Pricing under rough volatility

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.     

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.     

Volatility forecasting: Intra-day versus inter-day models

Volatility prediction is the key variable in forecasting the prices of options, value-at-risk and, in general, the risk that investors face. By estimating not only inter-day volatility models that capture the main characteristics of asset returns, but also intra-day models, we were able to investigate their forecasting performance for three European equity indices. A consistent relation is shown between the examined models and the specific purpose of volatility forecasts. Although researchers cannot apply one model for all forecasting purposes, evidence in favor of models that are based on inter-day datasets when their criteria based on daily frequency, such as value-at-risk and forecasts of option prices, are provided.     

Volatility conditional on price trends

The influence of the past price behaviour on the realized volatility is investigated, showing that trending (driftless) prices lead to increased (decreased) realized volatility. This ‘volatility induced by trend’ constitutes a new stylized fact. The past price behaviour is measured by the product of two non-overlapping returns (of the form r × L[r] where L is the lag operator), and is different from the usual heteroskedasticity. The effect is studied empirically using USD/CHF foreign exchange data, in a large range of time horizons. On the modelling side, a set of ARCH based processes are modified in order to include the ‘volatility induced by trend’ effect, and their forecasting performances are compared. The aim is to understand the role and importance of the various terms that can be included in such a model. For a better forecast, it is shown that the main factor is the shape of the memory kernel (i.e. power law), and the next most important factor is the trend effect. The subtle role of mean reversion is also discussed.     

Multiple time scales and the exponential Ornstein–Uhlenbeck stochastic volatility model

We study the exponential Ornstein–Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behaviour in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.     

Stochastic integrals driven by fractional Brownian motion and arbitrage: a tale of two integrals

Recent research suggests that fractional Brownian motion can be used to model the long-range dependence structure of the stock market. Fractional Brownian motion is not a semi-martingale and arbitrage opportunities do exist, however. Hu and Øksendal [Infin. Dimens. Anal., Quant. Probab. Relat. Top., 2003, 6, 1–32] and Elliott and van der Hoek [Math. Finan., 2003 Elliott, RJ and van de Hoek, J. 2003. A general fractional white noise theory applications to finance. Math. Finan., 13: 301–330. [Crossref], [Web of Science ®] , [Google Scholar], 13, 301–330] propose the use of the white noise calculus approach to circumvent this difficulty. Under such a setting, they argue that arbitrage does not exist in the fractional market. To unravel this discrepancy, we examine the definition of self-financing strategies used by these authors. By refining their definitions, a new notion of continuously rebalanced self-financing strategies, which is compatible with simple buy and hold strategies, is given. Under this definition, arbitrage opportunities do exist in fractional markets.     

Option pricing under short-lived arbitrage: theory and tests

Models in financial economics derived from no-arbitrage assumptions have found great favour among theoreticians and practitioners. We develop a model of option prices where arbitrage is short lived. The arbitrage process is Ornstein–Uhlenbeck with zero mean and rapid adjustment of deviations. We find that arbitrage correlated with the underlying can have sizeable impact on option prices. We use data from five large capitalization firms to test implications of the model. Consistent with the existence of arbitrage, we find that idiosyncratic factors significantly effect arbitrage model parameters.     

基于高频数据的金融市场波动溢出分析

在讨论"已实现"波动率、"已实现"协方差基础上,针对金融市场的高频数据,引入"已实现"波动变结构,分阶段计算"已实现"波动率的相关系数,检验"已实现"波动率相关系数,判断在变结构点前后是否发生显著变化,从而分析金融市场之间的波动溢出效应,并进行实证分析。     

Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process which is dynamically consistent to the power law. The volatility process of the model is driven by a fractional Brownian motion with Hurst parameter less than half. The fractional Brownian motion is correlated with a Brownian motion which drives the asset price process. We derive an asymptotic expansion of the implied volatility as the time to maturity tends to zero. For this purpose, we introduce a new approach to validate such an expansion, which enables us to treat more general models than in the literature. The local-stochastic volatility model is treated as well under an essentially minimal regularity condition in order to show such a standard model cannot be dynamically consistent to the power law.     

Are classical option pricing models consistent with observed option second-order moments? Evidence from high-frequency data

As a means of validating an option pricing model, we compare the ex-post intra-day realized variance of options with the realized variance of the associated underlying asset that would be implied using assumptions as in the Black and Scholes (BS) model, the Heston, and the Bates model. Based on data for the S&P 500 index, we find that the BS model is strongly directionally biased due to the presence of stochastic volatility. The Heston model reduces the mismatch in realized variance between the two markets, but deviations are still significant. With the exception of short-dated options, we achieve best approximations after controlling for the presence of jumps in the underlying dynamics. Finally, we provide evidence that, although heavily biased, the realized variance based on the BS model contains relevant predictive information that can be exploited when option high-frequency data is not available.