This article presents simple formulae for the local variance gamma model of Carr and Nadtochiy (2017), extended with a piecewise-linear local variance function. The new formulae allow us to calibrate the model efficiently to market option quotes. On a small set of quotes, exact calibration is achieved under one millisecond. This effectively results in an arbitrage-free interpolation of class C². The article proposes a good regularization when the quotes are noisy. Finally, it puts in evidence an issue of the model at-the-money, which is also present in the related one-step finite difference technique of Andreasen and Huge (2011), and gives two solutions for it.

This paper presents the Runge-Kutta-Legendre (RKL) finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the RKL scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank-Nicolson, with Rannacher time-stepping.

In the collocating volatility (CLV) model, the stochastic collocation technique is used as a convenient representation of the terminal distribution of the market option prices. A specific dynamic is added in the form of a stochastic driver process, which allows more control over the prices of forward starting options. This is reminiscent of the Markov functional models. (Grzelak uses a single-factor Ornstein–Uhlenbeck process as the driver for the CLV model, and Fries uses a single-factor Wiener process with time-dependent volatility in his equity Markov functional model. Van der Stoep et al consider a Heston stochastic volatility driver process and show that it offers more flexibility to capture the forward smile in the context of foreign exchange options.) In this paper, we discuss all aspects of derivative pricing under the Heston– CLV model: calibration with an efficient Fourier method; a Monte Carlo simulation with second-order convergence; and accurate partial differential equation pricing through implicit and explicit finite-difference methods.

This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.

The valuation of European options under the Heston model (or any other stochastic volatility model where the characteristic function is analytically known) involves the computation of a Fourier transform type of numerical integration. This paper describes how adaptive Filon and adaptive Flinn quadratures may be used to calculate this integral efficiently in accordance with a level of accuracy defined by the user. We then compare the accuracy and the performance of our quadratures with that of others commonly used for this problem, such as the optimal alpha method applied by Lord and Kahl. Finally, the paper concludes with a concrete case of calibration of the model on different sets of market data.

This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge.