Stochastic Calculus & Continuous-Time Models

We present a Monte Carlo approach to pairs trading on mean-reverting spreads modeled by Lévy-driven Ornstein-Uhlenbeck processes. Specifically, we focus on using a variance gamma driving process, an infinite activity pure jump process to allow for more flexible models of the price spread than is ava

A novel generative machine learning approach for the simulation of sequences of financial price data with drawdowns quantifiably close to empirical data is introduced. Applications such as pricing drawdown insurance options or developing portfolio drawdown control strategies call for a host of drawd

In this research work, we propose a high-order time adapted scheme for pricing a coupled system of fixed-free boundary constant elasticity of variance (CEV) model on both equidistant and locally refined space-grid. The performance of our method is substantially enhanced to improve irregularities in

This note explores in more details instabilities of explicit super-time-stepping schemes, such as the Runge-Kutta-Chebyshev or Runge-Kutta-Legendre schemes, noticed in the litterature, when applied to the Heston stochastic volatility model. The stability remarks are relevant beyond the scope of supe

Market making plays a crucial role in providing liquidity and maintaining stability in financial markets, making it an essential component of well-functioning capital markets. Despite its importance, there is limited research on market making in the Chinese stock market, which is one of the largest

We study the problem of optimally hedging the price exposure of liquidity positions in constant-product automated market makers (AMMs) when the hedge is funded by collateralized borrowing. A liquidity provider (LP) who borrows tokens to construct a delta-neutral position faces a trade-off: higher he

We develop a finite-horizon model in which liquid-asset returns exhibit Levy-stable scaling on a data-driven window [“tau_UV, tau_IR”] and aggregate into a finite-variance regime outside. The window and the tail index alpha are identified from the log-log slope of the central body and a two-segment

We explore a link between stochastic volatility (SV) and path-dependent volatility (PDV) models. Using assumed density filtering, we map a given SV model into a corresponding PDV representation. The resulting specification is lightweight, improves in-sample fit, and delivers robust out-of-sample for

We study an American option pricing problem with liquidity risks and transaction fees. As endogenous transaction costs, liquidity risks of the underlying asset are modeled by a mean-reverting process. Transaction fees are exogenous transaction costs and are assumed to be proportional to the trading

Following the foundational work of the Black–Scholes model, extensive research has been developed to price the option by addressing its underlying assumptions and associated pricing biases. This study introduces a novel framework for pricing European call options by modeling the underlying asset’s

We conduct modeling of the price dynamics following order flow imbalance in market microstructure and apply the model to the analysis of Chinese CSI 300 Index Futures. There are three findings. The first is that the order flow imbalance is analogous to a shock to the market. Unlike the common practi

A self-exciting point process with a continuous-time autoregressive moving average intensity process, named CARMA(p,q)-Hawkes model, has recently been introduced. The model generalizes the Hawkes process by substituting the Ornstein-Uhlenbeck intensity with a CARMA(p,q) model where the associated st

We develop diffusion models for time-varying correlation using stochastic processes defined on the unit circle. Specifically, we study Brownian motion on the circle and the von Mises diffusion, and propose their use as continuous-time models for correlation dynamics. The von Mises process, introduce

We derive a closed-form approximation for the credit default swap (CDS) spread in the two-dimensional shifted square-root diffusion (SSRD) model using asymptotic coefficient expansion technique to approximate solutions of nonlinear partial differential equations. Specifically, we identify the Cauchy

This research presents a comprehensive framework for transitioning financial diffusion models from the risk-neutral (RN) measure to the real-world (RW) measure, leveraging results from probability theory, specifically Girsanov’s theorem. The RN measure, fundamental in derivative pricing, is contrast

We study the continuous-time pre-commitment mean-variance portfolio selection in a time-varying financial market. By introducing two indexes which respectively express the average profitability of the risky asset (AP) and the current profitability of the risky asset (CP), the optimal portfolio selec

Algorithmic trading refers to executing buy and sell orders for specific assets based on automatically identified trading opportunities. Strategies based on reinforcement learning (RL) have demonstrated remarkable capabilities in addressing algorithmic trading problems. However, the trading patterns

We present a thorough empirical study on real interest rates by also including risk aversion through the introduction of the market price of risk. With the view of complex systems science and its multidisciplinary approach, we use the theory of bond pricing to study the long term discount rate. Cent

