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Articoli di riviste sul tema "Stochastic Volatility"

Autore: Grafiati

Pubblicato: 4 giugno 2021

Ultima modifica: 9 giugno 2025

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1

Blanco, Belen. "Capturing the volatility smile: parametric volatility models versus stochastic volatility models." Public and Municipal Finance 5, no. 4 (December 26, 2016): 15–22. http://dx.doi.org/10.21511/pmf.05(4).2016.02.

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Abstract (sommario):

Black-Scholes option pricing model (1973) assumes that all option prices on the same underlying asset with the same expiration date, but different exercise prices should have the same implied volatility. However, instead of a flat implied volatility structure, implied volatility (inverting the Black-Scholes formula) shows a smile shape across strikes and time to maturity. This paper compares parametric volatility models with stochastic volatility models in capturing this volatility smile. Results show empirical evidence in favor of parametric volatility models. Keywords: smile volatility, parametric, stochastic, Black-Scholes. JEL Classification: C14 C68 G12 G13

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2

SABANIS, SOTIRIOS. "STOCHASTIC VOLATILITY." International Journal of Theoretical and Applied Finance 05, no. 05 (August 2002): 515–30. http://dx.doi.org/10.1142/s021902490200150x.

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Hull and White [1] have priced a European call option for the case in which the volatility of the underlying asset is a lognormally distributed random variable. They have obtained their formula under the assumption of uncorrelated innovations in security price and volatility. Although the option pricing formula has a power series representation, the question of convergence has been left unanswered. This paper presents an iterative method for calculating all the higher order moments of volatility necessary for the process of proving convergence theoretically. Moreover, simulation results are given that show the practical convergence of the series. These results have been obtained by using a displaced geometric Brownian motion as a volatility process.

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3

Alghalith, Moawia, Christos Floros, and Konstantinos Gkillas. "Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility." Risks 8, no. 2 (April 11, 2020): 35. http://dx.doi.org/10.3390/risks8020035.

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We propose novel nonparametric estimators for stochastic volatility and the volatility of volatility. In doing so, we relax the assumption of a constant volatility of volatility and therefore, we allow the volatility of volatility to vary over time. Our methods are exceedingly simple and far simpler than the existing ones. Using intraday prices for the Standard & Poor’s 500 equity index, the estimates revealed strong evidence that both volatility and the volatility of volatility are stochastic. We also proceeded in a Monte Carlo simulation analysis and found that the estimates were reasonably accurate. Such evidence implies that the stochastic volatility models proposed in the literature with constant volatility of volatility may fail to approximate the discrete-time short rate dynamics.

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4

Veraart, Almut E. D., and Luitgard A. M. Veraart. "Stochastic volatility and stochastic leverage." Annals of Finance 8, no. 2-3 (May 21, 2010): 205–33. http://dx.doi.org/10.1007/s10436-010-0157-3.

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5

Sun, Ya, Meiyi Wang, and Hua Xie. "Volatility analysis of the flight block time based on the stochastic volatility model." Journal of Physics: Conference Series 2489, no. 1 (May 1, 2023): 012002. http://dx.doi.org/10.1088/1742-6596/2489/1/012002.

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Abstract To effectively predict the volatility of flight block time, this paper constructs a stochastic volatility model based on actual flight block time data, solves the model parameters by the Markov chain Monte Carlo method, and uses the standard stochastic volatility (SV-N) model and thick-tailed stochastic volatility (SV-T) model to characterize the volatility of flight block time. The results show that the thick-tailed stochastic volatility model is better than the standard stochastic volatility model in describing the volatility of the segment runtime, and the thick-tailed stochastic volatility model is chosen to predict the volatility of the flight block time. Predicting the flight block time volatility in real time can provide a theoretical basis for traffic traveler planning.

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6

Mahatma, Yudi, and Ibnu Hadi. "Stochastic Volatility Estimation of Stock Prices using the Ensemble Kalman Filter." InPrime: Indonesian Journal of Pure and Applied Mathematics 3, no. 2 (November 10, 2021): 136–43. http://dx.doi.org/10.15408/inprime.v3i2.20256.

