584 research outputs found
Quasi-Monte Carlo methods for the Heston model
In this paper, we discuss the application of quasi-Monte Carlo methods to the
Heston model. We base our algorithms on the Broadie-Kaya algorithm, an exact
simulation scheme for the Heston model. As the joint transition densities are
not available in closed-form, the Linear Transformation method due to Imai and
Tan, a popular and widely applicable method to improve the effectiveness of
quasi-Monte Carlo methods, cannot be employed in the context of path-dependent
options when the underlying price process follows the Heston model.
Consequently, we tailor quasi-Monte Carlo methods directly to the Heston model.
The contributions of the paper are threefold: We firstly show how to apply
quasi-Monte Carlo methods in the context of the Heston model and the SVJ model,
secondly that quasi-Monte Carlo methods improve on Monte Carlo methods, and
thirdly how to improve the effectiveness of quasi-Monte Carlo methods by using
bridge constructions tailored to the Heston and SVJ models. Finally, we provide
some extensions for computing greeks, barrier options, multidimensional and
multi-asset pricing, and the 3/2 model.Comment: 20 pages. Submitte
Fourth order compact scheme for option pricing under Merton and Kou jump-diffusion models
In this article, a three-time levels compact scheme is proposed to solve the
partial integro-differential equation governing the option prices under
jump-diffusion models. In the proposed compact scheme, the second derivative
approximation of unknowns is approximated by the value of unknowns and their
first derivative approximations which allow us to obtain a tri-diagonal system
of linear equations for the fully discrete problem. Moreover, consistency and
stability of the proposed compact scheme are proved. Due to the low regularity
of typical initial conditions, the smoothing operator is employed to ensure the
fourth-order convergence rate. Numerical illustrations for pricing European
options under Merton and Kou jump-diffusion models are presented to validate
the theoretical results
Numerical and optimal control methods for partial differential equations arising in computational finance
The chosen title for my PhD thesis is "Numerical and optimal control methods for partial differential equations arising in computational finance". The body of my research is divided into two parts. The first part is devoted to the application of an alternating direction implicit numerical method for solving stochastic volatility option pricing models. The second part focuses on a partial-integro differential equation constrained optimal control approach to parameter estimation for the forward jump-diffusion option pricing model. The body of the thesis is preceded by an extensive introduction, which seeks to contextualize my work with respect to the field of computational finance, this is followed by a brief conclusion. Finally, the thesis is completed by a list of refer ences. The first project proposes a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. The approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence. The second project proposes to solve a parameter calibration problem for the forward jump-diffusion option pricing model proposed by Andersen and Andreasen. A distributed optimal control approach is employed, with a partial-integro differential equation as our state equation. By approaching the problem from a functional analysis perspective, I investigate the necessary regularity conditions for our parameters of interest. Following this, the existence of optimal solutions is proven under certain analytical conditions. Furthermore, the first-order necessary conditions for optimality are also established. Finally, a projected-gradient optimization method is applied numerically to empirical market data and results are given
Revisiting the fractional cointegrating dynamics of implied-realized volatility relation with wavelet band spectrum regression
This paper revisits the fractional cointegrating relationship between ex-ante
implied volatility and ex-post realized volatility. We argue that the concept
of corridor implied volatility (CIV) should be used instead of the popular
model-free option-implied volatility (MFIV) when assessing the fractional
cointegrating relation as the latter may introduce bias to the estimation. For
the realized volatility, we use recently proposed methods which are robust to
noise as well as jumps and interestingly we find that it does not affect the
implied-realized volatility relation. In addition, we develop a new tool for
the estimation of fractional cointegrating relation between implied and
realized volatility based on wavelets, a wavelet band least squares (WBLS). The
main advantage of WBLS in comparison to other frequency domain methods is that
it allows us to work conveniently with potentially non-stationary volatility
due to the properties of wavelets. We study the dynamics of the relationship in
the time-frequency domain with the wavelet coherence confirming that the
dependence comes solely from the lower frequencies of the spectra. Motivated by
this result we estimate the relationship only on this part of the spectra using
WBLS and compare our results to the fully modified narrow-band least squares
(FMNBLS) based on the Fourier frequencies. In the estimation, we use the S&P
500 and DAX monthly and bi-weekly option prices covering the recent financial
crisis and we conclude that in the long-run, volatility inferred from the
