The price of granularity and fractional finance

1. Introduction

Theoretical Asset Prices are based on financial models that associate securities or other assets future states, say at time “1”, S j ( 1 ) , j = 1 , 2 , … , n , to “state prices” π j . On the basis of these, a security with known future state preferences, say n states, has an implied value given by the sum of its cash flows or ∑ j = 1 n π j S j ( 1 ) , j = 1 , 2 , … , n . This means that the current price of a security and its complete future state prices, are known. In this sense, the current price S ( 0 ) , implies future state prices priced and exchanged now. Such models assume that there exists an n-vector of positive state prices π j , such that the current price is unique but future implied prices need not be unique, or: (1) S ( 0 ) = ∑ j = 1 n S j ( 1 ) π j or S ( 0 ) ⇔ { π j ; S j ( 1 ) ; j = 1 , 2 , … , n } . Financial pricing is then an inverse problem defined by the hypothesis that future states and their prices are predictable and priced by a kernel (its probability measure). When markets are complete, expectation based on a kernel probability measure defines a pricing martingale (albeit there may be more than one Martingale that need not be pricing Martingales). When a pricing Martingale is defined, its current and future price are in expectation equal. In such circumstances, there is no profit in trading. The mathematical assumptions of a pricing Martingale and its properties define complete markets. For example, a pricing martingale implies no-arbitrage, no transaction costs, no information asymmetry, no dominant financial agent, etc. Financial pricing in a complete market model is therefore relative to a mathematical construct based on a shared rationality and an information set that underlies all financial agents’ exchanges and trades. In addition, it is defined relative to a model’s granularity. Thus, changing the “complete market” model granularity, may necessarily violate the fundamental assumptions of the uniqueness of a price implied in complete markets.

A financial pricing model is thus a reference model with respect to which financial traders, buy, sell, bet and value financial assets and exchange at a given price. It assumes further, that all future states and their prices are known and shared, summarized parametrically at a current time (its filtration) and changing over time as new information accrues and its meaning analyzed and interpreted. These assumptions, including the model’s granularity, may define a pricing martingale, with respect to which an equilibrium price is defined. Financial pricing models are therefore model specific, defined with respect to their assumptions, their known future states, relative to what we know, relative to the information shared by financial agents, etc. and relative to the model granularity, the data we use to analyze and predict prices, as well as the mathematical behavior of fractional models that financial granular models are based on.

Fractional models of different granularity (defined by a fractional index) define time scales, augmenting or reducing the relative time intervals at which data is measured and theoretical analyses are based on. In a financial sense, it alters (augmenting or reducing) the number of future states to account for in a financial model and therefore the information required by financial agents to price assets. In addition, it introduces a long run memory, contributing to a given price (or rates of return) to depend on its history (and not only on the infinitesimal neighboring past time) and finally on the speed of convergence of these time intervals implied by the fractional index. The purpose of this paper is to assess a fractional asset pricing model and define the price of granularity relative to that of a complete market model. For simplicity, we consider simple examples to highlight some of the fractional financial pricing problems we raise.

Empirical and theoretical studies based on fractional models in finance as well as R/S (range to standard volatility statistics) analyses abound. They measure the effects of fractional parameters estimates on risk measurement, on the “long run and autocorrelation memory” implied in fractional calculus (see [7,74,85–89,92–95] as well as [66]). From the financial viewpoint, there is an extensive literature on fractional financial models (see for example [5,6,9,14,18,20,23–25,40,68,78–82,90,97,98]). Although some studies have defined “pricing martingales” for fractional financial models, it is not clear that these are the market Pricing Model which is accepted by traders and used to assess their trading strategy. Further, since a market price is unique, defined by a unique Martingale, competing and fractional “pricing martingales” may in fact be incomplete.

2. Financial complete markets and fractional finance

A complete pricing model (whether fractional or not) is an inverse problem which defines a unique correspondence between a current expectation over all complete and known future prices. Practically it is defined by an “information relative probability measure”, for example, an “insurance” adapted probability measure, f Q ( · ) that translates future prospects into a risk free discounted expectation of known future state prices – also called the risk free probability measure. Their current price is then defined uniquely relative to a risk free spot bond price [72]. Thus, two assets with identical financial properties ought to have the same price, the price of the one defines the other and vice versa. For example, for a risk neutral probability measure, its martingale corresponds at time t to the following: S ( t ) ( 1 + R f ) t + k = E Q { S ˜ ( t + k ) ( 1 + R f ) t + k | ℑ t } or S ( t ) = ( 1 + R f ) − k E Q { S ˜ ( t + k ) | ℑ t } , where E Q ( S ˜ ( T ) | ℑ t ) is an expectation taken with respect to a model probability measure Q. The stock risk premium is then, by definition, defined by the probability measure defined with respect to a filtration ℑ t . In continuous time, assuming that future states preferences S ˜ ( T ) are completely known, we have: S ( t ) = e − R f ( T − t ) ∫ − ∞ ∞ S ( T ) d F Q ( S ( T ) | μ , σ ) with μ, σ, the distribution mean and its volatility while F Q ( S ( T ) | μ , σ ) is the Distribution Function of the probability measure that accounts for the “insurance contract” that has removed the risk consequences from the financial exchange (and thereby its pricing by a risk free bond).

A fractional financial pricing model may then assume several forms (and therefore violating the fundamental assumption of a unique price). It differs from a complete market price as it “alters the measure-scale of time”, introducing “auto-correlated-memory”, a parametrized quantitative time interval and its associated “speed of convergence” to its infinitesimal (continuous time) limit. As a result, the parameters defining the pricing model, may differ from a reference complete market model. In other words, even if a fractional martingale pricing model is constructed, it need not be a pricing martingale since pricing martingales are both unique (by definition) and based on informational and other assumptions commonly shared by all financial agents and based on rational exchanges between these agents, eventually leading to the unique market “pricing martingale”.

