Volatility Parameter Research Articles

Time variation in an optimal asymmetric preference monetary policy model

AbstractThis paper considers a time varying parameter extension of the Ruge-Murcia’s (Ruge-Murcia, F. J. 2003. “Does the Barro-Gordon Model Explain the Behavior of us Inflation? A Reexamination of the Empirical Evidence.”

The Fine Structure of Currency ReturnsImplied in One-Touch Options

Under Black-Scholes assumptions, the returns process for an option’s underlying asset follows a logarithmic diffusion. The volatility parameter determines all aspects of risk. But Black-Scholes falls short of explaining real world returns, as well as real world option prices. Extensions to the model add stochastic volatility, possibly through a stochastic time-change, or jumps; or might even replace the diffusion entirely with an infinity of jumps of different sizes as in the variance gamma (VG) model. These models are flexible enough that they all can fit a sample of returns and price plain vanilla options quite well. This article brings into consideration another type of option contract: the one-touch option. The payoffs on these path-dependent contracts are determined not by the level of the underlying at expiration but by the extreme values along the price path from the initial date to expiration. The option pays when, and only if, the underlying hits a prespecified level at some point during its lifetime. Buesser notes that the various returns processes have quite different path behavior, so calibrating against one-touch options provides a significant source of additional information about the market dynamics investors expect. Comparing relative performance across models for the FX market, plain vanilla options on USDEUR and JPYUSD appear to embed a substantially different returns process than is implied by one-touch options. <b>TOPICS:</b>Options, quantitative methods, developed

Measuring Credit Risk in a Large Banking System: Econometric Modeling and Empirics

We develop a novel high-dimensional non-Gaussian modeling framework to infer conditional and joint risk measures for many financial sector firms. The model is based on a dynamic Generalized Hyperbolic Skewed-t block-equicorrelation copula with time-varying volatility and dependence parameters that naturally accommodates asymmetries, heavy tails, as well as non-linear and time-varying default dependence. We demonstrate how to apply a conditional law of large numbers in this setting to define risk measures that can be evaluated quickly and reliably. We apply the modeling framework to assess the joint risk from multiple financial firm defaults in the euro area during the 2008-2012 financial and sovereign debt crisis. We document unprecedented tail risks during 2011-12, as well as their steep decline after subsequent policy actions.

An endogenous volatility approach to pricing and hedging call options with transaction costs

Standard delta hedging fails to exactly replicate a European call option in the presence of transaction costs. We study a pricing and hedging model similar to the delta hedging strategy with an endogenous volatility parameter for the calculation of delta over time. The endogenous volatility depends on both the transaction costs and the option strike prices. The optimal hedging volatility is calculated using the criterion of minimizing the weighted upside and downside replication errors. The endogenous volatility model with equal weights on the up and down replication errors yields an option premium close to the Leland [J. Finance, 1985, 40, 1283–1301] heuristic approach. The model with weights being the probabilities of the option's moneyness provides option prices closest to the actual prices. Option prices from the model are identical to the Black–Scholes option prices when transaction costs are zero. Data on S&P 500 index cash options from January to June 2008 illustrate the model.

Fisher Information for Fractional Brownian Motion Under High-Frequency Discrete Sampling

We investigate the issue of the validation of the local asymptotic normality property of three characterizing parameters of the fractional Brownian motion under high-frequency discrete sampling. We prove that the local asymptotic normality property holds true for the likelihood only when at least one of the volatility parameter and the Hurst exponent is known. We provide optimal rates of convergence of the three parameters and Fisher information matrix in closed form.

One Factor Gaussian Short Rate Model Implementation

We collect some results in Piterbarg, Interest Rate Modelling, needed for the implementation of a GSR model. We develop explicit formulas for piecewise constant volatility and reversion parameters under the forward measure.

On an implicit assessment of fuzzy volatility in the Black and Scholes environment

In this work we suggest a methodology to obtain the membership of a non-observable parameter through implicit information. To this aim we profit from the interpretation of membership functions as coherent conditional probabilities. We develop full details for the well known Black and Scholes pricing model where the membership of the volatility parameter is obtained from a sample of either asset prices or market prices for options written on that asset.

Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models

Bayesian inference for stochastic volatility models using MCMC methods highly depends on actual parameter values in terms of sampling efficiency. While draws from the posterior utilizing the standard centered parameterization break down when the volatility of volatility parameter in the latent state equation is small, non-centered versions of the model show deficiencies for highly persistent latent variable series. The novel approach of ancillarity-sufficiency interweaving has recently been shown to aid in overcoming these issues for a broad class of multilevel models. It is demonstrated how such an interweaving strategy can be applied to stochastic volatility models in order to greatly improve sampling efficiency for all parameters and throughout the entire parameter range. Moreover, this method of “combining best of different worlds” allows for inference for parameter constellations that have previously been infeasible to estimate without the need to select a particular parameterization beforehand.