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AbstractVolatility plays important role in options trading. In their seminal paper published in 1973, Black and Scholes assume that the stock price volatility, which is the underlying security volatility of a call option, is constant. But thereafter, researchers found that the return volatility was not constant but conditional to the information set available at the computation time. In this research, we improve a methodology to estimate volatility and interest rate using Ensemble Kalman Filter (EnKF). The price of call and put option used in the observation and the forecasting step of the EnKF algorithm computed using the solution of Black-Scholes PDE. The state-space used in this method is the augmented state space, which consists of static variables: volatility and interest rate, and dynamic variables: call and put option price. The numerical experiment shows that the EnKF algorithm is able to estimate accurately the estimated volatility and interest rates with an RMSE value of 0.0506.Keywords: stochastic volatility; call option; put option; Ensemble Kalman Filter. AbstrakVolatilitas adalah faktor penting dalam perdagangan suatu opsi. Dalam makalahnya yang dipublikasikan tahun 1973, Black dan Scholes mengasumsikan bahwa volatilitas harga saham, yang merupakan volatilitas sekuritas yang mendasari opsi beli, adalah konstan. Akan tetapi, para peneliti menemukan bahwa volatilitas pengembalian tidaklah konstan melainkan tergantung pada kumpulan informasi yang dapat digunakan pada saat perhitungan. Pada penelitian ini dikembangkan metodologi untuk mengestimasi volatilitas dan suku bunga menggunakan metode Ensembel Kalman Filter (EnKF). Harga opsi beli dan opsi jual yang digunakan pada observasi dan pada tahap prakiraan pada algoritma EnKF dihitung menggunakan solusi persamaan Black-Scholes. Ruang keadaan yang digunakan adalah ruang keadaan yang diperluas yang terdiri dari variabel statis yaitu volatilitas dan suku bunga, dan variabel dinamis yaitu harga opsi beli dan harga opsi jual. Eksperimen numerik menunjukkan bahwa algoritma ENKF dapat secara akurat mengestimasi volatiltas dan suku bunga dengan RMSE 0.0506.Kata kunci: volatilitas stokastik; opsi beli; opsi jual; Ensembel Kalman Filter.

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7

Guyon, Julien. "Stochastic Volatility Modeling." Quantitative Finance 17, no. 6 (April 18, 2017): 825–28. http://dx.doi.org/10.1080/14697688.2017.1309181.

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8

Bandi, Federico M., and Roberto Renò. "NONPARAMETRIC STOCHASTIC VOLATILITY." Econometric Theory 34, no. 6 (July 3, 2018): 1207–55. http://dx.doi.org/10.1017/s0266466617000457.

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Abstract (sommario):

We provide nonparametric methods for stochastic volatility modeling. Our methods allow for the joint evaluation of return and volatility dynamics with nonlinear drift and diffusion functions, nonlinear leverage effects, and jumps in returns and volatility with possibly state-dependent jump intensities, among other features. In the first stage, we identify spot volatility by virtue of jump-robust nonparametric estimates. Using observed prices and estimated spot volatilities, the second stage extracts the functions and parameters driving price and volatility dynamics from nonparametric estimates of the bivariate process’ infinitesimal moments. For these infinitesimal moment estimates, we report an asymptotic theory relying on joint in-fill and long-span arguments which yields consistency and weak convergence under mild assumptions.

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9

Capobianco, E. "Stochastic Volatility Systems." International Journal of Modelling and Simulation 17, no. 2 (January 1997): 137–42. http://dx.doi.org/10.1080/02286203.1997.11760322.

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10

Ilinski, Kirill, and Oleg Soloviev. "Stochastic volatility membrane." Wilmott 2004, no. 3 (May 2004): 74–81. http://dx.doi.org/10.1002/wilm.42820040317.

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11

Yang, Ben-Zhang, Jia Yue, Ming-Hui Wang, and Nan-Jing Huang. "Volatility swaps valuation under stochastic volatility with jumps and stochastic intensity." Applied Mathematics and Computation 355 (August 2019): 73–84. http://dx.doi.org/10.1016/j.amc.2019.02.063.

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12

Zhu, Song-Ping, and Guang-Hua Lian. "Analytically pricing volatility swaps under stochastic volatility." Journal of Computational and Applied Mathematics 288 (November 2015): 332–40. http://dx.doi.org/10.1016/j.cam.2015.04.036.

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13

Zhou, Yanli, Shican Liu, Shuang Li, and Xiangyu Ge. "The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach." Journal of Function Spaces 2021 (September 17, 2021): 1–14. http://dx.doi.org/10.1155/2021/1217665.

