Derivatives pricing is quantitative finance’s oldest deep-math tradition, and this hub holds the collection’s densest concentration of stochastic calculus. The research spans pricing models (jump diffusions, stochastic volatility, rough volatility), numerical methods (Monte Carlo variants, PDE solvers, ML surrogates for high-dimensional pricing), and the newer deep hedging literature that learns hedge ratios directly under transaction costs.
The quadrant system earns its keep here: much of this work is mathematically airtight but empirically untested (“Lab Rats” in our scoring), which is fine for pricing engines and dangerous for trading claims. When a paper proposes a new model, the questions that matter are calibration stability — does it fit today’s surface without absurd parameters tomorrow — and hedging performance out of sample, not merely in-sample fit to option quotes.
Related hubs: Volatility Modeling, Fixed Income & Rates, Machine Learning for Forecasting.
Financial market volatility is a crucial factor for investment planning, option pricing, and financial market regulation, and technology is widely recognized as a key driver of economic growth. In this project, we investigate the co-movement of technology and non-technology sectors over the last two
We synthesize and discuss some new developments in econophysics. In doing so, we focus on option pricing. We relax the assumptions of constant volatility and interest rate. In doing so, we rely on the square root of the Brownian motion. We also provide simple, closed-form pricing formulas for the Am
We consider a discrete-time incomplete multi-asset market model with continuous price jumps. For a wide class of contingent claims, including European basket call options, we compute the bounds of the interval containing the no-arbitrage prices. We prove that the lower bound coincides, in fact, with
We develop a practical framework for identifying and quantifying the hidden layers of risks and optionality embedded in American options by introducing stochasticity into one or more of their underlying determinants. The heuristic approach remedies the problems of conventional pricing systems, which
There exist two separate branches of finance that require advanced quantitative techniques: the “Q” area of derivatives pricing, whose task is to &quo
It is known that the price of call options in the Heston model is determined in a non-unique way. In this paper, this problem is analyzed from the point of view of the existing mathematical theory of uniqueness classes for degenerate parabolic equations. For the special case of degeneracy, a new exa
It is known that the value of a call option in the case of constant elasticity processes (CEV) with the indicator $α$ exceeding the critical $α=1$ is determined in a non-unique way. We show how, based on an already existing mathematical theory concerning the correctness of boundary conditions for de
Calculating true volatility is an essential task for option pricing and risk management. However, it is made difficult by market microstructure noise. Particle filtering has been proposed to solve this problem as it favorable statistical properties, but relies on assumptions about underlying market
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
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
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
The Black-Scholes framework is crucial in pricing a vast number of financial instruments that permeate the complex dynamics of world markets. Associated with this framework, we consider a second-order differential operator $L(x, {\partial_x}) := v^2(x,t) (\partial_x^2 -\partial_x)$ that carries a va
AI stocks trade at extraordinary valuations. We develop an asset pricing model in which investors use AI stocks to hedge against an AI singularity that displaces their consumption. Because markets are incomplete – investors cannot trade private AI capital – AI stocks command a premium. Market inco
Chitra et al. (2025) claim that Target Weight Mechanism (TWM) in Perpetual Demand Lending Pools (PDLPs) can lower the delta of the portfolio under certain condition. We prove that their condition is self-contradictory. Furthermore, we prove an impossibility result that no TWM can lower the delta uni
This paper implements an efficient numerical algorithm for the time-fractional Black-Scholes model governing European options. The proposed method comprises the Crank-Nicolson approach to discretize the time variable and exponential B-spline approximation for the space variable. The implemented meth
On April 22, 2020, the CME Group switched to Bachelier pricing for a group of oil futures options. The Bachelier model, or more generally the arithmetic Brownian motion (ABM), is not so widely used in finance, though. This paper provides the first comprehensive survey of options pricing under ABM. U
Environmental, Social, and Governance (ESG) finance is a cornerstone of modern finance and investment, as it changes the classical return-risk view of investment by incorporating an additional dimension of investment performance: the ESG score of the investment. We define the ESG price process and i
Based on the analog between the stochastic dynamics and quantum harmonic oscillator, we propose a market force driving model to generalize the Black-Scholes model in finance market. We give new schemes of option pricing, in which we can take various unexpected market behaviors into account to modify
We consider a financial market in which the short rate is modeled by a continuous time Markov chain (CTMC) with a finite state space. In this setting, we show how to price any financial derivative whose payoff is a function of the state of the underlying CTMC at the maturity date. We also show how t
This paper refutes the claim that the expected rate of return of the underlying asset plays no role in the Black-Scholes-Merton option pricing model.