Mean-Reverting Logarithmic Modeling of VIX

Since March 26, 2004, when the CBOE Futures Exchange (CFE) began trading futures written on S&P500 volatility index (VIX), volatility has become a widely accepted asset class as trading, diversifying and hedging vehicle by traders, investors and portfolio managers over the past few years. On February 24, 2006, CBOE introduced options written on VIX index and since then VIX option series has now become the most actively traded index option series on CBOE. This thesis focuses on mathematical modeling of spot VIX with standalone approach. Unlike the consistent modeling approach in literature, which starts with specifying joint dynamics for SPX index and its instantaneous stochastic volatility then derives expression for spot VIX and price VIX derivatives based on this expression, standalone approach starts with directly specifying dynamics for spot VIX and prices VIX derivatives in this simpler framework. Although there is work in literature that studies the mean-reverting logarithmic model (MRLR), no work has been done in considering stochastic volatility in MRLR to capture the positive implied volatility skew of VIX option, nor have they compared the pure diffusion version of MRLR with its jump and/or stochastic volatility extensions. Furthermore, most of the literature only focus on static pricing formulas for VIX future and VIX option, no work has been done in investigating the dynamic feature of VIX future, calibration and hedging strategies of mean-reverting logarithmic models, as well as the convexity adjustment of VIX future from forward variance swap, which has a liquid variance swap market to back out the vol-of-vol information in mean-reverting logarithmic models. In this thesis, I present four versions of MRLR models. The first model is a pure diffusion model where spot VIX follows a mean-reverting logarithmic dynamics. Then I extend this basic MRLR model by adding jump or stochastic volatility into spot VIX dynamics to get MRLRJ and MRLRSV models. Finally, I combine jump and stochastic volatility together and add them into dynamics of spot VIX to get the fully specified MRLRSVJ model. For all the four models, I derive either transition function or conditional characteristic function of spot VIX. Based on those results, the pricing formulas for VIX future and VIX option are derived. In order to calibrate to VIX future term structure, I make the long-term mean of spot VIX be a time-dependent function and use the diffusion, jump and/or stochastic volatility parameters to calibrate VIX implied volatility surface. Two types of calibration strategies are suggested in this thesis. On the first stage of calibration, we need to calibrate all vol-of-vol parameters to convexity of spot VIX or VIX future. One strategy is to calibrate those parameters to VIX option implied volatility surface. Another strategy is to calibrate them to convexity adjustment of VIX future from forward variance swap, which can be replicated by liquid variance swaps. On the second stage of calibration, the long-term mean function of spot VIX is used to fit VIX futuer term structure given the vol-of-vol parameters calibrated on the first stage. In addition to the static pricing formula, dynamics of VIX future is also derived under all mean-reverting logarithmic models. The analysis in this thesis shows that VIX future follows geometric Brownian motion under MRLR model, jump-diffusion dynamics under MRLRJ model, stochastic volatility dynamics under MRLRSV model and stochastic volatility with jump dynamics under MRLRSVJ model. I develop the hedging strategies of VIX future and VIX option under mean-reverting logarithmic models. As spot VIX is not tradable asset, investors are unable to take positions on this index. Instead, research in literature has shown that a shorter-term VIX future has good power in forecasting movements of the subsequent VIX future. Therefore, hedging VIX future with a shorter-term VIX future is expected to perform well. Moreover, as VIX option can also be regarded as an option on a VIX future contract that has same maturity as VIX option, using the shorter-term VIX future contract as hedging instrument is a natural choice. In this thesis, I derive hedging ratios of VIX future and VIX option under the above hedging strategy. At last, numerical analysis in this thesis compares the four models in fitting VIX implied volatility surface. The results show that MRLR is unable to create positive implied volatility skew for VIX option. In contrast, MRLRJ and MRLRSV models perform equally well in fitting positive skew. However, the fully specified MRLRSVJ model adds little value in fitting VIX skew but incurs additional cost of calibrating more parameters and is subject to less stable parameters over maturities and over time.

Heavy-tailed mixture GARCH volatility modeling and Value-at-Risk estimation

This paper presents a heavy-tailed mixture model for describing time-varying conditional distributions in time series of returns on prices. Student-t component distributions are taken to capture the heavy tails typically encountered in such financial data. We design a mixture MT(m)-GARCH(p,q) volatility model for returns, and develop an EM algorithm for maximum likelihood estimation of its parameters. This includes formulation of proper temporal derivatives for the volatility parameters. The experiments with a low order MT(2)-GARCH(1,1) show that it yields results with improved statistical characteristics and economic performance compared to linear and nonlinear heavy-tail GARCH, as well as normal mixture GARCH. We demonstrate that our model leads to reliable Value-at-Risk performance in short and long trading positions across different confidence levels.

