Options - 45 Years Since the Publication of the Black-Scholes-Merton Model: The Gershon Fintech Center Conference

Contents
Preface
About the Editors
Introduction
References
1. Using Option Pricing Information to Time Diversify Portfolio Returns
1. Average Returns are the Main Focus
2. Compound Return Facts: Tail Risks Dominate!
3. Compound Returns a Function of Risk
4. Big Problem: Tail Events More Frequent than Normal Distribution
5. Think Tails of the Distribution: Concentrate on Normal Events
6. Relative Performance Evaluation
7. Investment Strategies — Asset Allocation Static Constraints are Costly
8. Factors that Affect Terminal Wealth
8.1 Fact 1
8.2 Fact 2
8.3 Fact 3
8.4 Fact 4
8.5 Fact 5
8.6 Fact 6
9. Measuring Tail Risk Using Market Prices
10. Tail Gains/Losses from Opiton Prices
11. Enhancing Compound Returns Through Dynamic Risk Management (1996–2015)
12. Using Option Prices to Forecast Risk Changing Risk Proactively
13. Using Option Prices to Measure Risk
14. Uncertainty of the Distribution of Returns
15. Adaptive Strategy — Pre and Post “2008” Crisis
16. Issues
2. How Good is Black–Scholes–Merton, Really?
1. Introduction
2. Things I Like About BSM
3. Things I Don’t Like About BSM
4. Technically Speaking
4.1 Coping with new markets and contracts
4.2 Dropping assumptions
4.3 Robustness with respect to volatility
5. Derivations of BSM
6. The Marketing Department’s Derivation
7. Conclusion
References
3. Probabilistic Interpretation of Black Implied Volatility
1. Assumptions
2. Probabilistic Interpretation of the Implied Variance Rate
3. Financial Interpretations of the Implied Variance Rate
3.1 Fixed strike price
3.2 Floating strike price
4. Approximation for Short Maturity Options
5. Empirical Tests of the Implied Variance Rate Formula
6. Summary and Further Research
4. Probability-Free Models in Option Pricing: Statistically Indistinguishable Dynamics and Historical vs Implied Volatility
