Derivatives

Notes on the SWIFT method based on Shannon Wavelets for Option Pricing – Revisited ArXiv ID: 2401.01758 “View on arXiv” Authors: Unknown Abstract This note revisits the SWIFT method based on Shannon wavelets to price European options under models with a known characteristic function in 2023. In particular, it discusses some possible improvements and exposes some concrete drawbacks of the method. Keywords: Shannon Wavelets, Option Pricing, Characteristic Function, Spectral Methods, Numerical Methods, Derivatives ...

Representation of forward performance criteria with random endowment via FBSDE and its application to forward optimized certainty equivalent ArXiv ID: 2401.00103 “View on arXiv” Authors: Unknown Abstract We extend the notion of forward performance criteria to settings with random endowment in incomplete markets. Building on these results, we introduce and develop the novel concept of \textit{“forward optimized certainty equivalent (forward OCE)”}, which offers a genuinely dynamic valuation mechanism that accommodates progressively adaptive market model updates, stochastic risk preferences, and incoming claims with arbitrary maturities. In parallel, we develop a new methodology to analyze the emerging stochastic optimization problems by directly studying the candidate optimal control processes for both the primal and dual problems. Specifically, we derive two new systems of forward-backward stochastic differential equations (FBSDEs) and establish necessary and sufficient conditions for optimality, and various equivalences between the two problems. This new approach is general and complements the existing one for forward performance criteria with random endowment based on backward stochastic partial differential equations (backward SPDEs) for the related value functions. We, also, consider representative examples for both forward performance criteria with random endowment and for forward OCE. Furthermore, for the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measures. ...

Quantum-inspired nonlinear Galerkin ansatz for high-dimensional HJB equations ArXiv ID: 2311.12239 “View on arXiv” Authors: Unknown Abstract Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of time-dependent PDEs, the dominant paradigm involves casting the approximate solution in terms of stochastic minimization of an objective function given by the norm of the PDE residual, viewed as a function of the neural network parameters. Recently, advancements have been made in the direction of an alternative approach which shares aspects of nonlinearly parametrized Galerkin methods and variational quantum Monte Carlo, especially for high-dimensional, time-dependent PDEs that extend beyond the usual scope of quantum physics. This paper is inspired by the potential of solving Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences the exploration of nonlinearly parametrized trial functions for which the evolution equations are analytically tractable. As a precursor to the Neural Galerkin scheme, we present trial functions with evolution equations that admit closed-form solutions, focusing on time-dependent HJB equations relevant to finance. ...

Non-linear non-zero-sum Dynkin games with Bermudan strategies ArXiv ID: 2311.01086 “View on arXiv” Authors: Unknown Abstract In this paper, we study a non-zero-sum game with two players, where each of the players plays what we call Bermudan strategies and optimizes a general non-linear assessment functional of the pay-off. By using a recursive construction, we show that the game has a Nash equilibrium point. Keywords: Non-Zero-Sum Game, Bermudan Strategies, Nash Equilibrium, Recursive Construction, Non-Linear Assessment Functional, Derivatives/Contingent Claims ...

Quantum Computational Algorithms for Derivative Pricing and Credit Risk in a Regime Switching Economy ArXiv ID: 2311.00825 “View on arXiv” Authors: Unknown Abstract Quantum computers are not yet up to the task of providing computational advantages for practical stochastic diffusion models commonly used by financial analysts. In this paper we introduce a class of stochastic processes that are both realistic in terms of mimicking financial market risks as well as more amenable to potential quantum computational advantages. The type of models we study are based on a regime switching volatility model driven by a Markov chain with observable states. The basic model features a Geometric Brownian Motion with drift and volatility parameters determined by the finite states of a Markov chain. We study algorithms to estimate credit risk and option pricing on a gate-based quantum computer. These models bring us closer to realistic market settings, and therefore quantum computing closer the realm of practical applications. ...

Deeper Hedging: A New Agent-based Model for Effective Deep Hedging ArXiv ID: 2310.18755 “View on arXiv” Authors: Unknown Abstract We propose the Chiarella-Heston model, a new agent-based model for improving the effectiveness of deep hedging strategies. This model includes momentum traders, fundamental traders, and volatility traders. The volatility traders participate in the market by innovatively following a Heston-style volatility signal. The proposed model generalises both the extended Chiarella model and the Heston stochastic volatility model, and is calibrated to reproduce as many empirical stylized facts as possible. According to the stylised facts distance metric, the proposed model is able to reproduce more realistic financial time series than three baseline models: the extended Chiarella model, the Heston model, and the Geometric Brownian Motion. The proposed model is further validated by the Generalized Subtracted L-divergence metric. With the proposed Chiarella-Heston model, we generate a training dataset to train a deep hedging agent for optimal hedging strategies under various transaction cost levels. The deep hedging agent employs the Deep Deterministic Policy Gradient algorithm and is trained to maximize profits and minimize risks. Our testing results reveal that the deep hedging agent, trained with data generated by our proposed model, outperforms the baseline in most transaction cost levels. Furthermore, the testing process, which is conducted using empirical data, demonstrates the effective performance of the trained deep hedging agent in a realistic trading environment. ...