This paper develops a coupled model for day-ahead electricity prices and average daily temperature which allows to model quanto weather and energy derivatives. These products have gained on popularity as they enable to hedge against both volumetric and price risks. Electricity day-ahead prices and a

In this paper, we introduce a data-driven, single-parameter Markov diffusion model for the VIX. The volatility factor evolves in $(-1,1)$ with a uniform invariant distribution ensured by Legendre polynomials, mapped to the empirical distribution. We derive analytical series solutions for VIX futures

Deep hedging is a deep-learning-based framework for derivative hedging in incomplete markets. The advantage of deep hedging lies in its ability to handle various realistic market conditions, such as market frictions, which are challenging to address within the traditional mathematical finance framew

Motivated by the current global high inflation scenario, we aim to discover a dynamic multi-period allocation strategy to optimally outperform a passive benchmark while adhering to a bounded leverage limit. To this end, we formulate an optimal control problem to outperform a benchmark portfolio thro

We develop policy gradients methods for stochastic control with exit time in a model-free setting. We propose two types of algorithms for learning either directly the optimal policy or by learning alternately the value function (critic) and the optimal control (actor). The use of randomized policies

Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state variables. Exploiting the derivative information leads to fewer s

This paper considers a Markovian model of a limit order book where time-dependent rates are allowed. With the objective of understanding the mechanisms through which a microscopic model of an orderbook can converge to more general diffusion than a Brownian motion with constant coefficient, a simple

Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numeri

On April 20, 2020, the May front-month WTI oil futures contract, one day before its expiration date, opened near $$17/$barrel and dropped far below zero in a single trading day, reaching an intraday low of $-$40.32$ and settling at $-$37.63$. Such market behavior was unforeseen at the time. This

Despite the empirical success in modeling volatility of the rough Bergomi (rBergomi) model, it suffers from pricing and calibration difficulties stemming from its non-Markovian structure. To address this, we propose a comprehensive computational framework that enhances both simulation and calibratio

This paper investigates a time-inconsistent portfolio selection problem in the incomplete mar ket model, integrating expected utility maximization with risk control. The objective functional balances the expected utility and variance on log returns, giving rise to time inconsistency and motivating t

This study develops an inverse portfolio optimization framework for recovering latent investor preferences including risk aversion, transaction cost sensitivity, and ESG orientation from observed portfolio allocations. Using controlled synthetic data, we assess the estimator’s statistical properties

We introduce a novel rough Bergomi (rBergomi) model featuring a variance-driven exponentially weighted moving average (EWMA) time-dependent Hurst parameter $H_t$, fundamentally distinct from recent machine learning and wavelet-based approaches in the literature. Our framework pioneers a unified roug

We study two complementary methodologies for calibrating implied volatility surfaces: analytical approximations and data-driven models based on rough path theory. On the analytical side, we revisit a second-order asymptotic expansion for the Heston model, and we propose a new, VIX-based calibration

The cryptocurrency options market is notable for its high volatility and lower liquidity compared to traditional markets. These characteristics introduce significant challenges to traditional option pricing methodologies. Addressing these complexities requires advanced models that can effectively ca

Recognizing the importance of jump risk in option pricing, we propose a neural jump stochastic differential equation model in this paper, which integrates neural networks as parameter estimators in the conventional jump diffusion model. To overcome the problem that the backpropagation algorithm is n

We study optimal consumption and retirement using a Cobb-Douglas utility and a simple model in which an interesting bifurcation arises. With high wealth, individuals plan to retire. With low wealth they plan to never retire. At a critical level of initial wealth they may choose to defer this decisio

We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and

Classic stochastic volatility models assume volatility is unobservable. We use the Volatility Index: S&P 500 VIX to observe it, to easier fit the model. We apply it to corporate bonds. We fit autoregression for corporate rates and for risk spreads between these rates and Treasury rates. Next, we div

Deep generative models are becoming increasingly used as tools for financial analysis. However, it is unclear how these models will influence financial markets, especially when they infer financial value in a semi-autonomous way. In this work, we explore the interplay between deep generative models

\begin{“abstract”} In this paper, we integrated the statistical arbitrage strategy, pairs trading, into the Black-Litterman model and constructed efficient mean-variance portfolios. Typically, pairs trading underperforms under volatile or distressed market condition because the selected asset pairs