1. The Initial Question: Statistical Estimation and Valuation
2. Indistinguishable Models Leading to Different Option Prices
2.1 Matching margins
2.2 Matching the whole law on a Δ grid
2.3 A fundamental case: σt(y) = νy
2.4 A technical link with the rough volatility literature
2.5 Arbitrarily different option prices under indistinguishable models
3. Reconciling Historical and Implied Volatility with a Single Dynamics
4. Possible Explanation of Arbitrary Option Prices?
4.1 Rough paths and option pricing
5. Conclusions: 20 Years of Pathwise Pricing
Acknowledgments
References
5. VIX and Derivatives
1. Outline
2. Background
2.1 Brief history of derivatives markets
2.2 Size of markets (notional)-global
2.3 OTC derivative markets
3. History of VIX
4. Futures/Options on VIX (CBOE) (Figures 1–6)
5. VIX Research
6. VIX Derivatives
7. VIX Methodology (in Brief)
7.1 VIX-Methodology (CBOE/GS)
7.2 VIX-Methodology (Brenner–Galai)
8. VIX Settlement and Manipulation
9. February 5, 2018
10. Ambiguity (in Brief) (Figure 7)
6. Multivariate Fractional Brownian Motion and Generalizations of SABR Model
1. Introduction
2. Empirical Evidence
3. Fractional Brownian Motion
4. Self-Similarity
5. Covariance
6. Fractional SABR Model
7. At-the-Money Volatility Skew
References
7. Buy Rough, Sell Smooth
1. Introduction
2. Realized and Implied Roughness
2.1 Realized roughness
2.2 Implied roughness
2.3 Descriptive statistics of realized and implied roughness
3. Sorted Portfolios
4. Controlling for Other Factors
4.1 Liquidity
4.2 Implied volatility and skewness
4.3 Double sorts
4.4 Fama–MacBeth Regressions
5. Event Risk: Earnings Announcements and FOMC Meetings
5.1 Earnings announcements
5.1.1 Testing for earnings surprise predictability
5.1.2 Strategy performance near earnings announcements
5.2 Strategy performance near FOMC announcements
6. Conclusions
Appendix: Filtering of Option Data
References
8. Volatility is Rough
1. Introduction
1.1 Volatility modeling
1.2 Fractional volatility
1.3 The shape of the implied volatility surface
1.4 Main results and organization of the chapter
2. Smoothness of the Volatility: Empirical Results
2.1 Estimating the smoothness of the volatility process
2.2 DAX and Bund futures contracts
2.3 S&P and NASDAQ indices
2.4 Other indices
2.5 Distribution of the increments of the log-volatility
2.6 Does H vary over time?
3. A Simple Model Compatible with the Empirical Scaling of the Volatility
3.1 Specification of the RFSV model
3.2 RFSV model autocovariance functions
3.3 RFSV vs FSV again
3.4 Simulation-based analysis of the RFSV model
4. Spurious Long Memory of Volatility?
5. Forecasting Using the RFSV Model
5.1 Forecasting log-volatility
5.2 Variance prediction
6. Conclusion
Acknowledgments
Appendix A Proofs
A.1 Proof of Proposition 3.1
A.2 Proof of Corollary 3.1
Appendix B Estimations of H
B.1 On different indices
B.2 On different time intervalss
Appendix C The Effect of Smoothing
Numerical example
Acknowledgment
References
9. Things We Think We Know
1. Models of Asset Returns
2. Alternative Asset Models
3. Rough Volatility
4. A Simpler Alternative to Rough Volatility
5. High-Frequency Data
6. Conclusions
References
10. Cumulant Formulas for Implied Volatility
1. Implied Volatility and its Importance
2. Relate Implied Volatility Skew to Underlying Distribution
3. Notations
4. Exact Relationship: Distribution ↔ Implied Volatilities
5. Moments
6. Cumulants
7. Skewness and Kurtosis
8. How do Skewness/Kurtosis Relate to Implied Volatilty?
9. BFW Approach
10. Our Approach
11. Cumulant Expansion for Implied Volatility
12. Five Moments
13. The Book of Five Moments
14. Moment Formula
15. Implied Volatility: BFW vs Refined vs Exact (Figure 1)
16. Intuition: Jump–Diffusion Dynamics
17. Option Price Approximation
18. Implied Volatility Approximation
19. Conclusions
11. Implied Volatility Asymptotics: Black–Scholes and Beyond
1. Introduction
2. Implied Volatility Close to Maturity
2.1 Approximate inversion of the Black–Scholes formula
2.2 ATM skew
2.3 Diffusion models
2.4 Jump-diffusion and Lévy models
2.5 Universal parametrization of kt and the limiting smile
2.6 ATM skew in jump-diffusion/Lévy models
2.7 Rough volatility models
3. Implied Volatility Far from Maturity
3.1 Lévy processes and Cramer’s theorem
3.2 Renormalizing the smile
4. Extreme Strikes
5. Conclusion
References
12. The Smile of Stochastic Volatility Models
1. Outline
2. Motivation
3. Expansion of the Price of a Vanilla Option
3.1 The Perturbation Equations
3.2 The price at order 0
3.3 The price at order 1
3.4 The price at order 2
3.5 Expansion of the implied volatility
3.6 Comments
3.7 Another derivation which stays at the level of operators
4. Short Maturity: Structural Dependencies
5. Long-term Asymptotics of Implied Volatility
6. First Example: A Heston-like Model
7. Second Example: The Bergomi Model
8. Numerical Experiments
8.1 First order
8.2 Second order
9. Rederiving the Link Between Skew and Skewness of Log-Returns
10. Conclusion
11. Risk Magazine, May 2012
References
13. A Neural Network Approach to Understanding Implied Volatility Movements
1. Introduction
2. Neural Networks
3. Data
4. Model Selection Criteria
5. Results for Three-Feature Model
6. Results for Four-Feature Model
7. Conclusions
Acknowledgments
References
14. Modeling Volatility Risk in Equity Options Market: A Statistical Approach
1. Introduction
2. Principal Component Analysis of the Correlation Matrix of the Implied Volatility Surfaces
3. Implied Volatility Surfaces and Random Matrix Theory
