Market memory and fat tail consequences in option pricing on the expOU stochastic volatility model

The expOU stochastic volatility model is capable of reproducing fairly well most important statistical properties of financial markets daily data. Among them, the presence of multiple time scales in the volatility autocorrelation is perhaps the most relevant which makes appear fat tails in the return distributions. This paper wants to go further on with the expOU model we have studied in Ref. [J. Masoliver, J. Perelló, Quant. Finance 6 (2006) 423] by exploring an aspect of practical interest. Having as a benchmark the parameters estimated from the Dow Jones daily data, we want to compute the price for the European option. This is actually done by Monte Carlo, running a large number of simulations. Our main interest is to “see” the effects of a long-range market memory from our expOU model in its subsequent European call option. We pay attention to the effects of the existence of a broad range of time scales in the volatility. We find that a richer set of time scales brings the price of the option higher. This appears in clear contrast to the presence of memory in the price itself which makes the price of the option cheaper.     

Measuring statistical evenness: A panoramic overview

Motivated by the question “how equal is the distribution of wealth within a given human population?” economics devised an impressive toolbox of quantitative measures of societal egalitarianism including the Lorenz curve and the following indices: Gini, Pietra, Hoover, Amato, Hirschman, Theil and Atkinson. These quantitative measures-considered in the broader context of general data-sets with positive values-are, in effect, general gauges of statistical evenness. While the application of Gini’s index grew beyond economics and reached diverse fields of science, the aforementioned “evenness toolbox” has largely remained within the confines of the social sciences. The aim of this Paper is to expose this “evenness toolbox” to the physics community by presenting a comprehensive evenness-based approach to a fundamental problem in science—the measurement of statistical heterogeneity.     

A unified model for price return distributions used in econophysics

For a decade, a new theoretical movement called “econophysics” has been initiated by some physicists who began to publish articles devoted to the study of economic and financial phenomena. Since then, econophysicists have written a very prolific literature about the way of characterizing the evolution of financial prices. Today, there is an “extreme diversity” of models recently developed by econophysicists whose research is sometimes presented as an ill-defined field. The objective of this paper is precisely to provide a unified framework in order to contribute to unify econophysics and to base this new field on shared scientific standards.     

Non-parametric extraction of implied asset price distributions

We present a fully non-parametric method for extracting risk neutral densities (RNDs) from observed option prices. The aim is to obtain a continuous, smooth, monotonic, and convex pricing function that is twice differentiable. Thus, irregularities such as negative probabilities that afflict many existing RND estimation techniques are reduced. Our method employs neural networks to obtain a smoothed pricing function, and a central finite difference approximation to the second derivative to extract the required gradients.This novel technique was successfully applied to a large set of FTSE 100 daily European exercise (ESX) put options data and as an Ansatz to the corresponding set of American exercise (SEI) put options. The results of paired t-tests showed significant differences between RNDs extracted from ESX and SEI option data, reflecting the distorting impact of early exercise possibility for the latter. In particular, the results for skewness and kurtosis suggested different shapes for the RNDs implied by the two types of put options. However, both ESX and SEI data gave an unbiased estimate of the realised FTSE 100 closing prices on the options’ expiration date. We confirmed that estimates of volatility from the RNDs of both types of option were biased estimates of the realised volatility at expiration, but less so than the LIFFE tabulated at-the-money implied volatility.     

Accuracy-Risk Trade-Off Due to Social Learning in Crowd-Sourced Financial Predictions

A critical question relevant to the increasing importance of crowd-sourced-based finance is how to optimize collective information processing and decision-making. Here, we investigate an often under-studied aspect of the performance of online traders: beyond focusing on just accuracy, what gives rise to the trade-off between risk and accuracy at the collective level? Answers to this question will lead to designing and deploying more effective crowd-sourced financial platforms and to minimizing issues stemming from risk such as implied volatility. To investigate this trade-off, we conducted a large online Wisdom of the Crowd study where 2037 participants predicted the prices of real financial assets (S&P 500, WTI Oil and Gold prices). Using the data collected, we modeled the belief update process of participants using models inspired by Bayesian models of cognition. We show that subsets of predictions chosen based on their belief update strategies lie on a Pareto frontier between accuracy and risk, mediated by social learning. We also observe that social learning led to superior accuracy during one of our rounds that occurred during the high market uncertainty of the Brexit vote.     