In such cases, model differences arise also due to parameters measurements defined by the model granularity (whether it is the risk free rate, the asset’s rate of return or its volatility) or the stochastic calculus we apply to manipulate such models and establish their relationships over the small time intervals that their fractional derivatives imply (and in particular the long run memory embedded in fractional derivatives). Thus, in real financial time series (say, high-frequency, intraday, day, weekly, etc.) their granularity may lead to models that differ theoretically and practically from a complete market model defined relative to another model granularity. In a Physics sense, “relativity and quantum mechanics” differ in their approach to “big and small”, each of which is based on theoretical constructs that are not obviously reconciled. From a financial viewpoint, a fractional financial model is necessarily a model departing from the reference and uniformly accepted market price model. In this sense, it measures a fractional index premium price for departing from the fundamental market pricing model. For example, defining a risk neutral probability measure, pricing a financial asset relative to its risk free and observable interest rate, will require that we define a fractional interest rate to price the risk premium that such a model entails. In other words, a fractional risk neutral pricing model requires that it be defined relative to bond prices in a fractional bond market. Although it may lead to a semi-Martingale pricing model, it need not be the “true market pricing model”, providing thereby arbitrage opportunities by trading on multi-time scales models. Furthermore, a fractional price (see also [89]) might not have a complete fractional probability measure (i.e. with a sum of all probabilities equal 1) which lead necessarily to an incomplete (or overly complete) set of future state preferences that violate the assumption of complete markets pricing models.

In the real world, financial incompleteness abounds for a broad set of reasons (for example [3,4,11,19,64,91]). In many cases, however, information and mathematical constructs are used to price and account for their incompleteness. These risk premium models imply a departure from the complete market efficient price and therefore it is a measured risk premium of their “incompleteness”. Financial spreads for example, are a manifestation that point out to financial agents that are not homogenous, that may differ by the information they have, their risk preferences (utilities), their wealth but also by the manner in which they measure and calculate market prices (see for example, Tapiero [84] on the Assets Pricing and Economic Inequalities, Quantitative Finance 2015). Although in practice, arguments can be made for financial markets to be essentially incomplete, there are also arguments that provide a robust rationale to believe that markets cannot be incomplete “forever” and will therefore revert to an equilibrium state of complete markets (however flimsy, fast or slow, fractional or not, that such a return to equilibrium may be). Although in the real world of finance, theoretical market models are incomplete, complete market models provide an important anchor to theoretical finance and a model reference we may use to price the price of its incompleteness for any of the reasons that violate the assumptions of complete markets. The price of granularity is one such price.

3. Granularity and its financial implications

Continuous time models are based on infinitesimal time and state increments ( d t , d S ) while their derivatives are integer derivatives. These elements, assuming regular and continuous functions define Taylor series, integration and derivative operators, broadly referred to a Riemannian–Cauchy Calculus. Fractional calculus however is defined with respect to operators with time and state fractional increments ( ( d t ) H , d H S ) (Jumarie [44–53], Gray et al. [35], Kluppelberg and Kuhn [54]). For example, say that an asset is priced at time intervals d t = 0.01 . If 0 < H < 1 , and since d t < 1 , the fractional time interval is necessarily greater than d t , ( d t ) H > d t . Time scale has therefore implications to the financial model we use, parameters estimation and the mathematical tools we apply to such models. Similarly, they may-redefine future measurements of state preferences and therefore future risks implied in a particular model (for example, to missing data jumps in a time series due to granular time records, etc.). In other words, a “time clock-granularity” defined by a fractional parameter H contributes to a model’s properties and thereby to its statistical and computational properties. These differences are encountered in every-day life and in many of our products. A Patek–Phillipe watch for example, compared to a $10 watch, have a much smaller granular time movement. Recording the landscape from a fast moving High-Speed train will be far less complete than from a slow moving train that accumulates and memorizes the information that the landscape provides.