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Abstract (sommario):

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

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14

PFANTE, OLIVER, and NILS BERTSCHINGER. "VOLATILITY INFERENCE AND RETURN DEPENDENCIES IN STOCHASTIC VOLATILITY MODELS." International Journal of Theoretical and Applied Finance 22, no. 03 (May 2019): 1950013. http://dx.doi.org/10.1142/s0219024919500134.

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Stochastic volatility models describe stock returns [Formula: see text] as driven by an unobserved process capturing the random dynamics of volatility [Formula: see text]. The present paper quantifies how much information about volatility [Formula: see text] and future stock returns can be inferred from past returns in stochastic volatility models in terms of Shannon’s mutual information. In particular, we show that across a wide class of stochastic volatility models, including a two-factor model, returns observed on the scale of seconds would be needed to obtain reliable volatility estimates. In addition, we prove that volatility forecasts beyond several weeks are essentially impossible for fundamental information theoretic reasons.

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15

Aït-Sahalia, Yacine, Chenxu Li, and Chen Xu Li. "Implied Stochastic Volatility Models." Review of Financial Studies 34, no. 1 (March 30, 2020): 394–450. http://dx.doi.org/10.1093/rfs/hhaa041.

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Abstract This paper proposes “implied stochastic volatility models” designed to fit option-implied volatility data and implements a new estimation method for such models. The method is based on explicitly linking observed shape characteristics of the implied volatility surface to the coefficient functions that define the stochastic volatility model. The method can be applied to estimate a fully flexible nonparametric model, or to estimate by the generalized method of moments any arbitrary parametric stochastic volatility model, affine or not. Empirical evidence based on S&P 500 index options data show that the method is stable and performs well out of sample.

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16

FOUQUE, JEAN-PIERRE, GEORGE PAPANICOLAOU, and K. RONNIE SIRCAR. "MEAN-REVERTING STOCHASTIC VOLATILITY." International Journal of Theoretical and Applied Finance 03, no. 01 (January 2000): 101–42. http://dx.doi.org/10.1142/s0219024900000061.

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Abstract (sommario):

We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value, but it is fast mean-reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identified using a wavelet-based tool. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

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17

Ball, Clifford A., and Antonio Roma. "Stochastic Volatility Option Pricing." Journal of Financial and Quantitative Analysis 29, no. 4 (December 1994): 589. http://dx.doi.org/10.2307/2331111.

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18

Corlay, Sylvain, Joachim Lebovits, and Jacques Lévy Véhel. "MULTIFRACTIONAL STOCHASTIC VOLATILITY MODELS." Mathematical Finance 24, no. 2 (February 11, 2013): 364–402. http://dx.doi.org/10.1111/mafi.12024.

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19

Leisen, Dietmar P. J. "A Stochastic Volatility Lattice." IFAC Proceedings Volumes 31, no. 16 (June 1998): 75–80. http://dx.doi.org/10.1016/s1474-6670(17)40461-7.

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20

Asai, Manabu, and Michael McAleer. "Asymmetric Multivariate Stochastic Volatility." Econometric Reviews 25, no. 2-3 (September 2006): 453–73. http://dx.doi.org/10.1080/07474930600712913.

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21

Serletis, Apostolos, and Maksim Isakin. "Stochastic volatility demand systems." Econometric Reviews 36, no. 10 (October 7, 2015): 1111–22. http://dx.doi.org/10.1080/07474938.2014.977091.

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22

Ghysels, Eric, Christian Gouriéroux, and Joann Jasiak. "Stochastic volatility duration models." Journal of Econometrics 119, no. 2 (April 2004): 413–33. http://dx.doi.org/10.1016/s0304-4076(03)00202-1.

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23

Kurose, Yuta, and Yasuhiro Omori. "Dynamic equicorrelation stochastic volatility." Computational Statistics & Data Analysis 100 (August 2016): 795–813. http://dx.doi.org/10.1016/j.csda.2015.01.013.

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24

Le�n, �ngel, and Gonzalo Rubio. "Smiling under stochastic volatility." Spanish Economic Review 6, no. 1 (April 1, 2004): 53–75. http://dx.doi.org/10.1007/s10108-003-0077-8.

