On the calibration of fractional two-factor stochastic volatility model with non-Lipschitz diffusions

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.

Pricing forward-start variance swaps with stochastic volatility

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H < 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

Calibration of stochastic volatility models: A Tikhonov regularization approach

In this article we test the ability of the stochastic differential model proposed by Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362] of forecasting the returns of a long-short equity hedge fund index and of a market index, that is, of the Hedge Fund Research performance Index (HFRI)-Equity index and of the S&P 500 (Standard & Poor 500 New York Stock Exchange) index, respectively. The model is based on the assumptions that the value of the variation of the log-return of the hedge fund index (HFRI-Equity) is proportional up to an additive stochastic error to the value of the variation of the log-return of a market index (S&P 500) and that the log-return of the market index can be satisfactorily modelled using the Heston stochastic volatility model. The model consists of a system of three stochastic differential equations, two of them are the Heston stochastic volatility model and the third one is the equation that models the behaviour of the hedge fund index and its relation with the market index. The model is calibrated on observed data using a method based on filtering and maximum likelihood proposed by Mariani et al. [Maximum likelihood estimation of the Heston stochastic volatility model using asset and option prices: An application of nonlinear filtering theory, Opt. Lett., 2 (2008), pp. 177–222] and further developed in Fatone et al. [Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds, J. Inv. Ill-Posed Probl. 15 (2007), pp. 329–362; The calibration of the Heston stochastic volatility model using filtering and maximum likelihood methodsin Proceedings of Dynamic Systems and Applications, Vol. 5, G.S. Ladde, N.G. Medhin, C. Peng, and M. Sambandham, eds., Dynamic Publishers, Atlanta, USA, 2008, pp. 170–181]. That is, an inverse problem for the stochastic dynamical system representing the model is solved using the calibration procedure. The data analysed is from January 1990 to June 2007, and are monthly data. For each observation time, they consist of the value at the observation time of the log-returns of the HFRI-Equity and of the S&P 500 indices. The calibration procedure uses appropriate subsets of data, that is the data observed in a 6 months time period. The 6 months data time period used in the calibration is rolled through the time series generating a sequence of calibration problems. The values of the HFRI-Equity and S&P 500 indices forecasted using the calibrated models are compared to the values of the indices observed. The result of the comparison is very satisfactory. The website http://www.econ.univpm.it/recchioni/finance/w8 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of some of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

The objective of this article is to evaluate the performance of the option pricing model at the cross-sectional level. For that purpose, the authors propose a statistical framework, in which they in particular account for the uncertainty associated with the reported pricing performance. Instead of a single figure, the authors determine an entire probability distribution function for the loss function that is used to measure the performance of the option pricing model. This method enables them to visualize the effect of parameter uncertainty on the reported pricing performance. Using a data-driven approach, the authors confirm previous evidence that standard volatility models with clustering and leverage effects are sufficient for the option pricing purpose. In addition, they demonstrate that there is short-term persistence but long-term heterogeneity in cross-sectional option pricing information. This finding has two important implications. First, it justifies the practitioner’s routine to refrain from time series approaches and instead estimate option pricing models on a cross section by cross section basis. Second, the long-term heterogeneity in option prices pinpoints the importance of measuring, comparing, and testing the option pricing model for each cross section separately. To the authors’ knowledge no statistical testing framework has previously been applied to a single cross section of option prices. They propose a method that addresses that need. The proposed framework can be applied to a broad set of models and data. In the empirical part of the article, they show by means of example, an application that uses a discrete time volatility model on S&P 500 index options. TOPICS:Options, volatility measures Key Findings • Absolute pricing performance, measured by a loss-function, is an inappropriate criteria to benchmark competing option pricing models at the cross-sectional level. • The long-term heterogeneity in cross-sectional option pricing information and the uncertainty of reported pricing performance calls for the necessity to rely on an entire probability distribution function of the loss function when comparing models performance. • This article proposes a statistical framework, based on a data-driven approach, to compare model performance accounting for model uncertainty applicable to a single cross-section of option prices.

Factorization of European and American Option Prices under Complete and Incomplete Markets

<abstract><p>In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.</p></abstract>

Calibration of Stochastic Volatility Models: An Optimal Control Approach

It is widely noted that the Heston stochastic volatility model fails to capture the fat tails often observed in daily equity returns. Adding random jumps improves the model’s ability to capture extreme events. This extension is known as the Bates stochastic volatility jump (SVJ) model. The model parameters for the Heston and Bates SVJ models are generally calibrated to option prices inducing the so-called risk-neutral measure. However, in the absence of a sufficiently liquid options market, one has to resort to calibration under the realworld measure. In this paper, we calibrate the Heston and Bates SVJ models to historical equity returns in the United States and South Africa using the efficient method of moments(EMM). We then show how a real-world stochastic volatility model can be used in practice to test a simple volatility targeting strategy. Our findings suggest that stochastic volatility and jumps are both required to characterise equity returns in South Africa. Furthermore, volatility targeting is an effective strategy that allows investors to manage the downside risk of a portfolio.

We introduce a hybrid stochastic volatility model where the asset price process follows the Heston model and interest rates are governed by a two-factor stochastic model. Two cases are considered. First, it is assumed that interest rates and asset prices are uncorrelated. The characteristic function method is used to derive semi-analytical pricing formulae for plain vanilla options. In the second case we introduce a correlation between the asset price process and the short rate process and use Monte Carlo simulations for pricing options. To reduce the stochastic error, we implement the control variate method where an estimator of the option value for the uncorrelated case is used as a control variate. The options are priced with a varying correlation coefficient. We observe that the control variate method allows us to speed up Monte Carlo computations by a factor with the magnitude of several hundreds. The efficiency of the method is higher for smaller values of the correlation coefficient. We then study the impact a correlation between the two processes has on option prices. It has been noticed that the call option price is an increasing function of the correlation coefficient.

The reasonable pricing of options can effectively help investors avoid risks and obtain benefits, which plays a very important role in the stability of the financial market. The traditional single option pricing model often fails to meet the ideal expectations due to its limited conditions. Combining an economic model with a deep learning model to establish a hybrid model provides a new method to improve the prediction accuracy of the pricing model. This includes the usage of real historical data of about 10,000 sets of CSI 300 ETF options from January to December 2020 for experimental analysis. Aiming at the prediction problem of CSI 300ETF option pricing, based on the importance of random forest features, the Convolutional Neural Network and Long Short-Term Memory model (CNN-LSTM) in deep learning is combined with a typical stochastic volatility Heston model and stochastic interests CIR model in parameter models. The dual hybrid pricing model of the call option and the put option of CSI 300ETF is established. The dual-hybrid model and the reference model are integrated with ridge regression to further improve the forecasting effect. The results show that the dual-hybrid pricing model proposed in this paper has high accuracy, and the prediction accuracy is tens to hundreds of times higher than the reference model; moreover, MSE can be as low as 0.0003. The article provides an alternative method for the pricing of financial derivatives.

Barrier option pricing under the 2-hypergeometric stochastic volatility model