Monthly Archives: September 2014

Fifty shades of SABR simulation

We compare a large variety (not fifty, but a large amount) of simulation schemes for the SABR model due to Hagan et al. [2002]. Many schemes are inspired by the recent work of Islah [2009], who has shown that, conditional on the integrated variance, an asset in the SABR model can be approximated by, after a suitable transformation, a squared Bessel process. The…

Monte Carlo pricing in the Schöbel-Zhu model and its extensions

In this paper we propose a simulation algorithm for the Schöbel-Zhu (1999) model and its extension to include stochastic interest rates, the Schöbel-Zhu-Hull-White model as considered in Van Haastrecht et al. (2009). Both schemes are derived by analyzing the lessons learned from the Andersen (2008) scheme on how to avoid the so-called leaking correlation phenomenon in the simulation of the…

Complex logarithms in Heston-like models

The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions like the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier…

A comparison of biased simulation schemes for stochastic volatility models

Using an Euler discretisation to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretisation is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special…

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

In this paper we extend the stochastic volatility model of Schöbel and Zhu [1999] by including stochastic interest rates. Furthermore we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a correlation between the instantaneous interest rates, the volatilities and the underlying stock returns. By deriving the characteristic function of the log-asset price…

A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Lévy Processes

A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the…

Control variates for callable LIBOR exotics – a preliminary study

Monte Carlo simulation is currently the method of choice for the pricing of callable derivatives in LIBOR market models. Lately more and more papers are surfacing in which variance reduction methods are applied to the pricing of derivatives with early exercise features. We focus on one of the conceptually easiest variance reduction methods, control variates. The basis of our method…

Optimal Fourier inversion in semi-analytical option pricing

At the time of writing this article, Fourier inversion is the computational method of choice for a fast and accurate calculation of plain vanilla option prices in models with an analytically available characteristic function. Shifting the contour of integration along the complex plane allows for different representations of the inverse Fourier integral. In this article, we present the optimal contour…

Level-slope-curvature – fact or artefact?

The first three factors resulting from a principal components analysis of term structure data are in the literature typically interpreted as driving the level, slope and curvature of the term structure. Using slight generalisations of theorems from total positivity, we present sufficient conditions under which level, slope and curvature are present. These conditions have the nice interpretation of restricting the…