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25

Cavaliere, Giuseppe. "Stochastic Volatility: Selected Readings." Economic Journal 116, no. 512 (June 1, 2006): F326—F327. http://dx.doi.org/10.1111/j.1468-0297.2006.01102_1.x.

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26

Fouque, Jean-Pierre, George Papanicolaou, Ronnie Sircar, and Knut Solna. "Multiscale Stochastic Volatility Asymptotics." Multiscale Modeling & Simulation 2, no. 1 (January 2003): 22–42. http://dx.doi.org/10.1137/030600291.

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27

Le, Truc. "Stochastic market volatility models." Applied Financial Economics Letters 1, no. 3 (May 2005): 177–88. http://dx.doi.org/10.1080/17446540500101986.

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28

Aknouche, Abdelhakim. "Periodic autoregressive stochastic volatility." Statistical Inference for Stochastic Processes 20, no. 2 (June 14, 2016): 139–77. http://dx.doi.org/10.1007/s11203-016-9139-z.

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29

Cordis, Adriana S., and Chris Kirby. "Discrete stochastic autoregressive volatility." Journal of Banking & Finance 43 (June 2014): 160–78. http://dx.doi.org/10.1016/j.jbankfin.2014.03.020.

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30

Abraham, Bovas, N. Balakrishna, and Ranjini Sivakumar. "Gamma stochastic volatility models." Journal of Forecasting 25, no. 3 (2006): 153–71. http://dx.doi.org/10.1002/for.982.

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31

Javaheri, Alireza. "Inference and stochastic volatility." Wilmott 2004, no. 4 (July 2004): 56–63. http://dx.doi.org/10.1002/wilm.42820040415.

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32

PAN, MIN, and SHENGQIAO TANG. "OPTION PRICING AND EXECUTIVE STOCK OPTION INCENTIVES: AN EMPIRICAL INVESTIGATION UNDER GENERAL ERROR DISTRIBUTION STOCHASTIC VOLATILITY MODEL." Asia-Pacific Journal of Operational Research 28, no. 01 (February 2011): 81–93. http://dx.doi.org/10.1142/s0217595911003065.

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This article investigates the valuation of executive stock options when the stock return volatility is governed by the general error distribution stochastic volatility model, involving both the features of the stock return volatility and the abnormal fluctuations of the stock price at the expiration date. We estimate the parameters in the general error distribution stochastic volatility model using the Markov Chain Monte Carlo method with Shanghai & Shenzhen 300 Index in China as a sample, and compare the executive stock option values calculated by Black-Scholes option pricing model and the option pricing model under general error distribution stochastic volatility model. The results show that the general error distribution stochastic volatility model has greater veracity in describing the volatility of stock market returns, and there is divergence between the two values estimated by Black-Scholes option pricing model and the option pricing model under general error distribution stochastic volatility model. The divergence varies with the discrepancy between the price of underlying stock at the granting date and the strike price of the option.

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33

LEE, ROGER W. "IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY." International Journal of Theoretical and Applied Finance 04, no. 01 (February 2001): 45–89. http://dx.doi.org/10.1142/s0219024901000870.

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For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

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34

FONG, WAI MUN, and WING-KEUNG WONG. "THE STOCHASTIC COMPONENT OF REALIZED VOLATILITY." Annals of Financial Economics 02, no. 01 (June 2006): 0650004. http://dx.doi.org/10.1142/s2010495206500047.

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Abstract (sommario):

Volatility–volume regressions provide a convenient framework to study sources of volatility predictability. We apply this approach to the daily realized volatility of common stocks. We find that unexpected volume plays a more significant role in explaining realized volatility than expected volume, and accounts for about one-third of the non-persistent component in the volatility process. Contrary to the findings of Lamoureux and Lastrapes (1990), the ARCH effect is robust even in the presence of volume. However, this component explains only about half of the variations in realized volatility. Thus, large portion of realized volatility is clearly stochastic. This presents a significant challenge to the goal of achieving precise realized volatility forecasts.

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35

Zhu, Yingzi, and Marco Avellaneda. "A Risk-Neutral Stochastic Volatility Model." International Journal of Theoretical and Applied Finance 01, no. 02 (April 1998): 289–310. http://dx.doi.org/10.1142/s0219024998000163.