Idiosyncratic risk in the Dow Jones Eurostoxx50 Index

Recent evidence by Campbell et al. [J.Y. Campbell, M. Lettau B.G. Malkiel, Y. Xu, Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk, The Journal of Finance (February) (2001)] shows an increase in firm-level volatility and a decline of the correlation among stock returns in the US. In relation to the Euro-Area stock markets, we find that both aggregate firm-level volatility and average stock market correlation have trended upwards.We estimate a linear model of the market risk–return relationship nested in an EGARCH(1, 1)-M model for conditional second moments. We then show that traditional estimates of the conditional risk–return relationship, that use ex-post excess-returns as the conditioning information set, lead to joint tests of the theoretical model (usually the ICAPM) and of the Efficient Market Hypothesis in its strong form.To overcome this problem we propose alternative measures of expected market risk based on implied volatility extracted from traded option prices and we discuss the conditions under which implied volatility depends solely on expected risk. We then regress market excess-returns on lagged market implied variance computed from implied market volatility to estimate the relationship between expected market excess-returns and expected market risk.We investigate whether, as predicted by the ICAPM, the expected market risk is the main factor in explaining the market risk premium and the latter is independent of aggregate idiosyncratic risk.     

A contribution to the systematics of stochastic volatility models

We systematically compare several classes of stochastic volatility models of stock market fluctuations. We show that the long-time return distribution is either Gaussian or develops a power-law tail, while the short-time return distribution has generically a stretched-exponential form, but can also assume an algebraic decay, in the family of models which we call “GARCH” type. The intermediate regime is found in the exponential Ornstein-Uhlenbeck process. We also calculate the decay of the autocorrelation function of volatility.     

Vector financial rogue waves

The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black-Scholes option pricing model. In this Letter, we analytically propose vector financial rogue waves of the coupled nonlinear volatility and option pricing model without an embedded w-learning. Moreover, we exhibit their dynamical behaviors for chosen different parameters. The vector financial rogue wave (rogon) solutions may be used to describe the possible physical mechanisms for the rogue wave phenomena and to further excite the possibility of relative researches and potential applications of vector rogue waves in the financial markets and other related fields.     

The relationships between Shanghai stock market and CNY/USD exchange rate: New evidence based on cross-correlation analysis,structural cointegration and nonlinear causality test

This paper explores the co-movement of Shanghai stock market and China Yuan (CNY) exchange rates. First, we find that stock price and exchange rate are significantly cross-correlated. Second, employing a cointegration test allowing for a structural break, we find that the Shanghai Composite Index (SCI) is not cointegrated with the exchange rate of CNY/USD. The so-called “cointegration” found in previous studies is just caused by the shock of the recent financial crisis. Third, using linear and nonlinear Granger causality tests, we find no causality between stock prices and exchange rates during the period before the recent financial crisis. After the financial crisis, a unidirectional causality behavior running from exchange rates to stock index is present.     

Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black-Scholes model with transaction costs

This paper deals with the problem of discrete time option pricing using the fractional Black-Scholes model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained. In addition, the relation between scaling and implied volatility smiles is discussed.     

Pilot-Wave Theory and Financial Option Pricing

This paper tries to argue why pilot-wave theory could be of use in financial economics. We introduce the notion of information wave. We consider a stochastic guidance equation and part of the drift term of that equation makes reference to the phase of the wave. In order to embed information in financial option pricing we could use such a drift. We also briefly argue how we could embed information in the pricing kernel of the option price. PACS: 03, 89.65.Gh.     

Comment on “Time-changed geometric fractional Brownian motion and option pricing with transaction costs” by Hui Gu et al.

The purpose of this comment is to point out the inappropriate assumption of “3αH>1$3\alpha H>1$” and two problems in the proof of “Theorem 3.1” in section 3 of the paper “Time-changed geometric fractional Brownian motion and option pricing with transaction costs” by Hui Gu et al. [H. Gu, J.R. Liang, Y. X. Zhang, Time-changed geometric fractional Brownian motion and option pricing with transaction costs, Physica A 391 (2012) 3971–3977]. Then we show the two problems will be solved under our new assumption.     

Pricing currency options in the mixed fractional Brownian motion

This paper deals with the problem of pricing European currency options in the mixed fractional Brownian environment. Both the pricing formula and the mixed fractional partial differential equation for European call currency options are obtained. Some Greeks and the estimator of volatility are also provided. Empirical studies and simulation results confirm the theoretical findings and show that the mixed fractional Brownian pricing model is a reasonable one.     

The role of communication and imitation in limit order markets

In this paper we develop an order driver market model with heterogeneous traders that imitate each other on different network structures. We assess how imitations among otherway noise traders, can give rise to well known stylized facts such as fat tails and volatility clustering. We examine the impact of communication and imitation on the statistical properties of prices and order flows when changing the networks’ structure, and show that the imitation of a given, fixed agent, called “guru", can generate clustering of volatility in the model. We also find a positive correlation between volatility and bid-ask spread, and between fat-tailed fluctuations in asset prices and gap sizes in the order book.     

Does implied volatility of currency futures option imply volatility of exchange rates?

