Haastrecht, A. van, Lord, R. and A.A.J. Pelsser (2009). "Monte Carlo pricing in the Schöbel-Zhu model and its extensions", working paper, Delta Lloyd, University of Amsterdam, Cardano and Maastricht University. (abstract) (ssrn) (pdf)

Lord, R. and C. Kahl (2006). "Why the rotation count algorithm works", Tinbergen Institute Discussion Paper No. 2006-065/2. (abstract) (ssrn) (pdf)

Lord, R. (2006). "Pricing of baskets, Asians and swaptions under affine Lévy models", working paper, Erasmus University Rotterdam and Rabobank International. (abstract) (pdf)

Kahl, C. and R. Lord (2010). "Fourier inversion methods", forthcoming in: Handbook of Computational Finance (eds.: J-C. Duan, J.E. Gentle and W. Härdle).

Lord, R. (2009). "Fourier methods in option pricing", forthcoming in: Encyclopedia of Quantitative Finance (ed.: R. Cont).

Lord, R. and C. Kahl (2010). "Complex logarithms in Heston-like models", forthcoming in: Mathematical Finance. (abstract) (ssrn) (pdf)

Lord, R. (2010). "Comment on: a note on the discontinuity problem in Heston's stochastic volatility model", forthcoming in: Applied Mathematical Finance. (abstract) (ssrn) (pdf)

Lord, R., Koekkoek, R. and D. van Dijk (2010). "A comparison of biased simulation schemes for stochastic volatility models", Quantitative Finance, vol. 10, no. 2, pp. 177-194. (abstract) (ssrn) (pdf)

Haastrecht, A. van, Lord, R., Pelsser, A.A.J. and D.F. Schrager (2009). "Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility", Insurance: Mathematics and Economics, vol. 45, no. 3, pp. 436-448 (abstract) (ssrn) (pdf)

Lord, R., Fang, F., Bervoets, F. and C.W. Oosterlee (2008). "A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes", SIAM Journal on Scientific Computing, vol. 30, no. 4, pp. 1678-1705. (abstract) (ssrn) (pdf)

Lord, R. and C. Kahl (2007). "Optimal Fourier inversion in semi-analytical option pricing", Journal of Computational Finance, vol. 10, no. 4., pp. 1-30. (abstract) (ssrn) (pdf)

Lord, R. and A.A.J. Pelsser (2007). "Level-slope-curvature - fact or artefact?", Applied Mathematical Finance, vol. 14, no. 2, pp. 105-130. (abstract) (ssrn) (pdf)

Buitelaar, J. and R. Lord (2007). "Control variates for callable Libor exotics - a preliminary study", in: Proceedings of the 5th actuarial and financial mathematics day (eds.: M. Vanmaele et al.), pp. 21-31. (abstract) (pdf)

Lord, R. (2006). "Partially exact and bounded approximations for arithmetic Asian options", Journal of Computational Finance, vol. 10, no. 2, pp. 1-52. (abstract) (ssrn) (pdf)

Lord, R. (2005). A motivation for conditional moment matching", in: Proceedings of the 3rd actuarial and financial mathematics day (eds.: M. Vanmaele et al.), pp. 69-79. (abstract) (pdf)

Lord, R. and C. Kahl (2010). "Complex logarithms in Heston-like models", forthcoming in: Mathematical Finance. (abstract) (ssrn) (pdf)

Abstract: 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" (see further below) is the initial version of this paper.

Lord, R. (2010). "Comment on: a note on the discontinuity problem in Heston's stochastic volatility model", forthcoming in: Applied Mathematical Finance. (abstract) (ssrn) (pdf)

Abstract: 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.

Lord, R., Koekkoek, R. and D. van Dijk (2010). "A comparison of biased simulation schemes for stochastic volatility models", Quantitative Finance, vol. 10, no. 2, pp. 177-194. (abstract) (ssrn) (pdf)

Abstract: 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 Econometrics conference in Rotterdam and the Fourth World Congress of the Bachelier Finance Society in Tokyo.

Haastrecht, A. van, Lord, R., Pelsser, A.A.J. and D.F. Schrager (2009). "Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility", Insurance: Mathematics and Economics, vol. 45, no. 3, pp. 436-448 (abstract) (ssrn) (pdf)

Abstract: 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.

Lord, R., Fang, F., Bervoets, F. and C.W. Oosterlee (2008). "A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes", SIAM Journal on Scientific Computing, vol. 30, no. 4, pp. 1678-1705. (abstract) (ssrn) (pdf)

Abstract: 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.

Lord, R. and C. Kahl (2007). "Optimal Fourier inversion in semi-analytical option pricing", Journal of Computational Finance, vol. 10, no. 4., pp. 1-30. (abstract) (ssrn) (pdf)

Abstract: 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 of the Fourier integral, taking into account numerical issues such as cancellation and explosion of the characteristic function. This allows for robust and fast option pricing for almost all levels of strikes and maturities.

