Stochastic Calculus

‘P’ Versus ‘Q’: Differences and Commonalities between the Two Areas of QuantitativeFinance ArXiv ID: ssrn-1717163 “View on arXiv” Authors: Unknown Abstract There exist two separate branches of finance that require advanced quantitative techniques: the “Q” area of derivatives pricing, whose task is to &quo Keywords: Quantitative Finance, Derivatives Pricing, Stochastic Calculus, Fixed Income, Derivatives Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The paper delves deep into stochastic calculus, PDEs, and advanced stochastic processes (e.g., Ornstein-Uhlenbeck, Heston model), indicating high mathematical complexity. However, it is purely theoretical/conceptual with no data, code, backtests, or implementation details, resulting in very low empirical rigor. flowchart TD A["Research Question<br>Differences & Commonalities<br>between P & Q Finance"] --> B["Methodology<br>Literature Review & Comparative Analysis"] B --> C["Key Inputs<br>Stochastic Calculus Models &<br>Derivatives Pricing Frameworks"] C --> D{"Computational Process<br>Analysis of Methodologies"} D --> E["P Area<br>Pricing & Risk Management<br>(Stochastic Control, Calibration)"] D --> F["Q Area<br>Derivatives Pricing & Hedging<br>(Risk-Neutral Valuation)"] E & F --> G["Outcomes<br>Unified Quantitative Framework<br>Distinct Methodologies &<br>Common Mathematical Foundations"]

Error bound for the asymptotic expansion of the Hartman-Watson integral ArXiv ID: 2504.04992 “View on arXiv” Authors: Unknown Abstract This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $θ(r,t)$ in the regime $rt=ρ$ constant. The leading order term has the form $θ(ρ/t,t)=\frac{“1”}{“2πt”}e^{"-\frac{1"}{“t”} (F(ρ)-π^2/2)} G(ρ) (1 + \vartheta(t,ρ))$, where the error term is bounded uniformly over $ρ$ as $|\vartheta(t,ρ)|\leq \frac{“1”}{“70”}t$. ...

Stochastic Calculus for Option Pricing with Convex Duality, Logistic Model, and Numerical Examination ArXiv ID: 2408.05672 “View on arXiv” Authors: Unknown Abstract This thesis explores the historical progression and theoretical constructs of financial mathematics, with an in-depth exploration of Stochastic Calculus as showcased in the Binomial Asset Pricing Model and the Continuous-Time Models. A comprehensive survey of stochastic calculus principles applied to option pricing is offered, highlighting insights from Peter Carr and Lorenzo Torricelli’s ``Convex Duality in Continuous Option Pricing Models". This manuscript adopts techniques such as Monte-Carlo Simulation and machine learning algorithms to examine the propositions of Carr and Torricelli, drawing comparisons between the Logistic and Bachelier models. Additionally, it suggests directions for potential future research on option pricing methods. ...

Optimal execution and speculation with trade signals ArXiv ID: 2306.00621 “View on arXiv” Authors: Unknown Abstract We propose a price impact model where changes in prices are purely driven by the order flow in the market. The stochastic price impact of market orders and the arrival rates of limit and market orders are functions of the market liquidity process which reflects the balance of the demand and supply of liquidity. Limit and market orders mutually excite each other so that liquidity is mean reverting. We use the theory of Meyer-$σ$-fields to introduce a short-term signal process from which a trader learns about imminent changes in order flow. Her trades impact the market through the same mechanism as other orders. With a novel version of Marcus-type SDEs we efficiently describe the intricate timing of market dynamics at moments when her orders concur with that of others. In this setting, we examine an optimal execution problem and derive the Hamilton–Jacobi–Bellman (HJB) equation for the value function of the trader. The HJB equation is solved numerically and we illustrate how the trader uses the signals to enhance the performance of execution problems and to execute speculative strategies. ...

Risk-Neutral Probabilities Explained ArXiv ID: ssrn-1395390 “View on arXiv” Authors: Unknown Abstract All too often, the concept of risk-neutral probabilities in mathematical finance is poorly explained, and misleading statements are made. The aim of this paper Keywords: risk-neutral probabilities, martingales, stochastic calculus, derivatives pricing, Quantitative Finance Complexity vs Empirical Score Math Complexity: 7.0/10 Empirical Rigor: 2.0/10 Quadrant: Lab Rats Why: The paper focuses on theoretical foundations, including continuous-time stochastic processes like geometric Brownian motion and martingales, but lacks any empirical backtesting, data, or implementation details. flowchart TD A["Research Goal: Explain Risk-Neutral Probabilities clearly"] --> B["Methodology: Critical Review of Stochastic Calculus"] B --> C["Input: Misleading Statements in Texts"] C --> D["Computational Process: Martingale Measure Derivation"] B --> E["Input: Derivatives Pricing Models"] E --> D D --> F["Key Finding: Q-Measure vs. P-Measure"] D --> G["Key Finding: No-Arbitrage Pricing Framework"]

MathematicalFinanceIntroduction to Continuous Time Financial Market Models ArXiv ID: ssrn-976593 “View on arXiv” Authors: Unknown Abstract These are my Lecture Notes for a course in Continuous Time Finance which I taught in the Summer term 2003 at the University of Kaiserslautern. I am aware that t Keywords: continuous time finance, stochastic calculus, option pricing, martingales, stochastic differential equations, Derivatives / Quantitative Finance Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The paper presents dense, advanced mathematics centered on stochastic analysis, stochastic calculus, and derivations of the Black-Scholes model, with no empirical data or backtesting. flowchart TD A["Research Goal: Develop Continuous Time Financial Market Models"] --> B["Methodology: Stochastic Calculus & Martingales"] B --> C["Data: Geometric Brownian Motion SDE Inputs"] C --> D["Computation: Black-Scholes Option Pricing & PDE Solution"] D --> E["Outcome: Valuation of Derivatives & Risk Management Insights"]

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"]