Derivatives

Machine learning for option pricing: an empirical investigation of network architectures ArXiv ID: 2307.07657 “View on arXiv” Authors: Unknown Abstract We consider the supervised learning problem of learning the price of an option or the implied volatility given appropriate input data (model parameters) and corresponding output data (option prices or implied volatilities). The majority of articles in this literature considers a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification and recent advances in machine learning methods for PDEs, we investigate empirically whether and how the choice of network architecture affects the accuracy and training time of a machine learning algorithm. We find that the generalized highway network architecture achieves the best performance, when considering the mean squared error and the training time as criteria, within the considered parameter budgets for the Black-Scholes and Heston option pricing problems. Considering the transformed implied volatility problem, a simplified DGM variant achieves the lowest error among the tested architectures. We also carry out a capacity-normalised comparison for completeness, where all architectures are evaluated with an equal number of parameters. Finally, for the implied volatility problem, we additionally include experiments using real market data. ...

Efficient Learning of Nested Deep Hedging using Multiple Options ArXiv ID: 2305.12264 “View on arXiv” Authors: Unknown Abstract Deep hedging is a framework for hedging derivatives in the presence of market frictions. In this study, we focus on the problem of hedging a given target option by using multiple options. To extend the deep hedging framework to this setting, the options used as hedging instruments also have to be priced during training. While one might use classical pricing model such as the Black-Scholes formula, ignoring frictions can offer arbitrage opportunities which are undesirable for deep hedging learning. The goal of this study is to develop a nested deep hedging method. That is, we develop a fully-deep approach of deep hedging in which the hedging instruments are also priced by deep neural networks that are aware of frictions. However, since the prices of hedging instruments have to be calculated under many different conditions, the entire learning process can be computationally intractable. To overcome this problem, we propose an efficient learning method for nested deep hedging. Our method consists of three techniques to circumvent computational intractability, each of which reduces redundant computations during training. We show through experiments that the Black-Scholes pricing of hedge instruments can admit significant arbitrage opportunities, which are not observed when the pricing is performed by deep hedging. We also demonstrate that our proposed method successfully reduces the hedging risks compared to a baseline method that does not use options as hedging instruments. ...

Efficient inverse $Z$-transform: sufficient conditions ArXiv ID: 2305.10725 “View on arXiv” Authors: Unknown Abstract We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric Lévy models, the proof is straightforward. In the case of non-symmetric Lévy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-Lévy processes. ...

Finite-Difference Solution Ansatz approach in Least-Squares Monte Carlo ArXiv ID: 2305.09166 “View on arXiv” Authors: Unknown Abstract This article presents a simple but effective and efficient approach to improve the accuracy and stability of Least-Squares Monte Carlo. The key idea is to construct the ansatz of conditional expected continuation payoff using the finite-difference solution from one dimension, to be used in linear regression. This approach bridges between solving backward partial differential equations and Monte Carlo simulation, aiming at achieving the best of both worlds. In a general setting encompassing both local and stochastic volatility models, the ansatz is proven to act as a control variate, reducing the mean squared error, thereby leading to a reduction of the final pricing error. We illustrate the technique with realistic examples including Bermudan options, worst of issuer callable notes and expected positive exposure on European options under valuation adjustments. ...

Derivatives in IslamicFinance ArXiv ID: ssrn-1015615 “View on arXiv” Authors: Unknown Abstract Despite their importance for financial sector development, derivatives are few and far between in countries where the compatibility of capital market transactio Keywords: Derivatives, Emerging Markets, Capital Market Transparency, Financial Regulation, Derivatives Complexity vs Empirical Score Math Complexity: 4.0/10 Empirical Rigor: 3.0/10 Quadrant: Philosophers Why: The paper presents conceptual valuation models and legal analysis on Shari’ah-compliant derivatives but lacks empirical backtesting, statistical metrics, or implementation-heavy data analysis. flowchart TD A["Research Goal: Assess derivative market development in emerging Islamic finance (EMIF)"] --> B["Methodology: Qualitative Case Study Analysis"] B --> C["Data/Inputs: Regulatory reports, global financial benchmarks, EMIF policy reviews"] C --> D["Computational Process: Comparative analysis of legal frameworks vs. international standards"] D --> E["Key Finding: Low derivative adoption due to regulatory ambiguity & religious compliance"] E --> F["Outcome: Proposal for standardized Shariah-compliant derivative contracts (e.g., IW'adah)"]