Martingale expansion in mixed normal limit

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived by Yoshida [28] as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) are discussed.

On solvability of a two-sided singular control problem

We study a two-sided singular control problem in a general linear diffusion setting and provide a set of conditions under which an optimal control exists uniquely and is of singular control type. Moreover, under these conditions the associated value function can be written in a quasi-explicit form. Furthermore, we investigate comparative static properties of the solution with respect to the volatility and control parameters. Lastly we illustrate the results with two explicit examples.

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ+, and σ− required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.

Approximating distributional behaviour of LTI differential systems using Gaussian function and its derivatives

This article is concerned with the approximation of the distributional behaviour of linear, time-invariant (LTI) systems. First, we review the different types of approximations of distributions by smooth functions and explain their significance in characterising system properties. Second, we consider the problem of changing the state of controllable LTI differential systems in a very short time. Thus, we establish an interesting relation between the time and volatility parameters of the Gaussian function and its derivatives in the approximation of distributional solutions. An algorithm is then proposed for calculating the distributional input and its smooth approximation which minimises the distance to an arbitrary target state. The optimal choice of the volatility parameter for the state transition is also derived. Finally, some complementary distance problems are also considered. The main results of this article are illustrated by numerous examples.

Research on Fuzzy-PID Compound Control in Inverter-Driven Energy-Saving Technology for Injection Molding Machine

the modern Inverter-driven technology has been applied in energy-saving control for injection molding machine. Aimming at the complex control of system because of its multiple input multiple output, strongly nonlinearity, some volatility parameters and uncertainty factor , the adaptive fuzzy PID compound control is put forward from the viewpoint of energy-conditioning. Using the above method, the control system have better dynamic characteristics and stability, strong robustness and adaptability. Experiment results show that the method can effectively improve energy-saving technology for the injection molding machine.

Computation of the effects of uncertainty in volatility on option pricing and hedging

We quantify the effect of uncertainty in the volatility parameter σ on the Black–Scholes price of the European and American put. We apply probabilistic uncertainty analysis to the Black–Scholes model and compare the results with those of the Uncertain Volatility model. From historical data, we calibrate a probability distribution for the volatility. We then use Monte Carlo (MC) and a surrogate Polynomial Chaos (PC)/MC method to compute uncertainty bounds. The calibrated probability distribution is not one related to a standard orthogonal basis, so a basis is constructed numerically for the PC approximation. We show how to construct one stably from the probability distribution. We show that both methods give the same results, and quantify the relative speedup of the surrogate method. Finally, we investigate the effect of the parametric uncertainty, and show, for example, that the presence of uncertainty smoothes out the optimal exercise boundary of the American put.

Volatility of price indices for heterogeneous goods

Price indices for heterogenous goods such as real estate or fine art constitute crucial information for institutional or private investors considering alternative investments in times of financial markets turmoil. Classical mean-variance analysis of alternative investments has been hampered by the lack of a systematic treatment of volatility in these markets. This may seem surprising as derivatives on subsets of the traded goods require a precise modelling and estimation of the underlying volatility. For example, in art markets, auction houses often give price guarantees to the seller that resemble put options. In this paper we propose a hedonic regression framework which explicitly defines an underlying stochastic process for the price index, allowing to treat the volatility parameter as the object of interest. The model can be estimated using maximum likelihood in combination with the Kalman filter. We derive theoretical properties of the volatility estimator and show that it outperforms the standard estimator. We show that extensions to allow for time-varying volatility are straightforward using a local-likelihood approach. In an application to a large data set of international blue chip artists, we show that volatility of the art market, although generally lower than that of financial markets, has risen over the last years and, in particular, during the recent European debt crisis.

Modeling language evolution: Aromanian, an endangered language in Greece

Time evolution of the relative density of speakers of an endangered language, Aromanian, which is spoken by a bilingual community in North-Western Greece, is approached theoretically by means of a two-state model and a three-state model. The same prestige and volatility parameters are used in these two models. Furthermore, a culture parameter and a second exponent are introduced in the three-state model. The parameters of each model are fitted to the current status of Aromanian, on the basis of field evidence collected by us, and the first findings about the risk of the language’s extinction are presented.

Two Notes About Adaptive Filters for Volatility Parameter

This paper consider the adaptive filtering problem in a semi-martingale process for trend and volatility parameters under the stochastic volatility assumption. First, we consider the Black-Scholes model for volatility and then we extend our results to the other models for volatility process. A conclusion section is also given.

A decomposition formula for option prices in the Heston model and applications to option pricing approximation

By means of classical Ito calculus, we decompose option prices as the sum of the classical Black–Scholes formula, with volatility parameter equal to the root-mean-square future average volatility, plus a term due to correlation and a term due to the volatility of the volatility. This decomposition allows us to develop first- and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy for short maturities. Numerical examples are given.