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## Mean-Reverting Stochastic Volatility (2000)

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Citations: | 40 - 10 self |

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1510 | A closed-form solution for options with stochastic volatility with applications to bond and currency options
- HESTON
- 1993
(Show Context)
Citation Context ... [25] for detailed calculations in the former regime. 1.2.2 Separation of Scales There has been much analysis of specific Ito models in the literature by numerical and analytical methods, for example =-=[18, 35, 38]-=-, many of which have ignored correlation effects and/or the volatility risk premium for tractability. Our goal is to identify and estimate from market data the relevant parameters for derivative prici... |

1267 |
Conditional Heteroskedasticity in Asset Returns: A
- Nelson
- 1992
(Show Context)
Citation Context ... (f(y) = e y in (6-7)) stochastic volatility model [29], is the most relevant here because it contains skew, or nonzero correlation, whereas in ARCH/GARCH, there is none. In the original EGARCH paper =-=[30]-=-, Nelson used maximum likelihood estimation (with a non-Gaussian random variable replacing the second Brownian increment dZ), and subsequent studies, especially in the stochastic volatility literature... |

1002 | The pricing of options on assets with stochastic volatilities
- Hull, White
- 1987
(Show Context)
Citation Context ...33], called the smile curve and, more recently, negative or positive sloping [34], known as skew. This particular shortcoming is remedied by stochastic volatility models first studied by Hull & White =-=[21]-=-, Scott [35] and Wiggins [39] in 1987. The underlying asset price is modelled as a stochastic process which is now driven by a random volatility Ito process that may or may not be independent. It was ... |

704 | Empirical performance of alternative option pricing models
- Bakshi, Cao, et al.
- 1997
(Show Context)
Citation Context ... [9] find that implied volatilities contain additional information to that in historical volatilities, so that the out-of-sample predictive power of the former is greater. This motivates some authors =-=[4, 12] to calibr-=-ate their models using derivative data only, and some "hybrid" approaches [8, 35] get most parameters from underlying data and the remaining (usually today's volatility oe t , and/or the vol... |

640 |
Studies of Stock Price Volatility Changes.”
- Black
- 1976
(Show Context)
Citation Context ...ed distribution for the asset price that lognormal or zero-correlation stochastic volatility models do not exhibit. The skew is documented in empirical studies of historical stock prices, for example =-=[7]-=-, and past implied volatility data [5]. 1 This was suggested to us by Darrell Duffie. 1.3 Main Result 1. When the rate of volatility mean-reversion ff, defined in (7), is large (volatility persistence... |

601 | Bayesian analysis of stochastic volatility models
- Jacquier, Polson, et al.
- 1994
(Show Context)
Citation Context ...at indicates that ff is large, or that the half-life of mean-reversion is on the order of a trading day for our data. Such a fast meanreversion rate has previously been observed in exchange rate data =-=[24, 27]-=-, and in equities [8]. 4.4.1 Rate of mean reversion from spectra In this section we study the rate of mean reversion ff from a spectral analysis of the normalized fluctuation sequence D n given by (21... |

472 |
Binomial Trees
- Rubinstein
- 1998
(Show Context)
Citation Context ...isk can (theoretically) be perfectly hedged by the underlying, and there is no volatility risk premium to be estimated. Numerically inferred local volatility surfaces from market data by tree methods =-=[34]-=- or relative-entropy minimization [3] or interpolation [36] have yielded interesting qualitative properties of the (risk-neutral) probability distribution used by the market to price derivatives (such... |

362 | Stock price distributions with stochastic volatility: An analytical approach. - Stein, Stein - 1991 |

346 | Stochastic Volatility
- Ghysels, Harvey, et al.
- 1996
(Show Context)
Citation Context ...could be improved by using historical underlying data as well, though it is not clear how to implement this. We also refer the reader to recent surveys of the stochastic volatility literature such as =-=[16, 17, 20]. 1.2 Pres-=-ent Approach We concentrate on the "pure" stochastic volatility approach in which volatility oe t is modeled as an Ito process driven by a Brownian motion that has a component independent of... |

303 | Implied volatility functions: Empirical tests
- Dumas, Fleming, et al.
- 1998
(Show Context)
Citation Context ...tes are extremely useful for contemporaneous calibration of exotic securities, but this approach has not yet produced a stable surface that can be used consistently and with confidence over time. See =-=[13] for a det-=-ailed empirical study of this issue and [25] for a mathematical explanation of why these surface-fits are outperformed by "fixed smile" (projected) implied volatilities. Possibly this shortc... |

