4,173 research outputs found
USLV: Unspanned Stochastic Local Volatility Model
We propose a new framework for modeling stochastic local volatility, with
potential applications to modeling derivatives on interest rates, commodities,
credit, equity, FX etc., as well as hybrid derivatives. Our model extends the
linearity-generating unspanned volatility term structure model by Carr et al.
(2011) by adding a local volatility layer to it. We outline efficient numerical
schemes for pricing derivatives in this framework for a particular four-factor
specification (two "curve" factors plus two "volatility" factors). We show that
the dynamics of such a system can be approximated by a Markov chain on a
two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by
direct (Kroneker) products of values of pairs of curve and volatility factors,
respectively. The resulting Markov chain dynamics on such partly "folded" state
space enables fast pricing by the standard backward induction. Using a
nonparametric specification of the Markov chain generator, one can accurately
match arbitrary sets of vanilla option quotes with different strikes and
maturities. Furthermore, we consider an alternative formulation of the model in
terms of an implied time change process. The latter is specified
nonparametrically, again enabling accurate calibration to arbitrary sets of
vanilla option quotes.Comment: Sections 3.2 and 3.3 are re-written, 3 figures adde
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
No-Arbitrage Deep Calibration for Volatility Smile and Skewness
Volatility smile and skewness are two key properties of option prices that
are represented by the implied volatility (IV) surface. However, IV surface
calibration through nonlinear interpolation is a complex problem due to several
factors, including limited input data, low liquidity, and noise. Additionally,
the calibrated surface must obey the fundamental financial principle of the
absence of arbitrage, which can be modeled by various differential inequalities
over the partial derivatives of the option price with respect to the expiration
time and the strike price. To address these challenges, we have introduced a
Derivative-Constrained Neural Network (DCNN), which is an enhancement of a
multilayer perceptron (MLP) that incorporates derivatives in the output
function. DCNN allows us to generate a smooth surface and incorporate the
no-arbitrage condition thanks to the derivative terms in the loss function. In
numerical experiments, we apply the stochastic volatility model with smile and
skewness parameters and simulate it with different settings to examine the
stability of the calibrated model under different conditions. The results show
that DCNNs improve the interpolation of the implied volatility surface with
smile and skewness by integrating the computation of the derivatives, which are
necessary and sufficient no-arbitrage conditions. The developed algorithm also
offers practitioners an effective tool for understanding expected market
dynamics and managing risk associated with volatility smile and skewness.Comment: 9 pages, 7 figure
B-spline techniques for volatility modeling
This paper is devoted to the application of B-splines to volatility modeling,
specifically the calibration of the leverage function in stochastic local
volatility models and the parameterization of an arbitrage-free implied
volatility surface calibrated to sparse option data. We use an extension of
classical B-splines obtained by including basis functions with infinite
support. We first come back to the application of shape-constrained B-splines
to the estimation of conditional expectations, not merely from a scatter plot
but also from the given marginal distributions. An application is the Monte
Carlo calibration of stochastic local volatility models by Markov projection.
Then we present a new technique for the calibration of an implied volatility
surface to sparse option data. We use a B-spline parameterization of the
Radon-Nikodym derivative of the underlying's risk-neutral probability density
with respect to a roughly calibrated base model. We show that this method
provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch
a Galerkin method with B-spline finite elements to the solution of the partial
differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page
Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model
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Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model
The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model
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