In Riemannian calculus, a price change in a small time interval, d S ( t ) = S ( t + d t ) − S ( t ) is based on local past time increments d t (where higher order increments ( d t ) i , i ⩾ 2 are assumed negligible): (2) d S = S ( t + d t ) − S ( t ) = ∑ i = 1 n 1 Γ ( i + 1 ) ∂ i S ( t ) ∂ t i ( d t ) i + ℜ n + 1 ( t ) , where ℜ n + 1 ( t ) is a remainder accounting for neglected terms in the Taylor series expansion. Thus, for infinitesimal increments d t , a first derivative is defined by: (3) d S ( t ) = lim Δ t → 0 ( S ( t + Δ t ) − S ( t ) ) and d S ( t ) d t = lim Δ t → 0 S ( t + Δ t ) − S ( t ) Δ t . Higher order derivatives may be calculated as well by setting: (4) d n S ( t ) = lim Δ t → 0 ∑ i = 0 n ( − 1 ) i n i S ( t + ( n − i ) Δ t ) and (5) d n S ( t ) ( d t ) n = lim Δ t → 0 1 ( Δ t ) n × ∑ i = 0 n ( − 1 ) i n i S ( t + ( n − i ) Δ t ) . For example, for n = 2 , we have: (6) d 2 S ( t ) ( d t ) 2 = lim Δ t → 0 1 ( Δ t ) 2 × ∑ i = 0 2 ( − 1 ) i 2 i S ( t + ( 2 − i ) Δ t ) = lim Δ t → 0 2 0 S ( t + ( 2 − 0 ) Δ t ) + ( − 1 ) 2 1 S ( t + ( 2 − 1 ) Δ t ) + 2 2 S ( t + ( 2 − 2 ) Δ t ) / ( Δ t ) 2 = lim Δ t → 0 S ( t + 2 Δ t ) − 2 S ( t + Δ t ) + S ( t ) ( Δ t ) 2 = lim Δ t → 0 1 ( Δ t ) { Δ S ( t + Δ t ) ( Δ t ) − Δ S ( t ) ( Δ t ) } = lim Δ t → 0 Δ 2 S ( t ) ( Δ t ) 2 = d 2 S ( t ) ( d t ) 2 . A fractional derivative is defined instead by its non-integer (fractional) order, defined by the infinite polynomial: (7) d H S = ∑ i = 0 ∞ ( − 1 ) i H i S ( t + ( H − i ) ( Δ t ) H ) . For example, its first few terms are: d H S = ( − 1 ) i H i S ( t + ( H − i ) ( Δ t ) H ) = H 0 S ( t + H ( Δ t ) H ) − H 1 S ( t + ( H − 1 ) ( Δ t ) H ) (8) + H 2 S ( t + ( H − 2 ) ( Δ t ) H ) + ⋯ . Therefore, for a fractional index 0 < H < 1 , we have, a derivative defined by its past events we may not neglect and thereby its long run memory which is a function of past values t − ( 1 − H ) ( Δ t ) H , ( Δ t ) H > ( Δ t ) , or: (9) d H S = S ( t + H ( Δ t ) H ) − H 1 S ( t − ( 1 − H ) ( Δ t ) H ) + H 2 S ( t − ( 2 − H ) ( Δ t ) H ) + ⋯ . The fractional index clearly defines the memory effects on its fractional derivative S ( H ) ( t ) = d H S / ( d t ) H . If 0 < H < 1 then ( d t ) H > d t which we can interpret as a “loss of model information” due to measurements at intervals that would be greater than the presumed reference model with time intervals d t . This loss of information is compensated by its accounting of past events. If 1 < H , say H = 1.95 , then for small ( Δ t ) , d 1.95 S = S ( t + 1.95 ( Δ t ) 1.95 ) − H 1 S ( t + 0.95 ( Δ t ) 1.95 ) + ⋯ with other terms neglected. Therefore the derivative depends on both extremely small time increments ( Δ t ) 1.95 and future state prices S ( t + 0.95 ( Δ t ) 1.95 ) which are clearly not known at time t if it were a random variable. Of course, and as fractional calculus stipulates we may split the derivative order by defining instead d 1.95 S ( d t ) 1.95 = d d t ( d 0.95 S ( d t ) 0.95 ) where ( d 0.95 S ( d t ) 0.95 ) is a fractional derivative. In Ultra High Frequency Trading with ticks occurring in milliseconds, we note that price variations estimates will be both extremely small and dependent on their past accounted for by higher order derivatives (i.e. variations on variations)! In other words, as indicated above, setting 2 > H = 1 + α and 0 < α < 1 , we have: (10) d H S = d 1 + α S = d ( d α S ) . Similarly in Riemannian calculus, we have for the second derivative: (11) d 2 S ( t ) ( d t ) 2 = d d t ( d S ( t ) d t ) . In other words, 2 data sets, one based on estimates of time increments d t and ( d t ) H or data sets of various granularities need not lead to the same results – both theoretically and practically.

Liouville [56] fractional definition of the Cauchy remainder with a fractional index 0 < H < 1 is: (12) ℜ H ( t ) = 1 Γ ( H ) ∫ 0 t ( t − τ ) H − 1 S ( τ ) d τ , 0 < H < 1 and of course, for 1 < H < 2 : ℜ 1 + α ( t ) = d d t ℜ α ( t ) = 1 Γ ( H ) 1 d t ∫ 0 t ( t − τ ) H − 1 S ( τ ) d τ , (13) 0 < α < 1 , H = 1 + α . Thus, fundamental financial theories based on a different granularity, may or may not lead to the same results, nor would they necessarily lead to the same financial conclusions. The time scale of measurement in both financial theory and in practice are thus to be accounted for. For example, if rates of return models are measured with a data granularity with fractional parameter H while the model variance is measure at a granularity 2 H , their analysis has to account for their differences. This is particularly the case when we consider Brownian motion pricing models based on their fractional time variance while expected rates of returns and bonds rates of returns are measured and treated at a different time scale.

Models such as ARCH-GARCH that estimate stochastic volatility models based on long run memory (albeit estimated empirically rather than analytically) have been constructed to better explain the leptokurtic character of rates of returns distributions embedded in a fractional financial model due to their deviating from the standard normal probability distribution (as we shall see subsequently, [1,28,32,96]). Other studies have shown that distributions have tails fatter than the normal distribution and therefore pointing to financial models defaults and thus to their incompleteness. This fatness of the tails arises naturally from the convolution of stock prices distributions with a Power Law expressing both a loss of information to the model-data granularity or to incomplete information [89]. Further, stochastic volatility and fractional Brownian motion models have shown that non-linear volatility growth and non-linear dependence may be observed in fact. In such cases, the assumptions of a linear volatility time scale for a “normal” finance may in practice be doubtful. References to these models and the problems they deal with are numerous.

