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In this short note, using our geometric method introduced in a previous paper \cite{phl} and initiated by \cite{ave}, we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-…
We present a quantitative study of the markets and models evolution across the credit crunch crisis. In particular, we focus on the fixed income market and we analyze the most relevant empirical evidences regarding the divergences between Libor and OIS rates, the explosion of Basis Swaps spreads, and the diffusion of collateral agreements and CSA-discounting, in terms of credit and liquidity effects. We also review t…
This paper proposes a hybrid methodology to improve the approximation of SABR (Stochastic Alpha Beta Rho) implied volatility by combining analytical structure with machine learning. The approach augments the neural-network input representation with geometric features derived from the stochastic differential equations of the SABR model. Unlike approaches that fully replace analytical formulas with black-box models, th…
Accurately characterizing the implied volatility curves is a central challenge in option pricing and risk management. The classical SABR model by Hagan et al. has been widely adopted in practice due to its well-defined stochastic volatility structure and its tractable closed-form approximation for Black implied volatility. However, under complex market conditions, its fitting accuracy for implied volatility curves re…
Volatility Skew and Smile of Interest Rate products (Swaption and Caplet) are represented by SABR (Stochastic Alpha Beta Rho model). So, the Interest Rate derivatives model for pricing the callable exotic swaps should be comparable to the SABR volatility surface. In the interest rate derivatives models, Libor Market Model (LMM) (in a post-Libor world, Forward Market Model (FMM)) is one of the most popular models used…
The SABR model is a cornerstone of interest rate volatility modeling, but its practical application relies heavily on the analytical approximation by Hagan et al., whose accuracy deteriorates for high volatility, long maturities, and out-of-the-money options, admitting arbitrage. While machine learning approaches have been proposed to overcome these limitations, they have often been limited by simplified SABR dynamic…
In this paper, we consider three stochastic-volatility models, each characterized by distinct dynamics of instantaneous volatility: (1) a CIR process for squared volatility (i.e., the classical Heston model); (2) a mean-reverting lognormal process for volatility; and (3) a CIR process for volatility. Previous research has provided semi-analytical approximations for these models in the form of simple (non-mean-reverti…
In order to overcome the drawbacks of assuming deterministic volatility coefficients in the standard LIBOR market models to capture volatility smiles and skews in real markets, several extensions of LIBOR models to incorporate stochastic volatilities have been proposed. The efficient calibration to market data of these more complex models becomes a relevant target in practice. The main objective of the present work i…
The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular, the normal ($β=0$) SABR model is a popular model choice for interest rates because it allows negative asset values. The option price and delta under the SABR model are typically obtained via asymptotic implied volatility approximation, but these are often inaccurate and arbitrageable. Using a recently discovered pric…
We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ($β=1$) SABR model. In this model the option time-value can be represented as an integral of the form $V(T) = \int_{0}^\infty e^{-\frac{u^2}{2T}} g(u) du$ with $g(u)$ a "payoff function" which is given by an integral over the McKean kernel $G(s,t)$. We study the analyticity properties of the function $g…
Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant…
This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black-Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error in a way similar to the control variate method because the CEV model is the zero vol-of-vol limit of …
We introduce the simulation tool SABCEMM (Simulator for Agent-Based Computational Economic Market Models) for agent-based computational economic market (ABCEM) models. Our simulation tool is implemented in C++ and we can easily run ABCEM models with several million agents. The object-oriented software design enables the isolated implementation of building blocks for ABCEM models, such as agent types and market mechan…
The SABR model is a stochastic volatility model not admitting a closed form solution. Hagan, Kumar, Leniewski and Woodward have obtained an approximate solution by means of perturbative techniques. A more precise approximation was found by Henry-Labordère with the heat kernel expansion method. The latter relies on deep and hard theorems from Riemannian geometry which are almost totally unknown to the professionals of…
Loss-versus-Rebalancing (LVR) is the dominant adverse-selection cost borne by liquidity providers on automated market makers. Under geometric Brownian motion, arbitrage profit scales with the probability of a profitable block, which vanishes as the block time $Δt \to 0$; this is the standing argument for ever-shorter blocks. Modeling the reference price instead as a jump-diffusion, I show that the constant-product LV…
Financial portfolio management (PM) is one of the most applicable problems in reinforcement learning (RL) owing to its sequential decision-making nature. However, existing RL-based approaches rarely focus on scalability or reusability to adapt to the ever-changing markets. These approaches are rigid and unscalable to accommodate the varying number of assets of portfolios and increasing need for heterogeneous data. Al…
We consider a Kyle-type model where insider trading takes place among a potentially large population of liquidity traders and is subject to legal penalties. Insiders exploit the liquidity provided by the trading masses to "camouflage" their actions and balance expected wealth with the necessary stealth to avoid detection. Under a diverse spectrum of prosecution schemes, we establish the existence of equilibria for ar…
Large Language Models (LLMs) have recently been leveraged for asset pricing tasks and stock trading applications, enabling AI agents to generate investment decisions from unstructured financial data. However, most evaluations of LLM timing-based investing strategies are conducted on narrow timeframes and limited stock universes, overstating effectiveness due to survivorship and data-snooping biases. We critically ass…