251 |
ARCH models as diffusion approximations.
- Nelson
- 1990
(Show Context)
Citation Context ...ng formulas are structurally unchanged by different choices. The simplest example, f(y) = e y was proposed by Scott [35] and was also studied by Wiggins [39]. It is related to EGARCH models by Nelson =-=[29]-=-; the asymptotic analysis of Section 3 for this particular case appears in [14]. Figure 1 shows the estimate S&P 500 twenty-day transition probability density (from the high-frequency data using metho... |

249 |
The crash of ’87: Was it expected? The evidence from options markets
- Bates
- 1991
(Show Context)
Citation Context ...at lognormal or zero-correlation stochastic volatility models do not exhibit. The skew is documented in empirical studies of historical stock prices, for example [7], and past implied volatility data =-=[5]-=-. 1 This was suggested to us by Darrell Duffie. 1.3 Main Result 1. When the rate of volatility mean-reversion ff, defined in (7), is large (volatility persistence) , the implied volatility curve from ... |

240 |
Option Pricing When the Variance Changes Randomly: Theory, Estimation, and an Application,”
- Scott
- 1987
(Show Context)
Citation Context ...the smile curve and, more recently, negative or positive sloping [34], known as skew. This particular shortcoming is remedied by stochastic volatility models first studied by Hull & White [21], Scott =-=[35]-=- and Wiggins [39] in 1987. The underlying asset price is modelled as a stochastic process which is now driven by a random volatility Ito process that may or may not be independent. It was shown by Ren... |

230 | Pricing Foreign Currency Options with Stochastic Volatility,” - Melino, Turnbull - 1990 |

176 |
Option Values under Stochastic Volatilities,”
- Wiggins
- 1987
(Show Context)
Citation Context ...nd, more recently, negative or positive sloping [34], known as skew. This particular shortcoming is remedied by stochastic volatility models first studied by Hull & White [21], Scott [35] and Wiggins =-=[39]-=- in 1987. The underlying asset price is modelled as a stochastic process which is now driven by a random volatility Ito process that may or may not be independent. It was shown by Renault & Touzi [32]... |

173 |
Stock market volatility and the information content of stock index options.
- Day, Lewis
- 1992
(Show Context)
Citation Context ...at EGARCH models provide a better description than GARCH or CIR-based models (see equation (10)). In Merville & Piptea [28], implied volatility is found to be strongly mean-reverting, and Day & Lewis =-=[9]-=- find that implied volatilities contain additional information to that in historical volatilities, so that the out-of-sample predictive power of the former is greater. This motivates some authors [4, ... |

147 |
Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23
- Rubinstein
- 1976
(Show Context)
Citation Context ...ce of a nonflat implied volatility surface (or term-structure) has been noticed and become more pronounced, especially since the 1987 crash. This phenomenon, which is well-documented in, for example, =-=[23, 33]-=-, stands in empirical contradiction to the consistent use of a classical Black-Scholes (constant volatility) approach to pricing options and similar securities. However, it is clearly desirable to mai... |

141 |
GMM estimation of a stochastic volatility model: a Monte Casrlo study.
- Andersen, Sørensen
- 1996
(Show Context)
Citation Context ...ber of moments potentially better exploiting the data, but greatly reducing the accuracy with which the weighting matrix itself can be estimated. The detailed Monte Carlo study by Andersen & Sorensen =-=[2]-=- of this method applied to the expOU stochastic volatility model (which they refer to as the stochastic volatility model) gives some guidelines in this regard, but they strongly caution against using ... |

100 | The Mathematics of Financial derivatives. A Student Introduction. Press Syndicate of the University of Cambridge, - Dewynne - 1996 |

94 | Bounds of probability” - Shimko - 1993 |

85 | Adaptive covariance estimation of locally stationary processes
- Mallat, Papanicolaou, et al.
- 1998
(Show Context)
Citation Context ...the random variables and their samples to keep the notation simple - which is meant will be clear in context. 4.2.2 Segments of stationarity We use the code BBLCT (Best Basis Local Cosines Transform) =-=[26]-=- to locate segments of the data within which the Ds can be treated as stationary. Details are given in Appendix C. Our findings using this tool are summarized as follows: 1. In our first pass at the S... |

83 | Testing Option Pricing Models
- Bates
- 1996
(Show Context)
Citation Context ...ata by moments or likelihood methods in the ARCH-related literature, fitting implied volatility alone, or "hybrid" approaches using both underlying and derivative data. An extensive review a=-=ppears in [6] and we gi-=-ve only a brief summary. In the first category, the EGARCH models, whose continuous-time diffusion limit is the "expOU" (f(y) = e y in (6-7)) stochastic volatility model [29], is the most re... |

