Volatility, risk modeling and utility

Applied Mathematics and Computation 182 (2006) 1749–1754 www.elsevier.com/locate/amc

Volatility, risk modeling and utility Michael Nwogugu P.O. Box 170002, Brooklyn, NY 11217, USA

Abstract This article develops the foundations for several new models of risk and volatility. Methods used include utility analysis and mathematical psychology. Risk and volatility can be modeled as aggregations of preferences of market participants, and or optimization problems. Market risk assessment is a multi-criteria process that typically requires information processing, and thus cannot be defined accurately by rigid quantitative models. Existing market-risk models (GARCH, ARMA, S-V, VAR, etc.) are inaccurate. Its possible to quantify elements of trading dynamics and market psychology, and hence develop more accurate risk models. Areas for further research include: (a) development of dynamic market-risk models that incorporate psychology, more elements of trading dynamics, knowledge differences, information, and trading rules in each market; and (b) further development of concepts in belief systems.  2006 Elsevier Inc. All rights reserved. Keywords: Decision-making; Complexity; Risk and uncertainty; Artificial intelligence; Psychology; Non-linear dynamics

1. Introduction This article critiques GARCH/ARC/VAR models and develops the foundations for several new models of risk and volatility. The article builds on work in [44,45], Nwogugu (2005c). The relevant literature on volatility modeling include: ([40,14,42,23,25,26,8,9,60]; Janssen & Jager (2000); [38,24,29,7]; Gabaix, Gopikrishnan, Plerou & Stanley (2003); [39]; Benartzi & Thaler (1995); [43,31,46,49,36,6,30,2,1,3–5,10–12,15–20]; Haumer et al. (2002); [21,22,27,28,32–35,37,41,47,48,50–59]). 2. Volatility, risk modeling and utility Asset volatility can be modeled in various ways. 2.1. Method one In summary, volatility (of an asset or a market) can be modeled as the sum of all preferences of market participants over time. This is because volatility arises from prices changes which are a manifestation of E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter  2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.01.079

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changes in investor utilities, investor psychology and investor wealth, which in turn are manifestations of news and information processing capabilities. (Brocas and Carrillo, 2000); [13,49]; (Gonzalez-Rivera, Lee & Mishra, 2004). [6]; (Dreman and Lufkin, 2000); [31]; (Caginalp, Porter & Smith, 2000; Sorkin, Hays & West, 2001; McKenzie & Faff, 2003). These investor preferences can be expressed in terms of utilities. Thus, Z þ1 Z T X U i oP oT ; V ¼ 1

t

U i ¼ f ðm1 ; m2 ; . . . ; mn Þ; oV =oT 6¼ 0; oU =omi 6¼ 0; omi =oP 6¼ 0; omi =oP P oU i =oP ; 1 < V < 1; where: V T P Ui

Volatility in time horizon T Time horizon Price of asset Utility for market participant i

mi = forecasted variable i. Such variables include: L = liquidity (L = FT * VPT); PR = price, MS = Market Sentiment; IR = interest rates; IN = inflation; CER = currency exchange rates; PS = propensity to substitute assets in the same market; TC = transaction costs; IC = information processing capability; V = estimated volume of trading in the underlying asset in the period t; M = volume of margin debt; AC = available cash (cash in securities accounts at various brokerages and banks); ACI = cash held by institutions; AMC = amount of available margin credit; T = treasury market activity (measured in terms of currency/dollar volume of trades); FT = frequency of trading – average number of completed trades per unit of time (in period t); TR = trade arrival rates (estimated number of orders per unit of time); VPT = volume per trade (estimated average dollar volume per executed trade); IS = investor sentiment – psychology; N = estimated number of market participants; L = leverage ratios for the industry (in which the subject company operates in) or the asset; ICR = index of credit ratings in the industry, BD = bid-asked spread, RA = ‘rate of agreement’ among parties to trades, etc. Its assumed that the forecasted variable m1 represents the average of all investors utilities from m1, P e(m1) = ( m1)/N. • V is not sub-additive. • 1 6 V 6 1, but generally, 20 6 V 6 20. The negative and positive signs indicate direction of volatility. • V is not symmetrical about 0. That is, jVaj 5 jVaj. Its assumed that the market consists of three types of traders – professional traders (P), Institutional traders (II) and retail investors (RI). Each of these three groups react to information in similar ways. The activities of each group create utility U(P), U(II) and U(RI) which are all measures of value of volatility. In order to calculate forecasted volatility during time t, forecasts of each variable m1 for each group are prepared. The forecasted value of a variable mi for a group G, during time t, referred to as mi,G,t is then converted into a score on a scale. The scores for all m1,G are converted added to obtain U(P), U(II) and U(RI) – that is, X X X U ðPÞ ¼ mi;P;t ; U ðIIÞ ¼ mi;II;t ; and U ðRIÞ ¼ mi;RI;t :  1 6 ½V ¼ U ðPÞ þ U ðIIÞ þ U ðRIÞ 6 þ1;  1 6 U ðPÞ 6 1; 1 6 U ðIIÞ 6 1; 1 6 U ðRIÞ 6 þ1; U ðPÞ 6¼ 0;

