High Order Discretization Schemes for Stochastic Volatility Models

@article{Sbai2009HighOD,
  title={High Order Discretization Schemes for Stochastic Volatility Models},
  author={Mohamed Karim Sbai and Benjamin Jourdain},
  journal={ERN: Econometric Modeling in Financial Economics (Topic)},
  year={2009}
}
  • M. Sbai, B. Jourdain
  • Published 13 August 2009
  • Mathematics
  • ERN: Econometric Modeling in Financial Economics (Topic)
In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using Ito's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with… 
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References

SHOWING 1-10 OF 38 REFERENCES
Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes
TLDR
This paper suggests a method for the exact simulation of the stock price and variance under Hestons stochastic volatility model and other affine jump diffusion processes and achieves an O(s-1/2) convergence rate, where s is the total computational budget.
Fast strong approximation Monte Carlo schemes for stochastic volatility models
Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either…
A comparison of biased simulation schemes for stochastic volatility models
TLDR
The new full truncation scheme is introduced, tailored to minimize the positive bias found when pricing European options and outperforms all considered biased schemes in terms of bias and root-mean-squared error.
Convergence of discretized stochastic (interest rate) processes with stochastic drift term
For applications in finance, we study the stochastic differential equation dXs = (2βXs + δs) ds + g(Xs) dBs with β a negative real number, g a continuous function vanishing at zero which…
Efficient Simulation of the Heston Stochastic Volatility Model
Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo…
Stock Price Distributions with Stochastic Volatility: An Analytic Approach
We study the stock price distributions that arise when prices follow a diffusion process with a stochastically varying volatility parameter. We use analytic techniques to derive an explicit…
A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method
TLDR
It is proved that the classical Runge–Kutta method for ODEs is directly applicable to the ODE drawn from the random variable, and this is applied to the problem of pricing Asian options under the Heston stochastic volatility model.
The law of the Euler scheme for stochastic differential equations
SummaryWe study the approximation problem ofEf(XT) byEf(XTn), where (Xt) is the solution of a stochastic differential equation, (XTn) is defined by the Euler discretization scheme with stepT/n, andf…
Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application
Abstract In this paper, we examine the pricing of European call options on stocks that have variance rates that change randomly. We study continuous time diffusion processes for the stock return and…
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
TLDR
A new variance reduction method is introduced, which can be viewed as a statistical analogue of Romberg extrapolation method, which uses two Euler schemes with steps delta and delta(beta), which leads to an algorithm which has a complexity significantly lower than the complexity of the standard Monte Carlo method.
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