Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.     

The Dynamics of the S&P 500 Implied Volatility Surface

This empirical study is motivated by the literature on “smile-consistent” arbitrage pricing with stochastic volatility. We investigate the number and shape of shocks that move implied volatility smiles and surfaces by applying Principal Components Analysis. Two components are identified under a variety of criteria. Subsequently, we develop a “Procrustes” type rotation in order to interpret the retained components. The results have implications for both option pricing and hedging and for the economics of option pricing. This revised version was published online in June 2006 with corrections to the Cover Date.     

THE ECONOMIC VALUE OF USING REALIZED VOLATILITY IN FORECASTING FUTURE IMPLIED VOLATILITY

We examine the economic benefits of using realized volatility to forecast future implied volatility for pricing, trading, and hedging in the S&P 500 index options market. We propose an encompassing regression approach to forecast future implied volatility, and hence future option prices, by combining historical realized volatility and current implied volatility. Although the use of realized volatility results in superior performance in the encompassing regressions and out-of-sample option pricing tests, we do not find any significant economic gains in option trading and hedging strategies in the presence of transaction costs.     

投资者情绪、期权隐含信息与股市波动率预测——基于上证50ETF期权的经验研究

在异质自回归模型(HAR-RV)中引入中国上证50ETF期权隐含信息和投资者情绪,本文分别对中国股票市场未来日、周和月波动率进行预测。研究发现,期权隐含信息和投资者情绪能够提高HAR-RV模型对股票市场未来波动率的预测效果。投资者情绪对未来波动率的影响存在两种机制:在情绪高涨期间,月已实现波动率与未来波动率正相关,说明以个人投资者占主体所引起的价格信息机制,在中国股票市场交易中占主导作用;风险中性偏度与未来波动率负相关,说明以个人投资者占主体所引起的噪声交易机制占主导作用。     

Does the Hurst index matter for option prices under fractional volatility?

This study examines the effect of fractional volatility on option prices. To this end, we develop an approximation method for the pricing of European-style contingent claims when volatility follows a fractional Brownian motion. Through extensive numerical experiments, we confirm that the decrease in the smile amplitude under fractional volatility is much slower than that under the standard stochastic volatility model. We also show that the Hurst index under fractional volatility has a crucial impact on option prices when the maturity is short and speed of mean reversion is slow. On the contrary, the impact of the Hurst index on option prices reduces for long-dated options.     

A bias in the volatility smile

We show that even if options traded with Black–Scholes–Merton pricing under a known and constant volatility, meaning essentially in perfect markets, one would still obtain smiles, skews, and smirks. We detect this problem by pricing options with a known volatility and reverse engineering to back into the implied volatility from the model price that was derived from the assumed volatility. The returned volatilities follow distinctive patterns resulting from algorithmic choices of the user and the quotation unit of the option. In particular, the common practice of penny pricing on option exchanges results in a significant loss of accuracy in implied volatility. For the most common scenarios faced in practice, the problem primarily exists in short-term options, but it manifests for virtually all cases of moneyness of at least 10 % and often 5 %. While it is theoretically possible to almost eliminate the problem, practical limitations in trading prevent any realistic chance of avoiding this error. It is even more difficult to identify and control the problem when smiles also arise from market imperfections, as is widely accepted. We empirically estimate a very conservative lower bound of the effect at about 16 % of the observed smile for 30-day options. Thus, we document a previously unknown phenomenon that a portion of the volatility smile is not of an economic nature. We provide some best-practice recommendations, including the explicit specification of the algorithmic choices and a warning against using off-the-shelf routines.     

基于分数维随机利率的分数跳-扩散外汇期权定价

假设利率为分数维随机利率,外汇汇率服从分数跳一扩散过程,并且波动率为常数,期望收益率为时间的非随机函数,本文利用保险精算方法,得出了看涨、看跌外汇欧武期权的一般定价公式,并建立了平价公式。     

Long memory options: LM evidence and simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black–Scholes’ geometric Brownian motion assumption. Moreover, Elliott and van der Hoek [Elliott, R.J., van der Hoek, J., 2003. A general fractional white noise theory and applications to finance. Math. Finance 13, 301–330] provide a theoretical framework for incorporating these findings into the Black–Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black–Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company’s own shares for long expiration periods. It makes a significant difference in the valuation when an option is “blue” or when it is “red.” For a proper valuation of such stock options, the degrees of persistence of the companies’ share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.     

Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices

Implicit in the prices of traded financial assets are Arrow–Debreu prices or, with continuous states, the state-price density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and we derive its asymptotic sampling theory. This estimator provides an arbitrage-free method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, for example, negative skewness and excess kurtosis for asset returns, and volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.     