80 |
Pricing European Currency Options: A Comparison of the Modified Blacke-Scholes Model and a Random Variance Model.”
- Chesney, Scott
- 1989
(Show Context)
Citation Context ... Nelson used maximum likelihood estimation (with a non-Gaussian random variable replacing the second Brownian increment dZ), and subsequent studies, especially in the stochastic volatility literature =-=[8, 27, 35, 39]-=- use the Generalized Method of Moments (GMM). In using GMM, it is necessary to choose which moments to match and what weighting matrix to use. Indeed, there is a trade-off between a large number of mo... |

78 |
Option hedging and implied volatilities in a stochastic volatility model,
- Renault, Touzi
- 1996
(Show Context)
Citation Context ...[39] in 1987. The underlying asset price is modelled as a stochastic process which is now driven by a random volatility Ito process that may or may not be independent. It was shown by Renault & Touzi =-=[32]-=- that stochastic volatility European option prices produce the smile curve for any volatility process uncorrelated with the Brownian motion driving the price process, and this robustness to specific m... |

69 |
Dynamic Asset Pricing Theory, 2nd ed.
- Duffie
- 1996
(Show Context)
Citation Context ...amma r) f (Y s ) ds; ~ Z t = Z t + Z t 0 fl s ds; define independent Brownian motions ( ~ W ; ~ Z) under Q(fl), assuming for instance that ( ��\Gammar f(Y t ) ; fl t ) satisfies the Novikov condit=-=ion [11]-=-. Obviously this will not be the case with f(y) = e y and Y Gaussian. Nevertheless e y can be cutoff at 0 and the cutoff removed at the end to obtain the formula given as an example in Section 3. The ... |

69 |
Transform Analysis and Option Pricing for Affine Jump-Diffusions. Econometrica,
- Duffie, Pan, et al.
- 2000
(Show Context)
Citation Context ... [9] find that implied volatilities contain additional information to that in historical volatilities, so that the out-of-sample predictive power of the former is greater. This motivates some authors =-=[4, 12] to calibr-=-ate their models using derivative data only, and some "hybrid" approaches [8, 35] get most parameters from underlying data and the remaining (usually today's volatility oe t , and/or the vol... |

61 | Calibrating Volatility Surfaces Via Relative-Entropy Minimization.” Working Paper
- Avellaneda, Friedman, et al.
- 1996
(Show Context)
Citation Context ...edged by the underlying, and there is no volatility risk premium to be estimated. Numerically inferred local volatility surfaces from market data by tree methods [34] or relative-entropy minimization =-=[3]-=- or interpolation [36] have yielded interesting qualitative properties of the (risk-neutral) probability distribution used by the market to price derivatives (such as excess skew and leptokurtosis in ... |

56 | Derivative asset analysis in models with level-dependent and stochastic volatility’,
- Frey
- 1997
(Show Context)
Citation Context ...could be improved by using historical underlying data as well, though it is not clear how to implement this. We also refer the reader to recent surveys of the stochastic volatility literature such as =-=[16, 17, 20]. 1.2 Pres-=-ent Approach We concentrate on the "pure" stochastic volatility approach in which volatility oe t is modeled as an Ito process driven by a Brownian motion that has a component independent of... |

55 | Recovering probability distributions from contemporaneous security prices,
- Jackerth, Rubinstein
- 1996
(Show Context)
Citation Context ...ce of a nonflat implied volatility surface (or term-structure) has been noticed and become more pronounced, especially since the 1987 crash. This phenomenon, which is well-documented in, for example, =-=[23, 33]-=-, stands in empirical contradiction to the consistent use of a classical Black-Scholes (constant volatility) approach to pricing options and similar securities. However, it is clearly desirable to mai... |

23 |
Cracking the Smile
- Duan
- 1996
(Show Context)
Citation Context ...t most studies agree implied volatility is stationary and meanreverting. Note that it is difficult to directly compare numbers from previous empirical surveys such as those of Bakshi et al. [4], Duan =-=[10]-=-, or Melino & Turnbull [27] because those and many others use data on the coarse daily scale for which GARCH-type models are designed. Additionally, the fitting is usually using options data only, whe... |

17 | Stochastic volatility, smile & asymptotics
- Sircar, Papanicolaou
- 1999
(Show Context)
Citation Context ...[21] with volatility a geometric Brownian motion give a negative skew for negative correlation and positive skew for positive correlation. This is confirmed by small fluctuation asymptotic results in =-=[37]-=- for any Ito volatility process, and also the results of Section 3. The explicit formulas for the implied volatility curve are different in the limit of small fluctuations and in the limit of fast mea... |

