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A piecewise linear dynamical model is proposed for a stock price. The model considers the price is driven by three rather standard demand components: chartist, fundamental and market makers. The chartist demand component is related to the study of differences between moving averages. This generates a high order system characterized by a piecewise linear map not trivial to study. The model has been studied analytically in its fixed points and dynamics and then numerically. Results are in line with the related literature: the fundamental demand component helps the stability of the system and keeps prices bounded; market makers satisfy their role of restoring stability, while the chartist demand component produces irregularity and chaos. However, in some cases, the chartist demand component assumes the role to compensate the fundamental demand component, felt in an autogenerated loop, and pushes the dynamics to equilibrium. This fact suggests that the instability must not be searched into the nature of the different investment styles rather in the relative proportion of the contribution of market actors.

Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay. Commun.~Math.~Sci., 18(1):109-121, 2020](arXiv:1808.02341) proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method's efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

In this paper we study the optimal investment and reinsurance problem of an insurance company whose investment preferences are described via a forward dynamic exponential utility in a regime-switching market model. Financial and actuarial frameworks are dependent since stock prices and insurance claims vary according to a common factor given by a continuous time finite state Markov chain. We construct the value function and we prove that it is a forward dynamic utility. Then, we characterize the investment strategy and the optimal proportional level of reinsurance. We also perform numerical experiments and provide sensitivity analyses with respect to some model parameters., Comment: 32 pages, 6 figures, 1 table

This work aims to provide an efficient method to evaluate the Credit Value Adjustment (CVA) for a vulnerable European option, which is an option subject to some default event concerning the issuer solvability. Financial options traded in OTC markets are of this type. In particular, we compute the CVA in some popular stochastic volatility models such as SABR, Hull et al., which have proven to fit quite well market derivatives prices, admitting correlation with the default event. This choice covers the relevant case of Wrong Way Risk (WWR) when a credit deterioration determines an increase in the claim value. Contrary to the structural modeling adopted in [G. Wang, X. Wang & K. Zhu (2017) Pricing vulnerable options with stochastic volatility, Physica A 485, 91–103; C. Ma, S. Yue & Y. Ma (2020) Pricing vulnerable options with Stochastic volatility and Stochastic interest rate, Computational Economics 56, 391–429], we use the reduced-form intensity-based approach to provide an explicit representation formula for the vulnerable option price and related CVA. Later, we specialize the evaluation formula and construct its approximation for the three models mentioned above. Assuming a CIR model for the default intensity process, we run a numerical study to test our approximation, comparing it with Monte Carlo simulations. The results show that for moderate values of the correlation and maturities not exceeding one year, the approximation is very satisfactory as of accuracy and computational time.

In this paper we analyze the existence of cointegrating relationships between Bitcoin, S&P 500, and the quantity of money M2. We perform our analysis with and without applying time warping pre-processing. In all cases we find strong evidence that, in the period 2016-2021 the three time series show two cointegrating relationships and therefore share a common stochastic trend. In addition, a low correlation between Bitcoin and S&P 500 is detected. These finding justify the increased interest of investors in Bitcoin as an alternative asset class. The economic interpretation is that the stock valuation is primarily determined by financial phenomena, in particular the availability of large quantity of money. Money supporting investment is due both to the actions of Quantitative Easing and to the exchange of creditor/debtor role that took place between households and firms. The price of both Bitcoin and stocks is increasingly influenced by the amount of money in circulation and follows the same stochastic trend.

In this paper we test for regime changes and possible regime commonalities in the price dynamics of Bitcoin, Ethereum, Litecoin and Monero, as representatives of the cryptocurrencies asset class. Several parametric models are considered for the joint dynamics of the basket price where parameters are modulated through a Hidden Markov Chain with finite state space. Best specifications within Gaussian and Autoregressive models for price differences are selected by means of the AIC and BIC information criteria and through an out-of-sample forecasting performance. The empirical results, within the period January 2016 to October 2019, suggest that three or four states may be relevant to describe the dynamics of each individual cryptocurrency, depending on the selection criteria, while the entire basket displays at most three common states. Finally, we show how the identification of appropriate models may be exploited in order to build profitable investment strategies on the considered cryptocurrencies.

We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the diffusion limit depends on: (i) the link function, (ii) the conditional second moment of the score, (iii) the normalization of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tailed densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tailed models with a GARCH update, for which the volatility of volatility is explosive. We examine in simulations the implications of such results on approximate estimation and filtering of diffusion processes. An extension to models with a time-varying conditional mean and to conditional covariance models is also developed. We provide general conditions under which a class of discrete-time volatility models driven by the score of the conditional density converges in distribution to a stochastic differential equation as the interval between observations goes to zero. We show that the form of the diffusion limit depends on: (i) the link function, (ii) the conditional second moment of the score, (iii) the normalization of the score. Interestingly, the properties of the stochastic differential equation are strictly entangled with those of the discrete-time counterpart. Score-driven models with fat-tailed densities lead to continuous-time processes with finite volatility of volatility, as opposed to fat-tailed models with a GARCH update, for which the volatility of volatility is explosive. We examine in simulations the implications of such results on approximate estimation and filtering of diffusion processes. An extension to models with a time-varying conditional mean and to conditional covariance models is also developed.

