Transform analysis for Hawkes processes with applications in dark pool trading

Hawkes processes are a class of simple point processes that are self-exciting and have a clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed nonnegative random variables. This Hawkes model is non-Markovian in general. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyse various performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate of a resting dark order.     

A self-exciting switching jump diffusion: properties,calibration and hitting time

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options.     

Constant proportion portfolio insurance strategies in contagious markets

Constant Proportion Portfolio Insurance (CPPI) strategies are popular as they allow to gear up the upside potential of a stock index while limiting its downside risk. From the issuer’s perspective it is important to adequately assess the risks associated with the CPPI, both for correct ‘gap’ fee charging and for risk management. The literature on CPPI modelling typically assumes diffusive or Lévy-driven dynamics for the risky asset underlying the strategy. In either case the self-contagious nature of asset prices is not taken into account. In order to account for contagion while preserving analytical tractability, we introduce self-exciting jumps in the underlying dynamics via Hawkes processes. Within this framework we derive the loss probability when trading is performed continuously. Moreover, we estimate measures of the risk involved in the practical implementation of discrete-time rebalancing rules governing the CPPI product. When rebalancing is performed on a frequency less than weekly, failing to take contagion into account will significantly underestimate the risks of the CPPI. Finally, in order to mimic a situation with low liquidity, we impose a daily trading cap on the risky asset and find that the Hawkes process driven models give rise to the highest risk measures even under daily rebalancing.     

Jump-diffusion volatility models for variance swaps: An empirical performance analysis

This paper studies a class of tractable jump-diffusion models, including stochastic volatility models with various specifications of jump intensity for stock returns and variance processes. We employ the Markov chain Monte Carlo (MCMC) method to implement model estimation, and investigate the performance of all models in capturing the term structure of variance swap rates and fitting the dynamics of stock returns. It is evident that the stochastic volatility models, equipped with self-exciting jumps in the spot variance and linearly-dependent jumps in the central-tendency variance, can produce consistent model estimates, aptly explain the stylized facts in variance swaps, and boost pricing performance. Moreover, our empirical results show that large self-exciting jumps in the spot variance, as an independent risk source, facilitate term structure modeling for variance swaps, whilst the central-tendency variance may jump with small sizes, but signaling substantial regime changes in the long run. Both types of jumps occur infrequently, and are more related to market turmoils over the period from 2008 to 2021.     

Bond and option pricing for interest rate model with clustering effects

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.     

Quadratic Hawkes processes for financial prices

We introduce and establish the main properties of QHawkes (‘Quadratic’ Hawkes) models. QHawkes models generalize the Hawkes price models introduced in Bacry and Muzy [Quant. Finance, 2014, 14(7), 1147–1166], by allowing feedback effects in the jump intensity that are linear and quadratic in past returns. Our model exhibits two main properties that we believe are crucial in the modelling and the understanding of the volatility process: first, the model is time-reversal asymmetric, similar to financial markets whose time evolution has a preferred direction. Second, it generates a multiplicative, fat-tailed volatility process, that we characterize in detail in the case of exponentially decaying kernels, and which is linked to Pearson diffusions in the continuous limit. Several other interesting properties of QHawkes processes are discussed, in particular the fact that they can generate long memory without necessarily being at the critical point. A non-parametric fit of the QHawkes model on NYSE stock data shows that the off-diagonal component of the quadratic kernel indeed has a structure that standard Hawkes models fail to reproduce. We provide numerical simulations of our calibrated QHawkes model which is indeed seen to reproduce, with only a small amount of quadratic non-linearity, the correct magnitude of fat-tails and time reversal asymmetry seen in empirical time series.     

High-frequency financial data modeling using Hawkes processes

Intraday Value-at-Risk (VaR) is one of the risk measures used by market participants involved in high-frequency trading. High-frequency log-returns feature important kurtosis (fat tails) and volatility clustering (extreme log-returns appear in clusters) that VaR models should take into account. We propose a marked point process model for the excesses of the time series over a high threshold that combines Hawkes processes for the exceedances with a generalized Pareto distribution model for the marks (exceedance sizes). The conditional approach features intraday clustering of extremes and is used to calculate instantaneous conditional VaR. The models are backtested on real data and compared to a competitor approach that proposes a nonparametric extension of the classical peaks-over-threshold method. Maximum likelihood estimation is computationally intensive; we use a differential evolution genetic algorithm to find adequate starting values for the optimization process.     

