Market Price Of Volatility Risk Research Articles

Optimal DC pension investment with square-root factor processes under stochastic income and inflation risks

ABSTRACT This paper studies optimally defined contribution (DC) pension investment problems under the expected utility maximization framework with stochastic income and inflation risks. The member has access to a financial market consisting of a risk-free asset (money account), an inflation-indexed bond, and a stock. The market price of volatility risk is assumed to depend on an affine-form, Markovian, square-root factor process, while the return rate and the volatility of the stock are possibly given by general non-Markovian, unbounded stochastic processes. This financial framework recovers the Black–Scholes model, constant elasticity of variance (CEV) model, Heston model, 3/2 model, 4/2 model, and some non-Markovian models as exceptional cases. To tackle the potentially non-Markovian structures, we adopt a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, closed-form expressions for the optimal investment strategies and optimal value functions are obtained for the power, logarithmic, and exponential utility functions. Moreover, explicit solutions to some special cases of our portfolio model are provided. Finally, numerical examples are provided to illustrate the effects of model parameters on the optimal investment strategies under the 4/2 model.

Valuation of European Options Under an Uncertain Market Price of Volatility Risk

We propose a model to quantify the effect of parameter uncertainty on the option price in the Heston model. More precisely, we present a Hamilton–Jacobi–Bellman framework which allows us to evaluate best and worst-case scenarios under an uncertain market price of volatility risk. For the numerical approximation, the Hamilton–Jacobi–Bellman equation is reformulated to enable the solution with a finite element method. A case study with butterfly options exhibits how the dependence of Delta on the magnitude of the uncertainty is nonlinear and highly varied across the parameter regime.

Pricing the Volatility Risk Premium with a Discrete Stochastic Volatility Model

Investors’ decisions on capital markets depend on their anticipation and preferences about risk, and volatility is one of the most common measures of risk. This paper proposes a method of estimating the market price of volatility risk by incorporating both conditional heteroscedasticity and nonlinear effects in market returns, while accounting for asymmetric shocks. We develop a model that allows dynamic risk premiums for the underlying asset and for the volatility of the asset under the physical measure. Specifically, a nonlinear in mean time series model combining the asymmetric autoregressive conditional heteroscedastic model with leverage (NGARCH) is adapted for modeling return dynamics. The local risk-neutral valuation relationship is used to model investors’ preferences of volatility risk. The transition probabilities governing the evolution of the price of the underlying asset are adjusted for investors’ attitude towards risk, presenting the asset returns as a function of the risk premium. Numerical studies on asset return data show the significance of market shocks and levels of asymmetry in pricing the volatility risk. Estimated premiums could be used in option pricing models, turning options markets into volatility trading markets, and in measuring reactions to market shocks.

Stochastic Volatility and Asset Pricing Puzzles

This paper builds a real-options, term structure model of the …rm to shed new light on the value premium, …nancial distress, momentum, and credit spread puzzles. The model incorporates stochastic volatility in the …rm productivity process and a negative market price of volatility risk. Since the equity of growth …rms and …nancially distressed …rms have embedded options, such securities hedge against volatility risk in the market and thus command lower volatility risk premia than the equities of value or …nancially healthy …rms. Abnormal risk-adjusted momentum pro…ts are concentrated among low credit-rating …rms for similar reasons. Conversely, since increases in volatility generally reduce the value of debt, corporate debt will tend to command large volatility risk premia, allowing the model to generate higher credit spreads than existing structural models. The paper illustrates that allowing for endogenous default by equityholders is necessary for the model to account for the credit spreads of both investment grade and junk debt. The model is extended to include rare disasters and multiple time scales in volatility dynamics to better account for the expected default frequencies and credit spreads of short maturity debt. Finally, the paper uses a methodology based on asymptotic expansions to solve the model.

Bayesian Estimation of a Stochastic Volatility Model, Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a hybrid Markov Chain Monte Carlo sampling algorithm. Candidate draws for the unobserved volatilities are obtained by applying the Kalman filter and smoother to a linearization of a state-space representation of the model. The method is illustrated using the Heston (1993) stochastic volatility model applied to Australian News Corporation spot and option price data. Alternative models nested in the Heston framework are ranked via Bayes Factors and via fit, predictive and hedging performance.