Motivated by the empirical evidence of high-frequency lead-lag effects and cross-asset linkages, we introduce a multi-asset price formation model which generalizes standard univariate microstructure models of lagged price adjustment. Econometric inference on such model provides: (i) a unified statistical test for the presence of lead-lag correlations in the latent price process and for the existence of a multi-asset price formation mechanism; (ii) separate estimation of contemporaneous and lagged dependencies; (iii) an unbiased estimator of the integrated covariance of the efficient martingale price process that is robust to microstructure noise, asynchronous trading, and lead-lag dependencies. Through an extensive simulation study, we compare the proposed estimator to alternative approaches and show its advantages in recovering the true lead-lag structure of the latent price process. Our application to a set of NYSE stocks provides empirical evidence for the existence of a multi-asset price formation mechanism and sheds light on its market microstructure determinants. Supplementary materials for this article are available online.

We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance 30 (2020) 782–832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, Inahama–Kawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance 16 (2016) 887–904) and Forde–Zhang (SIAM J. Financial Math. 8 (2017) 114–145), one obtains precise asymptotics for European options which refine known large deviation asymptotics.

We study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. The estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. We solve the problem via the martingale method and, with a concavification procedure, we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy., 19 pages, 3 figures

In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small-noise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782–832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence.

This paper proposes a nonparametric theory for statistical inferences on zero returns of high-frequency asset prices. Using an infill asymptotic design, we derive limit theorems for the percentage of zero returns observed on a finite time interval and for other related quantities. Within this framework, we develop two nonparametric tests. First, we test whether intra-day zero returns are independent and identically distributed. Second, we test whether intra-day variation of the likelihood of occurrence of zero returns can be solely explained by a deterministic diurnal pattern. In an empirical application to ten representative stocks of the NYSE, we provide evidence that the null of independent and identically distributed intra-day zero returns can be conclusively rejected. We further find that a deterministic diurnal pattern is not sufficient to explain the intra-day variability of the distribution of zero returns.

We propose a class of score-driven realized covariance models where volatilities and correlations are separately estimated. We can thus combine univariate realized volatility models with a recently introduced class of score-driven realized covariance models based on Wishart and matrix- F distributions. Compared to the latter, the proposed models remain computationally simple at high dimensions and allow for higher flexibility in parameter estimation. Through a Monte-Carlo study, we show that the two-step maximum likelihood procedure provides accurate parameter estimates in small samples. Empirically, we find that the proposed models outperform those based on joint estimation, with forecasting gains that become more significant as the cross-section dimension increases.

The objective of this paper is to study the filtering problem for a system of partially observable processes $(X, Y)$, where $X$ is a non-Markovian pure-jump process representing the signal and $Y$ is a general jump-diffusion which provides observations. Our model covers the case where both processes are not necessarily quasi left-continuous, allowing them to jump at predictable stopping times. By introducing the Markovian version of the signal, we are able to compute an explicit equation for the filtering process via the innovations approach.

The analysis of the intraday dynamics of correlations among high-frequency returns is challenging due to the presence of asynchronous trading and market microstructure noise. Both effects may lead to significant data reduction and may severely underestimate correlations if traditional methods for low-frequency data are employed. We propose to model intraday log-prices through a multivariate local-level model with score-driven covariance matrices and to treat asynchronicity as a missing value problem. The main advantages of this approach are: (i) all available data are used when filtering correlations, (ii) market microstructure noise is taken into account, (iii) estimation is performed through standard maximum likelihood methods. Our empirical analysis, performed on 1-second NYSE data, shows that opening hours are dominated by idiosyncratic risk and that a market factor progressively emerges in the second part of the day. The method can be used as a nowcasting tool for high-frequency data, allowing to study the real-time response of covariances to macro-news announcements and to build intraday portfolios with very short optimization horizons., Comment: 30 pages, 10 figures, 7 tables

In this study we provide an analytical characterization of the impact of zero returns on the popular realized covariance estimator of Barndorff-Nielsen and Shephard [Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica, 2004, 72(3), 885–925]. In our framework, efficient price processes evolve as a semimartingale with some likelihood of repeated prices. We show that the standard realized covariance estimator is asymptotically affected by a downward bias, and the size of the bias depends on these likelihoods. We demonstrate that this result can be used to construct a consistent estimator of the integrated covariance of a vector semimartingale. The advantages with respect to other estimators are discussed with data.