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We construct a risk-neutral stochastic volatility model using no-arbitrage pricing principles. We then study the behavior of the implied volatility of options that are deep in and out of the money according to this model. The motivation of this study is to show the difference in the asymptotic behavior of the distribution tails between the usual Black–Scholes log-normal distribution and the risk-neutral stochastic volatility distribution. In the second part of the paper, we further explore this risk-neutral stochastic volatility model by a Monte-Carlo study on the implied volatility curve (implied volatility as a function of the option strikes) for near-the-money options. We study the behavior of this "smile" curve under different choices of parameter for the model, and determine how the shape and skewness of the "smile" curve is affected by the volatility of volatility ("V-vol") and the correlation between the underlying asset and its volatility.

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36

Tauchen, George. "Stochastic Volatility in General Equilibrium." Quarterly Journal of Finance 01, no. 04 (December 2011): 707–31. http://dx.doi.org/10.1142/s2010139211000237.

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The connections between stock market volatility and returns are studied within the context of a general equilibrium framework. The framework rules out a priori any purely statistical relationship between volatility and returns by imposing uncorrelated innovations. The main model generates a two-factor structure for stock market volatility along with time-varying risk premiums on consumption and volatility risk. It also generates endogenously a dynamic leverage effect (volatility asymmetry), the sign of which depends upon the magnitudes of the risk aversion and the intertemporal elasticity of substitution parameters.

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37

Lu, Xiang, Gunter Meissner, and Hong Sherwin. "A Unified Stochastic Volatility—Stochastic Correlation Model." Journal of Mathematical Finance 10, no. 04 (2020): 679–96. http://dx.doi.org/10.4236/jmf.2020.104039.

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38

JIANG, GEORGE J. "STOCHASTIC VOLATILITY AND JUMP-DIFFUSION — IMPLICATIONS ON OPTION PRICING." International Journal of Theoretical and Applied Finance 02, no. 04 (October 1999): 409–40. http://dx.doi.org/10.1142/s0219024999000212.

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Abstract (sommario):

This paper conducts a thorough and detailed investigation on the implications of stochastic volatility and random jump on option prices. Both stochastic volatility and jump-diffusion processes admit asymmetric and fat-tailed distribution of asset returns and thus have similar impact on option prices compared to the Black–Scholes model. While the dynamic properties of stochastic volatility model are shown to have more impact on long-term options, the random jump is shown to have relatively larger impact on short-term near-the-money options. The misspecification risk of stochastic volatility as jump is minimal in terms of option pricing errors only when both the level of kurtosis of the underlying asset return distribution and the level of volatility persistence are low. While both asymmetric volatility and asymmetric jump can induce distortion of option pricing errors, the skewness of jump offers better explanations to empirical findings on implied volatility curves.

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Yoon, Ji-Hun, Jeong-Hoon Kim, Sun-Yong Choi, and Youngchul Han. "Stochastic volatility asymptotics of defaultable interest rate derivatives under a quadratic Gaussian model." Stochastics and Dynamics 17, no. 01 (December 15, 2016): 1750003. http://dx.doi.org/10.1142/s0219493717500034.

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Abstract (sommario):

Stochastic volatility of underlying assets has been shown to affect significantly the price of many financial derivatives. In particular, a fast mean-reverting factor of the stochastic volatility plays a major role in the pricing of options. This paper deals with the interest rate model dependence of the stochastic volatility impact on defaultable interest rate derivatives. We obtain an asymptotic formula of the price of defaultable bonds and bond options based on a quadratic term structure model and investigate the stochastic volatility and default risk effects and compare the results with those of the Vasicek model.

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40

Barndorff-Nielsen, O. E., and A. E. D. Veraart. "Stochastic Volatility of Volatility and Variance Risk Premia." Journal of Financial Econometrics 11, no. 1 (August 16, 2012): 1–46. http://dx.doi.org/10.1093/jjfinec/nbs008.

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41

Woerner, Jeannette H. C. "Estimation of integrated volatility in stochastic volatility models." Applied Stochastic Models in Business and Industry 21, no. 1 (January 2005): 27–44. http://dx.doi.org/10.1002/asmb.548.

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42

Berestycki, Henri, J�r�me Busca, and Igor Florent. "Computing the implied volatility in stochastic volatility models." Communications on Pure and Applied Mathematics 57, no. 10 (2004): 1352–73. http://dx.doi.org/10.1002/cpa.20039.

