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We study the problems of consistency and the existence of finite-dimensional realizations for multi-curve interest rate models of Heath–Jarrow–Morton type, generalizing the geometric approach developed by T. Björk and co-authors for the classical single-curve setting. We characterize when a multi-curve interest rate model is consistent with a given parameterized family of forward curves and spreads and when a model can be realized by a finite-dimensional state process. We illustrate the general theory in a number of model classes and examples, providing explicit constructions of finite-dimensional realizations. Based on these theoretical results, we perform the calibration of a three-curve Hull–White model to market data and analyse the stability of the estimated parameters. [ABSTRACT FROM AUTHOR]

The Nelson–Siegel and the Svensson models are two widely used models for the term structure of interest rates. These models are quite simple and intuitive, but fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, we provide a novel mathematical analysis of the fitting problem based on parametric optimization. We formulate the fitting problem as a separable nonlinear least-squares problem, in which the linear parameters can be eliminated. We provide a thorough discussion on the conditioning of the inner part of the reformulated problem and show that many of the reported difficulties encountered when solving it are inherent to the problem formulation itself and cannot be tackled by choosing a particular optimization algorithm. Our stability analysis provides novel insights that we use to show that some of the ill-conditioning can be avoided, and that a suitably chosen penalty approach can be used to address the remaining ill-conditioning. Numerical results indicate that this approach has the expected impact while being independent of any choice of a particular optimization algorithm. We further establish smoothness properties of the reduced objective function, putting global optimization methods for the reduced problem on a sound mathematical basis. [ABSTRACT FROM AUTHOR]

We propose a new, data-driven approach for efficient pricing of – fixed- and floating-strike – discrete arithmetic Asian and Lookback options when the underlying process is driven by the Heston model dynamics. The method proposed in this article constitutes an extension of Perotti and Grzelak [Fast sampling from time-integrated bridges using deep learning, J. Comput. Math. Data Sci. 5 (2022)], where the problem of sampling from time-integrated stochastic bridges was addressed. The model relies on the Seven-League scheme [S. Liu et al. The seven-league scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations, Risks 10 (2022), p. 47], where artificial neural networks are employed to 'learn' the distribution of the random variable of interest utilizing stochastic collocation points [L.A. Grzelak et al. The stochastic collocation Monte Carlo sampler: Highly efficient sampling from expensive distributions, Quant. Finance 19 (2019), pp. 339–356]. The method results in a robust procedure for Monte Carlo pricing. Furthermore, semi-analytic formulae for option pricing are provided in a simplified, yet general, framework. The model guarantees high accuracy and a reduction of the computational time up to thousands of times compared to classical Monte Carlo pricing schemes. [ABSTRACT FROM AUTHOR]

This paper studies equity basket options – i.e. multi-dimensional derivatives whose payoffs depend on the value of a weighted sum of the underlying stocks – and develops a new and innovative approach to ensure consistency between options on individual stocks and the index comprising them. Specifically, we show how to resolve a well-known problem that when individual constituent distributions of an equity index are inferred from the single-stock option markets and combined in a multi-dimensional local/stochastic volatility model, the resulting basket option prices will not generate a skew matching that of the options on the equity index corresponding to the basket. [ABSTRACT FROM AUTHOR]

Exposure simulations are fundamental to many xVA calculations and are a nested expectation problem where repeated portfolio valuations create a significant computational expense. Sensitivity calculations which require shocked and unshocked valuations in bump-and-revalue schemes exacerbate the computational load. A known reduction of the portfolio valuation cost is understood to be found in polynomial approximations, which we apply in this article to interest rate sensitivities of expected exposures. We consider a method based on the approximation of the shocked and unshocked valuation functions, as well as a novel approach in which the difference between these functions is approximated. Convergence results are shown, and we study the choice of interpolation nodes. Numerical experiments with interest rate derivatives are conducted to demonstrate the high accuracy and remarkable computational cost reduction. We further illustrate how the method can be extended to more general xVA models using the example of CVA with wrong-way risk. [ABSTRACT FROM AUTHOR]

This paper deals with a copula-based stochastic dependence problem in the context of financial risks. We discuss the financial framework for assessing the theoretical up-front value of government guarantees on bank liabilities. EU States widely use these contracts to improve the financial system's stability and manage the banking sector in crisis situations; in Italy, they have also been used to address the consequences of the Covid-19 emergency. From a market viewpoint, we deal with a defaultable guarantee contract where the State-guarantor and the bank-borrower are both subject to default risk, and their risks are interconnected. We show that the classical Gaussian copula is not satisfactory for modeling the dependence among the considered risks. Indeed, using the benchmark market model for credit risk portfolio management, we highlight some contradictory results observed for the up-front values of the guarantee when the default intensity of the guarantor is smaller than that of the borrower. Then, we introduce a new family of modified Gaussian copulas that overcomes the limitations of the standard approach, allowing to determine realistic results in terms of the guarantees "mark-to-model" value when the benchmark market model does not work. Numerical simulations validate the theoretical proposal. [ABSTRACT FROM AUTHOR]

