Download Free PDF. Symposium on stochastic volatility: an introductory overview Annals of Finance, Frederi Viens. A short summary of this paper. Symposium on stochastic volatility: an introductory overview. Ann Finance — DOI These articles concentrate mainly on questions pertaining to option pricing under various uncertainty assumptions about market volatility.
Tools from stochastic analysis and statistical inference are used to present solutions via explicit computations or numerical methods, with model estima- tion and calibration based on market and simulated data. Since there is no specific canonical model of SV that can account simultaneously for all the features of financial volatility observed in markets, the quantitative study of SV has been and continues to be vigorously studied — and sometimes hotly disputed.
University St. Viens In this Symposium issue of the Annals of Finance, we look at some of the most recent advances in the quantitative study of SV, covering mathematical, statistical, econometric, and computational aspects, all with sound financial interpretation as the primary motivator. We will encounter a rich variety of topics and methods, whose common goal is to explain the complexity of swings in market data.
The main financial emphasis in this issue is on the pricing of options and other financial derivatives. In accordance with this belief, the authors of this Symposium propose a number of methods in which it is possible to let option prices speak for themselves, by calibrating models against them.
Among the methods encountered here, some allow explicit option prices, a good path to reconcile financial intuition and precise quantitative statements. Other methods require Monte-Carlo techniques, for solutions which cannot be approximated in closed form; this is useful particularly when Bayesian and other non-parametric statistics are invoked.
Some articles concentrate on developing statistical method- ology in parametric and non-parametric contexts, for discrete and continuous models. Others propose new investigative terrain, with uncertainty terms involving jumps, or long memory, in order to incorporate SV features which are difficult to describe using more classical models. Most methods are tested extensively using simulated or market data, and a number of topics have a definite statistical bend. We now give a technical summary of the articles in this issue, which starts with studies of basic and extended SV models, then covers statistical aspects of volatil- ity calibration and estimation, followed by studies of long-memory SV, and finally a treatment of portfolio optimization.
Stochastic volatility models have been advocated as a good modeling tool for option pricing because they are capable of explaining the volatility smiles, smirks, frowns, and other skews observed in the volatilities implied by option markets. However, not every SV model can be calibrated to explain every possible implied volatility skew.
It can be represented as merely a standard Black-Scholes model which is shifted in space by a constant value. It is chosen for its mathematical tractability and the fact that it exhibits a volatility skew.
Yet Roger Lee and Dan Wang provide a mathematical proof that this model has some undesirable features, including a mono- tonicity constraint and a bounded slope on the implied volatility, which makes it poorly suited for computing equity options. However, they show that one should not discard this model even in the case of equity derivatives, since it can be used as a control variate in the Monte-Carlo computation of option prices in more flexible SV models, such as the Constant Elasticity of Variance CEV , and especially its non-local generalization known as the SABR stochastic alpha-beta-rho.
This model can also be endowed with a stochastic market volatility component, just as in a stan- dard Hull-White model or any stochastic volatility framework, in order to account for implied volatility skews. Even when the underlying market volatilty is constant, this model results in stochastically switching asset volatility. Remarkably, in this model, a detailed probabilistic analysis reveals explicit expressions for option prices and rep- lication. When the market volatility is stochastic, option pricing approximations can be obtained in the fast mean reverting regime.
SV models are also capable of capturing the phenomenon by which volatility tends to increase when market prices decrease. This so-called leverage effect can be modeled by assuming that the noise term driving the volatility is correlated to the noise term driving the stock or market process; the leverage correlation coefficient would then typically be a negative constant, in order to describe the rise in volatility when bad news hits the market.
In some cases, it may be preferable to allow several different leverage coefficients. More specifically they consider standard SV models, including the Heston and Barndorff-Nielsen-Shepard models, and use correlation coefficients which are driven by so-called Jacobi processes. The other parameters in the SV models can then be adjusted to account for a number of different volatility skew shapes; the authors explain how to produce specific symmetries, asymmetries, convexities, and shifts in implied volatilities.
In their models, they also investigate the mathematical and statis- tical aspects of empirical leverage, empirical volatility feedback, and non-parametric realized variances and covariances. The estimation of volatility based on high-frequency data is at the core of very general questions in stochastic calculus, such as the representation of quadratic vari- ation, but it is also a topic of much current interest and empirical work in finance.
Viens financial econometrics, making simplifying modeling assumptions is often considered as a better idea than trying to described many features of financial data by developing highly complex models. A number of different statistical mathods are available in this situation, for which Mykland develops a theory based on an approximation tech- nique he calls locally constant volatility. It allows one to replace the market dynamics by ones in which volatilities remain constant in predetermined small intervals, in which the model break points need not have any relation with the market observation times beyond some mild assumptions.
The full volatility matrix of the approximate market, using all observation times, will typically have a number of unobserved val- ues. Nevertheless, Mykland shows a contiguity theorem: parameter consistency under the approximate model implies consistency under the original model; the price to pay for the approximation is a bias in the asymptotic distribution.
As applications of this theory, in which this bias is typically zero, Mykland proposes an ANOVA with multiple regression and finite smoothing, to estimate the residual quadratic variation of a given process after regressing on multiple others. He also shows how to construct a simple moments-based estimator of the variance in the maximum likelihood a. Hayashi-Yoshida estimator for the volatility matrix. Realized volatility based on historical data, and implied volatility based on current option prices and the Black-Scholes formula, are two aspects of the same quantitative question.
Practitioners generally regard implied volatility as containing more infor- mation than its realized counterpart, in the sense that it helps understand forecasting, which is not possible in a short-memory context using only historical data. Yet because the two concepts are tightly related, it is worth trying to explain this quantitatively. Lan Zhang does this in a general context, in which she shows in an SV frame- work that, in accordance with econometric intuition, realized volatility coincides with instantaneous implied volatility when the latter exists.
However, she also shows that it normally does not exist, because cumulative implied volatility is typically diffusive. In this case, she derives no-arbitrage identities and inequalities, and shows how to use diffusive implied volatility to hedge away some of the Gamma and Vega risk. Hedging using implied volatility is desirable in order to minimize reliance on raw option prices whose dependence on moneyness is high.
In order to deal with hedging in incomplete multifactor SV models, in which the goal is to minimize the hedging error, Zhang proposes two schemes for empirical model selection to choose an optimal number of hedging instruments. J Polit Econ — Google Scholar. Gerber HU An introduction to mathematical risk theory. Grandell J Aspects of risk theory. Pliska SR Introduction to mathematical finance. Blackwell, Oxford Google Scholar.
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