IFRS 9 Measurement of Financial Instruments 2018: Jameel’s Non- Normal Brownian Motion Models are Indeed IFRS 9 Complaint Models

The measurement of Financial Instruments under IFRS 9 requires the incorporation of forward-looking information and Economic forecasts of the future macroeconomic scenarios into the existing Accounting, Banking and Economic Models. In this paper, the author considered Geometric Brownian Motion, Biagin, Cox-IngersollRoss, Ornstein-Uhlenbeckprocess, Vasicek, Black-Karasinki, Chen, Kalotay-Williams-Fabozzi, LongstaffSchwatz, Ho-Lee, Hull and White, and Black-Derman-Toy Models for Pricing Stocks, Bitcoin, Indexes, ETFs, and Leveraged ETFs, Bonds, Interest Rate Movements, Caps, Floors, European Swaptions, and Bond Options thereby incorporating forward-looking information {WJB(t)} satisfying Jameel’s Criterion and Geometric average of only positive Economic forecasts of the future Macroeconomic parameters {(μA) and(σA)} using Jameel’s Contractional-Expansional Stress Methods and Jameel’s substitutions {(μA ± σAWJB(t))} , μA is POSITIVE INFINITESIMAL, σA ≥ 1 and define σA as Geometric Volatility of only positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters and μA as Geometric Means ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters. The paper replaces the Wiener Process {W(t): t ≥ 0} in the existing models with the following proposed NON-NORMAL STRESS CONDITIONS: (i) {(μA ± σAWJB(t))}, if σA > 1 , μA is positive infinitesimal; (ii) {(μA ± WJB(t))} , if σA = 1 , μA is positive infinitesimal; (iii) {(±σAWJB(t))}, ifσA > 1, μA = 0, and; (iv) {(±WJB(t))}, ifσA = 1, μA = 0. The paper tested the performances of only proposed STOCKS stressed closed form models using Chevron Corporation (CVX) Stock data extracted from yahoo finance, time series from 2014 – 1991. The results were fascinatingly interesting, impressive, viable and reliable, sophisticated, and complaint with IFRS 9 since they incorporated forward-looking information and Economic forecasts of the future macroeconomic parameters thereby minimizing the differences between market prices and models prices.


Introduction
The IASB in July, 2014 issued the final version of IFRS 9 Measurement of Financial Instruments beginning on or after 1 st January, 2018 with early adoption permitted.It replaces IAS 39 Financial Instruments: Recognition and Measurement.The major target of accounting standards is to provide financial information that stake-holders would find useful when making decisions.The most challenging aspects required by IFRS 9 are the treatment on Published by IDEAS SPREAD

Stochastic Process
A stochastic (uncertainty) process can be defined as a Mathematical object usually described as a collection of random variables or can be defined as numerical values of some system randomly changing over time, for instance movement of a gas molecule.

Random Walk
A cornerstone of the theory of stochastic processes is called a Random Walk

Normal Distribution
(a) The density of the normal distribution is expresses in the following way: , if and , one calls this density of the standard normal distribution.
(b) The distribution function of the standard normal distribution is expressed in the following manner: (c) If the random variable is normally distributed with parameters and , one write .A normally distributed random variable can adopt the values from the entire and it is true that: Expected Values: and the Variance : .

Brownian Movement
The stochastic process which is designated as the Brownian movement or also the Wiener Process.
It is determined by the fact that is NORMALLY DISTRIBUTED random variable with expected value zero and variance , therefore it is true that: .The Geometric Brownian Motion (GBM) assumes that stock prices are Log-Normally Distributed with a mean of the certain component and a standard deviation of the uncertain component that .Suppose that  is a Stochastic Process satisfying the Stochastic Differential Equation (SDE) given by: , where the last integral is defined as an Ito Integral.Such a process  is often called as Ito Process.Note that the process  appears on both sides of the above equation, but the value at  given on the left depends only on the values at times s and  ≤ .Assuming that  has continuous paths, it suffices to know () for all  <  on the right.Nevertheless, there is a need for supporting theory (which has been developed) about the existence and uniqueness of solution to the integral (or equivalently the SDE).