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43

Franco, Sebastian, and Anatoliy Swishchuk. "Pricing of Pseudo-Swaps Based on Pseudo-Statistics." Risks 11, no. 8 (August 3, 2023): 141. http://dx.doi.org/10.3390/risks11080141.

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Abstract (sommario):

The main problem in pricing variance, volatility, and correlation swaps is how to determine the evolution of the stochastic processes for the underlying assets and their volatilities. Thus, sometimes it is simpler to consider pricing of swaps by so-called pseudo-statistics, namely, the pseudo-variance, -covariance, -volatility, and -correlation. The main motivation of this paper is to consider the pricing of swaps based on pseudo-statistics, instead of stochastic models, and to compare this approach with the most popular stochastic volatility model in the Cox–Ingresoll–Ross (CIR) model. Within this paper, we will demonstrate how to value different types of swaps (variance, volatility, covariance, and correlation swaps) using pseudo-statistics (pseudo-variance, pseudo-volatility, pseudo-correlation, and pseudo-covariance). As a result, we will arrive at a method for pricing swaps that does not rely on any stochastic models for a stochastic stock price or stochastic volatility, and instead relies on data/statistics. A data/statistics-based approach to swap pricing is very different from stochastic volatility models such as the Cox–Ingresoll–Ross (CIR) model, which, in comparison, follows a stochastic differential equation. Although there are many other stochastic models that provide an approach to calculating the price of swaps, we will use the CIR model for comparison within this paper, due to the popularity of the CIR model. Therefore, in this paper, we will compare the CIR model approach to pricing swaps to the pseudo-statistic approach to pricing swaps, in order to compare a stochastic model to the data/statistics-based approach to swap pricing that is developed within this paper.

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44

Derman, Emanuel, and Iraj Kani. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility." International Journal of Theoretical and Applied Finance 01, no. 01 (January 1998): 61–110. http://dx.doi.org/10.1142/s0219024998000059.

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In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define the arbitrage-free evolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair. We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index. You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.

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45

Zhang, Luwen, and Li Wang. "Generalized Method of Moments Estimation of Realized Stochastic Volatility Model." Journal of Risk and Financial Management 16, no. 8 (August 16, 2023): 377. http://dx.doi.org/10.3390/jrfm16080377.

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The purpose of this paper is to study the generalized method of moments (GMM) estimation procedures of the realized stochastic volatility model; we give the moment conditions for this model and then obtain the estimation of parameters. Then, we apply these moment conditions to the realized stochastic volatility model to improve the volatility prediction effect. This paper selects the Shanghai Composite Index (SSE) as the original data of model research and completes the volatility prediction under a realized stochastic volatility model. Markov chain Monte Carlo (MCMC) estimation and quasi-maximum likelihood (QML) estimation are applied to the parameter estimation of the realized stochastic volatility model to compare with the GMM method. And the volatility prediction accuracy of these three different methods is compared. The results of empirical research show that the effect of model prediction using the parameters obtained by the GMM method is close to that of the MCMC method, and the effect is obviously better than that of the quasi-maximum likelihood estimation method.

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46

Kouritzin, Michael A. "Microstructure Models with Short-Term Inertia and Stochastic Volatility." Mathematical Problems in Engineering 2015 (2015): 1–17. http://dx.doi.org/10.1155/2015/323475.

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Partially observed microstructure models, containing stochastic volatility, dynamic trading noise, and short-term inertia, are introduced to address the following questions: (1) Do the observed prices exhibit statistically significant inertia? (2) Is stochastic volatility (SV) still evident in the presence of dynamical trading noise? (3) If stochastic volatility and trading noise are present, which SV model matches the observed price data best? Bayes factor methods are used to answer these questions with real data and this allows us to consider volatility models with very different structures. Nonlinear filtering techniques are utilized to compute the Bayes factor on tick-by-tick data and to estimate the unknown parameters. It is shown that our price data sets all exhibit strong evidence of both inertia and Heston-type stochastic volatility.

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47

VAN DER STOEP, ANTHONIE W., LECH A. GRZELAK, and CORNELIS W. OOSTERLEE. "COLLOCATING VOLATILITY: A COMPETITIVE ALTERNATIVE TO STOCHASTIC LOCAL VOLATILITY MODELS." International Journal of Theoretical and Applied Finance 23, no. 06 (September 2020): 2050038. http://dx.doi.org/10.1142/s0219024920500387.

