Extracting Information from Options Markets: Smiles, State–Price Densities and Risk Aversion

In this paper, recent techniques of estimating implied information from derivatives markets are presented and applied empirically to the French derivatives market. We determine nonparametric implied volatility functions, state–price densities and historical densities from a high–frequency CAC 40 stock index option dataset. Moreover, we construct an estimator of the risk aversion function implied by the joint observation of the cross–section of option prices and time–series of underlying asset value. We report a decreasing implied volatility curve with the moneyness of the option. The estimated relative risk aversion functions are positive and globally consistent with the decreasing relative risk aversion assumption.     

Option pricing under a stressed-beta model

Empirical studies have concluded that stochastic volatility is an important component of option prices. We introduce a regime-switching mechanism into a continuous-time Capital Asset Pricing Model which naturally induces stochastic volatility in the asset price. Under this Stressed-Beta model, the mechanism is relatively simple: the slope coefficient—which measures asset returns relative to market returns—switches between two values, depending on the market being above or below a given level. After specifying the model, we use it to price European options on the asset. Interestingly, these option prices are given explicitly as integrals with respect to known densities. We find that the model is able to produce a volatility skew, which is a prominent feature in option markets. This opens the possibility of forward-looking calibration of the slope coefficients, using option data, as illustrated in the paper.     

Using Richardson extrapolation techniques to price American options with alternative stochastic processes

In this paper the authors investigate the performance of the original and repeated Richardson extrapolation methods for American option pricing by implementing both the original and modified Geske?CJohnson approximation formulae. A comprehensive numerical comparison includes alternative stochastic processes of the underlying asset price. The numerical results show that whether the original or modified formula is implemented, the Richardson extrapolation techniques work very well. The repeated Richardson extrapolation strongly outperforms the original, especially when the underlying asset price follows a stochastic volatility process. Moreover, this study verifies the feasibility of the estimated error bounds of the American option prices under alternative stochastic processes by applying the repeated Richardson extrapolation method and estimating the interval of true American option values, as well as determining the number of options needed for an approximation to achieve a desired accuracy level.     

The Pricing of Options on Assets with Stochastic Volatilities

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity.     

Non-parametric estimation of historical volatility

Evolving volatility is a dominant feature observed in most financial time series and a key parameter used in option pricing and many other financial risk analyses. A number of methods for non-parametric scale estimation are reviewed and assessed with regard to the stylized features of financial time series. A new non-parametric procedure for estimating historical volatility is proposed based on local maximum likelihood estimation for the t-distribution. The performance of this procedure is assessed using simulated and real price data and is found to be the best among estimators we consider. We propose that it replaces the moving variance historical volatility estimator.     

Recovering the probability density function of asset prices using garch as diffusion approximations

This paper uses Garch models to estimate the objective and risk-neutral density functions of financial asset prices and by comparing their shapes, recover detailed information on economic agents' attitudes toward risk. It differs from recent papers investigating analogous issues because it uses Nelson's result that Garch schemes are approximations of the kind of differential equations typically employed in finance to describe the evolution of asset prices. This feature of Garch schemes usually has been overshadowed by their well-known role as simple econometric tools providing reliable estimates of unobserved conditional variances. We show instead that the diffusion approximation property of Garch gives good results and can be extended to situations with (i) non-standard distributions for the innovations of a conditional mean equation of asset price changes and (ii) volatility concepts different from the variance. The objective PDF of the asset price is recovered from the estimation of a nonlinear Garch fitted to the historical path of the asset price. The risk-neutral PDF is extracted from cross-sections of bond option prices, after introducing a volatility risk premium function. The direct comparison of the shapes of the two PDF_s reveals the price attached by economic agents to the different states of nature. Applications are carried out with regard to the futures written on the Italian 10-year bond.     

When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

Pricing double barrier options under a volatility regime-switching model with psychological barriers

The prices of lots of assets have been proved in literature to exhibit special behaviors around psychological barriers, which is an important fact needed to be considered when pricing derivatives. In this paper, we discuss the valuation problem of double barrier options under a volatility regime-switching model where there exist psychological barriers in the prices of underlying assets. The volatility can shift between two regimes, that is to say, when the asset price rises up or falls down through the psychological barrier, the volatility takes two different values. Using the Laplace transform approach, we obtain the price of the double barrier knock-out call option as well as its delta. We also provide the eigenfunction expansion pricing formula and examine the effect of the psychological barrier on the option price and delta, finding that the gamma of the option is discontinuous at such barriers.     

Hedging volatility risk

Volatility risk plays an important role in the management of portfolios of derivative assets as well as portfolios of basic assets. This risk is currently managed by volatility “swaps” or futures. However, this risk could be managed more efficiently using options on volatility that were proposed in the past but were never introduced mainly due to the lack of a cost efficient tradable underlying asset.The objective of this paper is to introduce a new volatility instrument, an option on a straddle, which can be used to hedge volatility risk. The design and valuation of such an instrument are the basic ingredients of a successful financial product. In order to value these options, we combine the approaches of compound options and stochastic volatility. Our numerical results show that the straddle option is a powerful instrument to hedge volatility risk. An additional benefit of such an innovation is that it will provide a direct estimate of the market price for volatility risk.     

Modelling the Stochastic Dynamics of Volatility for Equity Indices

The paper develops a class of continuous timestochastic volatility models, which generate asset price returnsthat are approximately Student t distributed. Using thecriterion of local risk minimisation in an incomplete marketsetting, option prices are computed. It is shown that impliedvolatility smile and skew patterns of the type often observed inthe markets can be obtained from this class of stochasticvolatility models.     

Volatility surfaces: theory,rules of thumb,and empirical evidence

Implied volatilities are frequently used to quote the prices of options. The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset's volatility surface. Traders monitor movements in volatility surfaces closely. In this paper we develop a no-arbitrage condition for the evolution of a volatility surface. We examine a number of rules of thumb used by traders to manage the volatility surface and test whether they are consistent with the no-arbitrage condition and with data on the trading of options on the S&P 500 taken from the over-the-counter market. Finally we estimate the factors driving the volatility surface in a way that is consistent with the no-arbitrage condition.     

Testing the local volatility assumption: a statistical approach

In practice, the choice of using a local volatility model or a stochastic volatility model is made according to their respective ability to fit implied volatility surfaces. In this paper, we adopt a different point of view. Indeed, using a purely statistical methodology, we design new procedures aiming at testing the assumption of a local volatility model for the price dynamics, against the alternative of a stochastic volatility model. These test procedures are based only on historical data and do not require any calibration procedures via option prices. We also provide a convincing simulation study and an empirical analysis on future contracts on interest rates.     

Firm size and volatility analysis in the Spanish stock market

Using Spanish stock market data, this paper examines volatility spillovers between large and small firms and their impact on expected returns. By using a conditional capital asset pricing model (CAPM) with an asymmetric multivariate GARCH-M covariance structure, it is shown that there exist bidirectional volatility spillovers between both types of companies, especially after bad news. After estimating the model, a positive and significant price of risk is obtained. This result is consistent with the volatility feedback effect, one of the most popular explanations of the asymmetric volatility phenomenon, and explains why risk premiums are much more sensitive to negative return shocks coming from the whole market or other related markets.     

Debt rollover-induced local volatility model

This paper introduces a structural scenario-based model with debt rollover risk and a higher-fidelity treatment of the bankruptcy procedure. The emerging stock price process is a generalized Brownian motion with state-dependent local volatility, and the resultant implied volatility smile is due exclusively to structural features (debt rollover and credit risks). Therefore, the model reinforces structural foundations of local volatility option pricing models. The paper advocates a joint modeling and calibration framework for multiple classes of derivatives on the firm’s asset value. In particular, an empirical application to Solar City equity and stock option valuation demonstrates the versatility and efficiency gains of the suggested model.

     

Maturity cycles in implied volatility

The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.Received: December 2003, Mathematics Subject Classification (2000): 91B70, 60F05, 60H30JEL Classification: C13, G13Jean-Pierre Fouque: Work partially supported by NSF grant DMS-0071744.Ronnie Sircar: Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance.We thank a referee for his/her comments which improved the paper.     

Optimal delta hedging for options

As has been pointed out by a number of researchers, the normally calculated delta does not minimize the variance of changes in the value of a trader's position. This is because there is a non-zero correlation between movements in the price of the underlying asset and movements in the asset's volatility. The minimum variance delta takes account of both price changes and the expected change in volatility conditional on a price change. This paper determines empirically a model for the minimum variance delta. We test the model using data on options on the S&P 500 and show that it is an improvement over stochastic volatility models, even when the latter are calibrated afresh each day for each option maturity. We also present results for options on the S&P 100, the Dow Jones, individual stocks, and commodity and interest-rate ETFs.     

关于资产价格与货币政策问题的一些思考

在全球金融危机的大背景下,货币政策是否应该对资产价格膨胀作出反应引起关注。本文对相关理论进行了归纳,并从通货膨胀机理的角度对资产价格与货币政策的关系进行了探讨,提出了建立和完善更加关注资产价格的货币政策框架的建议。     

An analytical approximation to the option formula for the GARCH model

This article derives an analytical approximation to the option formula for a spot asset price whose conditional variance equation follows a nonlinear asymmetric GARCH (NGARCH) process. The approximate option formula, which is just a volatility adjustment in comparison to the Black-Scholes (BS) formula, is very simple and provides the volatility term structure of spot asset prices. Also, the formula shows that the most characteristic feature of an NGARCH model appears in the vega of a European option, which depends on both the spread between the long-run variance and the current one and a parameter reproduced from the stationary property of the conditional variance. This methodology can be easily extended to an option formula for the generalized GARCH process.     

American options under stochastic volatility: control variates,maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of an American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow-scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price an American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where the scaling parameter value is equal to unity, fast and slow-scale approximations are equally accurate.