Exact Analytical Solution for the Normal SABR Model

Pricing Continuously Monitored Barrier Options Under the Sabr Model: A Closed-Form Approximation

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance.

In this work, we present a multiple time-step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho (SABR) model. The proposed method is an extension of the one time-step Monte Carlo method that we proposed in Leitao et al. (Appl. Math. Comput. 293: 461–479, 2017). We call it the mSABR method. A highly efficient method results, with many interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time-step Monte Carlo method is especially useful for long-term or for exotic options. This paper is a short version of an already published paper (Leitao et al. On an efficient multiple time-step Monte Carlo simulation of the SABR model. Quantitative Finance.

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

This paper reviews 17 studies addressing dynamic option hedging in frictional markets through Deep Reinforcement Learning (DRL). Specifically, this work analyzes the DRL models, state and action spaces, reward formulations, data generation processes and results for each study. It is found that policy methods such as DDPG are more commonly employed due to their suitability for continuous action spaces. Despite diverse state space definitions, a lack of consensus exists on variable inclusion, prompting a call for thorough sensitivity analyses. Mean-variance metrics prevail in reward formulations, with episodic return, VaR and CvaR also yielding comparable results. Geometric Brownian motion is the primary data generation process, supplemented by stochastic volatility models like SABR (stochastic alpha, beta, rho) and the Heston model. RL agents, particularly those monitoring transaction costs, consistently outperform the Black–Scholes Delta method in frictional environments. Although consistent results emerge under constant and stochastic volatility scenarios, variations arise when employing real data. The lack of a standardized testing dataset or universal benchmark in the RL hedging space makes it difficult to compare results across different studies. A recommended future direction for this work is an implementation of DRL for hedging American options and an investigation of how DRL performs compared to other numerical American option hedging methods.

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

Hyperbolic Normal Stochastic Volatility Model

Exact analytical solution of a nonlinear reaction–diffusion model in porous catalysts

A Note on the Option Price and 'Mass at Zero in the Uncorrelated SABR Model and Implied Volatility Asymptotics'

Granular materials are extensively used throughout many industries. In countries like Australia and South Africa, national economies are critically dependent on the agricultural and mining industries, and it is important to understand bulk material behaviour in order to handle these materials properly. There are many diverse approaches to modelling granular materials. In this thesis, the Coulomb-Mohr continuum mechanical theory of granular materials is utilised, and some exact analytical solutions are presented for both stress and velocity fields. These solutions are formally valid provided sin φ = 1, where φ is the angle of internal friction, and they provide limiting solutions for so called highly frictional granular materials, for which sin φ ≈ 1. The solutions so obtained are exploited for three real industrial granular problems, namely gravity flow from converging hoppers, the stress distribution in stockpiles and finally, the stress distribution within rat-holes. The flow of granular materials, in the presence of gravity, through a converging hopper is an important problem that occurs in many industrial processes. Here, for highly frictional granular materials for both wedge and conical hopper flows, exact analytical solutions are obtained to determine the stress and velocity fields within the neighborhood of the hopper outlet. For the same problem, these exact solutions are exploited as the leading order terms in a regular perturbation series, where the correction terms are of order 1− sin φ. In this way, approximate analytical solutions are obtained that can be used to describe the behaviour of more realistic granular materials. The practice of storing granular materials in stockpiles occurs throughout the ii world in many industrial situations. In this thesis, exact solutions for highly frictional granular materials are used for the determination of the stress distributions within two-dimensional and axially symmetric stockpiles. Stockpile profiles considered here include the standard wedge and cone, as well as two-dimensional parabolic and axially symmetric cubic curved shaped stockpiles. We emphasise that curved stockpiles have not previously been considered in the literature. Our model assumes that the stockpile is composed of two regions, namely an inner rigid region and an outer yield region. The exact solution applies in the outer yield region, which is then extended continuously into the inner rigid region. A rat-hole is the general term used to describe those stable cavities, which frequently occur in storage hoppers and stockpiles, whose formation prevents further material falling through the outlet. In this thesis, we present exact solutions for both two-dimensional and axially symmetric rat-holes, where it is assumed that the highly frictional granular material has flowed through a central outlet, until the point is reached when there is no further flow of material. The remaining material is assumed to be in limiting equilibrium, that is static but on the point of yield. In both two dimensions and axial symmetry, two types of rat-holes are studied. The first type has a sloping upper surface of infinite extent, where the material is resting on a sloping rigid base containing an infinitesimal central outlet. The second type has a curved upper surface, where the material is resting between vertical walls and a horizontal rigid plane containing an infinitesimal central outlet. Again, we emphasise that neither of these rat-hole problems have previously been investigated in the literature.

This paper presents an exact analytical solution of one-dimensional transport of a contaminant undergoing advection, dispersion, sorption, and first-order decay, subject to unique boundary conditions, a first-order decaying contaminant concentration in time at the source with a constant concentration at infinity. This exact solution provides a method to study a new area of physical processes and to evaluate approximate methods and models that have been developed to analyze these problems. Recently two fields have developed interest in modeling physical processes that are best conceptualized using a decaying source boundary condition. These two fields are long-term modeling of recalcitrant non-aqueous phase liquid (NAPL) spills and radioactive waste disposal. Both of which can be approximated using a decaying source term. This paper briefly introduces some general field transport problems that can conceptually be described using a decaying source boundary condition. The paper then describes the governing equations, boundary and initial conditions, and the solution techniques used to develop the analytical solution. Next the analytic solution is provided and discussed along with some example cases. The results of the exact analytical solution are compared to another similar solution from the literature.

When polymeric liquids undergo large‐amplitude shearing oscillations, the shear stress responds as a Fourier series, the higher harmonics of which are caused by fluid nonlinearity. Previous work on large‐amplitude oscillatory shear flow has produced analytical solutions for the first few harmonics of a Fourier series for the shear stress response (none beyond the fifth) or for the normal stress difference responses (none beyond the fourth) [JNNFM,166, 1081 (2011)], but this growing subdiscipline of macromolecular physics has yet to produce an exact solution. Here, we derive what we believe to be the first exact analytical solution for the response of the extra stress tensor in large‐amplitude oscillatory shear flow. Our solution, unique and in closed form, includes both the normal stress differences and the shear stress for both startup and alternance. We solve the corotational Maxwell model as a pair of nonlinear‐coupled ordinary differential equations, simultaneously. We choose the corotational Maxwell model because this two‐parameter model (η0andλ) is the simplest constitutive model relevant to large‐amplitude oscillatory shear flow, and because it has previously been found to be accurate for molten plastics (when multiple relaxation times are used). Byrelevantwe mean that the model predicts higher harmonics. We find good agreement between the first few harmonics of our exact solution, and of our previous approximate expressions (obtained using the Goddard integral transform). Our exact solution agrees closely with the measured behavior for molten plastics, not only at alternance, but also in startup.

Exact quantum-mechanical solution for the one-dimensional harmonic oscillator model asymmetrically confined into the infinite well

In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7.72, and 40.25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.

The exploding reflector model has been suggested as a way to simplify calculation of wave propagation. Previous work has produced numerical solutions to a model where two media are in contact and a signal is generated at the surface of the upper layer. Scatter from the interface was modelled as an exploding reflector. The strength and shape of the signal that arrives at the base of the lower layer were calculated following the scatter from the interface. These numerical solutions gave results that were comparable to finite difference solutions of the wave equation.The physical meaning and an extension of the exploding reflector model are discussed here. Exact analytical solutions to a two layer model separated by an incoherent scattering layer are derived by two distinct methods, one using the mathematical computer package Maple and then by simple geometrical arguments. Exact analytical solutions could be found for the case where the layers had identical thickness and sound velocities and the interface scattered the waves incoherently.Two different laws for the reduction in intensity of the signal as a function of distance were examined being the inverse unity power law and inverse square law. Two different laws for scattering from the interface were also examined being an isotropic scatter for which scattering intensity is independent of scattering angle, and a Lambertian scatter which obeys Lambert's Cosine Law.Analytical solutions to the model with unequal thicknesses or unequal acoustic velocities were investigated. Analytical solutions could be found in some cases but could not usually be reduced to simple expressions that could be presented here. The general solution is possible but required the solution to the general quartic equation. Solution expressions are very long and can only reasonably be given as process flows in a mathematical computing package such as Maple.The models described here and the solutions found previously may find a use in describing the incoherent scatter from interfaces between layers.