Considerable theoretical and empirical research [13,37,41,57–63,65,66] have also pointed out that estimates of the fractional parameter H (also called the Hurst index), may be expressed in terms of a sample range to volatility ratio statistic (albeit based on infinite sample series and therefore, defined by their data-granularity). They point out, for simplicity to ( R / s ) ∝ ( T ) H where R / s is the sample range to the sample volatility estimate and T is the sample time series on the basis of which estimates of the Hurst index are obtained. The fractional parameter H is thus a function of two variability estimates, both dependent as defined by the same statistical time records. From a financial viewpoint, an interpretation of the fractional parameter as that relating to volatility has to be used carefully, as it points out to two interaction and dependent statistics: one underlying the series noise (the volatility) and the other “outliers” or jumps (see also Irwin [43], Barnett and Lewis [2] for a statistical measure of outliers by range statistics). Explicitly, let S ( i ) be the ith ordered statistic (largest) price in a time interval T (say trades within a day), then R ( T ) / σ T = ( S ( T ) − S ( 1 ) ) / σ T with σ T the underlying sample volatility may be used as a measure of the sample propensity of an outlier statistic. Neymann and Scott [70] introduced further the concept of outlier prone and outlier resistant families of distributions which are based on the ratio of two extremes of a sample and the range of that sample. Although the distributions of the range are difficult to compute [27] their distribution and their inverse distribution for random walks Brownian motion and birth and death processes have been computed explicitly (for example, [42,93–95]). Further approximate volatility estimates based and range statistics have profusely been used albeit with little success. Parkinson [76] (see also [31]) using a Feller’s result (1957) provides a volatility estimate given by: (14) σ ˆ 2 = 1 T n ( 4 ln 2 ) ∑ i = 1 n R i . Of course, when a time series underlying noise has a greater volatility, its range statistics will tend to increase as well since both are co-dependent. For example, using an intraday data set or a day data sets, will lead necessarily to different results. For these reasons, theoretical and practical financial models based on discrete time estimates (however small the time interval, since all financial time series are in fact discrete) ought to be far more concerned about the measurement of these series and their interpretation. For references on the range process and process volatility (see for example, [1,8,21,34,65–67]).

These extensive and simplistic developments of known results, were motivated by a common neglect of fractional finance in the finance literature which has emphasized its mathematical properties and less its financial implications. Their implications to finance are nonetheless important and potentially useful. Below, we consider their application to both a fractional Bond pricing model to highlight the equivalence and the differences between bond prices of different granularity and to pricing a granular Black–Scholes pricing model.

4. The risk free spot and fractional discounts

Say that the price of a bond is defined in terms of an agreed or legislated spot (continuous time) risk free rate given by R f ( t ) whose price at maturity, say time T, is B ( T ) . The risk free pricing model is thus: (15) d B ( t ) B ( t ) = R f ( t ) d t , d B ( t ) B ( t ) = B ( T ) > 0 or equivalently d ln B ( t ) = R f ( t ) d t . Its current price is thus: (16) 0 < B ( 0 ) = B ( T ) exp ( − ∫ 0 T R f ( τ ) d τ ) . In this case, the bond terminal value and legislated interest rate defines the current bond price. Consider instead a fractional bond defined by B H ( t ) = R f H ( t ) B ( t ) , B H ( T ) > 0 where R f H denotes a risk free discount rate corresponding to a fractional bond. Further, by definition, its fractional derivative is: B ( H ) ( t ) = d ( H ) B ( t ) / ( d t ) ( H ) (often expressed by D H ( B ) ( t ) ) and let B H ( T ) be its price at its maturity at T.

In this case, (17) d B H ( t ) B ( t ) = R f H ( t ) ( d t ) H , d B H ( t ) B ( t ) = B H ( T ) > 0 or d ln H ( B H ( t ) ) = R f H ( t ) ( d t ) H , d ln H ( B H ( t ) ) = B H ( T ) > 0 , where d ln H ( B H ( t ) ) is a fractional logarithmic function corresponding to the inverse of the Mittag–Leffler function E H ( . . ) defined by the polynomial E H ( h ) = ∑ i = 0 ∞ h i Γ ( H + i ) (note that E 1 ( h ) = e h ). Thus, (18) d ln H ( B H ( t ) ) = R f H ( t ) ( d t ) H , d ln H ( B H ( t ) ) = B H ( T ) > 0 or ∫ 0 t ln H ( B H ( t ) ) = ∫ 0 t R f H ( τ ) ( d τ ) H , ∫ 0 t ln H ( B H ( t ) ) = B H ( T ) > 0 . Since this is a deterministic function, and 0 < H < 1 , we have by the Riemann–Liouville function: (19) B H ( t ) = B H ( 0 ) E H ( ∫ 0 t ( t − τ ) H − 1 R f H ( τ ) d τ ) , 0 < H < 1 . And therefore at maturity, assuming that both the fractional and the non-fractional bonds have the same price B ( T ) , we have: (20) B H ( 0 ) = B ( T ) E H ( ∫ 0 T ( t − τ ) H − 1 R f H ( τ ) d τ ) , B H ( 0 ) = 0 < H < 1 and B 1 ( 0 ) = e − ∫ 0 T ( R f H ( τ ) d τ ) B ( T ) . Their prices are thus different although their ratio is defined uniquely by: B ( 0 ) B H ( 0 ) = e − ∫ 0 T R f ( τ ) d τ (21) × E H ( ∫ 0 T ( t − τ ) H − 1 R f H ( τ ) d τ ) . If both bonds are sold at the same price, then B H ( 0 ) = B ( 0 ) . Nonetheless, they need be unique since they can both have the same price but with spot rates that may differ as the equation below indicates: (22) ∫ 0 T R f ( τ ) d τ = ln { E H ( ∫ 0 T ( t − τ ) H − 1 R f H ( τ ) d τ ) } . For example, let the fractional spot rate be a constant R ¯ f H , then, Eq. (22) is reduced to: (23) ∫ 0 T R f ( τ ) d τ = ln { E H ( R ¯ f H T H ) } = ln ∑ i = 1 ∞ ( R ¯ f H T H ) i Γ ( H + i ) . Similarly, say that a bank borrows funds from the Fed with terms transferred to individual borrowers as fractional bonds. Arbitrage profits may then be made by the Banks, defined by the fractional interest charged consumers-borrowers providing a spread (granular premium) ln { E H ( R ¯ f H T H ) } − ∫ 0 T R f ( τ ) d τ ⩾ 0 .