A model for interest rates with clustering effects

We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for the characteristic function for interest rates and their integral. We introduce a class of equivalent measures under which the features of the process are preserved. We infer the prices of bonds and their dynamics under a risk-neutral measure. The question of derivatives pricing is developed under a forward measure, and a numerical algorithm is proposed to evaluate caplets and floorlets. The model is fitted to EONIA rates from 2004 to 2014 using a peaks-over-threshold procedure. From observation of swap curves over the same period, we filter the evolution of risk premiums for Brownian and jump components. Finally, we analyse the sensitivity of implied caplet volatility to parameters defining the level of self-excitation.     

Exchange options under clustered jump dynamics

Exchange options are one of the most popular exotic options, and have important implications for many common financial arrangements and for implied beta as a measure of systematic risk. In this study, we extend the existing literature on exchange options to allow for clustered jump contagion dynamics in each single asset, as well as across assets, using the Hawkes jump-diffusion model. We derive the analytical pricing formulae, the Greeks, and the optimal hedging strategy via Fourier transforms. Using an illustrative numerical analysis, we present the relationship between the exchange option price and clustered jump intensities and jump sizes in the underlying assets. We discuss the managerial insights on financial arrangements with exchange option characteristics. Furthermore, we discuss the implications of incorporating clustered jumps into the estimation of implied beta with exchange options, in which the applications can be insightful and useful in finance practice.     

Linear-quadratic term structure models – Toward the understanding of jumps in interest rates

We study linear-quadratic term structure models with random jumps in the short rate process where the jump arrival rate follows a stochastic process. Empirical results based on the US data show that incorporating stochastic jump intensity significantly improves model fit to the dynamics of both interest rate and volatility term structure. Our results also show that jump intensity is negatively correlated with interest rate changes and the average size is larger on the downside than upside. Examining the relation between jump intensity and macroeconomic shocks, we find that at monthly frequency, jumps are neither triggered by nor predictive of changes in macroeconomic variables. At daily frequency, however, we document interesting patterns for jumps associated with information shocks.     

An intensity model for credit risk with switching Lévy processes

We develop a switching regime version of the intensity model for credit risk pricing. The default event is specified by a Poisson process whose intensity is modeled by a switching Lévy process. This model presents several interesting features. First, as Lévy processes encompass numerous jump processes, our model can duplicate the sudden jumps observed in credit spreads. Also, due to the presence of jumps, probabilities do not vanish at very short maturities, contrary to models based on Brownian dynamics. Furthermore, as the parameters of the Lévy process are modulated by a hidden Markov chain, our approach is well suited to model changes of volatility trends in credit spreads, related to modifications of unobservable economic factors.     

Structural changes in foreign investors’ trading behavior and the corresponding impact on Taiwan's stock market

This study investigates the impact of the expected and unexpected trading behavior of foreign investors on return volatilities during structural change periods. And the jump intensity model pinpoints crucial events that have influenced the stock market. The empirical results find that there has been a stabilizing effect of foreign investment on Taiwan's stock market as restrictions on foreign trading have been gradually relaxed, as opposed to there being a complete relaxation of the restrictions imposed on Qualified Foreign Institutional Investors (QFIIs).     

Modelling jumps in electricity prices: theory and empirical evidence

Objective of this paper is to enhance the understanding of modelling jumps and to analyse the model risk based on the jump component in electricity markets. We provide a common modelling framework that allows to incorporate the main jump patterns observed in electricity spot prices and compare the effectiveness of different jump specifications. To this end, we calibrate the models to daily European Energy Exchange (EEX) market data through Markov Chain Monte Carlo based methods. To assess the quality of the estimated jump processes, we analyse their trajectorial and statistical properties. Moreover, even when the models are calibrated to a cross-section of derivative prices substantial model risk remains.      