Liquidity is a risk factor of primary relevance that can significantly affect the asset allocation decisions of investors. In this paper, we introduce the concept of portfolio staleness and propose a simple framework to manage portfolio liquidity, intended as the cost needed to liquidate the portfolio. Within this framework, the traditional minimum variance problem is solved under the additional constraint that portfolio staleness must be smaller than a given threshold. We show that a dynamic asset allocation strategy based on the staleness constrained portfolio can significantly enhance portfolio liquidity over the standard minimum variance solution. Meanwhile, the increase in portfolio risk is limited, generating large liquidity gains per unit of risk.

We study a dynamic portfolio optimization problem related to convergence trading, which is an investment strategy that exploits temporary mispricing by simultaneously buying relatively underpriced assets and selling short relatively overpriced ones with the expectation that their prices converge in the future. We build on the model of Liu and Timmermann (2013) and extend it by incorporating unobservable Markov-modulated pricing errors into the price dynamics of two co-integrated assets. We characterize the optimal portfolio strategies in full and partial information settings both under the assumption of unrestricted and beta-neutral strategies. By using the innovations approach, we provide the filtering equation that is essential for solving the optimization problem under partial information. Finally, in order to illustrate the model capabilities, we provide an example with a two-state Markov chain., Comment: 25 pages, 5 figures

In this paper we investigate the pricing problem of a pure endowment contract when the insurer has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative via the solutions of suitable backward stochastic differential equations., 36 pages

Asset prices can be stale. We define price staleness as a lack of price adjustments yielding zero returns (i.e., zeros). The term idleness (respectively, near idleness) is, instead, used to define staleness when trading activity is absent (respectively, close to absent). Using statistical and pricing metrics, we show that zeros are a genuine economic phenomenon linked to the dynamics of trading volume and, therefore, liquidity. Zeros are, in general, not the result of institutional features, like price discreteness. In essence, spells of idleness or near idleness are stylized facts suggestive of a key, omitted market friction in the modeling of asset prices. We illustrate how accounting for this friction may generate sizable risk compensations in short-dated option returns. This paper was accepted by Kay Giesecke, finance.

This paper presents an optimal allocation problem in a financial market with one risk-free and one risky asset, when the market is driven by a stochastic market price of risk. We solve the problem in continuous time, for an investor with a Constant Relative Risk Aversion (CRRA) utility, under two scenarios: when the market price of risk is observable (the {\em full information case}), and when it is not (the {\em partial information case}). The corresponding market models are complete in the partial information case and incomplete in the other case, hence the two scenarios exhibit rather different features. We study how the access to more accurate information on the market price of risk affects the optimal strategies and we determine the maximal price that the investor would be willing to pay to get such information. In particular, we examine two cases of additional information, when an exact observation of the market price of risk is available either at time $0$ only (the {\em initial information case}), or during the whole investment period (the {\em dynamic information case})., 34 pages, 9 figures

Reinsurance counterparty credit risk (RCCR) is the risk of a loss arising from the fact that a reinsurance company is unable to fulfill her contractual obligations towards the ceding insurer. RCCR is an important risk category for insurance companies which, so far, has been addressed mostly via qualitative approaches. In this paper we therefore study value adjustments and dynamic hedging for RCCR. We propose a novel model that accounts for contagion effects between the default of the reinsurer and the price of the reinsurance contract. We characterize the value adjustment in a reinsurance contract via a partial integro-differential equation (PIDE) and derive the hedging strategies using a quadratic method. The paper closes with a simulation study which shows that dynamic hedging strategies have the potential to significantly reduce RCCR., 27 pages, 8 figures

Despite their effectiveness, linear models for realized variance neglect measurement errors on integrated variance and exhibit several forms of misspecification due to the inherent nonlinear dynamics of volatility. We propose new extensions of the popular approximate long-memory heterogeneous autoregressive (HAR) model apt to disentangle these effects and quantify their separate impact on volatility forecasts. By combining the asymptotic theory of the realized variance estimator with the Kalman filter and by introducing time-varying HAR parameters, we build new models that account for: (i) measurement errors (HARK), (ii) nonlinear dependencies (SHAR) and (iii) both measurement errors and nonlinearities (SHARK). The proposed models are simply estimated through standard maximum likelihood methods and are shown, both on simulated and real data, to provide better out-of-sample forecasts compared to standard HAR specifications and other competing approaches.