Granularity arbitrage thus arises when a lender borrows funds on one set of (advantageous) terms and then lends these funds to other agents using a different time scale while maintaining apparently the same terms. Granularity and its implications to fixed income pricing and the calculation of fair interest rates settings provides therefore an arbitrage advantage to those capable to calculate the fair terms of exchange (see also Jumarie’s book [53] as well as the many references to his prior work in the book).

Consider instead an investor in risk free bonds planning to retire with a set amount B ( T ) at a date T. The investor’s plan is then defined by how much to invest initially at the spot rate and how much to add in savings S ( t ) over time. The latter defined by fractional investments at a rate ( d t ) H > d t , 0 < H < 1 . For simplicity, we assume that all investments have a spot risk free rate of return R f 1 on the outstanding bonds investment. This is reduced to the following mixed time scales fractional model defined by: (24) d B ( t ) = R f B ( t ) d t + S ( t ) ( d t ) H , B ( 0 ) > 0 , 0 < H < 1 , B ( T ) = B ¯ , where S ( t ) are savings made at the fractional rate. Its solution is a straightforward exercise that yields: (25) B ( T ) = B ( 0 ) e R f T + H e R f T E H ( Γ ( 1 + H ) × ∫ 0 T ( T − τ ) H S ( τ ) d τ ) , where B ( 0 ) is the initial investment and S ( τ ) is an investment schedule in 0 < τ < T . The proof is straightforward. We let y ( t ) = B ( t ) e − R f t then, elementary manipulations lead to: d y ( t ) y ( t ) = d B ( t ) B − R f d t or y ( t ) = B ( t ) e − R f t − B ( 0 ) which yields: d y ( t ) y ( t ) = S ( t ) ( d t ) H , y ( 0 ) = B ( 0 ) > 0 , 0 < H < 1 , y ( T ) = B ¯ e R f T and d H y ( t ) y ( t ) = Γ ( 1 + H ) S ( t ) ( d t ) H , y ( 0 ) = B ( 0 ) > 0 , 0 < α < 1 , y ( T ) = B ¯ e R f T . Thus, y ( t ) − y ( 0 ) = ∫ 0 y ( t ) d H y ( t ) y ( t ) = ln H ( y ( t ) ) − ln H ( y ( 0 ) ) and E H ( ln H ( y ( t ) ) − ln H y ( 0 ) ) = y ( t ) − y ( 0 ) .

Or y ( t ) = E H ( ∫ 0 y ( t ) d H ξ ( t ) ξ ( t ) ) = H E H ( Γ ( 1 + H ) ∫ 0 t ( t − τ ) H − 1 S ( τ ) d τ ) − y ( 0 ) and therefore, B ( t ) e − R f t − B ( 0 ) (26) = H E H ( Γ ( 1 + H ) ∫ 0 t ( t − τ ) H − 1 S ( τ ) d τ ) . And at time T, we have Eq. (25) as indicated above. Thus, given a target saving for time T, B ( T ) , the investment plan of the investor planning a fixed income investment with a rate of return R f for his retirement at time T is given in terms of the parameters { H , B ( 0 ) , S ( τ ) } . Say that S ( τ ) = S ¯ H is a constant investment schedule made at the rate ( d t ) H , then: B ( T ) = ( B ( 0 ) + H E H ( Γ ( 1 + H ) S ¯ H T H ) ) e R f T , (27) 0 < H < 1 , where H E H ( Γ ( 1 + H ) S ¯ H T H ) is the current sum of money the investor would have to pay initially to avoid future payments. Of course, if H = 1 , then: B ( T ) = ( B ( 0 ) + e S ¯ 1 T ) e R f T as expected.

When H > 1 we will obtain, as indicated earlier, results that differ from the case H < 1 treated above, essentially due to the definition of the fractional derivative as indicated in the previous section.

The definition of a risk free spot rate with respect to which risk free investments are made is thus dependent on the model granularity and is an important element to determine the price of complete market models. For fractional models, selecting such a rate is therefore more complex as it ought to be defined in terms of the fractional parameter of the model. These requirements alter necessarily the pricing model under an assumption of complete markets.

Below, we consider the lognormal pricing model commonly used in class rooms and in pricing options (for example, [10,69] and an extensive financial literature extending this model we have referred to). We first derive a complete financial market pricing model which serves as a reference model – with respect to which we value a price based on a fractional lognormal model. The result of such an analysis will provide a granularity premium price for the fractional and incomplete financial market model. The model we use differs from related papers in the fractional financial literature that sought to study and implement fractional models for asset prices and the mathematical conditions to their being no-arbitrage models. Instead, we view fractional financial models as computational models that need not be a financial pricing model.