Pure jump Lévy processes for asset price modelling

The goal of the paper is to show that some types of Lévy processes such as the hyperbolic motion and the CGMY are particularly suitable for asset price modelling and option pricing. We wish to review some fundamental mathematic properties of Lévy distributions, such as the one of infinite divisibility, and how they translate observed features of asset price returns. We explain how these processes are related to Brownian motion, the central process in finance, through stochastic time changes which can in turn be interpreted as a measure of the economic activity. Lastly, we focus on two particular classes of pure jump Lévy processes, the generalized hyperbolic model and the CGMY models, and report on the goodness of fit obtained both on stock prices and option prices.     

Wavelet Galerkin pricing of American options on Lévy driven assets

The price of an American-style contract on assets driven by a class of Markov processes containing, in particular, Lévy processes of pure jump type with infinite jump activity is expressed as the solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented. Log-linear complexity in each time-step is achieved by wavelet compression of the moment matrix of the price process’ jump measure and by wavelet preconditioning of the large matrix LCPs at each time-step. Efficiency is demonstrated by numerical experiments for pricing American put contracts on various jump-diffusion and pure jump models. Failure of the smooth pasting principle is observed for American put contracts for certain finite variation pure jump price processes.     

The evolution of the January effect

This paper extends recent studies of the January effect by investigating the evolution of the daily pattern of the effect across size deciles. Our evidence documents a sizable mean reverting component beginning in the latter part of January and a shorter duration of the seasonal effect. Despite lower abnormal returns in the second part of January, higher abnormal returns in the first part of January keep the magnitude of the January effect unchanged. Further analysis of daily trading volumes suggests a stable trading volume intensity in the first part of January and a substantial decline in trading volume intensity in the second part of January.     

转融通基本问题研究

转融通实质上就是证券金融公司对证券公司的融资融券。证券金融公司是转融通业务的唯一主体,具有一定的垄断地位,但证券金融公司在开展转融通业务时,与证券公司是交易对手,处于平等的民事主体地位,证券金融公司不应定位为交易的中央对手方或者市场组织者。在《转融通业务监督管理试行办法》规定的框架下,证券金融公司可以根据实际情况和需要设计出不同的业务操作模式,但效率与安全应当是考虑设计转融通操作模式的两大基本出发点。从长远看,比照现有的融资融券业务操作模式构建证券金融公司的转融通业务操作模式,是一种可取的选择,值得作进一步的思考。     

Estimating the probability of informed trading—does trade misclassification matter?

Easley et al. [1996. Journal of Finance 51, 1405–1436] have proposed an empirical methodology to estimate the probability of informed trading (PIN). This approach has been employed in a wide range of applications in market microstructure, corporate finance, and asset pricing. To estimate the model, a researcher only needs the number of buyer- and seller-initiated trades. This information, however, is generally unobservable and has to be inferred from trade-classification algorithms, which are known to be inaccurate. In this paper, we show analytically that inaccurate trade classification leads to downward-biased PIN estimates and that the magnitude of the bias is related to a security's trading intensity. Simulation results and empirical evidence based on order and transaction data from the New York Stock Exchange are consistent with this argument. We propose a data-based adjustment procedure that substantially reduces the misclassification bias.     

Modeling of Contagious Credit Events and Risk Analysis of Credit Portfolios

We present a new model of the occurence of credit events such as rating changes and defaults for risk analyses of some portfolio credit derivatives. The framework of our model is based on a so-called top-down approach. Specifically, we first consider modeling the point process of each type of credit event in the whole economy using a self-exciting intensity process. Next, we characterize the point processes of credit events in the underlying sub-portfolio using random thinning processes specified by the distribution of credit ratings in the sub-portfolio. One of the main features of our model is that the model can capture credit risk contagion simultaneously among several credit portfolios. We present a credit event simulation algorithm based on our model and illustrate an application of the model to risk analyses of loan portfolios.     

PRICING CURRENCY FUTURES OPTIONS WITH LOGNORMALLY DISTRIBUTED JUMPS

We find that a mixed diffusion-jump process fits most daily currency futures price series better than a mixture of normal densities and, especially, an asymmetric stable Paretian model. We also find that Merton's (1976) mixed diffusion-jump option pricing model outperforms Black's (1 976) model for valuing currency futures options. Our results suggest that researchers should begin to consider the possibility of jump processes as time-independent models of other futures price series.