Papers, wiki, Option Blackboard, encyclopedia, and cards.
Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well-known models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we develop simple representations of PDs and…
Banks are required to use long-term default probabilities (PDs) of their portfolios when calculating credit risk capital under internal ratings-based (IRB) models. However, the calibration models and historical data typically reflect prevailing market conditions. According to Basel recommendations, averaging annual PDs over a full economic cycle should yield the long-term PD. In practice, the available data are often…
This paper proposes a simple technical approach for the analytical derivation of Point-in-Time PD (probability of default) forecasts, with minimal data requirements. The inputs required are the current and future Through-the-Cycle PDs of the obligors, their last known default rates, and a measurement of the systematic dependence of the obligors. Technically, the forecasts are made from within a classical asset-based …
With the reform of interest rate benchmarks, interbank offered rates (IBORs) like LIBOR have been replaced by risk-free rates (RFRs), such as the Secured Overnight Financing Rate (SOFR) in the U.S. and the Euro Short-Term Rate (\euro STR) in Europe. These rates exhibit characteristics like jumps and spikes that correspond to specific market events, driven by regulatory and liquidity constraints. To capture these char…
Estimating lifetime probabilities of default (PDs) under IFRS~9 and CECL requires projecting point--in--time transition matrices over multiple years. A persistent weakness is that macroeconomic forecast errors compound across horizons, producing unstable and volatile PD term structures. This paper reformulates the problem in a state--space framework and shows that a direct Kalman filter leaves non--vanishing variabil…
Starting from the classic result of Wentzell, we derive a conditional forward equation and an associated stochastic Dupire PDE for a local-stochastic-volatility model (LSV). As an application, we obtain a density-weighted Rao--Blackwell estimator for the leverage function in LSV. We also derive an SPDE for a rolling expiry vanilla option, in the spirit of the Musiela parametrization in interest rate modeling.
Multi-asset option pricing under local- and stochastic-volatility models leads naturally to high-dimensional parabolic PDEs. We develop an end-to-end quantum PDE framework for European option pricing under local-volatility Black--Scholes and Heston models. The framework takes classical contract and model data as input and returns classical estimates of selected option values. We solve the pricing PDEs after finite-di…
We introduce Entropy-Guided Multiplicative Updates (EGMU), a convex optimization framework for constructing multi-factor target-exposure portfolios by minimizing Kullback-Leibler divergence from a benchmark under linear factor constraints. We establish feasibility and uniqueness of strictly positive solutions when the benchmark and targets satisfy convex-hull conditions. We derive the dual concave formulation with ex…
Classical portfolio models degrade under structural breaks, whereas flexible machine-learning allocation methods often lack arbitrage consistency and interpretability. We propose Causal PDE-Control Models (CPCMs), a framework that integrates structural causal drivers, nonlinear filtering, and forward-backward PDE control to produce robust and transparent allocation rules under partial information. We construct driver…
Joint calibration to SPX and VIX market data is a delicate task that requires sophisticated modeling and incurs significant computational costs. The latter is especially true when pricing of volatility derivatives hinges on nested Monte Carlo simulation. One such example is the 4-factor Markov Path-Dependent Volatility (PDV) model of Guyon and Lekeufack (2023). Nonetheless, its realism has earned it considerable atte…
The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses a…
In this article we present a novel and general methodology for building second order finite volume implicit-explicit (IMEX) numerical schemes for solving two dimensional financial parabolic PDEs with mixed derivatives. In particular, applications to basket and Heston models are presented. The obtained numerical schemes have excellent properties and are able to overcome the well-documented difficulties related with nu…
In this article we derive partial differential equations (PDEs) for pricing interest rate derivatives under the generalized Forward Market Model (FMM) recently presented by A. Lyashenko and F. Mercurio in \cite{lyashenkoMercurio:Mar2019} to model the dynamics of the Risk Free Rates (RFRs) that are replacing the traditional IBOR rates in the financial industry. Moreover, for the numerical solution of the proposed PDEs…