Discrete TimeFinance ArXiv ID: ssrn-976589 “View on arXiv” Authors: Unknown Abstract These are my Lecture Notes for a course in Discrete Time Finance which I taught in the Winter term 2005 at the University of Leeds. I am aware that the notes ar Keywords: Discrete Time Finance, Derivatives Pricing, Risk Management, Stochastic Calculus, Derivatives Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The content is heavily theoretical, focused on rigorous mathematical derivations and proofs common in academic finance courses, while there is no mention of data, backtests, or practical implementation. flowchart TD A["Research Goal: Pricing & Hedging in<br>Discrete Time Models"] --> B["Key Inputs: Probability Space,<br>Adapted Processes, Filtration"] B --> C["Methodology: Dynamic Programming<br>& Martingale Representation"] C --> D["Computational Process:<br>Recursive Pricing Algorithms"] D --> E["Key Outcome 1: Fundamental<br>Theorem of Asset Pricing"] D --> F["Key Outcome 2: Optimal<br>Discrete Hedging Strategies"]

A Primer on StructuredFinance ArXiv ID: ssrn-832184 “View on arXiv” Authors: Unknown Abstract Regulatory concerns about the impact of structured claims on financial stability in times of stress are frequently too sweeping and indistinct for a judicious a Keywords: Structured claims, Financial stability, Regulatory concerns, Derivatives Complexity vs Empirical Score Math Complexity: 3.5/10 Empirical Rigor: 2.0/10 Quadrant: Philosophers Why: The paper defines structured finance concepts with moderate conceptual modeling but lacks advanced mathematical derivations, while its empirical rigor is low as it focuses on definitions and regulatory concerns without backtesting, datasets, or implementation details. flowchart TD A["Research Goal: Assess validity of regulatory concerns regarding structured claims"] --> B["Methodology: Multi-tier analysis of financial stability"] B --> C["Data: 2008 Financial Crisis & 2020 Pandemic stress events"] C --> D["Computation: Segregating structured vs. unstructured market impacts"] D --> E{"Analysis of Derivatives & Structured Claims"} E --> F["Key Finding: Regulatory concerns are often sweeping and indistinct"] E --> G["Key Finding: Structured claims do not universally threaten stability"] F --> H["Outcome: Advocacy for judicious, precise regulation"] G --> H

Risk Management and Corporate Governance: The Case of Enron ArXiv ID: ssrn-468168 “View on arXiv” Authors: Unknown Abstract Enron Board’s Finance Sub-Committee’s approval of the first bankrupting Raptor transaction, Talon, is examined in as much detail as published documents allow. Keywords: Corporate Governance, Risk Management, Enron, Derivatives, Equities Complexity vs Empirical Score Math Complexity: 0.5/10 Empirical Rigor: 0.5/10 Quadrant: Philosophers Why: This is a qualitative legal and organizational analysis of Enron’s corporate governance, focusing on board oversight and risk management, with no mathematical modeling or data-driven empirical testing. flowchart TD A["Research Goal"] --> B{"Methodology"} B --> C["Document Analysis"] C --> D["Input: SEC Filings &<br/>Board Meeting Minutes"] D --> E["Computational Process:<br/>Raptor Transaction Reconstruction"] E --> F["Key Findings/Outcomes"] F --> G["Governance Failure:<br/>Lack of Independent Oversight"] F --> H["Risk Failure:<br/>Inadequate Risk Management<br/>& Derivative Controls"]