5. The Black–Scholes model: Complete markets and fractional models

The financial lognormal model is defined by a stock rate of return μ and an adapted Brownian motion W ( t ) process whose variance is σ 2 t : (28) d S ( t ) S ( t ) = μ d t + σ d W ( t ) , S ( 0 ) = S 0 > 0 . To remove the risk implied by the Brownian motion, we price the risky asset relative to a risk free Bond whose interest rate is R f and pay an insurance-premium ( μ − R f ) / σ per unit volatility to “remove the risk consequences” of holding the risk asset. In this case, the risk premium of such an asset measured relative to the risk free rate of return R f is given by: ( μ − R f ) / σ , resulting in the following stock price process: (29) d S ( t ) S ( t ) = R f d t + σ d W Q ( t ) , d W Q ( t ) = [ d W ( t ) − μ − R f σ d t ] , where W Q ( t ) is a risk neutral Q-probability measure. The complete market financial pricing model is thus (under the Q probability measure) defined in expectation by: (30) S ( 0 ) = e − R f t E Q ( S ( t ) ) , where the Brownian Motion stochastic integral is interpreted in the sense of an Ito calculus and not in terms of a Stratonovich calculus [83]. As we shall see subsequently, applications of Stochastic integral definitions, will affect as well the fractional stochastic calculus applied. The Q probability measure accounts for the future price of risk by a risk premium and therefore under such a distribution, the stock price is riskless and its rate of return is a risk free rate and its price is an expected future price discounted at the risk free rate. In this model, the price of volatility risk has already been hedged by its risk premium μ − R f . In other words, the probability measure W Q ( t ) redefines for the investor an economic environment without financial risk consequences. This economic environment does not negate the occurrence of future random prices but renders an investor oblivious to their consequences and thus the asset price is by definition discounted by the risk free bond interest rate.

Financial pricing models are thus relative to an observed process such as a observed bond prices or to a reference portfolio it may be correlated to. Just as a risk neutral price is defined relative a risk free bond, a fractional financial pricing model may be measured relative to that of a unique and complete market pricing model. In other words, a fractional pricing model, is not a pricing model but a relative pricing model as it will, necessarily violate a number of assumption associated to financial complete models.

The introduction of granularity in a financial model has, as indicated previously, theoretical, computational, statistical and risk implications for financial modeling. [15–17] indicate that granularity introduces a potential for arbitrage and therefore markets are incomplete. Hu and Oksendahl [40] provided a solution to a fractional Black and Scholes model to be a pricing martingale and therefore with no formal arbitrage – although a martingale need to be a pricing model. Numerous papers have also provided numerous financial pricing applications (based on [22], seminal paper on fractional stochastic calculus), Elliott and Van der Hoek [26] provide a general fractional white-noise theory based on the Wicks–Ito–Skorohod (WIS) calculus applied to finance (see also Nualart and Taqqu [71]). Jumarie [45–47,50,51] applies and modifies the Riemann–Liouville derivative and fractional Taylor series to non-differentiable functions [48,56]. Financial pricing models are however “statistical” in the sense that futures are defined by constructing a complete probability distribution that defines future state prices. Their fractional transformation thus ought to result as well in a fractional probability distribution (while the Liouville kernel is essential a functional transformation). In other words, letting K H ( t ) be a fractional kernel, a fractional Brownian motion model W H ( t ) , may be defined relative to a non-fractional one W ( τ ) : (31) W H ( t ) = C H ∫ 0 t K H ( t , τ ) d W ( τ ) . The fractional kernel K H ( t , τ ) is said to define the FBM { W H ( t ) , t ⩾ 0 } , ∀ t ∈ ℜ + with: Pr { W H ( 0 ) = 0 } = 1 as a measurable random variable such that it has a null mean and a covariance given by: E { W H ( t ) W H ( τ ) } = σ H 2 C H ( t 2 H + τ 2 H − | t − τ | 2 H ) and a variance of the order C H t 2 H growing faster or slower than a linear time due to its fractional index. This behavior, as indicated above is due to the dependence of a current price on all previous prices that can either tamper or increase a stock linear time variance price. Using the Riemann–Liouville function for a fractional parameter 0 < H < 1 , [66] fractional kernel definition of a fractional Brownian motion is then defined by the convolution with a kernel: (32) K H ( t − τ ) = ( t − τ ) ( H − 1 ) Γ ( H + 1 ) and therefore , W H ( t ) = 1 Γ ( 2 H ) ∫ 0 t ( t − τ ) ( 2 H − 1 ) d W ( τ ) . When H = 1 / 2 the Brownian motion results W ( t ) . Note that at an infinite time, the variance is infinite and therefore the fractional distribution is not a normal probability distribution. Further, note that the fractional Brownian motion (FBM) results from a deterministic fractioning of the variance and not the normal probability distribution which is defined in the time interval ( − ∞ , + ∞ ) . However, over any finite time interval (with a compact support of the distribution) it is defined by a probability distribution [89]. Further, at the limit, it may be approximated by a Power Law. Explicitly, say that a Brownian motion is defined over a time interval Δ τ . Its fractional probability distribution is then a normal probability with a zero mean and a variance which is a function of the fractional index and its time interval. An intuitive proof is as follows. Let Δ W ( τ ) = W ( τ ) − W ( τ − Δ τ ) which are normally distributed random variables with mean zero and variance Δ τ . As a result, using an discrete and approximate Liouville kernel, we have (since 0 < H < 1 ): W H ( t ) = H ∑ i = 0 n [ ( n − i ) Δ τ ] ( H − 1 ) Δ ( W ( τ ) ) , (33) n = t Δ τ , τ = i Δ τ and W H ( t ) = H [ Δ ( W ( τ ) ) ] ∑ i = 0 n [ ( n − i ) Δ τ ] ( H − 1 ) = H [ Δ ( W ( τ ) ) ] (34) × [ Δ τ ] ( H − 1 ) ∑ i = 0 n [ ( n − i ) ] ( H − 1 ) . Assuming approximately that: (35) ∑ i = 0 n [ ( n − i ) ] ( H − 1 ) ≈ ∫ 0 n ( n − τ ) ( H − 1 ) d τ = n H / H . We obtain: (36) W H ( t ) = n H [ Δ τ ] ( H − 1 ) ( Δ ( W ( τ ) ) ) = t H [ Δ τ ] ( H − 1 ) Δ ( W ( τ ) ) ( Δ τ ) H and therefore, (37) W H ( t ) = t H Δ ( W ( τ ) ) Δ ( τ ) . Since Δ ( W ( τ ) ) Δ ( τ ) is a random normally distributed random variable with mean zero and variance 1, we have the mean and the variance for the fractional Brownian motion: (38) E ( W H ( t ) ) = 0 and var ( W H ( t ) ) = t 2 H . Note that when H = 1 / 2 , then: var ( W 1 ( t ) ) = t . Of course, when time tends to infinity, the variance tends to infinity non-linearly for all H > 1 / 2 .

Consider next the continuous time risk neutral probability measure for a lognormal pricing model with a fractional discount rate and a fractional Brownian motion: (39) d H S ( t ) S ( t ) = R f H ( d t ) H + σ H d W H Q ( t ) , S ( 0 ) = S 0 > 0 , where W H Q ( t ) is a fractional measure of the BM probability measure W Q ( t ) while R f H and σ H are two parameters associated to the FBM process. As indicated in the previous section, R f H is a function of the fractional index and the fraction-less risk free bond price (spot interest rate). We assume its solution to be of the form S H ( t ) = S 1 ( t ) S 2 ( t ) where (40) d H S 1 ( t ) S 1 ( t ) = R f H ( t ) ( d t ) H , S 1 ( 0 ) > 0 and d H S 2 ( t ) S 2 ( t ) = σ H d W H Q ( t ) , S 2 ( 0 ) > 0 . The first part is: (41) d ln H S 1 ( t ) − d ln H S 1 ( 0 ) = R f H ( t ) ( d t ) H and therefore (42) S 1 H ( t ) = S 1 ( 0 ) E H ( ∫ 0 t R f H ( τ ) ( d τ ) H ) . As stated earlier for a fractional bond, or, S 1 H ( t ) = S 1 ( 0 ) E H ( ∫ 0 t R f H ( τ ) ( d τ ) H ) . Its second part is the fractional stochastic differential equation: d H S 2 ( t ) S 2 ( t ) = σ H d W H Q ( t ) , (43) S ( 0 ) = S 0 > 0 or d H ln S 2 = ( − 1 2 ( σ H ) 2 ( d t ) 2 H ) (44) + σ H d W H ( t ) , 2 H > 1 . Consisting of two parts S 2 = S 2 , 1 S 2 , 2 . For S 2 , 2 ( t ) , we have: d H S 2 , 2 ( t ) S 2 , 2 ( t ) = d ln H ( S 2 , 2 ( t ) ) − d ln H ( S 2 , 2 ( 0 ) ) (45) = ( σ H ) ∫ 0 t d W H Q ( t ) or (46) S 2 , 2 H ( t ) = S 2 , 2 H ( 0 ) E H ( σ H W H Q ( t ) ) , where E H ( σ H W H Q ( t ) ) is the Mittag–Leffler function of σ H W H Q ( t ) , the polynomial sum of ( σ H W H Q ( t ) ) i , i = 0 , 1 , 2 , … which is not a normal probability distribution and therefore, S 2 , 2 H ( t ) is not a normally distributed random variable or the exponential of a normally distributed random variable (unless H = 1 / 2 , the reference Complete Market Probability Measure). For the deterministic part, S 2 , 1 ( t ) , we have: (47) ln 2 H S 2 , 1 2 H ( t ) − ln 2 H S 2 , 1 2 H ( 0 ) = ( − 1 2 ( σ H ) 2 2 H ∫ 0 t ( d t ) 2 H ) or S 2 , 1 2 H ( t ) (48) = S 2 , 1 2 H ( 0 ) E 2 H ( − 1 2 ( σ H ) 2 2 H ( t ) 2 H ) . In this case, S H ( t ) = S H ( 0 ) E H ( ∫ 0 t R f H ( τ ) ( d τ ) H ) × E 2 H ( − H ( σ H ) 2 ( t ) 2 H ) (49) × E H ( σ H W H Q ( t ) ) . And therefore the price of a fractional volatility Brownian motion lognormal pricing model S H ( t ) is defined by a Mittag–Leffler function of powered normally distributed random variables. Its price defined by its expectation of the distribution under the fractional risk neutral pricing measure (since it is based on both a risk free fractional bond price and therefore a fractional risk premium) is then: (50) S H ( 0 ) = E H Q ( S H ( t ) ) × { E H ( ∫ 0 t R f H ( τ ) ( d τ ) H ) × E 2 H ( − H ( σ H ) 2 ( t ) 2 H ) × E H ( σ H W H Q ( t ) ) } − 1 . Of course, when H = 1 / 2 , we obtain the risk neutral probability measure price, or: S 1 / 2 ( 0 ) = { E Q e − ∫ 0 t R f ( τ ) d τ − 1 2 ( σ ) 2 ( t ) + σ W Q ( t ) } × E Q ( S ( t ) ) (51) = e − ∫ 0 t R f ( τ ) d τ E Q ( S ( t ) ) . The price of granularity is therefore a price measured relative to that of a complete market pricing model and given by: S H ( 0 ) − S 1 / 2 ( 0 ) . This price is a function of the time scale fractional index H, expressing the mean returns trend as well as its volatility given by σ H ( d t ) H whose variance is σ H 2 ( d t ) 2 H in which case, the price of granularity (when 2 H > 1 ) will necessarily increase due to its volatility growth and tail risk due to the power laws implied in the Mittag–Leffler function summing powers of the fractional Brownian. Both the effects of volatility and tail risks can be verified by considering the fractional moment generating function of the fractional distribution.

