In this paper, a numerical technique is developed to discretize variable-order fractional Heston differential equation. The proposed strategy is followed by an optimization technology, genetic algorithm, for tuning the unknown parameters in the proposed model. The performance of the model is analyzed to profit and loss 500 close index from the US stock markets. Simulations illustrate the application of the proposed technique.

/pdf/computational-technique-for-simulating-variable-order-35qb735e2t.pdf

Computational technique for simulating variable-order fractional Heston model with application in US stock market

In this paper, we investigate the problem of estimating the volatility from the underlying asset price for discrete-time observations. This topic has attracted much research interest due to the key role of the volatility in finance. In this paper, we consider the Heston stochastic volatility model with jumps and we develop a new polynomial filtering method for the estimation of the volatility. The method relies on a linear filter which uses a polynomial state-space formulation of the discrete version of the continuous-time model. We demonstrate that a higher-order polynomial filtering method can be efficiently applied in the context of stochastic volatility models. Then, we compare our approach with some, well-established, techniques in the literature.

On parameter estimation of Heston’s stochastic volatility model: a polynomial filtering method

We consider assets for which price $X_t$ and squared volatility $Y_t$ are jointly driven by Heston joint stochastic differential equations (SDEs). When the parameters of these SDEs are estimated from $N$ sub-sampled data $(X_{nT}, Y_{nT})$, estimation errors do impact the classical option pricing PDEs. We estimate these option pricing errors by combining numerical evaluation of estimation errors for Heston SDEs parameters with the computation of option price partial derivatives with respect to these SDEs parameters. This is achieved by solving six parabolic PDEs with adequate boundary conditions. To implement this approach, we also develop an estimator $\hat \lambda$ for the market price of volatility risk, and we study the sensitivity of option pricing to estimation errors affecting $\hat \lambda$. We illustrate this approach by fitting Heston SDEs to 252 daily joint observations of the S\&P 500 index and of its approximate volatility VIX, and by numerical applications to European options written on the S\&P 500 index.

Option Pricing Accuracy for Estimated Heston Models

The Benchmark Regulation (BMR) imposes a necessity of the conversion of quote-based financial indices to transactional-based ones. The reform is a challenge for administrators of indices that perform feasibility studies of the conversion process. The article analyses pros and cons of various methods of the index reform indicating the optimal path of such activity as far as money market is concerned. A choice for a “parallel listing” path seems to be the safest one if one takes into account legal and economic risk embedded in various transition models.

/pdf/a-vicious-circle-of-the-benchmark-reform-4t4zvsko7g.pdf

A Vicious Circle of the Benchmark Reform

In this thesis, we consider some popular stochastic differential equation models used in finance, such as the Vasicek Interest Rate model, the Heston model and a new fractional Heston model. We discuss how to perform inference about unknown quantities associated with these models in the Bayesian framework. We describe sequential importance sampling, the particle filter and the auxiliary particle filter. We apply these inference methods to the Vasicek Interest Rate model and the standard stochastic volatility model, both to sample from the posterior distribution of the underlying processes and to update the posterior distribution of the parameters sequentially, as data arrive over time. We discuss the sensitivity of our results to prior assumptions. We then consider the use of Markov chain Monte Carlo (MCMC) methodology to sample from the posterior distribution of the underlying volatility process and of the unknown model parameters in the Heston model. The particle filter and the auxiliary particle filter are also employed to perform sequential inference. Next we extend the Heston model to the fractional Heston model, by replacing the Brownian motions that drive the underlying stochastic differential equations by fractional Brownian motions, so allowing a richer dependence structure across time. Again, we use a variety of methods to perform inference. We apply our methodology to simulated and real financial data with success. We then discuss how to make forecasts using both the Heston and the fractional Heston model. We make comparisons between the models and show that using our new fractional Heston model can lead to improve forecasts for real financial data.

Bayesian Stochastic Differential Equation Modelling with Application to Finance

Estimating the volatility from the underlying asset price history for the discrete observations case is a challenging inference problem. Yet it has attracted much research interest due to the key role of volatility in many areas of finance. In this paper we consider the Heston stochastic volatility model and propose an accurate analytic approximation for the volatility likelihood function. The model is based on considering the joint probability density of the asset and the volatility, and integrating out past volatility variables. The likelihood simplifies to a product of T terms, where T is the length of the past history considered. An extension to the problem of fixed parameter estimation is also presented. Simulation results indicate the effectiveness and accuracy of the proposed method.

An analytic approximation of the likelihood function for the Heston model volatility estimation problem

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An analytic approximation of the likelihood function for the Heston model volatility estimation problem