This paper introduces the $σ$-Cell, a novel Recurrent Neural Network (RNN) architecture for financial volatility modeling. Bridging traditional econometric approaches like GARCH with deep learning, the $σ$-Cell incorporates stochastic layers and time-varying parameters to capture dynamic volatility

We study a speculative trading problem within the exploratory reinforcement learning (RL) framework of Wang et al. [2020]. The problem is formulated as a sequential optimal stopping problem over entry and exit times under general utility function and price process. We first consider a relaxed versio

We present a differential machine learning method for zero-days-to-expiry (0DTE) options under a stochastic-volatility jump-diffusion model. To handle the ultra-short-maturity regime, we express the option price in Black-Scholes form with a maturity-gated variance correction, combining supervision o

High-dimensional option pricing and hedging present significant challenges in quantitative finance, where traditional PDE-based methods struggle with the curse of dimensionality. The BSDE framework offers a computationally efficient alternative to PDE-based methods, and recently proposed deep BSDE s

We propose and analyze a continuous-time robust reinforcement learning framework for optimal stopping under ambiguity. In this framework, an agent chooses a robust exploratory stopping time motivated by two objectives: robust decision-making under ambiguity and learning about the unknown environment

Traditional stochastic control methods in finance rely on simplifying assumptions that often fail in real world markets. While these methods work well in specific, well defined scenarios, they underperform when market conditions change. We introduce FinFlowRL, a novel framework for financial stochas

Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well understood, hindering their real-world applicability. In this rese

We address the problem of asset pricing in a market where there is no risky asset. Previous work developed a theoretical model for a shadow riskless rate (SRR) for such a market in terms of the drift component of the state-price deflator for that asset universe. Assuming asset prices are modeled by

We propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for $L^p-$functionals on rough path-spaces, using linear functionals of robust, rough path signatures. I

This paper addresses the importance of incorporating various risk measures in portfolio management and proposes a dynamic hybrid portfolio optimization model that combines the spectral risk measure and the Value-at-Risk in the mean-variance formulation. By utilizing the quantile optimization techniq

Monte Carlo simulation is widely used to numerically solve stochastic differential equations. Although the method is flexible and easy to implement, it may be slow to converge. Moreover, an inaccurate solution will result when using large time steps. The Seven League scheme, a deep learning-based nu

We propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential

Options with maturities below one week, hereafter “ultra-short-term” options, have seen a sharp increase in trading activity in recent years. Yet, these instruments are difficult to price jointly using classical pricing models due to the pronounced oscillations observed in the at-the-money implied-v

Modeling time series with long- or short-memory characteristics is a fundamental challenge in many scientific and engineering domains. While fractional Brownian motion has been widely used as a noise source to capture such memory effects, its incompatibility with Itô calculus limits its applicabilit

Reinforcement Learning (RL) applied to financial problems has been the subject of a lively area of research. The use of RL for optimal trading strategies that exploit latent information in the market is, to the best of our knowledge, not widely tackled. In this paper we study an optimal trading prob

This paper studies a joint stochastic optimal control and stopping (JCtrlOS) problem motivated by aquaculture operations, where the objective is to maximize farm profit through an optimal feeding strategy and harvesting time under stochastic price dynamics. We introduce a simplified aquaculture mode

Market making is a popular trading strategy, which aims to generate profit from the spread between the quotes posted at either side of the market. It has been shown that training market makers (MMs) with adversarial reinforcement learning allows to overcome the risks due to changing market condition

Large language models (LLMs) are increasingly deployed in agentic frameworks, in which prompts trigger complex tool-based analysis in pursuit of a goal. While these frameworks have shown promise across multiple domains including in finance, they typically lack a principled model-building step, relyi

We present a robust Deep Hedging framework for the pricing and hedging of option portfolios that significantly improves training efficiency and model robustness. In particular, we propose a neural model for training model embeddings which utilizes the paths of several advanced equity option models w

As the developed world replaces Defined Benefit (DB) pension plans with Defined Contribution (DC) plans, there is a need to develop decumulation strategies for DC plan holders. Optimal decumulation can be viewed as a problem in optimal stochastic control. Formulation as a control problem requires sp