Reconciling Open Interest with Traded Volume in Perpetual Swaps ArXiv ID: 2310.14973 “View on arXiv” Authors: Unknown Abstract Perpetual swaps are derivative contracts that allow traders to speculate on, or hedge, the price movements of cryptocurrencies. Unlike futures contracts, perpetual swaps have no settlement or expiration in the traditional sense. The funding rate acts as the mechanism that tethers the perpetual swap to its underlying with the help of arbitrageurs. Open interest, in the context of perpetual swaps and derivative contracts in general, refers to the total number of outstanding contracts at a given point in time. It is a critical metric in derivatives markets as it can provide insight into market activity, sentiment and overall liquidity. It also provides a way to estimate a lower bound on the collateral required for every cryptocurrency market on an exchange. This number, cumulated across all markets on the exchange in combination with proof of reserves, can be used to gauge whether the exchange in question operates with unsustainable levels of leverage, which could have solvency implications. We find that open interest in Bitcoin perpetual swaps is systematically misquoted by some of the largest derivatives exchanges; however, the degree varies, with some exchanges reporting open interest that is wholly implausible to others that seem to be delaying messages of forced trades, i.e., liquidations. We identify these incongruities by analyzing tick-by-tick data for two time periods in $2023$ by connecting directly to seven of the most liquid cryptocurrency derivatives exchanges. ...

Differential Quantile-Based Sensitivity in Discontinuous Models ArXiv ID: 2310.06151 “View on arXiv” Authors: Unknown Abstract Differential sensitivity measures provide valuable tools for interpreting complex computational models used in applications ranging from simulation to algorithmic prediction. Taking the derivative of the model output in direction of a model parameter can reveal input-output relations and the relative importance of model parameters and input variables. Nonetheless, it is unclear how such derivatives should be taken when the model function has discontinuities and/or input variables are discrete. We present a general framework for addressing such problems, considering derivatives of quantile-based output risk measures, with respect to distortions to random input variables (risk factors), which impact the model output through step-functions. We prove that, subject to weak technical conditions, the derivatives are well-defined and derive the corresponding formulas. We apply our results to the sensitivity analysis of compound risk models and to a numerical study of reinsurance credit risk in a multi-line insurance portfolio. ...

Integration of Fractional Order Black-Scholes Merton with Neural Network ArXiv ID: 2310.04464 “View on arXiv” Authors: Unknown Abstract This study enhances option pricing by presenting unique pricing model fractional order Black-Scholes-Merton (FOBSM) which is based on the Black-Scholes-Merton (BSM) model. The main goal is to improve the precision and authenticity of option pricing, matching them more closely with the financial landscape. The approach integrates the strengths of both the BSM and neural network (NN) with complex diffusion dynamics. This study emphasizes the need to take fractional derivatives into account when analyzing financial market dynamics. Since FOBSM captures memory characteristics in sequential data, it is better at simulating real-world systems than integer-order models. Findings reveals that in complex diffusion dynamics, this hybridization approach in option pricing improves the accuracy of price predictions. the key contribution of this work lies in the development of a novel option pricing model (FOBSM) that leverages fractional calculus and neural networks to enhance accuracy in capturing complex diffusion dynamics and memory effects in financial data. ...

A new adaptive pricing framework for perpetual protocols using liquidity curves and on-chain oracles ArXiv ID: 2308.16256 “View on arXiv” Authors: Unknown Abstract This whitepaper introduces an innovative mechanism for pricing perpetual contracts and quoting fees to traders based on current market conditions. The approach employs liquidity curves and on-chain oracles to establish a new adaptive pricing framework that considers various factors, ensuring pricing stability and predictability. The framework utilizes parabolic and sigmoid functions to quote prices and fees, accounting for liquidity, active long and short positions, and utilization. This whitepaper provides a detailed explanation of how the adaptive pricing framework, in conjunction with liquidity curves, operates through mathematical modeling and compares it to existing solutions. Furthermore, we explore additional features that enhance the overall efficiency of the decentralized perpetual protocol. ...