U ðIIÞ 6¼ 0;

U ðRIÞ 6¼ 0:

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This implies that combinations of U(P) + U(II) + U(RI) are not linear, because the change in the mix of types traders immediately changes market psychology and volatility. If U(P) = x, and U(II) = x, and U(RI) = x, then U ðPÞ þ U ðIIÞ þ U ðRIÞ 6¼ 3x: Hence,

n p 3

 p p U ðPÞ þ 3 U ðIIÞ þ 3 U ðRIÞ ; ½ðxÞU ðPÞ þ ðyÞU ðIIÞ hp io 3 3 3 þ ðzÞU ðRIÞ; 3 U ðPÞ þ U ðIIÞ þ U ðRIÞ :

U ðPÞ þ U ðIIÞ þ U ðRIÞ ¼ Max

where: 1 6 x 6 +1; 1 6 y 6 +1; 1 6 z 6 +1; p p And generally, U a þ U b ¼ Maxf½2 U a þ 2 U b Þ; ½ðxÞU a þ ðyÞU b g. Thus, in this model, volatility is a direct function of the number of each type of traders in the market, and the proportion of each group of traders relative to the total number of traders. 2.2. Method two Volatility can also be obtained using simulations/optimization. Volatility can be modeled as the percentage rate that simultaneously minimizes (or maximizes) the pre-specified deviations from the projected average of each variable mi over a period T, where each market participant’s utility Ui is a direct function of various variables – i.e. U i ¼ f ðm1 ; m2 ; . . . ; mn Þ: The cutoff deviation for each mI will vary across assets and can be determined by forecasts of each variable and analysis of future and historical trends. The objective function is: Min Dm1 ; Dm2 ; . . . ; Dmn ; s:t: Dm1 ; Dm2 ; . . . ; Dmn 6¼ 0; m1 ; m2 ; . . . ; mn ; are real numbers: V T P Ui

Volatility in time T Time horizon Price of asset Utility for market participant i

mn = forecasted variables such as liquidity, price, sentiment, interest rates, inflation, currency exchange, coefficient of variation, rates, propensity to substitute, transaction costs, information processing capability, volume, margin, available cash (cash in securities accounts at various brokerages and banks), treasury market activity, frequency of trading, investor psychology, number of market participants, investor sentiment, market sentiment, etc. 1 < V < 1: The underlying theory is that each variable has a ‘steady-state’, and is an indicator, and any positive or negative deviations from the ‘steady-state’ of a certain magnitude triggers trading activity which in turn creates volatility. This approach will produce estimated minimum and maximum volatilities (Koopman and Bos, 2004; Desroches and Gosselin, 2004; Najand, 2002); [6,46].

