For models that have an analytically available characteristic function, Fourier inversion is an important computational method for a fast and accurate calculation of plain vanilla option prices. In order to improve the numerical stability of the Fourier inversion, Lord and Kahl [2007] suggested a method to find an optimal contour of integration, taking into account numerical issues such as cancellation and explosion. Rather than having a problem-dependent contour, Joshi and Yang [2011] built on Andersen and Andreasen’s [2002] suggestion, and showed how to use the Black-Scholes formula as a control variate. In this paper we demonstrate that combining both methods can lead to improved accuracy, and that using the optimal contour is highly effective for out-of-the-money and in-the-money options.

Slides presented at the 9th World Congress of the Bachelier Finance Society, in New York, can be found here.

Slides presented at the 10th World Congress of the Bachelier Finance Society, in Dublin, can be found here.

]]>We present a comprehensive overview of derivative pricing in Gaussian affine asset pricing models. Gaussian affine asset pricing models are widely used in practice for pricing and scenario analysis due to their tractable pricing implications and easy estimation. This tractability is essential to efficiently evaluate portfolios of derivatives within many scenarios and time periods. We present efficient closed-form pricing formulas for the most common derivative instruments used by pension funds and insurance companies, such as interest rate swaps, swaptions, inflation-linked swaps, equity options, based on results from the literature. The pricing formulas are presented in a comprehensive computable form by utilising results based on the matrix exponential. Next, we show how some models commonly used in practice fit in the Gaussian affine framework, so that the pricing formulas can be applied to these cases. In particular, we discuss the KNW model by Koijen, Nijman and Werker (2010), which is widely used in the pension industry. Finally we discuss how our results can be applied to a time-inhomogeneous extension of the model that allows perfect calibration to the observed yield curve.

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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 result is exact for the zero correlation case. In Chen et al. this result has been utilised to arrive at a low-bias simulation scheme for the SABR model.

We show how a scheme for the underlying asset can be combined in a modular way with a scheme for the integrated variance. For the latter we resort to conditioning techniques used in Asian option pricing, but we also look at more simple schemes. For the zero correlation scheme the resulting scheme is almost exact, as we use recent work of Makarov and Glew [2010] on the simulation of squared Bessel processes with absorption at zero. If the correlation deviates significantly from zero, we have to resort to simpler schemes. All schemes are compared in numerical examples, also to the Ninomiya-Victoir with drift scheme recently devised by Bayer, Friz and Loeffen [2012].

Since the initial work, we have applied the insights gained from the SABR model to the free-boundary SABR model, introduced by Antonov, Spector and Konikov [2015]. This can be of particular relevance in the current low-rate environment.

This work has been presented at the 8th World Congress of the Bachelier Finance Society in Brussels, at the 10th Fixed Income Conference in Barcelona (slides here) as well as at the Global Derivatives Trading & Risk Management conference in Amsterdam. The latter presentation can be found here.

]]>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 Heston (1993) model. All introduced schemes are Exponentially Affine in Expectation (EAE), which greatly facilitates the derivation of a martingale correction. In addition we study the regularity of each scheme. The numerical results indicate that our scheme consistently outperforms the Euler scheme. For a special case of the Schöbel-Zhu model which coincides with the Heston model, our scheme performs similarly to the QE-M scheme of Andersen (2008). The results reaffirm that when simulating stochastic volatility models it is of the utmost importance to match the correlation between the asset price and the stochastic volatility process.

]]>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 inversion. In this paper we prove without any restrictions that there is a formulation of the characteristic function in which the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than the so-called rotation count algorithm of Kahl and Jäckel [2005], we solely focus on its stability in this article. The remainder of this paper shows how complex discontinuities can be avoided in the Variance Gamma and Schöbel-Zhu models, as well as in the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya.

“Why the rotation count algorithm works” is the initial version of this paper.

]]>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 case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretisation, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimise the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and Jäckel and Ninomiya and Victoir, as well as to the exact scheme of Broadie and Kaya. The choice of fix is found to be extremely important. The full truncation scheme outperforms all considered biased schemes in terms of bias and root-mean-squared error.

The presentation of this paper given at the 42nd Dutch Mathematical Congress can be found here. A poster presentation of this paper has been given at the 50 years of Econometricsconference in Rotterdam and the Fourth World Congress of the Bachelier Finance Society in Tokyo.

]]>Guo and Hung [2007] recently studied the complex logarithm present in the characteristic function of Heston’s stochastic volatility model. They proposed an algorithm for the evaluation of the characteristic function which is claimed to preserve its continuity. We show their algorithm is correct, although their proof is not.

]]>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 distribution, we are able to price European stock options in closed-form by Fourier inversion. Furthermore we present a foreign exchange generalization and show how the pricing of forward-starting options like cliquets can be performed. Additionally we discuss the practical implementation of these new models.

]]>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 Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponential Lévy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.

This paper has been presented at the Computational Finance minisymposium of the Jahrestagung der Deutschen Mathematiker-Vereinigung 2006 in Bonn, the Fourth World Congress of the Bachelier Finance Society in Tokyo, the AMaMeF workshop on financial modelling with jump processes in Palaiseau and the Frankfurt MathFinance Workshop 2007.

]]>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 is an upper bound of the callable contract in terms of plain vanilla contracts, which is found to be a highly effective control variate. Several examples of callable LIBOR exotics demonstrate the effectiveness and wide applicability of the method.

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