Abstract We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. prove that these are well-posed in $$(L^1-L^\infty )$$ ( L 1 - ∞ <mml:...

We propose, formalise and analyse the deep parametric PDE method to solve high-dimensional partial differential equations with a focus on financial applications. A single neural network approximates solution of whole family PDEs after being trained without need sample solutions. As practical application, we compute option prices Greeks in multivariate Black–Scholes model as there is an urgent f...

The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations for the adjoint process in stochastic control, as ...

After the discovery of fractal structures financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by non-standard stochastic process. The resultant mode...

Abstract We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima a geometric Brownian motion. It is shown that times are th first at which underlying process reaches lower or upper stochastic boundaries depending on current values its maximum minimum. The proof based reduction original sequences single...

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating actual financial market pricing, such as risk-free interest rate, stock price, underlying and security price volatility may be inaccurate real world. Therefore, it great practical significance to study fractional fuzzy model. In this p...

Abstract Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred PINNs in approximating a large class linear parabolic PDEs, namely Kolmogorov equations that include heat equation and Black-Scholes option pricing, as examples. construct networks, whose PINN residual (generalization error) can be made smal...

Stochastic volatility models lead to more realistic option prices than the Black-Scholes model which uses a constant volatility. Based on such models a two-dimensional parabolic partial differential equation can derived for option prices. Due to the early exercise possibility of American option contracts the arising pricing problems are free boundary problems. In this paper we consider the nume...

Option valuation has been a challenging issue of financial engineering and optimization for a long time. The increasing complexity of market conditions requires utilization of advanced models that, commonly, do not lead to closed-form solutions. Development of novel numerical procedures, which prove to be efficient within various option valuation problems, is therefore worthwhile. Notwithstan...

This paper uses risk-adjusted lognormal probabilities to derive the BlackScholes formula and explain the factors N(d1) and N(d2). It also shows how the one-period and multi-period binomial option pricing formulas can be restated so that they involve analogues of N(d1) and N(d2) which have the same interpretation as in the Black-Scholes model. Cet article utilise les probabilités lognormaux corr...