This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the relat
ed collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.
There is no exact closed form formula for pricing of European options with discrete cash dividends under the model where the underlying asset price follows a piecewise lognormal process with jumps at dividend ex-dates. This paper presents alternative
expansions based on the technique of Etore and Gobet, leading to more robust first, second and third-order expansions across the range of strikes and the range of dividend dates.
This paper presents the Runge-Kutta-Legendre 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 Runge-Kutta-Legendre 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.
The Mersenne-Twister is one of the most popular generators of uniform pseudo-random numbers. It is used in many numerical libraries and software. In this paper, we look at the Komolgorov entropy of the original Mersenne-Twister, as well as of more mo
dern variations such as the 64-bit Mersenne-Twisters, the Well generators, and the Melg generators.
This note shows that the cosine expansion based on the Vieta formula is equivalent to a discretization of the Parseval identity. We then evaluate the use of simple direct algorithms to compute the Shannon coefficients for the payoff. Finally, we expl
ore the efficiency of a Filon quadrature instead of the Vieta formula for the coefficients related to the probability density function.
We present an alternative formula to price European options through cosine series expansions, under models with a known characteristic function such as the Heston stochastic volatility model. It is more robust across strikes and as fast as the original COS method.
This paper presents simple formulae for the local variance gamma model of Carr and Nadtochiy, extended with a piecewise-linear local variance function. The new formulae allow 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 $C^2$. The paper 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, and gives two solutions for it.