6. Conclusion and discussion

Application of fractional calculus to financial models abound. Recently Rostek and Schöbel [80] provided a note on how to use and interpret fractional Brownian motion models for financial modeling. Prior and related studies include Mandelbrot and van Ness [66], Cheridito [15–17], Elliott and Van Der Hoek [26], Hu and Øksendal [40], Dung [23,24], Dung and Tien [25]. Bjork and Hult [9] uses the Wick product to derive a fractional approach to the Black–Scholes model. Such an approach however does not recognize the economic mechanism that renders a stochastic model, a stochastic “risk free” financial pricing model (since the risk consequences of the underlying price have been accounted for by an appropriate risk premium). Other approaches based on fractional models and their calculus are provided by Gu, Liang, Zhang [38], Jumarie [45,50], Meng and Wang [68], Rogers [78], Rostek [79], Sottinen and Valkeila [82]. These are mostly generalization of financial models to include the granularity implied in fractional models. This paper unlike previous approaches provided a simplistic presentation of pricing models based on their granularity and the price implied in granular models.

Financial models in general combine both stochastic and non-stochastic elements and their pricing consist therefore in how we account for their implied risk. For example, the risk premium paid for an asset’ volatility allows one to remove the consequences of the asset risk and thereby, price such an asset using a risk neutral probability measure. This paper recognized explicitly this risk and constructed a fractional model consistent with the price (the risk-premium, spread) of this risk. In this framework, unlike previous financial fractional models, a price is obtained relative to a complete market model. In other words, while a fractional model may be complete in a mathematical (Martingale) sense, it remains incomplete if another pricing model is considered to be the one to be complete. In this case, the fractional pricing model has a price which is measured relative to a complete financial pricing model which we chose to interpret as the price of granularity.

A fractional model can also be a complete markets price, in which case, all other financial fractional (or not) models may be priced relative to that complete market financial model. This presumption is reinforced by the fact that prices are relative. Not least of course are numerous papers by Mandelbrot [59], on the limits of arbitrage, in [60], on the covariance and R/S analysis, with Taqqu [65], on Robust R/S analysis of long run serial correlation, with Van Ness [66], on Fractional Brownian motion, with Wallis [62], on Noah, Joseph effects as well as in 1969 on computer experiments with fractional noises.

Although fractional models assume that model events’ time intervals are constant (although parametrized by a fractional parameter), extensions based on random time intervals (jumps) could be considered. Typically, Poisson Jump models assume that the time interval between events is exponential in which case, granularity over a number of integer time intervals has a Gamma distribution while over a non-integer it turns out to be of the fractional Poisson type. In this case, events (whether stochastic or not) do occur randomly with probability distributions that account for their “fractional randomness” [89], providing a far greater wealth of potential models that account for their real occurrence. For example, interest rate models do not vary continuously and may follow (often intractable) random and discrete time patterns. Continuous time models are in this case a convenience that has a price.

Applications of fractional models to finance and related problems abound. Cajueiro and Tabak [12] tested for time-varying long-range dependence of volatility for emerging markets. In 2007, they test the hypothesis that crude oil markets are weakly efficient over time. To do so a time-varying long-range dependence in prices and a volatility model was considered. Subsequently, in 2008, they tested for time-varying long-range dependence in real states equity returns. Ozdemir [75], uses a long run memory model to test the linkage between international stock markets. Willinger, Taqqu and Teverovsky [99] similarly studied stock market prices and long-range dependence, Xiao et al. [100] consider in a series of three papers problems spanning currency options in a fractional Brownian motion with jumps as well as European options with transaction costs using a fractional Brownian motion as well as a mixed fractional system (to obtain a “no-arbitrage” model). For earlier papers see for example [29], examining the dependency in intra-day stock index futures, [30], testing for long run memory in interest rate futures, [33], on Long memory relationships and the aggregation of dynamic models, [36], on long term dependence in common stock returns and in 1980, on long term dependence and least squares regression in investment analysis, [39], on Memory in commodity futures contracts, [57] on long term memory in stock market prices, and in 1997, on Fat tails, long memory and the stock market (a review paper since the 1960’s).

Ramirez et al. [77], consider a time varying fractional exponent Hurst index for US stock markets; Kumar and Nivedita [55], similarly consider the Multifractal properties of the Indian financial market, Gu et al. [38] consider a Time-changed geometric fractional Brownian motion and option pricing with transaction costs. By the same token, Gray et al. provide a generalized fractional processes, while Osler [73], provides generalized Taylor’s series for fractional derivatives and outlines as well some of their applications.

The essential intent of this paper, unlike previous contributions was meant to provide a fractional finance approach based on the fundamental principles we use in constructing financial complete market models. Therefore, is does not negate one or the other approach but emphasized their complementarity, based on their similarities and their differences, both of which are derived by the manner in which we define a financial models and treat time intervals to be larger or smaller, converging theoretically to a limit slower or faster.