16 | Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment.
- Fouque, Papanicolaou, et al.
- 1999
(Show Context)
Citation Context ...nd oe is the long-run historical asset price volatility. The terminal condition is ~ C 1 (T; x) = 0 and any boundary conditions are zero also. The example of a knock-out barrier option is computed in =-=[15]-=-. The table below then distinguishes the model parameters, defined in Section 2, from the parameters that are actually needed for the theory. The latter can be written as groupings of the former by th... |

12 |
Stock-price volatility, mean-reverting diffusion, and noise, _I.
- Merville, Pieptea
- 1989
(Show Context)
Citation Context ... al. [19] study implied volatility as a proxy for real volatility and conclude that EGARCH models provide a better description than GARCH or CIR-based models (see equation (10)). In Merville & Piptea =-=[28]-=-, implied volatility is found to be strongly mean-reverting, and Day & Lewis [9] find that implied volatilities contain additional information to that in historical volatilities, so that the out-of-sa... |

12 | Dynamic Asset Pricing Theory, 2nd Ed - Du¢, D - 1996 |

10 | Asymptotics of a two-scale stochastic volatility model,”
- Fouque, Papanicolaou, et al.
- 1998
(Show Context)
Citation Context ...le, f(y) = e y was proposed by Scott [35] and was also studied by Wiggins [39]. It is related to EGARCH models by Nelson [29]; the asymptotic analysis of Section 3 for this particular case appears in =-=[14]-=-. Figure 1 shows the estimate S&P 500 twenty-day transition probability density (from the high-frequency data using methods described in Section 4 and 5). It is shown in comparison to the correspondin... |

10 | A risk-neutral stochastic volatility model,
- Zhu, Avellaneda
- 1997
(Show Context)
Citation Context ... = sign(ae) for (in their case) lognormal volatility. The same relationship is reflected in the small fluctuation formulas for any correlated Ito stochastic volatility model in [37]. Zhu & Avellaneda =-=[41]-=- also work with a lognormal stochastic volatility and derive an explicit volatility risk premium assuming that short-term at-the-money calls are correctly priced by Black-Scholes. Their risk-neutral v... |

7 | Stochastic Volatility
- Hobson
- 1996
(Show Context)
Citation Context ...could be improved by using historical underlying data as well, though it is not clear how to implement this. We also refer the reader to recent surveys of the stochastic volatility literature such as =-=[16, 17, 20]. 1.2 Pres-=-ent Approach We concentrate on the "pure" stochastic volatility approach in which volatility oe t is modeled as an Ito process driven by a Brownian motion that has a component independent of... |

3 | models as di usion approximations - Arch - 1990 |

2 |
Nonparametric esimation of state-price densities implicit in financial asset prices
- it-Sahalia, Lo
- 1998
(Show Context)
Citation Context ...hese periods are relatively insignificant, especially as far as option prices, which come from an average of a functional of possible paths of the volatility, are concerned. Many authors, for example =-=[1]-=-, have proposed nonparametric estimation of the pricing measure for derivatives. The analysis in [37] is independent of specific modeling of the volatility process, but results in bands for option pri... |

2 |
Local volatilities in stochastic volatility models
- Lee
- 1998
(Show Context)
Citation Context ...on of exotic securities, but this approach has not yet produced a stable surface that can be used consistently and with confidence over time. See [13] for a detailed empirical study of this issue and =-=[25] for a mat-=-hematical explanation of why these surface-fits are outperformed by "fixed smile" (projected) implied volatilities. Possibly this shortcoming could be improved by using historical underlying... |

2 | Estimation of local power law processes
- Papanicolaou, Solna
- 1998
(Show Context)
Citation Context .... Note that we do not expect parameters of the volatility process to be constant across the segments of stationarity. 3 A similar analysis achieves this separation for atmospheric temperature data in =-=[31]-=-. ffl Our method is well-adapted to identifying fast mean-reversion and this is what we find for the S&P 500 index. Some negative aspects are: ffl The lack of a precise quantitative estimator for the ... |

2 | Stock-price volatility, mean-reverting di usion, and noise - Merville, Pieptea - 1989 |

1 |
Analysis of the term strucuture of implied volatilities
- Heynen, Kemna, et al.
- 1994
(Show Context)
Citation Context ...edure (especially estimation of the optimal weighting matrix) extremely complicated and computationally intensive, so we go after the parameters as we need them. In the second category, Heynen et al. =-=[19]-=- study implied volatility as a proxy for real volatility and conclude that EGARCH models provide a better description than GARCH or CIR-based models (see equation (10)). In Merville & Piptea [28], imp... |

1 | Transform analysis and option pricing for a ne jumpdi usions - e, Pan, et al. - 1998 |

1 | Adaptivecovariance estimation of locally stationary processes - Mallat, Papanicolaou, et al. - 1998 |