Stochastic differential equations and diffusion processes

Summation of Large Corrections to Short Distance Hadronic Cross-Sections

Resummation of the QCD perturbative series for hard processes

Factorization for short distance hadron-hadron scattering

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault [Long memory in continuous-time stochastic volatility models. Math. Finance, 1998, 8(4), 291–323]. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, . We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not a long memory process in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatil...

Volatility is rough

List of Figures. List of Tables. Foreword. Preface. Acknowledgments. Chapter 1: Stochastic Volatility and Local Volatility. Stochastic Volatility. Derivation of the Valuation Equation, Local Volatility, History, A Brief Review of Dupire's Work, Derivation of the Dupire Equation, Local Volatility in Terms of Implied Volatility, Special Case: No Skew, Local Variance as a Conditional Expectation of Instantaneous Variance. Chapter 2: The Heston Model. The Process. The Heston Solution for European Options. A Digression: The Complex Logarithm in the Integration (2.13). Derivation of the Heston Characteristic Function. Simulation of the Heston Process. Milstein Discretization. Sampling from the Exact Transition Law. Why the Heston Model Is so Popular. Chapter 3: The Implied Volatility Surface. Getting Implied Volatility from Local Volatilities. Model Calibration. Understanding Implied Volatility. Local Volatility in the Heston Model. Ansatz. Implied Volatility in the Heston Model. The Term Structure of Black-Scholes Implied Volatility in the Heston Model. The Black-Scholes Implied Volatility Skew in the Heston Model. The SPX Implied Volatility Surface. Another Digression: The SVI Parameterization. A Heston Fit to the Data. Final Remarks on SV Models and Fitting the Volatility Surface. Chapter 4: The Heston-Nandi Model. Local Variance in the Heston-Nandi Model. A Numerical Example. The Heston-Nandi Density. Computation of Local Volatilities. Computation of Implied Volatilities. Discussion of Results. Chapter 5: Adding Jumps. Why Jumps are Needed. Jump Diffusion. Derivation of the Valuation Equation. Uncertain Jump Size. Characteristic Function Methods. L-evy Processes. Examples of Characteristic Functions for Specific Processes. Computing Option Prices from the Characteristic Function. Proof of (5.6). Computing Implied Volatility. Computing the At-the-Money Volatility Skew. How Jumps Impact the Volatility Skew. Stochastic Volatility Plus Jumps. Stochastic Volatility Plus Jumps in the Underlying Only (SVJ). Some Empirical Fits to the SPX Volatility Surface. Stochastic Volatility with Simultaneous Jumps in Stock Price and Volatility (SVJJ). SVJ Fit to the September 15, 2005, SPX Option Data. Why the SVJ Model Wins. Chapter 6: Modeling Default Risk. Merton's Model of Default. Intuition. Implications for the Volatility Skew. Capital Structure Arbitrage. Put-Call Parity. The Arbitrage. Local and Implied Volatility in the Jump-to-Ruin Model. The Effect of Default Risk on Option Prices. The CreditGrades Model. Model Setup. Survival Probability. Equity Volatility. Model Calibration. Chapter 7: Volatility Surface Asymptotics. Short Expirations. The Medvedev-Scaillet Result. The SABR Model. Including Jumps. Corollaries. Long Expirations: Fouque, Papanicolaou, and Sircar. Small Volatility of Volatility: Lewis. Extreme Strikes: Roger Lee. Example: Black-Scholes. Stochastic Volatility Models. Asymptotics in Summary. Chapter 8: Dynamics of the Volatility Surface. Dynamics of the Volatility Skew under Stochastic Volatility. Dynamics of the Volatility Skew under Local Volatility. Stochastic Implied Volatility Models. Digital Options and Digital Cliquets. Valuing Digital Options. Digital Cliquets. Chapter 9: Barrier Options. Definitions. Limiting Cases. Limit Orders. European Capped Calls. The Reflection Principle. The Lookback Hedging Argument. One-Touch Options Again. Put-Call Symmetry. QuasiStatic Hedging and Qualitative Valuation. Out-of-the-Money Barrier Options. One-Touch Options. Live-Out Options. Lookback Options. Adjusting for Discrete Monitoring. Discretely Monitored Lookback Options. Parisian Options. Some Applications of Barrier Options. Ladders. Ranges. Conclusion. Chapter 10: Exotic Cliquets. Locally Capped Globally Floored Cliquet. Valuation under Heston and Local Volatility Assumptions. Performance. Reverse Cliquet. Valuation under Heston and Local Volatility Assumptions. Performance. Napoleon. Valuation under Heston and Local Volatility Assumptions. Performance. Investor Motivation. More on Napoleons. Chapter 11: Volatility Derivatives. Spanning Generalized European Payoffs. Example: European Options. Example: Amortizing Options. The Log Contract. Variance and Volatility Swaps. Variance Swaps. Variance Swaps in the Heston Model. Dependence on Skew and Curvature. The Effect of Jumps. Volatility Swaps. Convexity Adjustment in the Heston Model. Valuing Volatility Derivatives. Fair Value of the Power Payoff. The Laplace Transform of Quadratic Variation under Zero Correlation. The Fair Value of Volatility under Zero Correlation. A Simple Lognormal Model. Options on Volatility: More on Model Independence. Listed Quadratic-Variation Based Securities. The VIX Index. VXB Futures. Knock-on Benefits. Summary. Postscript. Bibliography. Index.

The Volatility Surface: A Practitioner's Guide

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

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.

/pdf/pricing-under-rough-volatility-3y8dqhrukw.pdf

Pricing under rough volatility

Exponentiation of eikonal cross sections in nonabelian gauge theories

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The Volatility Surface: A Practitioner's Guide

Volatility is rough

Volatility is rough

Pricing under rough volatility

Exponentiation of eikonal cross sections in nonabelian gauge theories