Portfolio's optimal drivers for diversification are common causes of the constituents' correlations. A closed-form formula for the conditional probability of the portfolio given its optimal common drivers is presented, with each pair constituent-common driver joint distribution modelled by Gaussian copulas. A conditional risk-neutral PDE is obtained for this conditional probability as a system of copulas' PDEs, allow…
In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional Itô formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools …
In this paper is proposed a 2 factor structural PDE model of pricing puttable bond with credit risk and derived the analytical pricing formula. To this end, first, a 2 factor structural (PDE) model of pricing zero coupon bond with credit risk is provided, the analytical pricing formula is derived under some conditions for default boundary and default recovery, and the strict monotonicity of the bond price function wi…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semi-linear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han and al. [32]. Our algorithm is based on …
Periodic Double Auctions (PDAs) are commonly used in the real world for trading, e.g. in stock markets to determine stock opening prices, and energy markets to trade energy in order to balance net demand in smart grids, involving trillions of dollars in the process. A bidder, participating in such PDAs, has to plan for bids in the current auction as well as for the future auctions, which highlights the necessity of g…
GRU is a lighter gated RNN with reset and update gates, often matching LSTM quality at lower cost on medium-length sequences.
Gradient descent updates parameters against the gradient of a loss: θ ← θ − η ∇_θ L. Stochastic and mini-batch variants make the method tractable on large datasets.
A GNN updates each node from its neighbors. Message passing lets the model use relational structure — markets, molecules, citation graphs — instead of forcing a grid.
In-context learning is when a frozen language model improves at a task from examples placed in the prompt, without weight updates.
PPO is a policy-gradient algorithm that clips the probability ratio so each update stays close to the previous policy, giving much of TRPO’s stability with first-order SGD.
RAG retrieves relevant documents first, then conditions a language model on that evidence so answers can be grounded, cited, and updated without retraining.
Vanishing gradients are when backprop multiplies many |Jacobian| < 1 factors so early layers receive ~0 update — the reason plain deep RNNs and tanh stacks died.
Default risk is the chance the issuer misses a contractual payment — the event credit spread is trying, noisily, to price.
PD is the probability a name defaults over a horizon — real-world for books, risk-neutral for CDS.
Bayesian inference updates a prior distribution over parameters with data via Bayes’ rule to get a posterior — beliefs as probabilities, not just a point estimate.
Itô’s lemma is the chain rule for stochastic calculus: a smooth function of an Itô process picks up a second-order dt term from (dW)² = dt.
The Kalman filter is the recursive Bayesian update for a linear-Gaussian state-space model: predict the hidden state, then correct with the new observation.
Bayesian inference updates a prior distribution over parameters with data via Bayes’ rule to get a posterior — beliefs as probabilities, not just a point estimate.
GRU is a lighter gated RNN with reset and update gates, often matching LSTM quality at lower cost on medium-length sequences.
Gradient descent updates parameters against the gradient of a loss: θ ← θ − η ∇_θ L. Stochastic and mini-batch variants make the method tractable on large datasets.
A GNN updates each node from its neighbors. Message passing lets the model use relational structure — markets, molecules, citation graphs — instead of forcing a grid.
In-context learning is when a frozen language model improves at a task from examples placed in the prompt, without weight updates.
The Kalman filter is the recursive Bayesian update for a linear-Gaussian state-space model: predict the hidden state, then correct with the new observation.
PD is the probability a name defaults over a horizon — real-world for books, risk-neutral for CDS.
PPO is a policy-gradient algorithm that clips the probability ratio so each update stays close to the previous policy, giving much of TRPO’s stability with first-order SGD.
RAG retrieves relevant documents first, then conditions a language model on that evidence so answers can be grounded, cited, and updated without retraining.
Vanishing gradients are when backprop multiplies many |Jacobian| < 1 factors so early layers receive ~0 update — the reason plain deep RNNs and tanh stacks died.