We present a systematic, trend-following strategy, applied to commodity futures markets, that combines univariate trend indicators with cross-sectional trend indicators that capture so-called {"\em momentum spillover"}, which can occur when there is a lead-lag relationship between the trending behav

Previous research on option strategies has primarily focused on their behavior near expiration, with limited attention to the transient value process of the portfolio. In this paper, we formulate Iron Condor portfolio optimization as a stochastic optimal control problem, examining the impact of the

The Capital Asset Pricing Model (CAPM) relates a well-diversified stock portfolio to a benchmark portfolio. We insert size effect in CAPM, capturing the observation that small stocks have higher risk and return than large stocks, on average. Dividing stock index returns by the Volatility Index makes

In this project, we propose to explore the Kalman filter’s performance for estimating asset prices. We begin by introducing a stochastic mean-reverting processes, the Ornstein-Uhlenbeck (OU) model. After this we discuss the Kalman filter in detail, and its application with this model. After a demons

We consider the Ornstein-Uhlenbeck (OU) process, a stochastic process widely used in finance, physics, and biology. Parameter estimation of the OU process is a challenging problem. Thus, we review traditional tracking methods and compare them with novel applications of deep learning to estimate the

We propose that the liquidity of an asset includes two components: liquidity jump and liquidity diffusion. We show that liquidity diffusion has a higher correlation with crypto wash trading than liquidity jump and demonstrate that treatment on wash trading significantly reduces the level of liquidit

The aim of this paper is to discuss an estimation and a simulation method in the \textsf{“R”} package YUIMA for a linear regression model driven by a Student-$t$ Lévy process with constant scale and arbitrary degrees of freedom. This process finds applications in several fields, for example finance,

We propose a new model for the forecasting of both the implied volatility surfaces and the underlying asset price. In the spirit of Guyon and Lekeufack (2023) who are interested in the dependence of volatility indices (e.g. the VIX) on the paths of the associated equity indices (e.g. the S&P 500), w

In this work, we present a novel machine learning approach for pricing high-dimensional American options based on the modified Gaussian process regression (GPR). We incorporate deep kernel learning and sparse variational Gaussian processes to address the challenges traditionally associated with GPR.

Central clearing counterparty houses (CCPs) play a fundamental role in mitigating the counterparty risk for exchange traded options. CCPs cover for possible losses during the liquidation of a defaulting member’s portfolio by collecting initial margins from their members. In this article we analyze t

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best-fitting model. Long-range memory and self-similarity estimators, commonly used

Prediction markets cannot exist without market makers, arbitrageurs, and other non-retail liquidity providers, yet the supply-side microstructure of Polymarket-class venues has not been characterized at on-chain pseudonymous-address scale. This paper studies non-retail participation on Polymarket us

In the context of time-subordinated Brownian motion models, Fourier theory and methodology are proposed to modelling the stochastic distribution of time increments. Gaussian Variance-Mean mixtures and time-subordinated models are reviewed with a key example being the Variance-Gamma process. A non-pa

We propose an approach to applying neural networks on linear parabolic variational inequalities. We use loss functions that directly incorporate the variational inequality on the whole domain to bypass the need to determine the stopping region in advance and prove the existence of neural networks wh

We introduce a modular framework that extends the signature method to handle American option pricing under evolving volatility roughness. Building on the signature-pricing framework of Bayer et al. (2025), we add three practical innovations. First, we train a gradient-boosted ensemble to estimate th

In this paper, we adopt the least squares Monte Carlo (LSMC) method to price time-capped American options. The aforementioned cap can be an independent random variable or dependent on asset price at random time. We allow various time caps. In particular, we give an algorithm for pricing the American

One the one hand, rough volatility has been shown to provide a consistent framework to capture the properties of stock price dynamics both under the historical measure and for pricing purposes. On the other hand, market price of volatility risk is a well-studied object in Financial Economics, and em

In this paper, we focus on the estimation of historical volatility of asset prices from high-frequency data. Stochastic volatility models pose a major statistical challenge: since in reality historical volatility is not observable, its current level and, possibly, the parameters governing its dynami

We consider the interaction between centralized trading and decentralized Proof of Stake (PoS) blockchain ecosystems. Motivated by the increasing dominance of centralized exchanges and the institutionalization of crypto markets, we study how trading activities on centralized exchanges affect staking