option prices using the corridor implied volatility (CIV) provides an unbiased
forecast of the realized volatility
On the efficient application of the repeated Richardson extrapolation technique to option pricing
Richardson extrapolation (RE) is a commonly used technique in financial applications for accelerating the convergence of numerical methods. Particularly in option pricing, it is possible to refine the results of several approaches by applying RE, in order to avoid the difficulties of employing slowly converging schemes. But the effectiveness of such a technique is fully achieved when its repeated version (RRE) is applied. Nevertheless, its application in financial literature is pretty rare. This is probably due to the necessity to pay special attention to the numerical aspects of its implementation, such as the choice of both the sequence of the stepsizes and the order of the method. In this contribution, we consider several numerical schemes for the valuation of American options and investigate the possibility of an appropriate application of RRE. As a result, we find that, in the analyzed approaches in which the convergence is monotonic, RRE can be used as an effective tool for improving significantly the accuracy.Richardson extrapolation, repeated Richardson extrapolation, American options, randomization technique, flexible binomial method
A First Option Calibration of the GARCH Diffusion Model by a PDE Method
Time-series calibrations often suggest that the GARCH diffusion model could
also be a suitable candidate for option (risk-neutral) calibration. But unlike
the popular Heston model, it lacks a fast, semi-analytic solution for the
pricing of vanilla options, perhaps the main reason why it is not used in this
way. In this paper we show how an efficient finite difference-based PDE solver
can effectively replace analytical solutions, enabling accurate option
calibrations in less than a minute. The proposed pricing engine is shown to be
robust under a wide range of model parameters and combines smoothly with
black-box optimizers. We use this approach to produce a first PDE calibration
of the GARCH diffusion model to SPX options and present some benchmark results
for future reference.Comment: 29 pages, 7 figure
The forecast ability of risk-neutral densities of foreign exchange
We estimate the process underlying the pricing of American options by using higher-order lattices combined with a multigrid method. This paper also tests whether the risk-neutral densities given from American options provide a good forecasting tool. We use a nonparametric test of the densities that is based on the inverse probability functions and is modified to account for correlation across time between our random variables, which are uniform under the null hypothesis. We find that the densities based on the American option markets for foreign exchange do quite well for the forecasting period over which the options are thickly traded. Further, simple models that fit the densities do about as well as more sophisticated models.Foreign exchange futures ; Options (Finance) ; Economic forecasting
Joint Modelling and Calibration of SPX and VIX by Optimal Transport
This paper addresses the joint calibration problem of SPX options and VIX
options or futures. We show that the problem can be formulated as a
semimartingale optimal transport problem under a finite number of discrete
constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE
formulation along with its dual counterpart. The solution, a calibrated
diffusion process, can be represented via the solutions of
Hamilton-Jacobi-Bellman equations arising from the dual formulation. The method
is tested on both simulated data and market data. Numerical examples show that
the model can be accurately calibrated to SPX options, VIX options and VIX
futures simultaneously
Numerical schemes and Monte Carlo techniques for Greeks in stochastic volatility models
The main objective of this thesis is to propose approximations to option sensitivities in stochastic volatility models. The first part explores sequential Monte Carlo techniques for approximating the latent state in a Hidden Markov Model. These techniques are applied to the computation of Greeks by adapting the likelihood ratio method. Convergence of the Greek estimates is proved and tracking of option prices is performed in a stochastic volatility model.
The second part defines a class of approximate Greek weights and provides high-order approximations and justification for extrapolation techniques. Under certain regularity assumptions on the value function of the problem, Greek approximations are proved for a fully implementable Monte Carlo framework, using weak Taylor discretisation schemes. The variance and bias are studied for the Delta and Gamma, when using such discrete-time approximations.
The final part of the thesis introduces a modified explicit Euler scheme for stochastic differential equations with non-Lipschitz continuous drift or diffusion; a strong rate of convergence is proved. The literature on discretisation techniques for stochastic differential equations has been motivational for the development of techniques preserving the explicitness of the algorithm. Stochastic differential equations in the mathematical finance literature, including the Cox-Ingersoll-Ross, the 3/2 and the Ait-Sahalia models can be discretised, with a strong rate of convergence proved, which is a requirement for multilevel Monte Carlo techniques.Open Acces
- …