804 research outputs found
Smart Monte Carlo: Various tricks using Malliavin calculus
Current Monte Carlo pricing engines may face computational challenge for the Greeks, because of not only their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournié et al. (1999), we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ration method of Broadie and Glasserman (1996). We show on numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model.Monte-Carlo, Quasi-Monte Carlo, Greeks,Malliavin Calculus, Wiener Chaos.
Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme
We present recent advances on Dirichlet forms methods either to extend
financial models beyond the usual stochastic calculus or to study stochastic
models with less classical tools. In this spirit, we interpret the asymptotic
error on the solution of an sde due to the Euler scheme in terms of a Dirichlet
form on the Wiener space, what allows to propagate this error thanks to
functional calculus.Comment: 15
Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus
This study introduces computation of option sensitivities (Greeks) using the
Malliavin calculus under the assumption that the underlying asset and interest
rate both evolve from a stochastic volatility model and a stochastic interest
rate model, respectively. Therefore, it integrates the recent developments in
the Malliavin calculus for the computation of Greeks: Delta, Vega, and Rho and
it extends the method slightly. The main results show that Malliavin calculus
allows a running Monte Carlo (MC) algorithm to present numerical
implementations and to illustrate its effectiveness. The main advantage of this
method is that once the algorithms are constructed, they can be used for
numerous types of option, even if their payoff functions are not
differentiable.Comment: Published at https://doi.org/10.15559/18-VMSTA100 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Multidimensional Quasi-Monte Carlo Malliavin Greeks
We investigate the use of Malliavin calculus in order to calculate the Greeks
of multidimensional complex path-dependent options by simulation. For this
purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the
multidimensional case. The multidimensional setting shows the convenience of
the Malliavin Calculus approach over different techniques that have been
previously proposed. Indeed, these techniques may be computationally expensive
and do not provide flexibility for variance reduction. In contrast, the
Malliavin approach exhibits a higher flexibility by providing a class of
functions that return the same expected value (the Greek) with different
accuracies. This versatility for variance reduction is not possible without the
use of the generalized integral by part formula of Malliavin Calculus. In the
multidimensional context, we find convenient formulas that permit to improve
the localization technique, introduced in Fourni\'e et al and reduce both the
computational cost and the variance. Moreover, we show that the parameters
employed for variance reduction can be obtained \textit{on the flight} in the
simulation. We illustrate the efficiency of the proposed procedures, coupled
with the enhanced version of Quasi-Monte Carlo simulations as discussed in
Sabino, for the numerical estimation of the Deltas of call, digital Asian-style
and Exotic basket options with a fixed and a floating strike price in a
multidimensional Black-Scholes market.Comment: 22 pages, 6 figure
Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations
In this article we consider the problem of pricing and hedging
high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We
assume a Black-Scholes market with time-dependent volatilities and show how to
compute the deltas by the aid of the Malliavin Calculus, extending the
procedure employed by Montero and Kohatsu-Higa (2003). Efficient
path-generation algorithms, such as Linear Transformation and Principal
Component Analysis, exhibit a high computational cost in a market with
time-dependent volatilities. We present a new and fast Cholesky algorithm for
block matrices that makes the Linear Transformation even more convenient.
Moreover, we propose a new-path generation technique based on a Kronecker
Product Approximation. This construction returns the same accuracy of the
Linear Transformation used for the computation of the deltas and the prices in
the case of correlated asset returns while requiring a lower computational
time. All these techniques can be easily employed for stochastic volatility
models based on the mixture of multi-dimensional dynamics introduced by Brigo
et al. (2004).Comment: 16 page
Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs
We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999), as well as Ma and Zhang (2000), using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others
- …