In this paper we introduce and solve a class of optimal stopping problems of recursive type. In particular, the stopping payoff depends directly on the value function of the problem itself. In a multi-dimensional Markovian setting we show that the problem is well posed, in the sense that the value is indeed the unique solution to a fixed point problem in a suitable space of continuous functions, and an optimal stopping time exists. We then apply our class of problems to a model for stock trading in two different market venues and we determine the optimal stopping rule in that case., 36 pages, 2 figures. In this version, we provide a general analysis of a class of recursive optimal stopping problems with both finite-time and infinite-time horizon. We also discuss other applications and provide a comparison with a non-recursive model

The objective of this paper is to give conditions ensuring that the backward partial integro differential equation associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution; that is the solution is continuous, twice differentiable in the diffusion component and differentiable in time. Our proof uses a probabilistic arguments and extends the results of Pham (1998) to processes with a pure jump component where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of financial and actuarial instruments, and we provide an example to pricing of CAT bond., 17 pages

This note is commenting on Hasbrouck (2018). The paper investigates the problem of price discovery on markets with trades recorded at sub-millisecond frequencies. The application of the popular information share measure of Hasbrouck (1995) to such data faces several difficulties, as the underlying vector error correction models would need a huge number of lags to capture dynamics at different time-scales. The problem is handled by imposing a set of restrictions on parameters inspired by the Heterogeneous Autoregressive model for realized volatility. We illustrate some potential drawbacks of the information share measure adopted in the paper and propose a modeling strategy aimed at dealing with such limitations. In particular, we introduce a structural multi-market model with a lagged adjustment mechanism describing lagged absorption of information across markets. The advantages of the method are shown in simulations.

We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at the money, we establish exact pricing formulas for European call options and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew which explodes as $T^{-1/2}$ , reproducing the empirical steep short end of the smile. This behaviour is a consequence of the singularity of the local volatility at the money. Finally, we look at continuous, non-differentiable versions of such a model. We still find, in simulations, exploding implied skews.

This paper proposes an accurate, parsimonious and fast-to-estimate forecasting model for integer-valued time series with long memory and seasonality. The modelling is achieved through an autoregressive Poisson process with a predictable stochastic intensity that is determined by two factors: a seasonal intraday pattern and a heterogeneous autoregressive component. We call the model SHARP, which is an acronym for seasonal heterogeneous autoregressive Poisson. We also present a mixed-data sampling extension of the model, which adopts the historical information flow more efficiently and provides the best (among all the models considered) forecasting performances, empirically, for the bid–ask spreads of NYSE equity stocks. We conclude by showing how bid–ask spread forecasts based on the SHARP model can be exploited in order to reduce the total cost incurred by a trader who is willing to buy or sell a given amount of an equity stock.

Bitcoin is a digital currency traded on different exchanges for different prices; this feature implies important issues about arbitrage opportunities. In this paper we investigate whether strong or weak form of arbitrage strategies are indeed possible by trading across different Bitcoin Exchanges. Our investigation, both theoretically and practically, gives as a result that arbitrage is indeed possible.

A growing body of studies on systemic risk in financial markets has emphasized the key importance of taking into consideration the complex interconnections among financial institutions. Much effort has been put in modeling the contagion dynamics of financial shocks, and to assess the resilience of specific financial markets - either using real network data, reconstruction techniques or simple toy networks. Here we address the more general problem of how shock propagation dynamics depends on the topological details of the underlying network. To this end we consider different realistic network topologies, all consistent with balance sheets information obtained from real data on financial institutions. In particular, we consider networks of varying density and with different block structures, and diversify as well in the details of the shock propagation dynamics. We confirm that the systemic risk properties of a financial network are extremely sensitive to its network features. Our results can aid in the design of regulatory policies to improve the robustness of financial markets.

This chapter illustrates a continuous time model for the dynamics of Bitcoin price, which depends on an attention or sentiment factor. The model is proven arbitrage-free under mild conditions and a quasi-closed pricing formula for European style derivatives is provided.

We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors' estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions., 31 pages, 1 figure

We investigate by means of agent-based simulations the influence of convex incentives, e.g. option-like compensation, on financial markets. We propose an agent based model already developed in Fabretti et al (2015), where the model was build with the aim of studying convex contract effect using the results of a laboratory experiment performed by Holmen et al. (2014) as benchmark. Here we replicate their results studying prices dynamics, volatility, volumes and risk preference effect. We show that convex incentives produces higher prices, lower liquidity and higher volatility when agents are risk averse, while, differently from Fabretti et al (2015), their effect is less evident if agents are risk lovers. This appears related to the fact that prices in the long run converge more likely to the equilibrium when agents are risk averse.

We introduce a class of stochastic volatility models ( X t ) t ≥ 0 for which the absolute moments of the increments exhibit anomalous scaling: E ( ∣ X t + h − X t ∣ q ) scales as h q / 2 for q q ∗ , but as h A ( q ) with A ( q ) q / 2 for q > q ∗ , for some threshold q ∗ . This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean reversion is superlinear.