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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.

Stochastic volatility, Heston, square root process, CEV process, Euler-Maruyama, discretisation, strong convergence, weak convergence, boundary behaviour

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.

Stochastic volatility, Stochastic interest rates, Schöbel-Zhu, Hull-White, Foreign Exchange, Forward starting options

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. Firstly we consider the partial differential equation approach to the pricing of Asian options. We show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Vecer [2001]. For the latter two formulations 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 Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we sharpen Thompson's [1999a,b] upper bound for the value of an Asian option. This is important for the practically relevant case of options with long maturities. Numerical results show that when the strike price is not extremely high, the resulting upper bound is tighter than recently introduced upper bounds in studies by Nielsen and Sandmann [2003] and Vanmaele et al. [2005].

Finally, we consider analytical approximations for the value of an Asian option. A much heard criticism on moment-matching approaches is that the error in the approximation is not known beforehand. We combine the traditional moment-matching approaches (e.g. Levy [1992]) with the conditioning approaches (e.g. Curran [1994]) and introduce a class of analytical approximations, which can be proven to lie between a sharp lower and upper bound. In numerical examples the accuracy of these new approximations is demonstrated. The approximations are found to outperform all of the current state-of-the-art upper bounds and approximations.

Asian option, average price option, basket option, lower bound, upper bound, analytical approximation, moment matching

Abstract:
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 virtually all levels of strikes and maturities.

Option pricing, Fourier inversion, Carr-Madan, Heston, stochastic volatility, characteristic function, damping, saddlepoint approximations

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.

Complex logarithm, affine jump-diffusion, stochastic volatility, Heston, characteristic function, moment stability, option pricing

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. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Option pricing , Bermudan options, American options, convolution, Lévy processes, Fast Fourier Transform

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 complexd iscontinuities 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.

Complex logarithm, affine jump-diffusion, stochastic volatility, Heston, characteristic function, option pricing, Fourier inversion, Variance Gamma, Schöbel-Zhu,exact simulation

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.

Principal components analysis, correlation matrix, total positivity, oscillation matrix, Schoenmakers-Coffey matrix

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.

Complex logarithm, stochastic volatility, Heston, characteristic function

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.

LIBOR market model, variance reduction, control variates, Monte Carlo, Bermudans, callable derivatives, Longstaff-Schwartz

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 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.

Stochastic volatility, Stochastic interest rates, Schöbel-Zhu, Heston, Hull-White, discretisation