Dong-Young Lim

This paper discusses the problem of valuation and risk management of structured products, which have been popular in recent financial markets. We propose a recursive method based on static replication for a variety of structured products, and, in particular, focus on products with autocallable and barrier features under a general Markovian diffusion with killing. The core idea of the proposed algorithm is to recursively utilize the strike-spread approach and calendar-spread approach in the liter

Abstract A Parisian option is a variant of a barrier option such that its payment is activated or deactivated only if the underlying asset remains above or below a barrier over a certain amount of time. We show that its complex payoff feature can cause dynamic hedging to fail. As an alternative, we investigate a quasi‐static hedge of Parisian options under a more general jump‐diffusion process. Specifically, we propose a strategy of decomposing a Parisian option into the sum of other contingent

Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an alternative approach, utilizing neural networks combined with ODE solvers to learn continuous latent representations through parameterized vector fields. Neural Stochastic Differential Equations (Neural SDEs) extend Neural ODEs by incorporating a diffusion term,

Abstract We consider nonconvex stochastic optimization problems where the objective functions have super-linearly growing and discontinuous stochastic gradients. In such a setting, we provide a nonasymptotic analysis for the tamed unadjusted stochastic Langevin algorithm (TUSLA) introduced in Lovas et al. (2020). In particular, we establish nonasymptotic error bounds for the TUSLA algorithm in Wasserstein-1 and Wasserstein-2 distances. The latter result enables us to further derive nonasymptotic

Real-world time series analysis faces significant challenges when dealing with irregular and incomplete data. While Neural Differential Equation (NDE) based methods have shown promise, they struggle with limited expressiveness, scalability issues, and stability concerns. Conversely, Neural Flows offer stability but falter with irregular data. We introduce 'DualDynamics', a novel framework that synergistically combines NDE-based method and Neural Flow-based method. This approach enhances expressi

In this paper, we investigate market behaviors at high-frequency using neural networks trained with order book data. Experiments are done intensively with 110 asset pairs covering 97% of spot-futures pairs in the Korea Exchange. An efficient training scheme that improves the performance and training stability is suggested, and using the proposed scheme, the lead–lag relationship between spot and futures markets are measured by comparing the performance gains of each market data set for predictin

Time series modeling and analysis have become critical in various domains. Conventional methods such as RNNs and Transformers, while effective for discrete-time and regularly sampled data, face significant challenges in capturing the continuous dynamics and irregular sampling patterns inherent in real-world scenarios. Neural Differential Equations (NDEs) represent a paradigm shift by combining the flexibility of neural networks with the mathematical rigor of differential equations. This paper pr

This study provides a systematic and unified approach for constructing exact and static replications for exotic options, using the theory of integral equations. In particular, we focus on barrier-type options including standard, double and sequential barriers. Our primary approach to static option replication is the DEK method proposed by [Derman, E., Ergener, D. and Kani, I., Static options replication. J. Derivat., 1994, 2, 78–95]. However, our solution approach is novel in the sense that we s

We present a new class of Langevin based algorithms, which overcomes many of\nthe known shortcomings of popular adaptive optimizers that are currently used\nfor the fine tuning of deep learning models. Its underpinning theory relies on\nrecent advances of Euler's polygonal approximations for stochastic differential\nequations (SDEs) with monotone coefficients. As a result, it inherits the\nstability properties of tamed algorithms, while it addresses other known\nissues, e.g. vanishing gradients

Time series modeling and analysis have become critical in various domains. Conventional methods such as RNNs and Transformers, while effective for discrete-time and regularly sampled data, face significant challenges in capturing the continuous dynamics and irregular sampling patterns inherent in real-world scenarios. Neural Differential Equations (NDEs) represent a paradigm shift by combining the flexibility of neural networks with the mathematical rigor of differential equations. This paper pr

Physics-informed neural networks (PINNs) have emerged as a prominent approach for solving partial differential equations (PDEs) by minimizing a combined loss function that incorporates both boundary loss and PDE residual loss. Despite their remarkable empirical performance in various scientific computing tasks, PINNs often fail to generate reasonable solutions, and such pathological behaviors remain difficult to explain and resolve. In this paper, we identify that PINNs can be adversely trained

We consider non-convex stochastic optimization problems where the objective functions have super-linearly growing and discontinuous stochastic gradients. In such a setting, we provide a non-asymptotic analysis for the tamed unadjusted stochastic Langevin algorithm (TUSLA) introduced in Lovas et al. (2020). In particular, we establish non-asymptotic error bounds for the TUSLA algorithm in Wasserstein-1 and Wasserstein-2 distances. The latter result enables us to further derive non-asymptotic esti

We introduce a new Langevin dynamics based algorithm, called the extended tamed hybrid ε-order polygonal unadjusted Langevin algorithm (e-THεO POULA), to solve optimization problems with discontinuous stochastic gradients, which naturally appear in real-world applications such as quantile estimation, vector quantization, conditional value at risk (CVaR) minimization, and regularized optimization problems involving rectified linear unit (ReLU) neural networks. We demonstrate both theoretically an