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Option contracts can be valued by using the Black-Scholes equation, a partial differential equation with initial conditions. An exact solution for European style options is known. The computation time and the error need to be minimized…

Black-Scholes implied volatility is a quantile. The insight follows from the normalized option price being a probability on the variance scale, with the inverse Gaussian distribution providing the link. It enables analytically exact and…

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift,…

We establish an explicit approximation formula for European put option prices within a general stochastic volatility model with time-dependent parameters. Our methodology is based on expansions of the mixing representation of the put option…

In the paper written by Klibanov et al, it proposes a novel method to calculate implied volatility of a European stock options as a solution to ill-posed inverse problem for the Black-Scholes equation. In addition, it proposes a trading…

We use modifications of the Adams method and very fast and accurate sinh-acceleration method of the Fourier inversion (iFT) (S.Boyarchenko and Levendorski\u{i}, IJTAF 2019, v.22) to evaluate prices of vanilla options; for options of…

The Black-Scholes model gives vanilla Europen call option prices as a function of the volatility. We prove Lipschitz stability in the inverse problem of determining the implied volatility, which is a function of the underlying asset, from a…

In the previous paper (Inverse Problems, 32, 015010, 2016), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1-2 trading days ahead of the present one. This new technique uses the…

We use the expectation of the range of an arithmetic Brownian motion and the method of moments on the daily high, low, opening and closing prices to estimate the volatility of the stock price. The daily price jump at the opening is…

This research addresses accurate option pricing by employing models beyond the traditional Black-Scholes framework. While Black-Scholes provides a closed-form solution, it is limited by assumptions of constant volatility, no dividends, and…

We analyze the empirical performance of several non-parametric estimators of the pricing functional for European options, using historical put and call prices on the S&P500 during the year 2012. Two main families of estimators are…

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black-Scholes price through the so-called `probability of survival'…

A new mathematical model for the Black-Scholes equation is proposed to forecast option prices. This model includes new interval for the price of the underlying stock as well as new initial and boundary conditions. Conventional notions of…

In this paper, a new numerical method based on adaptive gradient descent optimizers is provided for computing the implied volatility from the Black-Scholes (B-S) option pricing model. It is shown that the new method is more accurate than…

The incorporation of a dividend yield in the classical option pricing model of Black- Scholes results in a minor modification of the Black-Scholes formula, since the lognormal dynamic of the underlying asset is preserved. However, market…

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European…

Pricing financial derivatives, in particular European-style options at different time-maturities and strikes, means a relevant problem in finance. The dynamics describing the price of vanilla options when constant volatilities and interest…

This paper introduces the Inverse Gamma (IGa) stochastic volatility model with time-dependent parameters, defined by the volatility dynamics $dV_{t}=\kappa_{t}\left(\theta_{t}-V_{t}\right)dt+\lambda_{t}V_{t}dB_{t}$. This non-affine model is…

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a…

Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local-stochastic volatility setting. Our price approximations require only a normal CDF and…