The volatility term structure in a lognormal process for the short rate

One-factor Markov models are widely used by practitioners for pricing financial options. Their simplicity facilitates their calibration to the intial conditions and permits fast computer Implementations. Nevertheless, the danger remains that such models behave unrealistically, if the calibration of the volatility is not properly done. Here, we study a lognormal process and investigate how to specify the volatility constraints in such a way that the term structure of volatility at future times, as implied by the short rate process, has a realistic and stable shape. However, the drifting down of the volatility term structure is unavoidable. As a result, there is a tendency to underestimate option prices.     

Interest Rate Caps “Smile” Too! But Can the LIBOR Market Models Capture the Smile?

Using 3 years of interest rate caps price data, we provide a comprehensive documentation of volatility smiles in the caps market. To capture the volatility smiles, we develop a multifactor term structure model with stochastic volatility and jumps that yields a closed‐form formula for cap prices. We show that although a three‐factor stochastic volatility model can price at‐the‐money caps well, significant negative jumps in interest rates are needed to capture the smile. The volatility smile contains information that is not available using only at‐the‐money caps, and this information is important for understanding term structure models.     

Target volatility option pricing in the lognormal fractional SABR model

We examine in this article the pricing of target volatility options in the lognormal fractional SABR model. A decomposition formula of Itô's calculus yields an approximation formula for the price of a target volatility option in small time by the technique of freezing the coefficient. A decomposition formula in terms of Malliavin derivatives is also provided. Alternatively, we also derive closed form expressions for a small volatility of volatility expansion of the price of a target volatility option. Numerical experiments show the accuracy of the approximations over a reasonably wide range of parameters.     

Long memory in volatility and trading volume

We use fractionally-integrated time-series models to investigate the joint dynamics of equity trading volume and volatility. Bollerslev and Jubinski (1999) show that volume and volatility have a similar degree of fractional integration, and they argue that this evidence supports a long-run view of the mixture-of-distributions hypothesis. We examine this issue using more precise volatility estimates obtained using high-frequency returns (i.e., realized volatilities). Our results indicate that volume and volatility both display long memory, but we can reject the hypothesis that the two series share a common order of fractional integration for a fifth of the firms in our sample. Moreover, we find a strong correlation between the innovations to volume and volatility, which suggests that trading volume can be used to obtain more precise estimates of daily volatility for cases in which high-frequency returns are unavailable.     

A chaos expansion approach under hybrid volatility models

In this paper, we propose an approximation method based on the Wiener–Ito chaos expansion for the pricing of European contingent claims. Our method is applicable to widely used option pricing models such as local volatility models, stochastic volatility models and their combinations. This method is useful in practice since the resulting approximation formula is not computationally expensive, hence it is suitable for calibration purposes. We will show through some numerical examples that our approximation remains quite good even for the long maturity and/or the high volatility cases, which is a desired feature. As an example, we propose a hybrid volatility model and apply our approximation formula to the JPY/USD currency option market obtaining very accurate results.     

Locally Complete Markets,Exchange Rates and Currency Options

This paper extends the Heath, Jarrow and Morton model (1992) to atwo country setup. In the presence of common shocks and country specificshocks, we retrieve each country's pricing kernel implied by itsterm structure dynamics and show that the pricing kernels impose a constrainton the exchange rate to be the ratio of the pricing kernels. Under therisk neutral measure, the drift of the exchange rate is the interest ratedifferential, and the volatility reflects the forward rate risk-premiumdifferential of the two countries. The result implies that the risk premiumwill enter the currency option pricing model through the volatility term.Under the assumption of non-stochastic forward rate drift and volatility,we are able to derive closed-form solutions for currency options.     

The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.     

Buy rough,sell smooth

Recent work has documented roughness in the time series of stock market volatility and investigated its implications for option pricing. We study a strategy for trading stocks based on measures of their implied and realized roughness. A strategy that goes long the roughest-volatility stocks and short the smoothest-volatility stocks earns statistically significant excess annual returns of 6% or more, depending on the time period and strategy details. The profitability of the strategy is not explained by standard factors. We compare alternative measures of roughness in volatility and find that the profitability of the strategy is greater when we sort stocks based on implied rather than realized roughness. We interpret the profitability of the strategy as compensation for near-term idiosyncratic event risk.     

Pseudospectral methods for pricing options

Models with two or more risk sources have been widely applied in option pricing in order to capture volatility smiles and skews. However, the computational cost of implementing these models can be large—especially for American-style options. This paper illustrates how numerical techniques called ‘pseudospectral’ methods can be used to solve the partial differential and partial integro-differential equations that apply to these multifactor models. The method offers significant advantages over finite-difference and Monte Carlo simulation schemes in terms of accuracy and computational cost.