Stocks Pricing for IFRS 9 Compliance
Let define Brownian movement (Motion).If some quantities are constantly undergoing small, random fluctuations then we say it is undergoing a Brownian motion or in Physics can be defined as a random movement of particles in a fluid due to their collision with other atoms or molecules.This can be expressed as: ; , , ; , , 0 (a) Difficulties in identifying the right tail distribution (process) whether to use power-type or exponential-type distributions; (b) The stable distributions generalize normal distribution; (c) Geometric Brownian Motion (GBM) can only be used to forecast maximum of two weeks closing prices; (d) Geometric Brownian Motion (GBM) does not include cyclical or seasonal effects; and ( ) ( ) .

Ornstein -Uhlenbeck Process for IFRS 9 Compliance
Ornstein -Uhlenbeck Process can be used to model Interest Rates, Currency Exchange Rates, and Commodity Prices stochastically.It can also be used in Trading Strategy known as PAIRS TRADE.An Ornstein -Uhlenbeck Process,   , satisfies the following stochastic differential equation:   = (µ −   ) +   , where  > 0, µ and  > 0 are parameters and   denotes the Weiner process.µ is the mean value supported by fundamentals,  is the degree of volatility aroundit caused by shocks,  is the rate by which these shocks dissipate and variable reverts towards the mean.
The Closed Form Solution of Ornstein -Uhlenbeck Process is given by: ; where the parameters and are constants.The volatility of the short rate is assumed constant across time periods as in the Ho-Lee Model.
2.2.17 Black-Derman-Toy(1990) Model for IFRS 9 Compliance The Black-Derman-Toy (1990) Model term structure model, unlike the previous models discussed, assumes that the short rate distribution is lognormal instead of normal and is given by: , where satisfied Jameel's Criterion, is the volatility at time .is the natural logarithm of the short rate, and is the time varying drift parameter.

Heston Volatility Model for IFRS 9 Compliance
Heston Model is a financial model use to describe the evolution of the volatility of an underlying asset.The model assumes that  , the price of the asset is determine by a stochastic process:   = µ   + √       , where   is the instantaneous variance and given by:   = ( −   ) + √     and    ,    are Wiener Processes with correlation ρ or equivalently, with variance ρdt.Where, µ is the rate of return of the asset, θ is the long variance or long run average price variance as  tends to infinity, the expected value of   tends to θ,  is the rate at which   reverts to θ,  is the volatility of the volatility or Vol of Vol and determines the variance of   .If the parameters obey the following condition (known as the feller condition) then the process   is strictly positive that 2θ >  2 .

ETFs and Leveraged ETFs Pricing for IFRS 9 Compliance
The dynamics of ETF using stochastic calculus can be written as: While, the evolution of the underlying index (  ) ≥0 of Leveraged ETF is given by a Geometric Brownian Motion (GBM):   =   (µ +   ), where W is a standard Brownian Motion under the historical measure.µ is the ex-dividend annualized growth rate and  > 0 is the constant volatility.Thus, the dynamics of Leveraged ETF using stochastic calculus can be written as: LETF: 2.2.20 Propose Jameel's Stressed Closed FormModels presented from 2.2.8 to 2.2.19.for IFRS 9 Compliance Generally, using Jameel's Criterion and Jameel's Contractional-Expansional Stressed Methods, we replaces the WIENER PROCESSES (NORMAL and or LOG-NORMAL) terms appears in the CLOSED FORM SOLUTIONS of Ornstein -Uhlenbeck Process, Cox-Ingersoll-Ross (1985) Model, Vasicek Model, Black-Karasinki (1991) Model, Chen (1994) Model, Kalotay -Williams -Fabozzi (1993) Model, Longstaff -Schwatz (1992) Lee Model (1986) Model, Hull-White (1990) Model, Black-Derman-Toy(1990) Model, Heston Volatility Model andETFs and Leveraged ETFs Models by JAMEEL'S SUBSTITUTIONS FOR IFRS 9 COMPLIANCE as : (i)

Results
To test the performances of the proposed Sixteen ( 16 where B is Brownian Motion, by which we mean () = ((), ) + ((), )(), where a and b are real-valued functions on  2 , by which we mean the  satisfies the integral equation: (

Figure 1 .
Figure 1.Jameel's Contractional-Expansional Stressed Methods 0 , and; (iv) {(±  ())}, if  = 1, µ  = 0. Define   as Geometric Volatility ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters and µ  as Geometric Means ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters, { () } is a Non-Normal Brownian Motion variable fat-tail stochastic probability distribution of the considered Financial Instrument Return Satisfying Jameel's Criterion then we have the following Propose Jameel's Stressed Closed Form Stocks Pricing Models TYPESfor IFRS ())  =  0 exp (µ +  (µ  ±     ())) ,whenever   > 1, µ  is positive infinitesimal; TYPE 2: (  ())  =  0 exp (µ +  (µ  ±   ())),whenever   = 1, µ  is positive infinitesimal; TYPE 3: (  ())  =  0 exp (µ +  (±    ())) ,whenever   > 1, µ  = 0; TYPE 4: (  ())  =  0 exp (µ +  (±  ())) ,whenever  = 1, µ  = 0; 2.2.4 Indexes Pricing for IFRS 9 Compliance (b) Prediction of Future Market Indexes: as in (a) above, the result can be extended to predict future market indexes prices for instance S&P500 (composed of 118 companies from NASDAQ and 382 companies from NYSE), Wilshire 5000, NASDAQ composite Index, Russell 2000, Bottom Line, Nikkei 225, FTSE 100, S&P 100, Dow Figure 2. Jameel's Transformational Diagram for IFRS 9 Compliance 0 , and; (iv) {(±  ())} , if   = 1 , µ  = 0 , where µ  is POSITIVE INFINITESIMAL,   ≥ 1 and define   as Geometric Volatility ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters and µ  as Geometric Means ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters, { () } is a Non-Normal Brownian Motion variable fat-tail stochastic probability distribution of the considered Financial Instrument Return Satisfying Jameel's Criterion then we have the following Propose Jameel's Stressed Closed Form Bitcoin Pricing Models TYPESfor IFRS 0, and; (iv) {(±  ())}, if  = 1, µ  = 0, whereµ  is POSITIVE INFINITESIMAL,   ≥ 1 and define   as Geometric Volatility ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters and µ  as Geometric Means ofonly positive Arithmetic Means of the Underlying Asset Return and Returns of the future economic forecasts of macroeconomic parameters, { () } is a Non-Normal Brownian Motion variable fat-tail stochastic probability distribution of the considered Financial Instrument Return Satisfying Jameel's Criterion to obtain their Stressed Closed Models TYPESfor IFRS 9 Compliance.
) Jameel's Stressed Closed Form Solutions considering Stocks Geometric Brownian Model, the Author considered Chevron Corporation (CVX) Stock data extracted from yahoo finance using Time Series from 2014 -1991.Thus, the data distribution Mean equal 0.000326, Standard Deviation equal 0.015761, the Annual drift of the year preceding 2014 (2013) equal 0.000466 and the Annual Volatility of the year preceding 2014 (2013) equal 0.008325.Hence, µ  = 0 Figure 5.

