Turbocharging Monte Carlo pricing for the rough Bergomi model

The rough Bergomi model, introduced by Bayer et al. [Quant. Finance, 2016, 16(6), 887–904], is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model’s calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions—roughly 20 times on average, across different correlation regimes.     

Trader positions in VIX futures

We investigate the dynamic changes in trader positions of market participants in the VIX futures markets. We find that in a low-VIX period, below the 23.81 threshold determined by our model, changes in VIX futures affect the trading decisions of dealers and leveraged fund managers, but in an opposite direction. During a high-VIX period, dealers and leveraged fund managers would then alter their trading strategies. We highlight the important role of exchange-traded products trading in hedging demand of dealers and show the impact on VIX futures. Trader positions are determinants of VIX futures prices, basis, and VIX premium.     

Tracking performance of VIX futures ETPs

We find that, at the daily level, none of the VIX futures Exchange-Traded Products (ETPs) track their intrinsic values perfectly, although the tracking improves at the weekly and monthly frequency. We show that all products tracking deviations are driven by mainly driven by limits to arbitrage, measured by market conditions, liquidity and transaction costs. Exchange-Traded Funds’ (ETFs’) deviations are more significantly related to fund flows, which shows authorized participants taking advantage of arbitrage opportunities by creating/redeeming ETF units. The Exchange-Traded Notes (ETNs) do not have this mechanism, creation and redemption is reliant on the issuers and delayed, so their deviations are less related to flows.     

Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model

The rough Bergomi (rBergomi) model, introduced recently in Bayer et al. [Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on: (i) adaptive sparse grids quadrature (ASGQ), and (ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method. They reach a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e. to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.     

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.     

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.     

VIX futures term structure and the expectations hypothesis

Tests of the expectations hypothesis reveal that the slope of the VIX futures term structure predicts the direction but not the magnitude of the evolution of the short-end of the curve, but predicts neither the direction nor the magnitude of short-term changes in the long-end of the curve. Relative value seeking spread trades, constructed to exploit such violations, deliver excess returns with annualized Sharpe ratios equal or greater than those of volatility-writing strategies deployed by VIX ETN's for a majority of the 32 spread trade combinations tested. I demonstrate that profits from beta-neutral variations of the spread trades, which are not compensation for taking on equity downside risk by design, are propagated by inflows of capital into VIX futures markets, after controlling for factors that measure changes in the availability of hedge fund capital, risk appetite, and momentum. At the heart of profits, and by extension the term structure anomalies, is a disproportionally elevated basis propagated by long VIX demand that enters the futures market through ETN channels.     

An empirical comparison of continuous-time models of implied volatility indices

We explore the ability of alternative popular continuous-time diffusion and jump-diffusion processes to capture the dynamics of implied volatility indices over time. The performance of the various models is assessed under both econometric and financial metrics. To this end, data are employed from major European and American implied volatility indices and the rapidly growing CBOE volatility futures market. We find that the addition of jumps is necessary to capture the evolution of implied volatility indices under both metrics. Mean reversion is of second-order importance though. The results are consistent across the various metrics, markets, and construction methodologies.     

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.     

News sentiment and the investor fear gauge

This note examines the relationship between aggregate news sentiment and changes in the implied volatility index (VIX). A significant negative contemporaneous relationship between changes in VIX and news sentiment is discovered. The relationship is asymmetric whereby changes in VIX are larger following the release of negative news items.     

Explaining asset pricing puzzles associated with the 1987 market crash

The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options, despite minimal changes in aggregate consumption. We explain these events within a general equilibrium framework in which expected endowment growth and economic uncertainty are subject to rare jumps. The arrival of a jump triggers the updating of agents' beliefs about the likelihood of future jumps, which produces a market crash and a permanent shift in option prices. Consumption and dividends remain smooth, and the model is consistent with salient features of individual stock options, equity returns, and interest rates.     

Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

We study the mass at the origin in the uncorrelated stochastic alpha, beta, rho stochastic volatility model and derive several tractable expressions, in particular when time becomes small or large. As an application—in fact the original motivation for this paper—we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.     

Basis variations and regime shifts in the oil futures market

The conditional volatility of crude oil futures returns is modelled as a regime switching process. The model features transition probabilities that are functions of the basis. Consistent with the theory of storage, in volatile periods, an increase in backwardation is associated with an increase in the likellihood of switching to or remaining in the high-volatility state. Conditional on regimes, GARCH persistence is significantly reduced. Out-of-sample tests show that incorporating regime shifts improves the accuracy of short-term volatility forecasts.     

Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach

In this paper we study the ruin probability at a given time for liabilities of diffusion type, driven by fractional Brownian motion with Hurst exponent in the range (0.5, 1). Using fractional Itô calculus we derive a partial differential equation the solution of which provides the ruin probability. An analytical solution is found for this equation and the results obtained by this approach are compared with the results obtained by Monte-Carlo simulation.     

Trading and Investing in Volatility Products

Since the banking crisis the market for volatility exchange‐traded products has developed rapidly as it opens to clients beyond the large institutional investor pool. Speculation is driven by increasingly complex leveraged and inverse exposures including those that attempt to trade on significant roll costs in volatility futures curves. Longer‐term investors use these products for the purposes of equity diversification, driven by fears of an ongoing Eurozone crisis. We survey the burgeoning academic literature in this area and present a comprehensive and up‐to‐date comparison of the market and statistical characteristics of European and US exchange‐traded volatility products.     

Stock index futures hedging in the emerging Malaysian market

The paper investigates hedging effectiveness of dynamic and constant models in the emerging market of Malaysia where trading information is not readily available and market liquidity is lower compared to the developed equity markets. Using daily data from December 1995 to April 2001 and bivariate GARCH(1,1) and TGARCH models, the paper uses differing variance–covariance structures to obtain hedging ratios. Performance of models is compared in terms of variance reduction and expected utility levels for the full sample period as well as the three sub-periods which encompass the Asian financial crisis and introduction of new capital control measures in Malaysia. Findings show that rankings of the hedging models change for the in-sample period depending on evaluation criteria used. TGARCH based models provide better hedging performance but only in the period of higher information asymmetry following the imposition of capital controls in Malaysia. Overall, despite the structural breaks caused by the Asian financial crisis and new capital control regulations, out of sample hedging performance of dynamic GARCH models in the Malaysian emerging market is as good as the one reported for the highly developed markets in the previous literature. The findings suggest that changes in the composition of market agents caused by large scale retreat of foreign investors following the imposition of capital control regulations do not seem to have any material impact on the volatility characteristics of the Malaysian emerging market.