The Market Price of Volatility Risk and the Dynamics of Market and Actuarial Implied Volatilities

Option prices in the market embed a probability distribution over the option’s payoff, which can be fairly easily extracted. But in this risk-neutral distribution, the market’s true expectations about future returns are distorted by risk preferences, in particular, aversion to volatility risk. To separate the returns forecasts from risk aversion, one needs another way to estimate the market’s probability estimates, which the author calls “actuarial probabilities.” This difficult problem becomes even harder in the swaptions market, because the risk exposure for a swaption depends on two different time horizons. For example, a 2 × 5 swaption has two years (“expiry”) of optionality culminating with the option to enter into a swap with a maturity (“tail”) of five years. The volatility surface is three-dimensional, with two time variables and the strike interest rate (the “volatility cube”), and it is difficult to estimate statistically using available data on long-lived contracts. The author develops a procedure based on extracting a small number of common factors of the volatility process using principal components. Since all swaptions are exposed to these factors, with different loadings, modeling the evolution of the principal components, rather than the specific swaption volatilities, allows a more efficient use of the data. Applying this approach to about 10 years of swaption data, the article reports several interesting results, including the fact that three principal components capture a large proportion of the variation in actuarial volatilities across the volatility cube, while a couple extra PCs are needed to achieve the same level of explanatory power for market implied volatilities. One clear result is that the market price of volatility risk is time-varying, and it moves rather quickly, with rates of mean-reversion that seem much faster than for actuarial volatility, particularly for short expiry–long tail swaptions. Another interesting result is that the volatility risk premium reverses sign, being largely positive (market IV > actuarial IV) for contracts with short expiry, but becoming negative for long expiry contracts.

The Market Price of Volatility Risk and the Dynamics of Market and Actuarial Implied Volatilities

The Market Price of Volatility Risk and the Dynamics of Market and Actuarial Implied Volatilities

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility with Rough Market Price of Volatility Risk

In the option pricing literature, it is well known that: (i) the decrease in the smile amplitude is much slower than standard stochastic volatility models and, (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models and, while: (i) has been explained by using fractional volatility model with Hurst index H>1/2, (ii) is proved to be satisfied by a {\it rough} volatility model with H<1/2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a novel fractional volatility model with rough market price of volatility risk and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle.

Pricing European Put Option in a Geometric Brownian Motion Stochastic Volatility Model

Stochastic volatility models were introduced because option prices have been mis-priced using Black-Scholes model. In this work, focus is made on pricing European put option in a Geometric Brownian Motion (GBM) stochastic volatility model with uncorrelated stock and volatility. The option is priced using two numerical methods (Crank-Nicolson and Alternating Direction Implicit (ADI) finite difference). Numerical schemes were considered because the closed form solution to the model could not be obtained. The change in option value due to changes in volatility, maturity time and market price of volatility risk are considered and comparison between the efficiency of the numerical methods by computing the CPU time was made.

An analytic expansion method for the valuation of double-barrier options under a stochastic volatility model

In this paper, we study a double-barrier option with a stochastic volatility model whose volatility is driven by a fast mean-reverting process, where the option's payoff is extinguished as the underlying asset crosses one of two barriers. By using an asymptotic analysis and Mellin transform techniques, we derive semi-analytic option pricing formulas with the sum of a leading-order term and a correction-order term, and then the accuracy of the first approximation price of the double-barrier option is verified by using Monte Carlo simulation. Moreover, we analyze the impact of stochastic volatility on the double-barrier option prices. Finally, we demonstrate that our results enhance the existing double-barrier option price structures in view of flexibility and applicability through the market price of volatility risk.

Non-Gaussian GARCH option pricing models and their diffusion limits

This paper investigates the weak convergence of general non-Gaussian GARCH models together with an application to the pricing of European style options determined using an extended Girsanov principle and a conditional Esscher transform as the pricing kernel candidates. Applying these changes of measure to asymmetric GARCH models sampled at increasing frequencies, we obtain two risk neutral families of processes which converge to different bivariate diffusions, which are no longer standard Hull–White stochastic volatility models. Regardless of the innovations used, the GARCH implied diffusion limit based on the Esscher transform can be obtained by applying the minimal martingale measure under the physical measure. However, we further show that for skewed GARCH driving noise, the risk neutral diffusion limit of the extended Girsanov principle exhibits a non-zero market price of volatility risk which is proportional to the market price of the equity risk, where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution. Our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes.

Investigating the Market Price of Volatility Risk for Options in a Regime-Switching Market

To bridge the gap between the output of theoretical option pricing models and observed option prices on exchanges, it is necessary to price the volatility risk inherent in financial markets. Non zero market risk premia have been found in previous financial literature through an exploration of market data, quantifying the relationship between implied and realized volatility. Building upon previous work by Mielkie and Davison (2013) where an approximate solution was derived for options written on underlying assets with regime-switching volatility, we analyze the impact of the market price of volatility risk on theoretical option prices. Using financially intuitive constraints, we prove the necessity of placing restrictions on the market prices of volatility risk in order to get reasonable option prices. In particular, we show that negative state-dependent market prices of volatility risk are necessary in order for the option prices and corresponding hedge ratios to be financially rational. An exploration of the regime-switching option prices and their implied volatilities is given, as well as numerical results and intuition supporting our mathematical proofs.

On the Extended Girsanov Principle for GARCH Models and Their Diffusion Limits

This paper investigates the weak convergence of general non-Gaussian GARCH models together with an application to the pricing of European style options determined using an extended Girsanov principle and a conditional Esscher transform as the pricing kernel candidates. Applying these changes of measure to asymmetric GARCH models sampled at increasing frequencies, we obtain two risk neutral families of processes which converge to different bivariate diffusions, which are no longer standard Hull-White stochastic volatility models. Regardless of the innovations used, the GARCH implied diffusion limit based on the Esscher transform can be obtained by applying the minimal martingale measure under the physical measure. However, we further show that for skewed GARCH driving noise, the risk neutral diffusion limit of the extended Girsanov principle exhibits a non-zero market price of volatility risk which is proportional to the market price of the equity risk, where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution. Our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes.

Sequential Monte Carlo pricing of American-style options under stochastic volatility models

We introduce a new method to price American-style options on underlying investments governed by stochastic volatility (SV) models. The method does not require the volatility process to be observed. Instead, it exploits the fact that the optimal decision functions in the corresponding dynamic programming problem can be expressed as functions of conditional distributions of volatility, given observed data. By constructing statistics summarizing information about these conditional distributions, one can obtain high quality approximate solutions. Although the required conditional distributions are in general intractable, they can be arbitrarily precisely approximated using sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes, is potentially heavy computational demand. We present two variants of the algorithm, one closely related to the well-known least-squares Monte Carlo algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001) 113-147], and the other solving the same problem using a "brute force" gridding approach. We estimate an illustrative SV model using Markov chain Monte Carlo (MCMC) methods for three equities. We also demonstrate the use of our algorithm by estimating the posterior distribution of the market price of volatility risk for each of the three equities.

The CBOE S&amp;P 500 three‐month variance futures

AbstractIn this article, we study the market of the Chicago Board Options Exchange S&amp;P 500 three‐month variance futures that were listed on May 18, 2004. By using a simple mean‐reverting stochastic volatility model for the S&amp;P 500 index, we present a linear relation between the price of fixed time‐to‐maturity variance futures and the VIX2. The model prediction is supported by empirical tests. We find that a model with a fixed mean‐reverting speed of 1.2929 and a daily‐calibrated floating long‐term mean level has a good fit to the market data between May 18, 2004, and August 17, 2007. The market price of volatility risk estimated from the 30‐day realized variance and VIX2 has a mean value of −19.1184. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:48–70, 2010

Can Tests Based on Option Hedging Errors Correctly Identify Volatility Risk Premia?

This paper provides an in-depth analysis of the properties of popular tests for the existence and the sign of the market price of volatility risk. These tests are frequently based on the fact that for some option pricing models under continuous hedging the sign of the market price of volatility risk coincides with the sign of the mean hedging error. Empirically, however, these tests suffer from both discretization error and model mis-specification. We show that these two problems may cause the test to be either no longer able to detect additional priced risk factors or to be unable to identify the sign of their market prices of risk correctly. Our analysis is performed for the model of Black and Scholes (1973) (BS) and the stochastic volatility (SV) model of Heston (1993). In the model of BS, the expected hedging error for a discrete hedge is positive, leading to the wrong conclusion that the stock is not the only priced risk factor. In the model of Heston, the expected hedging error for a hedge in discrete time is positive when the true market price of volatility risk is zero, leading to the wrong conclusion that the market price of volatility risk is positive. If we further introduce model mis-specification by using the BS delta in a Heston world we find that the mean hedging error also depends on the slope of the implied volatility curve and on the equity risk premium. Under parameter scenarios which are similar to those reported in many empirical studies the test statistics tend to be biased upwards. The test often does not detect negative volatility risk premia, or it signals a positive risk premium when it is truly zero. The properties of this test furthermore strongly depend on the location of current volatility relative to its long-term mean, and on the degree of moneyness of the option. As a consequence tests reported in the literature may suffer from the problem that in a time-series framework the researcher cannot draw the hedging errors from the same distribution repeatedly. This implies that there is no guarantee that the empirically computed t-statistic has the assumed distribution.

The Pricing of Volatility Risk Across Asset Classes and the Fama and French Three Factor Model

We investigate the pricing of aggregate volatility risk across sets of diverse portfolios. In particular, we show that a two factor model comprising a measure of stock market volatility risk and the market excess return factor performs well in explaining the cross-section of returns across portfolios of stocks, corporate bonds and stock options. While previous studies have mainly confined themselves to a single asset class when estimating the market price of volatility risk, our study shows that volatility risk is priced \textit{consistently} (with a negative risk premium) across asset classes.

Computing the market price of volatility risk in the energy commodity markets

In this paper, we demonstrate the need for a negative market price of volatility risk to recover the difference between Black–Scholes [Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654]/Black [Black, F., 1976. Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the Business and Economics Statistics Section, American Statistical Association, pp. 177–181] implied volatility and realized-term volatility. Initially, using quasi-Monte Carlo simulation, we demonstrate numerically that a negative market price of volatility risk is the key risk premium in explaining the disparity between risk-neutral and statistical volatility in both equity and commodity-energy markets. This is robust to multiple specifications that also incorporate jumps. Next, using futures and options data from natural gas, heating oil and crude oil contracts over a 10 year period, we estimate the volatility risk premium and demonstrate that the premium is negative and significant for all three commodities. Additionally, there appear distinct seasonality patterns for natural gas and heating oil, where winter/withdrawal months have higher volatility risk premiums. Computing such a negative market price of volatility risk highlights the importance of volatility risk in understanding priced volatility in these financial markets.

The CBOE S&amp;P 500 Three-Month Variance Futures

In this paper, we study the market of the CBOE S&P 500 three-month variance futures that were listed on May 18, 2004. By using a simple mean-reverting stochastic volatility model for the S&P 500 index, we present a linear relation between the price of fixed time-to-maturity variance futures and the VIX^2. The model prediction is supported by empirical tests. We find that a model with a fixed mean-reverting speed of 1.2929 and a daily-calibrated floating long-term mean level has a good fit to the market data between May 18, 2004 and August 17, 2007. The market price of volatility risk estimated from the 30-day realized variance and VIX^2 has a mean value of -19.1184.

American Options Under Stochastic Volatility

The problem of pricing an American option written on an underlying asset with constant price volatility has been studied extensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatility models are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature for pricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American options under stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformation procedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence. Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of option prices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset and stochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.