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it. [ABSTRACT FROM AUTHOR]

The minimum (worst case) value of a long-only portfolio of bonds, over a convex set of yield curves and spreads, can be estimated by its sensitivities to the points on the yield curve. We show that sensitivity based estimates are conservative, i.e., underestimate the worst case value, and that the exact worst case value can be found by solving a tractable convex optimization problem. We then show how to construct a long-only bond portfolio that includes the worst case value in its objective or as a constraint, using convex–concave saddle point optimization. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for predicting electricity prices in fuzzy and stochastic environments. First, we analyze historical data on daily PJM Western Hub Real-Time Peak electricity prices and extract relevant features for modeling. Then, we introduce a comprehensive model for predicting the price of electricity within the PJM market. To adapt the model to the electricity price data, we implement algorithms based on the least squares algorithm to calibrate the model’s parameters. However, considering the model’s purpose in forecasting future electricity prices, we apply the parameters as fuzzy numbers. After that, acknowledging the significant losses that severe electricity fluctuations can incur for producers, consumers, and industrial owners, we develop a formula to assess the probability of such fluctuations occurring in both random and fuzzy spaces. Finally, because electricity transactions are conducted through contracts, we calculate the price of these contracts across various time frames and use the fuzzy numbers to obtain a margin for the price of the contracts. [ABSTRACT FROM AUTHOR]

We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed Lévy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the volatilities of FX rates exhibit self-exciting dynamics. By relying on the theory of affine processes, we show that our approach is analytically tractable and that the model structure is invariant under a suitable class of risk-neutral measures. A semi-closed pricing formula for currency options is obtained by Fourier methods. We propose two calibration methods, also by relying on deep-learning techniques, and show that a simple specification of the model can achieve a good fit to market data on a currency triangle. [ABSTRACT FROM AUTHOR]

In this paper, we study the pricing of contracts in fixed income markets under volatility uncertainty in the sense of Knightian uncertainty or model uncertainty. The starting point is an arbitrage-free bond market under volatility uncertainty. The uncertainty about the volatility is modeled by a G-Brownian motion, which drives the forward rate dynamics. The absence of arbitrage is ensured by a drift condition. Such a setting leads to a sublinear pricing measure for additional contracts, which yields either a single price or a range of prices and provides a connection to hedging prices. Similar to the forward measure approach, we define the forward sublinear expectation to simplify the pricing of cashflows. Under the forward sublinear expectation, we obtain a robust version of the expectations hypothesis, and we show how to price options on forward prices. In addition, we develop pricing methods for contracts consisting of a stream of cashflows, since the nonlinearity of the pricing measure implies that we cannot price a stream of cashflows by pricing each cashflow separately. With these tools, we derive robust pricing formulas for all major interest rate derivatives. The pricing formulas provide a link to the pricing formulas of traditional models without volatility uncertainty and show that volatility uncertainty naturally leads to unspanned stochastic volatility. [ABSTRACT FROM AUTHOR]

We propose an efficient computational method based on continuous-time Markov chain (CTMC) approximation to compute the distributions of the speed and duration of drawdown for general one-dimensional (1D) time-homogeneous Markov processes. We derive linear systems for the Laplace transforms of drawdown quantities and show how to solve them efficiently by recursion. In addition, we prove the convergence of our method and obtain a sharp estimate of the convergence rate under some assumptions. As applications, we consider pricing two financial options that provide protection against drawdown risk. Finally, we demonstrate the accuracy and efficiency of our method through extensive numerical experiments. [ABSTRACT FROM AUTHOR]

We establish theoretical properties of the solution to a two-variance-driven interest rate model with super-linear coefficient terms. Since this model is not tractable analytically, we construct an implementable numerical method to approximate it and prove the finite-time strong convergence theory under the local Lipschitz condition. Finally, we provide simulation examples to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the L p -distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes. [ABSTRACT FROM AUTHOR]