This paper has been presented at the World Congress on Computational Finance in London, March 26th 2007.

Lord, R. and A.A.J. Pelsser (2007). "Level-slope-curvature - fact or artefact?", Applied Mathematical Finance, vol. 14, no. 2, pp. 105-130. (abstract) (ssrn) (pdf)

Abstract: 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 level, slope and curvature of the correlation surface. It is proven that the Schoenmakers-Coffey correlation matrix also brings along such factors. Finally, we formulate and corroborate our conjecture that the order present in correlation matrices causes slope.

Presentation of this paper given at the QMF 2005 conference in Sydney can be found here.

Buitelaar, J. and R. Lord (2007). "Control variates for callable Libor exotics - a preliminary study", in: Proceedings of the 5th actuarial and financial mathematics day (eds.: M. Vanmaele et al.), pp. 21-31. (abstract) (pdf)

Abstract: 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.

Lord, R. (2006). "Partially exact and bounded approximations for arithmetic Asian options", Journal of Computational Finance, vol. 10, no. 2, pp. 1-52. (abstract) (ssrn) (pdf)

Abstract: This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the link between the approaches of Rogers and Shi (1995), Andreasen (1999), Hoogland and Neumann (2000) and Vecer (2001). For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Curran’s and Rogers and Shi’s lower bound for the general case of multiple underlyings. Thirdly, we considerably sharpen Thompson’s (1999a,b) upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran’s conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. In numerical examples we demonstrate that they outperform all current state-of-the-art bounds and approximations.

This paper has been presented at the 13th European Conference on Mathematics for Industry in Eindhoven.

Lord, R. (2005). A motivation for conditional moment matching", in: Proceedings of the 3rd actuarial and financial mathematics day (eds.: M. Vanmaele et al.), pp. 69-79. (abstract) (pdf)

Abstract: One can find approaches galore in the literature for the valuation of Asian basket options. When the number of underlyings is large one has to resort to bounds or approximations to value these options. In this respect, Curran (1994) and Rogers and Shi (1995) very successfully applied a conditioning approach. Recently, Lord (2005) combined their approach with the traditional ad-hoc moment matching approaches, to obtain an approximation which is extremely accurate and has an analytical bound on its error. Here we review this approach and extend the results to multiple conditioning variables, along the lines of Vanmaele, Deelstra and Liinev (2004).

Haastrecht, A. van, Lord, R. and A.A.J. Pelsser (2009). "Monte Carlo pricing in the Schöbel-Zhu model and its extensions", working paper, Delta Lloyd, University of Amsterdam, Cardano and Maastricht University. (abstract) (ssrn) (pdf)

Abstract: 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.

Lord, R. and C. Kahl (2006). "Why the rotation count algorithm works", Tinbergen Institute Discussion Paper No. 2006-065/2. (abstract) (ssrn) (pdf)

Abstract: The characteristic functions of many affine jump-diffusion models, such as Heston’s stochastic volatility model and all of its extensions, involve multivalued functions such as 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 under non-restrictive conditions on the parameters that the rotation count algorithm of Kahl and Jäckel chooses the correct branch of the complex logarithm. Under the same restrictions we prove that in an alternative formulation of the characteristic function the principal branch is the correct one. Seen as this formulation is easier to implement and numerically more stable than Heston’s formulation, it should be the preferred one. The remainder of this paper shows how complex discontinuities can be avoided in the Schöbel-Zhu model and the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya. Finally, we show that Matytsin’s SVJJ model has a closed-form characteristic function, though the complex discontinuities that arise there due to the branch switching of the exponential integral cannot be avoided under all circumstances.

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 Fourth World Congress of the Bachelier Finance Society in Tokyo.

Lord, R. (2006). "Pricing of baskets, Asians and swaptions under affine Lévy models", working paper, Erasmus University Rotterdam and Rabobank International. (abstract) (pdf)

Abstract: In this paper we revisit Curran's (1994) and Rogers' & Shi's (1995) lower bounds for the value of an Asian option, and show how to apply it to the valuation of basket options, Asian options and swaptions in a setting where the underlyings are exponentially affine in the state variables, and where we know the characteristic function hereof. Examples of models for which the techniques apply are models where the logarithm of the underlying asset is in the affine Lévy class, which consists of the affine jump-diffusion class, the Lévy market models, and various extensions and mixtures hereof. Numerical results for swaptions and Asians demonstrate that the lower bound is the most accurate approximation considered for these more general models. For swaptions we also come up with an alternative, faster approximation, which is heavily inspired on the Singleton-Umantsev (2002) approximation.

A presentation of this paper at the 5th Winter school on Financial Mathematics can be found here.