematical term used to represent an operator, whereas the term stochastic discount factor has roots in financial economics and extends the concept of the kernel to include adjustments for risk.

The APK summarizes investor preferences for payoffs over different market scenarios (states of the world). In the absence of arbitrage, all asset prices can be expressed as the expected value of the product of the pricing kernel and the asset payoff. Thus, the pricing kernel used with ℙ gives a complete description of asset prices, expected returns, and risk premia. The question is to estimate the pricing kernel.

The method introduced by Aït-Sahalia and Lo (2000)²⁸⁹ to construct empirical pricing kernels is of common use in the literature: a pricing kernel K is defined as a ratio

economic risk . statistical risk

The economic risk which contains the preferences of investors and the statistical risk which provides information on the dynamics of the data generating process (DGP). The economic risk is well assessed by Arrow-Debreu prices and can be estimated by the risk neutral density q obtained from the derivative market. Thus, obtaining an accurate estimator of q is a crucial step for pricing kernel estimation. How to obtain that estimator, is the asset pricing kernel problem.

288 Nejadmalayeri, Ali. “Basics of Asset Pricing”. Lecture notes. Oklahoma State University. January, 2007.

289 Aït-Sahalia, Y. and Lo, A. W. “Nonparametric risk management and implied risk aversion”. Journal of Econometrics 94, pp. 9-51. 2000.

96 Belomestny, et. al. (2000) ²⁹⁰ propose to estimate the pricing kernel nonparametrically by controlling the ratio of the risk-neutral density and the subjective density. Rosenberg, et. al. (2002)²⁹¹ propose estimate the kernel using current asset prices and a predicted asset payoff density.

A study of empirical pricing kernel methods is beyond this Master thesis. But what matters is the computational interpretation of the results. The asset pricing kernel, also known as the stochastic discount factor (SDF), is the random variable that satisfies the function used in computing the price of an asset²⁹². Let’s assume that an APK has been defined and it is available. The computational implementation of that APK is known as a pricing engine (PE). This engine represents the core of the pricing algorithm. However, since the construction of the APK, the same as the selection of ℙ in the pricing process discussed earlier, is not an objective choice, it is not possible to expect that the pricing engine will give a reliable output from the beginning. It is necessary then to calibrate the engine, using real actual data, so the parameters inside the PE can adjust properly. It isn’t until this point that we get a functional pricing algorithm:

pricing algorithm = pricing engine + calibration module

An essential characteristic of PEs is that they exhibit non-determinacy. Nondeterminacy is a property of the systems, regarding their predictability. In the realm of software, a nondeterminate program gives different results on different runs, given the same input²⁹³. It has been studied since a long time ago, see Hartmanis (1972) for an example.²⁹⁴. In the programming language context, nondeterminacy can be of two types: demonic and angelic²⁹⁵. Be aware that non-determinacy should not be confused with the property of being non deterministic, in the sense that a system can be deterministic, but nondeteminable at the same time.

What is interesting for our purposes, is that a simple and tentative taxonomy of computational finance environments can be constructed based on PE non-determinacy.

Class 0 Environments with completely determinate PEs in the sense that every event occurring triggers one or more actions, which if taken will result in a certain price movement (up, down, stay). Pricing mechanisms are quite simple. The pricing engine is somehow a reflection of the environment, the resulting pricing algorithms are very simple (basically a case-switch statement), no calibration is needed and several pricing measures would be possible. This kind of environments are mainly theoretical. An example can be a complete market under perfect competition with standardized products and no information asymmetries among the agents.

Class 1 Environments with completely determinate PEs in which for only a subset of events trigger actions resulting in certain price movements. The pricing mechanism remains simple, but they no longer are precisely accurate. This environment is conceptually equivalent with a class 0 environment with noise, but incorporating a variability measure, some prediction capabilities.

293 Jianxin, Xiong and Dingxing, Wang. “Analyzing Nondeterminacy of Message Passing Programs”.

Proceedings of the Second International Symposium on Parallel Architectures, Algorithms, and Networks. © IEEE,June, 1996.

294 Hartmanis, J. “On non-determinancy in simple computing devices”. Acta Informatica, Vol. 1, Issue 4, pp. 336-344. Springer. December, 1972.

295 Moris, Joseph N. “Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy”.

In: Kozen, Dexter & Shankland, Carron (Editors). Mathematics of Program Construction: 7th International Conference, MPC 2004, Proceedings, Scotland, UK, Springer-Verlag, July, 2004.

97 Class 2 Environments with partially non-determinate PEs. In contrast to classes 0 and 1, reliable prediction can no longer be based on input information. This environment resembles a highly-controlled financial environments, for example, through extensive policies and control.

Class 3 Environments with non-determinate PEs. Regular financial market models found in the real world.

Fundamental Theorem of Asset Pricing

The Fundamental Theorem of Asset Pricing (FTAP) is the theoretical foundation underlying the use of martingale methods in derivative pricing. In rough terms, the theorem states that in a market where no arbitrage opportunities exist, it is possible to build a probability measure ℙ * equivalent to ℙ such that the discounted value Vt (H) of any contingent claim with terminal payoff H-is represent by:

Vt (H) = E ^{ℙ *} [ H |

𝓕

t ] ²⁹⁶

In simple words, the FTAP, also known as the Representation Theorem says that the following statements are equivalent:

(1) The existence of a positive linear pricing rule.

(2) The existence of positive ‘risk neutral’ probabilities and an associated riskless rate.

(3) The existence of a positive state price density.

The proof can be found in Dybvig & Ross (1987)²⁹⁷. The upshot of the proof of their proof is that in complete arbitrage-free markets there exists a unique linear pricing rule, q, such that the price vector p = {p1, ..., pn} = qΠ. The linear pricing rule is then given by q = pΠ⁻¹. Π is the “Arrow-Debreu” tableau of prices.²⁹⁸

The Black-Scholes framework

Also known as the “Midas equation”, the Black–Scholes equation set the foundations for the spectacular growth of the financial sector, the ever increasing complexity of financial instruments and both the prosperity and crises cycles in the financial markets.

where:

V is the price of the financial derivative

σ represents the volatility (variance) of the stock S is the price of the commodity

r represents the risk-free interest rate

In order to understand properly the formula, a rearrangement of term might prove beneficial:

∂V / ∂t = c0 V ‒ c1 ∂V / ∂S ‒ c2 ∂²V / ∂S²

So what matters here is how the price of the derivative behaves over time (∂V / ∂t). The price of the derivative over time equals is expressed as a linear combination of 3 terms: the price of the derivative

296 Biagini, Sara and Cont, Rama. “Model-free Representation of Pricing Rules as Conditional Expectations”. In: Akahori, Jiro; Ogawa, Shigeyoshi & Watanabe, Shinzo. Stochastic Processes and Applications to Mathematical Finance: Proceedings of the 6th Ritsumeikan International Symposium, Ritsumeikan University, Japan, 6-10 March 2006, pp. 53-66. World Scientific, 2007.

297 Dybvig and Ross (1987) coined the terms “Fundamental Theorem” to describe these basic results and “Representation Theorem” to describe the principal equivalent forms for the pricing operator.

298Nejadmalayeri (2007) op. cit.

98 itself, how fast that price changes relative to the stock price, and how that change accelerates. This resembles the heat equation in physics, describing how the price of the derivative diffuses through the stock‒price space.²⁹⁹

Solving the Black-Scholes equation

Now, let’s explain traditional derivative pricing starting from the Black-Scholes equation, which is a partial differential equation (PDE). There are some direct methods available to solve the equation (e.g. Gauss elimination or LU factorization) and find the price of an asset, but their high computational complexity (a system with n unknown variables requires O(n³) operations), poor properties in memory usage and the fact that they require a complete representation (linearization) of the problem, prevent their practical use³⁰⁰. Because of this, the equation needs to be discretised in order to be solved. The way to do this is through multigrid methods. Multigrid methods³⁰¹ are iterative, in which each iteration goes from a fine grid to a coarse grid. The purpose of alternating between fine and coarse grids, is to achieve error reduction. That is essentially the idea behind multigrid methods. In coarser grids “smooth becomes rough” and low frequencies act like higher frequencies³⁰², meaning that error is more easily removed.

In order to construct such a method, mechanisms are needed that transfer the information in an appropriate way between the grids³⁰³. Only a few iterations are needed before changing from fine to coarse and coarse to fine. From a computational perspective, the interpretation of the mechanics of multigrid methods is precisely that, information management between girds³⁰⁴. To transfer

299 Stewart, Ian. 17 Equations that Changed the World. First edition. Profile Books, Ltd. 2017.

300 Van Brummelen, Harald. ‘Basics of Multigrid Methods’. Technische Universiteit Eindhoven. SINTEF winter school 2011. Available at:

https://static.tue.nl/fileadmin/content/faculteiten/wtb/Onderzoek/Onderzoeksgroepen/multiscale-engineering-fluid-dynamics/vanBrummelen_TALK2011a.pdf.

301 A grid is a small-sized geometrical shape that represents the domain described by the PDE, whose objective is to identify the discrete elements where the stable measurable properties of the original domain hold or can be applied. Grid generation is the first process involved in computing numerical solutions to the equations that describe a physical process, or in this case, a financial one.

The result of the solution depends upon the quality of grid. Source: Wikipedia.

302 Strang, Gilbert. ”Multigrid Methods”. Readings from the course Mathematical Methods for Engineers II, Section 6.3. Spring, 2006. Available at: https://ocw.mit.edu/courses/mathematics/18-086-mathematical-methods-for-engineers-ii-spring-2006/.

303 Volker, John. “Multigrid Methods”. Lecture Notes. Weierstrass Institute for Applied Analysis and Stochastics, Winter Semester 2013/14

304 The basic idea is to start with a coarse grid and get a solution quickly. Then with a finer grid, that solution is used as a starting guess and thus optimize the solution in a few iterations until the solution is a good enough approximation to the problem:

Source of image: EU Regional School 1st Short Course. 2013. Available at:

99 information between grids, suitable operators are needed³⁰⁵. A restriction matrix transfers vectors from the fine grid to the coarse grid. The return step to the fine grid is by an interpolation matrix. The remarkable result is that multigrid can solve many sparse and realistic systems to high accuracy in a fixed number of iterations, not growing with n (that is, complexity is O(N)).³⁰⁶ Besides being optimal, multigrid methods are of great interest because they are some of the very few scalable algorithms,

99 information between grids, suitable operators are needed³⁰⁵. A restriction matrix transfers vectors from the fine grid to the coarse grid. The return step to the fine grid is by an interpolation matrix. The remarkable result is that multigrid can solve many sparse and realistic systems to high accuracy in a fixed number of iterations, not growing with n (that is, complexity is O(N)).³⁰⁶ Besides being optimal, multigrid methods are of great interest because they are some of the very few scalable algorithms,

In document Pricing algorithms and computational price theory : the building blocks of computational finance and IT business applications (sivua 108-114)