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zer

This paper develops a robust parametric framework for jump detection in discretely observed CKLS-type jump-diffusion processes with high-frequency asymptotics, based on the minimum density power divergence estimator (MDPDE). The methodology exploits the intrinsic asymptotic scale separation between

We study reinforcement learning for controlled diffusion processes with unbounded continuous state spaces, bounded continuous actions, and polynomially growing rewards: settings that arise naturally in finance, economics, and operations research. To overcome the challenges of continuous and high-dim

The classic semi-Markov disability model is expanded with individual and collective health claims to improve its explanatory and predictive power – in particular in the context of group experience rating. The inclusion of collective health claims leads to a computationally challenging many-body pro

In this paper, we investigate a portfolio selection problem with transaction costs under a two-factor stochastic volatility structure, where volatility follows a mean-reverting process with a stochastic mean-reversion level. The model incorporates both proportional exogenous transaction costs and en

Prediction markets, such as Polymarket, aggregate dispersed information into tradable probabilities, but they still lack a unifying stochastic kernel comparable to the one options gained from Black-Scholes. As these markets scale with institutional participation, exchange integrations, and higher vo

This paper advances interest rate modeling in the post-LIBOR era by introducing rough stochastic volatility into the Forward Market Model (FMM). We establish a rigorous asymptotic expansion of swaption implied volatility, connecting the FMM to a rough Bergomi-type framework for forward swap rates. T

This paper considers the often overlooked fact that electricity spot prices in individual European generation zones evolve as a high dimensional panel structure. A general continuous time framework is developed by formulating the panel as an ambit field indexed by a cylinder surface, where the cross

We investigate the mechanisms behind the power-law distribution of stock returns using artificial market simulations. While traditional financial theory assumes Gaussian price fluctuations, empirical studies consistently show that the tails of return distributions follow a power law. Previous resear

We introduce matrix H theory, a framework for analyzing collective behavior arising from multivariate stochastic processes with hierarchical structure. The theory models the joint distribution of the multiple variables (the measured signal) as a compound of a large-scale multivariate distribution wi

An empirical analysis, suggested by optimal Merton dynamics, reveals some unexpected features of asset volumes. These features are connected to traders’ belief and risk aversion. This paper proposes a trading strategy model in the optimal Merton framework that is representative of the collective beh

It is a well-documented fact that the correlation function of the returns on two “related” assets is generally increasing as a function of the horizon $h$ of these returns. This phenomenon, termed the Epps Effect, holds true in a wide variety of markets, and there is a large body of literature devot

We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajec

The Barndorff-Nielsen and Shephard model is a representative jump-type stochastic volatility model. Still, no method exists to compute option prices numerically for the non-martingale case with infinite active jumps. We develop two simulation methods for such a case under change of measure and condu

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0, $$ with deterministic functions $F,G_1,…,G_d$ and independent Lévy processes of infinite variation $Z_1,…,Z_d$ with regularly varying Laplac

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of

We consider an optimal trading problem under a market impact model with endogenous market resistance generated by a sophisticated trader who (partially) detects metaorders and trades against them to exploit price overreactions induced by the order flow. The model features a concave transient impact

Motivated by practical applications, we explore the constrained multi-period mean-variance portfolio selection problem within a market characterized by a dynamic factor model. This model captures predictability in asset returns driven by state variables and incorporates cone-type portfolio constrain

Using Malliavin calculus techniques, we obtain formulas for computing Greeks under different rough Volterra stochastic volatility models. Due to the fact that underlying prices are not always square integrable, we extend the classical integration by parts formula to integrable but not necessarily sq

Denoising diffusion probabilistic models (DDPMs) have emerged as powerful generative models for complex distributions, yet their use in arbitrage-free derivative pricing remains largely unexplored. Financial asset prices are naturally modeled by stochastic differential equations (SDEs), whose forwar

We establish dual attainment for the multimarginal, multi-asset martingale optimal transport (MOT) problem, a fundamental question in the mathematical theory of model-independent pricing and hedging in quantitative finance. Our main result proves the existence of dual optimizers under mild regularit

This dissertation develops and justifies a novel method for deriving approximate formulas to estimate two parameters in stochastic volatility diffusion models with exponentially-affine characteristic functions and single- or two-factor variance. These formulas aim to improve the accuracy of option p