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We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) framework, introduced in [L. A. Grzelak (2019) The CLV framework — A fresh look at efficient pricing with smile, International Journal of Computer Mathematics 96 (11), 2209–2228]. The CV framework consists of two elements, a “kernel process” that can be efficiently evaluated and a local volatility function. The latter, based on stochastic collocation — e.g. [I. Babuška, F. Nobile & R. Tempone (2007) A stochastic collocation method for elliptic partial differential equations with random input Data, SIAM Journal on Numerical Analysis 45 (3), 1005–1034; B. Ganapathysubramanian & N. Zabaras (2007) Sparse grid collocation schemes for stochastic natural convection problems, Journal of Computational Physics 225 (1), 652–685; J. A. S. Witteveen & G. Iaccarino (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces, SIAM Journal on Scientific Computing 34 (2), A814–A838; D. Xiu & J. S. Hesthaven (2005) High-order collocation methods for differential equations with random inputs, SIAM Journal on Scientific Computing 27 (3), 1118–1139] — connects the kernel process to the market and allows the CV framework to be perfectly calibrated to European-type options. In this paper, we consider three different kernel process choices: the Ornstein–Uhlenbeck (OU) and Cox–Ingersoll–Ross (CIR) processes and the Heston model. The kernel process controls the forward smile and allows for an accurate and efficient calibration to exotic options, while the perfect calibration to liquid market quotes is preserved. We confirm this by numerical experiments, in which we calibrate the OU-CV, CIR-CV and Heston-CV frameworks to FX barrier options.

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48

Dhifaoui, Zouhaier, and Faicel Gasmi. "Linear and nonlinear linkage of conditional stochastic volatility of interbank interest rates: Empirical evidence of the BRICS countries." BRICS Journal of Economics 2, no. 2 (July 30, 2021): 4–16. http://dx.doi.org/10.38050/2712-7508-2021-2-1.

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The purpose of this article is to detect a possible linear and nonlinear causal relationship between the conditional stochastic volatility of log return of interbank interest rates for the BRICS countries in the period from January 2015 to October 2018. To extract the volatility of the analyzed time series, we use a stochastic volatility model with moving average innovations. To test a causal relationship between the estimated stochastic volatilities, two steps are applied. First, we used the Granger causality test and a vector autoregressive model (VAR). Secondly, we applied the nonlinear Granger causality test to the raw data to explore a new nonlinear causal relationship between stochastic volatility time series, and also applied it to the residual of the VAR model to confirm the causality detected in the first step. This study demonstrates the existence of some unidirectional/bidirectional linear/nonlinear causal relationships between the studied stochastic volatility time series.

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49

Liu, Jia. "A Bayesian Semiparametric Realized Stochastic Volatility Model." Journal of Risk and Financial Management 14, no. 12 (December 19, 2021): 617. http://dx.doi.org/10.3390/jrfm14120617.

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Abstract (sommario):

This paper proposes a semiparametric realized stochastic volatility model by integrating the parametric stochastic volatility model utilizing realized volatility information and the Bayesian nonparametric framework. The flexible framework offered by Bayesian nonparametric mixtures not only improves the fitting of asymmetric and leptokurtic densities of asset returns and logarithmic realized volatility but also enables flexible adjustments for estimation bias in realized volatility. Applications to equity data show that the proposed model offers superior density forecasts for returns and improved estimates of parameters and latent volatility compared with existing alternatives.

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50

Ma, Chaoqun, Shengjie Yue, and Yishuai Ren. "Pricing Vulnerable European Options under Lévy Process with Stochastic Volatility." Discrete Dynamics in Nature and Society 2018 (October 23, 2018): 1–16. http://dx.doi.org/10.1155/2018/3402703.

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Abstract (sommario):

This paper considers the pricing issue of vulnerable European option when the dynamics of the underlying asset value and counterparty’s asset value follow two correlated exponential Lévy processes with stochastic volatility, and the stochastic volatility is divided into the long-term and short-term volatility. A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility is governed by a mean-reverting process. Based on the proposed model, the joint moment generating function of underlying log-asset price and counterparty’s log-asset value is explicitly derived. We derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions. Finally, numerical simulations are provided to illustrate the effects of stochastic volatility, jump risk, and counterparty credit risk on the vulnerable option price.

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