In this paper, we concentrate on the European option pricing problem in incomplete markets by taking Vasicek stochastic interest rate model into account and expanding it in accordance with a Lévy process named Variance Gamma (VG) process. In doing so, first, we prove the existence and uniqueness of the solution to the stochastic differential equation of this model. Then, by the means of the hedging and replicating techniques, pricing formulas for the bond and, in succession, for European options are established. Finally, from a numerical point of view, we will analyze the value of a bond based on our consideration model and also, we will study a European call option bond price. All these results are reported through the Monte-Carlo simulation method. [ABSTRACT FROM AUTHOR]

We study Markowitz's mean-variance portfolio selection problem in a continuous-time Black–Scholes market with different borrowing and saving rates. The associated Hamilton–Jacobi–Bellman equation is fully nonlinear. Using a delicate partial differential equation and verification argument, the value function is proven to be C 3 , 2 smooth. It is also shown that there are a borrowing boundary and a saving boundary which divide the entire trading area into a borrowing-money region, an all-in-stock region, and a saving-money region in ascending order. The optimal trading strategy turns out to be a mixture of continuous-time strategy (as suggested by most continuous-time models) and discontinuous-time strategy (as suggested by models with transaction costs): one should put all the wealth in the stock in the middle all-in-stock region and continuously trade it in the other two regions in a feedback form of wealth and time. It is never optimal to short sale the stock. Numerical examples are also presented to verify the theoretical results and to give more financial insights beyond them. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generic interest rate market in the presence of roll-over risk, which generates spreads in spot/forward term rates. We do not require classical absence of arbitrage and rely instead on a minimal market viability assumption, which enables us to work in the context of the benchmark approach. In a Markovian setting, we extend the control-theoretic approach of Gombani and Runggaldier (Math. Finance23 (2013) 659–686) and derive representations of spot/forward spreads as value functions of suitable stochastic optimal control problems, formulated under the real-world probability and with power-type objective functionals. We determine endogenously the funding–liquidity spread by relating it to the risk-sensitive optimisation problem of a representative investor. [ABSTRACT FROM AUTHOR]

This article celebrates the legacy of Tomas Björk. As we reflect on Tomas' life and career, we explore his personal and professional journey, highlighting his most significant contributions to mathematical finance. [ABSTRACT FROM AUTHOR]

Discount is the difference between the face value of a bond and its present value. We propose an arbitrage-free dynamic framework for discount models, which provides an alternative to the Heath–Jarrow–Morton framework for forward rates. We derive general consistency conditions for factor models, and discuss affine term structure models in particular. There are several open problems, and we outline possible directions for further research. [ABSTRACT FROM AUTHOR]

We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given market. Trading is subject to nonproportional transaction costs and portfolio constraints, and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterisation requires an appropriate extension of the classical fundamental theorem of asset pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk–reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume. [ABSTRACT FROM AUTHOR]

We propose a method based on continuous-time Markov chain (CTMC) approximation to compute the distribution of Parisian stopping times and to price options of Parisian style under general one-dimensional Markov processes. We prove the convergence of the method under a general setting and obtain sharp estimates of the convergence rate for diffusion models. Our theoretical analysis reveals how to design the grid of the CTMC to achieve faster convergence. Numerical experiments are conducted to demonstrate the accuracy and efficiency of our method for both diffusion and jump models. To show the versatility of our approach, we develop extensions for multi-sided Parisian stopping times, the joint distribution of Parisian stopping times and first passage times, Parisian bonds, regime-switching models and stochastic volatility models. [ABSTRACT FROM AUTHOR]

In this paper we are dealing with a robust investment-consumption optimization problem in an incomplete market with a switching regime stochastic interest rate. Our methodology combines duality approach with stochastic control techniques (applied to the dual problem) specific to a non-Markovian setting, such as dynamic programming principle (initiated in Karoui and Quenez (SIAM J Control Optim 33(1):29–66, 1995)) and Backward Stochastic Differential Equations (BSDEs) theory. An auxiliary dual problem is established by means of infinite-dimensional convex duality. We derive explicit formulas for the optimal trading strategy and consumption rate in terms of the solution of some nonstandard BSDE with jumps. Links to other significant results in the domain are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments. [ABSTRACT FROM AUTHOR]

In this paper, we study the distance between the solution of stochastic Volterra integral equations and that of ordinary stochastic differential equations. This problem arises naturally in the recent studies. Based on the techniques of Malliavin calculus, we first provide explicit bounds on total variation distance between the solutions. Then, we apply our bounds to estimate the rate of convergence in the Smoluchowski-Kramers approximation and to investigate noise sensitivity of some stochastic equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