By investigating currency futures options, this paper provides an alternative economic implication for the result reported by Stein [Overreactions in the options market, Journal of Finance 44 (1989) 1011–1023] that long-maturity options tend to overreact to changes in the implied volatility of short-maturity options. When a GARCH process is assumed for exchange rates, a continuous-time relationship is developed. We provide evidence that implied volatilities may not be the simple average of future expected volatilities. By comparing the term–structure relationship of implied volatilities with the process of the underlying exchange rates, we find that long-maturity options are more consistent with the exchange rates process. In sum, short-maturity options overreact to the dynamics of underlying assets rather than long-maturity options overreacting to short-maturity options.     

Self-consistent asset pricing models

We discuss the foundations of factor or regression models in the light of the self-consistency condition that the market portfolio (and more generally the risk factors) is (are) constituted of the assets whose returns it is (they are) supposed to explain. As already reported in several articles, self-consistency implies correlations between the return disturbances. As a consequence, the alphas and betas of the factor model are unobservable. Self-consistency leads to renormalized betas with zero effective alphas, which are observable with standard OLS regressions. When the conditions derived from internal consistency are not met, the model is necessarily incomplete, which means that some sources of risk cannot be replicated (or hedged) by a portfolio of stocks traded on the market, even for infinite economies. Analytical derivations and numerical simulations show that, for arbitrary choices of the proxy which are different from the true market portfolio, a modified linear regression holds with a non-zero value α_i at the origin between an asset i's return and the proxy's return. Self-consistency also introduces “orthogonality” and “normality” conditions linking the betas, alphas (as well as the residuals) and the weights of the proxy portfolio. Two diagnostics based on these orthogonality and normality conditions are implemented on a basket of 323 assets which have been components of the S&P500 in the period from January 1990 to February 2005. These two diagnostics show interesting departures from dynamical self-consistency starting about 2 years before the end of the Internet bubble. Assuming that the CAPM holds with the self-consistency condition, the OLS method automatically obeys the resulting orthogonality and normality conditions and therefore provides a simple way to self-consistently assess the parameters of the model by using proxy portfolios made only of the assets which are used in the CAPM regressions. Finally, the factor decomposition with the self-consistency condition derives a risk-factor decomposition in the multi-factor case which is identical to the principal component analysis (PCA), thus providing a direct link between model-driven and data-driven constructions of risk factors. This correspondence shows that PCA will therefore suffer from the same limitations as the CAPM and its multi-factor generalization, namely lack of out-of-sample explanatory power and predictability. In the multi-period context, the self-consistency conditions force the betas to be time-dependent with specific constraints.     

Measuring statistical heterogeneity: The Pietra index

There are various ways of quantifying the statistical heterogeneity of a given probability law: Statistics uses variance — which measures the law’s dispersion around its mean; Physics and Information Theory use entropy — which measures the law’s randomness; Economics uses the Gini index — which measures the law’s egalitarianism. In this research we explore an alternative to the Gini index-the Pietra index-which is a counterpart of the Kolmogorov-Smirnov statistic. The Pietra index is shown to be a natural and elemental measure of statistical heterogeneity, which is especially useful in the case of asymmetric and skewed probability laws, and in the case of asymptotically Paretian laws with finite mean and infinite variance. Moreover, the Pietra index is shown to have immediate and fundamental interpretations within the following applications: renewal processes and continuous time random walks; infinite-server queueing systems and shot noise processes; financial derivatives. The interpretation of the Pietra index within the context of financial derivatives implies that derivative markets, in effect, use the Pietra index as their benchmark measure of statistical heterogeneity.     

Exchange rate regimes, saving glut and the Feldstein-Horioka puzzle: The East Asian experience

This paper investigates whether the recent experience of the emerging East Asian countries with current account surpluses is consistent with the “saving glut” hypothesis and the Feldstein and Horioka puzzle. The evidence suggests that the saving retention coefficients declined substantially in most of the countries after an endogenous break date coinciding with a major exchange rate regime change with the 1997-1998 crisis. Exchange rate flexibility appears to be enhancing financial integration. The results are consistent with an “investment slump” explanation rather than the “saving glut” postulation.     

The endogenous dynamics of financial markets: Interaction and information dissemination

We investigate the process that different interactions between investors will prompt information to propagate along a differentiated path and construct a financial market model. As information spreads, increasingly investors are attracted to participate in trading, then the “herding effect” is magnified gradually, which will induce the topology of market network to change and the price to fluctuate. Especially, under different initial conditions or parameters, the peak and fat-tail property is produced and the obtained statistic values coincide with empirical results: the power-law exponents between the peak value of return probability distribution and the time scales range from 0.579 to 0.747, and the exponents between the accumulation distribution and the return on the tail are close to 3. Besides, the extent of volatility clustering in our produced price series is close to that of S&P 500 and locates between NASDAQ and HSI. All the results obtained here indicate that the continuous variation of the “herding effect” resulting from information propagation among interacting investors may be the origin of stylized facts of price fluctuations.