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2.3. Method three Volatility can be modeled by calculating the minimum percentage rate change in the price of the asset that simultaneously causes a change of v standard deviations in the magnitude of all variables (mi, . . . , mn) over a period T, (or alternatively, simultaneously causes a change of c in the coefficient of variation of all variables – mi, . . . , mn – over a period T), where each market participant’s utility Ui is a direct function of various variables. Each variable mi must have a minimum pre-specified positive correlation (e.g. 70%) with the price of the underlying asset. v could be constant for all variables, or v could vary for all variables. The relevant vi for variable mi can be determined by calculating the standard deviation of variable mi. Volatility can be also calculated by simulating the percentage rate change in the price of the asset that simultaneously causes a change of v standard deviations in the magnitude of all variables (mi, . . . , mn) over a period T, where each market participant’s utility Ui is a direct function of various variables. Each variable mi must have a correlation of more than 70% with the price of the underlying asset. v could be constant for all variables. Alternatively, v could vary for all variables, and the relevant vi for variable mi can be determined by calculating the standard deviation of variable mi. 3. Conclusion Prospect theory, Cumulative prospect theory, Value-at-risk, GARCH/ARCH models are conceptually the same as expected utility theory. VAR/GARCH/ARCH models are structurally deficient and static. These models do not represent actual patterns of risk or decision-making in finance and markets. The coefficient of variation (of the first differences of the price series) is a better measure of risk than standard deviation. References [1] C. Acerbi, Spectral measures of risk: a coherent representation of subjective risk aversion, Journal of Banking and Finance 26 (5) (2002) 1505–1518. [2] V. Alanagar, R. Bhar, An international study of causality-in-variance: Interest rate and financial sector returns, Journal of Economics and Finance 27 (1) (2003) 39–56. [3] T. Andersen, T. Bollerslev, Heterogenous information arrivals and return volatility dynamics: Uncovering the long run in high frequency returns, Journal of Finance 52 (3) (1997) 975–995. [4] A. Atkins, E. Dyl, Transaction costs and hedging periods from common stocks, Journal of Finance 52 (1) (1997) 309–326. [5] C. Atkinson, M. Papakokkinou, Theory of optimal consumption and portfolio selection under a capital-at-risk (CaR) and a value-atrisk (VaR) constraint, IMA Journal of Management Mathematics 16 (1) (2005) 37. [6] K. Au, F. Chan, D. Wang, I. Vertinsky, Mood in foreign exchange trading: Cognitive processes and performance, Organizational Behavior and Human Decision Processes 91 (2) (2003) 322–338. [7] T. Bali, An extreme value approach to estimating volatility and value at risk, Journal of Business 76 (2003) 83–108. [8] N. Barberis, A . Shleifer, R. Vishny, A model of investor sentiment, Journal of Financial Economics XLIX (1998) 307–343. [9] N. Barberis, A. Shleifer, R. Vishny, Percolation models of financial markets, Advances in Complex Systems 4 (1) (1998) 19–27. [10] T. Barnhill, W. Maxwell, Modeling correlated market and credit risk in fixed income portfolios, Journal of Banking and Finance 26 (2002) 347–374. [11] R. Batchelor, I. Orakcioglu, Event-related GARCH: The impact of stock dividends in Turkey, Applied Financial Economics 13 (4) (2003) 295–305. [12] N. Bushueva, Finance Without Price Dynamics, PhD Thesis, Department of Mathematics, MIT, 2003. [13] B. Caillaud, B. Jullien, Modeling time-inconsistent preferences, European Economic Review 44 (2000) 1116–1124. [14] G. Caporale, N. Spittis, N. Spagnalo, Testing for causality-in-variance: An application to the east Asian markets, International Journal of Finance and Economics 7 (2002) 235–245. [15] M. Chernov, R. Gallant, E. Ghysels, G. Tauchen, Alternative models for stock price dynamics, Journal of Econometrics 116 (12) (2003) 225–235. [16] A. Christie, Stochastic behavior of common stock variances: Value, leverage and interest rate effects, Journal of Financial Economics 10 (1982) 407–432. [17] L. Cox, Mathematical Foundation of Risk Measurement, PhD Thesis, MIT Operations Research Center, June 1986. [18] K. Daniel, S. Titman, Evidence in the characteristics of cross-sectional variations in stock returns, Journal of Finance 52 (1) (1997) 1– 34. [19] R. Dawes, The prediction of the future versus an understanding of the past: A basic asymmetry, American Journal of Psychology 106 (1) (1993) 1–24.

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