We propose a bottom-up quantitative reverse stress testing framework that identifies forward-looking fragilities tailored to a bank's portfolio, credit and funding strategies, models, and calibration constraints. Thus, instead of relying on historical events, we run a Monte Carlo simulation, and we mine those future states that contribute the most to a bank's cost of capital expressed in terms of scenario differential. This approach allows identifying both the systemic and idiosyncratic weaknesses of the bank's portfolio, with applications that include solvency risk, extreme events hedging, liquidity risk management, trading and credit limits, model validation and model risk management. [ABSTRACT FROM AUTHOR]

Exponential Lévy processes provide a natural and tractable generalization of the classic Black–Scholes–Merton model which account for several stylized features of financial markets, including jumps and kurtosis. In the existing literature, closed-form option pricing formulas are sparse for exponential Lévy models, outside of special cases such as Merton's jump diffusion, and complex numerical techniques are required even to price European options. To bridge the gap, this work provides a comprehensive and unified pricing framework for vanilla and exotic path independent options under the Variance Gamma (VG), Finite Moment Log Stable (FMLS), one-sided Tempered Stable (TS), and Normal Inverse Gaussian (NIG) models. We utilize the Mellin Transform and residue calculus to obtain closed-form series representations for the price of several options, including vanillas (European), digitals, power, and log options. These formulas provide nice theoretical representations, but are also efficient to evaluate in practice, as numerous numerical experiments demonstrate. The closed-form nature of these option pricing formulas makes them ideal for adoption in practical settings, as they do not require complicated pricing methods to achieve high-accuracy prices, and the resulting pricing error is reliably controllable. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new connection between Cox–Ingersoll–Ross (CIR) and reflected Ornstein–Uhlenbeck (ROU) models driven by either a standard Wiener process or a fractional Brownian motion with H > 1 2 . We prove that, with probability 1, the square root of the CIR process converges uniformly on compacts to the ROU process as the mean reversion parameter tends to either σ 2 / 4 (in the standard case) or to 0 (in the fractional case). This also allows to obtain a new representation of the reflection function of the ROU as the limit of integral functionals of the CIR processes. The results of the paper are illustrated by simulations. [ABSTRACT FROM AUTHOR]

Abrupt happenings in financial markets have resulted to the need to adopt Lévy processes such as a variance gamma process in modelling financial derivatives since it has the ability to capture jumps that occur in such scenario. Sensitivity analysis in such market scenarios having characteristics of Lévy processes is made easier by adopting the integration by part techniques of Malliavin calculus. Thus, we apply the tools of the Malliavin calculus to obtain the special types of the greek vega required in sensitivity analysis in an interest rate market driven by the variance gamma process.

We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on λ units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean–Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity λ such that the players agree on the price. Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price. [ABSTRACT FROM AUTHOR]

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylised facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive, in two different ways, an explicit analytic expression for the joint characteristic function of the log-price and its integrated variance in general Gaussian stochastic volatility models. That analytic expression can be approximated by closed-form matrix expressions. This opens the door to fast approximation of the joint density and pricing of derivatives on both the stock and its realised variance by using Fourier inversion techniques. In the context of rough volatility modelling, our results apply to the (rough) fractional Stein–Stein model and provide the first analytic formulas for option pricing known to date, generalising that of Stein–Stein, Schöbel–Zhu and a special case of Heston. [ABSTRACT FROM AUTHOR]

We propose a new method to efficiently price swap rate derivatives under the LIBOR market model with stochastic volatility and displaced diffusion. This method applies series expansion techniques built around Gaussian (Gram–Charlier) or Gaussian mixture densities to polynomial processes. The standard pricing method for the considered model relies on dynamics freezing to recover a Heston-type model for which analytical formulas are available. This approach is time-consuming, and efficient approximations based on Gram–Charlier expansions have been proposed recently. In this article, we first discuss the fact that for a class of stochastic volatility model, including the Heston one, the classical sufficient condition ensuring the convergence of Gram–Charlier series is not satisfied. Then we propose an approximating model based on a Jacobi process for which we can prove the stability of Gram–Charlier-type expansions. For this approximation, we have been able to prove a strong convergence towards the original model; moreover, we give an estimate of the convergence rate. We also prove a new result on the convergence of the Gram–Charlier series when the volatility factor is not bounded from below. We finally illustrate our convergence results with numerical examples. [ABSTRACT FROM AUTHOR]

The paper establishes a related differential equation model about changes in financial interest rates. It uses information related to liquidity to feedback the law and stability of differential equations in interest rate changes. The article applies stochastic processes and partial differential equations to complex financial networks to confirm node yields in financial market networks. It confirms the existence of interest rate stickiness in Chinese financial markets. The advantage of this interest rate model is that when the external economic environment changes, the state of interest rates will also change accordingly.