Criterion Enhancement Axiom:Thode (2012) intensively discussed about the Best Goodness of Fit Tests
vii.Last Unit Axiom: let be such that it satisfied axioms (i) to (iv).Let be the ranks of fitness test of obtained from the tests respectively then if , regardless of the Time Series, Company and so on.Consequently, if for all fitness test runs, turn out to be the ( ) Published by IDEAS SPREAD same then the PREDICTED PRICE PATH will finitely coincides many times with the REAL PRICE PATH of the stock under consideration.2.1.9Top Fat-Tailed Probability Functions using Jameel's Criterion as of 2015 Using Jameel's Criterion, Jamilu (2015) considered Eleven (11) out of Fifty (50) World's Biggest Public Companies by FORBES as of 2015 Ranking regardless of the platform in which they are listed, Number of the Research Companies, Time Series (Short or Long), Old or Recently listed Companies using the time series from 2014 -2009 with the aim of finding the Best Fitted Fat -Tailed Stocks Probability Distributions.However, in this research paper, the Author considered Top Two (2) and 4th Stocks Fat-Tailed Probability Functions thereby comparing the performances of the Proposed Jameel's Stressed Closed Form Prices, Normal (Standard Brownian Motion) Prices with Market (Real) Prices as shown below: by IDEAS SPREAD (e) Geometric Brownian Motion (GBM) does not account for periods of constant values, they observed periods where prices stay on the same level, particularly true for asset with low liquidity.Also, Levy processes provide a natural generalization of the sum of independent and identically distributed (iid) random variables.The simplest possible levy processes are the standard Brownian motion tail distributions although, as some people favor power-type distributions other exponential-type distribution, although as pointed out by Kou (2002, P.1090), the power-type right tails cannot be use in models with continuous compounding as they lead to infinite expectation for the asset price.In view of the foregoing, we be a Fat-Tail Stochastic or Random Probability Function satisfied JAMEEL'S CRITERION then the NON-NORMAL BROWNIAN MOTION STOCK PRICE can be expressed as: are adaptive parameters and the same as in the Normal Model above.
that this is a sum of deterministic terms and an integral of a Model (CIR) describes Interest Rate Movements as driven by only one source of market risk and Interest Rate Derivatives.Also, under the no-arbitrage assumption, a BOND could be priced using this interest rate process and to Price Default Free Zero-Coupon Bonds.The CIR Model is given by the following stochastic differential equation:   = ( −   ) + √    , where   is a Wiener Process (modeling the random walk market risk factor) and a, b and  are the parameters.The parameter a corresponds to the speed of adjustment, b, the mean and  is Published by IDEAS SPREAD the expected change, or drift, in the short rate and a stochastic term which models the random component (volatility) of the short rate.The parameter is the volatility of the short rate and it is assumed to be constant, that is it does not change with time.

Table 1 .
CVX Stressed Prices with Log-Logistic (3P) compared with Real and Normal Prices Published by IDEAS SPREAD The Author uses  0 = 108.87 as of 11/28/2014 (a day before 12/1/2014) as the Initial Stock Price with intention to Predict Twenty One (21) working days (from 12/1/ 2014 to 12/30/ 2014) Chevron Corporation (CVX) Stock Prices thereby comparing the REAL PRICES, NORMAL PRICES with that ofSIXTEEN (16) PROPOSED JAMEEL'S STRESSED CLOSED FORM PRICESusing Top two and 4th fat-tailed Non-Normal Probability Distribution Functions satisfied Jameel's Criterion and are thus: LOG-LOGISTIC (3P), CAUCHY and BURR(4P).The Author performs the PREDICTION Using MICROSOFT EXCEL and obtained the following RESULTS as shown in Tablesand Charts below:Note that in Table1, the notation LL (3P)   (+  −),  = , , , means Positive or Negative Jameel's Stressed Closed Form Prices TYPES 1 to 4 with respect to LOG-LOGISTIC (3P), in Table2, the notation Cauchy   (+  −),  = , , , means Positive or Negative Jameel's Stressed Closed Form Prices TYPES 1 to 4 with respect to CAUCHY, InTable 3, the notation Burr (4P)   (+  −),  = , , , means Positive or Negative Jameel's Stressed Closed Form Prices TYPES 1 to 4 with respect to BURR (4P).Table 2. CVX Stressed Prices with Cauchy compared with Real and Normal Prices

Table 3 .
CVX Stressed Prices with Burr (4P) compared with Real and Normal Prices