Quantifying bid-ask spreads in the Chinese stock market using limit-order book data

The statistical properties of the bid-ask spread of a frequently traded Chinese stock listed on the Shenzhen Stock Exchange are investigated using the limit-order book data. Three different definitions of spread are considered based on the time right before transactions, the time whenever the highest buying price or the lowest selling price changes, and a fixed time interval. The results are qualitatively similar no matter linear prices or logarithmic prices are used. The average spread exhibits evident intraday patterns consisting of a big L-shape in morning transactions and a small L-shape in the afternoon. The distributions of the spread with different definitions decay as power laws. The tail exponents of spreads at transaction level are well within the interval (2,3) and that of average spreads are well in line with the inverse cubic law for different time intervals. Based on the detrended fluctuation analysis, we found the evidence of long memory in the bid-ask spread time series for all three definitions, even after the removal of the intraday pattern. Using the classical box-counting approach for multifractal analysis, we show that the time series of bid-ask spread do not possess multifractal nature.     

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.     

Continuous time Black-Scholes equation with transaction costs in subdiffusive fractional Brownian motion regime

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(S_α(t)), 0<α<1, here dX(τ)=μX(τ)(dτ)^2H+σX(τ)dB_H(τ), as a model of asset prices, which captures the subdiffusive characteristic of financial markets. We find the corresponding subdiffusive Black-Scholes equation and the Black-Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.     

Scaling and long range dependence in option pricing, IV: Pricing European options with transaction costs under the multifractional Black-Scholes model

This paper deals with the problem of discrete time option pricing using the multifractional Black-Scholes model with transaction costs. Using a mean self-financing delta hedging 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, we show that scaling and long range dependence have a significant impact on option pricing.     

Empirical regularities of order placement in the Chinese stock market

Using ultra-high-frequency data extracted from the order flows of 23 stocks traded on the Shenzhen Stock Exchange, we study the empirical regularities of order placement in the opening call auction, cool period and continuous auction. The distributions of relative logarithmic prices against reference prices in the three time periods are qualitatively the same with quantitative discrepancies. The order placement behavior is asymmetric between buyers and sellers and between the inside-the-book orders and outside-the-book orders. In addition, the conditional distributions of relative prices in the continuous auction are independent of the bid-ask spread and volatility. These findings are crucial to build an empirical behavioral microscopic model based on order flows for Chinese stocks.     

Studies of the limit order book around large price changes

We study the dynamics of the limit order book of liquid stocks after experiencing large intra-day price changes. In the data we find large variations in several microscopical measures, e.g., the volatility the bid-ask spread, the bid-ask imbalance, the number of queuing limit orders, the activity (number and volume) of limit orders placed and canceled, etc. The relaxation of the quantities is generally very slow that can be described by a power law of exponent ≈ 0.4. We introduce a numerical model in order to understand the empirical results better. We find that with a zero intelligence deposition model of the order flow the empirical results can be reproduced qualitatively. This suggests that the slow relaxations might not be results of agents’ strategic behaviour. Studying the difference between the exponents found empirically and numerically helps us to better identify the role of strategic behaviour in the phenomena.     

A precursor of market crashes: Empirical laws of Japan's internet bubble

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as X(t) = S(t)/S(0), where S(0), is the stock price for an onset time of the bubble. We selected approximately 3200 stocks traded on the Japanese Stock Exchange, and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which internet Bubble formed and crashed in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, P(S>x) ∼x^-α , with an exponent that moves over time. Furthermore we found that as the power-law exponents α approached two, the bubble burst. It is reasonable to suppose that it indicates that internet bubble is about to burst.     

Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

This paper deals with the problem of discrete-time option pricing by the mixed Brownian–fractional Brownian model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. In particular, the minimal pricing c_min(t,s_t) of an option under transaction costs is obtained, which shows that timestep δt and Hurst exponent H play an important role in option pricing with transaction costs. In addition, we also show that there exists fundamental difference between the continuous-time trade and discrete-time trade and that continuous-time trade assumption will result in underestimating the value of a European call option.     

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.     

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.     

From market games to real-world markets

This paper uses the development of multi-agent market models to present a unified approach to the joint questions of how financial market movements may be simulated, predicted, and hedged against. We first present the results of agent-based market simulations in which traders equipped with simple buy/sell strategies and limited information compete in speculatory trading. We examine the effect of different market clearing mechanisms and show that implementation of a simple Walrasian auction leads to unstable market dynamics. We then show that a more realistic out-of-equilibrium clearing process leads to dynamics that closely resemble real financial movements, with fat-tailed price increments, clustered volatility and high volume autocorrelation. We then show that replacing the `synthetic' price history used by these simulations with data taken from real financial time-series leads to the remarkable result that the agents can collectively learn to identify moments in the market where profit is attainable. Hence on real financial data, the system as a whole can perform better than random. We then employ the formalism of Bouchaud in conjunction with agent based models to show that in general risk cannot be eliminated from trading with these models. We also show that, in the presence of transaction costs, the risk of option writing is greatly increased. This risk, and the costs, can however be reduced through the use of a delta-hedging strategy with modified, time-dependent volatility structure. Received 30 August 2000     

Artificial Neural Networks Performance in WIG20 Index Options Pricing

In this paper, the performance of artificial neural networks in option pricing was analyzed and compared with the results obtained from the Black–Scholes–Merton model, based on the historical volatility. The results were compared based on various error metrics calculated separately between three moneyness ratios. The market data-driven approach was taken to train and test the neural network on the real-world options data from 2009 to 2019, quoted on the Warsaw Stock Exchange. The artificial neural network did not provide more accurate option prices, even though its hyperparameters were properly tuned. The Black–Scholes–Merton model turned out to be more precise and robust to various market conditions. In addition, the bias of the forecasts obtained from the neural network differed significantly between moneyness states. This study provides an initial insight into the application of deep learning methods to pricing options in emerging markets with low liquidity and high volatility.     

The emergence of coordination in public good games

In physical models it is well understood that the aggregate behaviour of a system is not in one to one correspondence with the behaviour of the average individual element of that system. Yet, in many economic models the behaviour of aggregates is thought of as corresponding to that of an individual. A typical example is that of public goods experiments. A systematic feature of such experiments is that, with repetition, people contribute less to public goods. A typical explanation is that people “learn to play Nash” or something approaching it. To justify such an explanation, an individual learning model is tested on average or aggregate data. In this paper we will examine this idea by analysing average and individual behaviour in a series of public goods experiments. We analyse data from a series of games of contributions to public goods and as is usual, we test a learning model on the average data. We then look at individual data, examine the changes that this produces and see if some general model such as the EWA (Expected Weighted Attraction) with varying parameters can account for individual behaviour. We find that once we disaggregate data such models have poor explanatory power. Groups do not learn as supposed, their behaviour differs markedly from one group to another, and the behaviour of the individuals who make up the groups also varies within groups. The decline in aggregate contributions cannot be explained by resorting to a uniform model of individual behaviour. However, the Nash equilibrium of such a game is a total payment for all the individuals and there is some convergence of the group in this respect. Yet the individual contributions do not converge. How the individuals “self-organsise” to coordinate, even in this limited way remains to be explained.     

Interest rates in quantum finance: Caps, swaptions and bond options

The prices of the main interest rate options in the financial markets, derived from the Libor (London Interbank Overnight Rate), are studied in the quantum finance model of interest rates. The option prices show new features for the Libor Market Model arising from the fact that, in the quantum finance formulation, all the different Libor payments are coupled and (imperfectly) correlated.Black’s caplet formula for quantum finance is given an exact path integral derivation. The coupon and zero coupon bond options as well as the Libor European and Asian swaptions are derived in the framework of quantum finance. The approximate Libor option prices are derived using the volatility expansion.The BGM-Jamshidian (Gatarek et al. (1996) [1], Jamshidian (1997) [2]) result for the Libor swaption prices is obtained as the limiting case when all the Libors are exactly correlated. A path integral derivation is given of the approximate BGM-Jamshidian approximate price.     

Quantitative model of price diffusion and market friction based on trading as a mechanistic random process

We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.     

Pricing European option with transaction costs under the fractional long memory stochastic volatility model

This paper deals with the problem of discrete time option pricing using the fractional long memory stochastic volatility model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained.     

Network breakdown “at the edge of chaos” in multi-agent traffic simulations

Traffic is highly influenced by network structure and human behaviour. Small changes in the human behaviour can lead to huge changes in the load of a traffic network. Current transportation models do not, and most of them cannot, research such random behaviour but always calculate a steady state. In our multi-agent transport simulation, we frequently observe seemingly random “network breakdowns”, huge traffic jams that spread over a big part of the network, making a normal traffic flow impossible. This paper describes the investigations that were performed on the results of our large-scale multi-agent transport simulations in an attempt to contribute to the better understanding of the dynamic processes in such simulations and, hopefully, better understanding and modelling of the real-world.     

Tick size and stock returns

Tick size is an important aspect of the micro-structural level organization of financial markets. It is the smallest institutionally allowed price increment, has a direct bearing on the bid-ask spread, influences the strategy of trading order placement in electronic markets, affects the price formation mechanism, and appears to be related to the long-term memory of volatility clustering. In this paper we investigate the impact of tick size on stock returns. We start with a simple simulation to demonstrate how continuous returns become distorted after confining the price to a discrete grid governed by the tick size. We then move on to a novel experimental set-up that combines decimalization pilot programs and cross-listed stocks in New York and Toronto. This allows us to observe a set of stocks traded simultaneously under two different ticks while holding all security-specific characteristics fixed. We then study the normality of the return distributions and carry out fits to the chosen distribution models. Our empirical findings are somewhat mixed and in some cases appear to challenge the simulation results.     

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.