4. Classification of Optionable Stocks into “Systemic” and “Idiosyncratic”
5. First Dimensional Reduction: The Pivot Method
6. Second Dimensional Reduction: Cross-Sectional Analysis of Correlations via RMT
7. Conclusion
A. Appendix
References
15. A General Theory of Option Pricing
1. Introduction
2. Integral Representation for European Option Pricing
2.1 General formalism
2.2 No-arbitrage condition
2.3 Using the density function to calculate the integrals
2.4 The PDE of the density function to maturity
2.5 An Example: Deterministic time dependent volatility yields the BSM model
3. Evidence that the Volatility Smile is Independent of the Term Structure
3.1 The methodology
3.2 The results of our analysis
3.3 Conclusions
4. Calculating the Price of European Options
4.1 Homogeneous density function
4.2 The consistency condition for gT (s)
4.3 Calculating gT(s) by iteration for the continues time process
4.4 A Numerical method to calculate gt1,t2(s1,s2) from gT(sT ) for time homogeneous density function
5. Obtaining the Conditional Probability Transfer Density by Bootstrapping the European Options Term Structure
6. Empirical Evidence from the Option Markets
6.1 Representations of ΣT in different asset classes
6.2 Methodology
6.3 Year-end rates: Comparing the model to the most accurate mid-market
7. Conclusions
Acknowledgments
References
16. Old Problems, Classical Methods, New Solutions
1. Introduction
2. Mathematical Preliminaries
2.1 The method of heat potentials
2.2 Extensions
2.3 Generalizations
2.4 Numerics
3. The Structural Default Model
3.1 Preliminaries
3.2 Formulation
3.3 Governing system of integral equations
3.4 The choice of bτ
3.5 Default boundaries
3.6 Main conjecture
4. Mean-Field Banking System
4.1 Preliminaries
4.2 Interconnected banking system
4.3 Governing system of integral equations
4.4 Numerical solution
5. Hitting Time Probability Distribution for an Ornstein–Uhlenbeck Process
5.1 Preliminaries
5.2 Main equations
5.3 Particular case, b = 0
5.4 General case
5.5 The governing system of integral equations
5.6 Flat boundary
5.7 Abel integral equation
6. The Supercooled Stefan Problem
7. The Integrate-and-Fire Neuron Excitation Model
7.1 Governing equations
7.2 The stationary problem
7.3 The nonstationary problem
8. Conclusions
Acknowledgment
References
17. 25 Years of Local Volatility and Beyond
1. Outline
2. Local Volatility Model
2.1 Barrier option prices
2.2 Link European options/path-dependent (exotic) options
2.3 One simple model that fits market smiles
2.4 The risk-neutral solution
2.5 Implied and local volatility surfaces (see Figure 3)
2.6 Forward equation
3. Forward Equations: A Simpler Derivation of the BSM PDE
3.1 Local volatility model forward equation
3.2 A simpler proof of Black–Scholes–Merton PDE
3.3 The Beauty of Forward Equations
3.4 Another Beauty of Forward Equations
3.5 Stochastic Volatility
3.6 Stochastic local volatility
3.7 Summary of local volatility model properties
3.8 Path-dependent (or exotic) options
4. Functional Itô Calculus: A Framework for Path Dependence
4.1 Review of Itô calculus (see Figure 4)
4.2 Functional derivatives
4.3 Examples of functionals and their derivatives
4.4 Results and applications
4.4.1 BSM/LVM PDE for path-dependent options
4.4.2 Volatility risk management for path-dependent options
4.4.3 A risk scale to compare options
5. Conclusion
References
18. Swap Rate à la Stock: Bermudan Swaptions Made Easy
1. Introduction and Motivation
2. Notation and Modeling Framework
3. Swap Rate à la Stock
4. Bermudan Swaption Pricing
5. Discussion
References
19. Thirty years of Derivatives Market: Originality of the French Experience
1. Introduction
20. Option Prices in the Equity, Index and Commodity Markets: The “Message from Markets”
1. Introduction
2. 2008 Black–Scholes–Merton 35th Anniversary, Conference on Financial Innovation, Vanderbilt University
2.1 Nobel laureates discuss financial innovation
3. VIX: S&P 500 Implied Volatility
3.1 Definition of implied volatility
3.2 VIX: The implied volatility of the S&P 500 Index
3.3 Conclusion: VIX under informationally-efficient financial markets
4. Merton’s [7] Jump-Diffusion Model
4.1 The commodity markets version of the Merton (1976) model
4.2 Fitting the Merton model to crude-oil futures and option prices
5. Conclusions
Acknowledgments
References
21. Options Markets in China: The New Frontier
1. Introduction
2. Exchange-Traded Options
2.1 Financial options
2.2 Commodity options
2.3 Market participants
3. Pricing Anomalies
3.1 Pricing efficiency
3.2 Violation of the put-call parity
3.3 Negative time value of money
3.4 Correlation between stock price and implied volatility
3.5 Risk premium of commodity options
4. Conclusion
Acknowledgment
References
22. Risk Exposure Valuation Using Measure Distortions: An Overview
1. Introduction
2. Issues with the Law of One Price
3. The Two Price Framework
4. Conservative Valuation and Distorted Expectations
5. Risk as Exposure and Its Valuation
6. Applications
6.1 A hedging example
6.2 Portfolio allocation example
7. Conclusion
References
23. Insider Trading
1. Motivation
1.1 New results with Neufcourt
1.2 Convergence of the information drifts
1.3 A new example
Acknowledgment
References
24. Contingent Claims Analysis in Corporate Finance
1. Introduction
2. CCA has Revolutionized the Field of Corporate Finance
3. Corporate Finance
4. Agenda
4.1 Foundations
4.2 The pricing of debt and equity
4.3 The pricing of warrants, convertible securities, preferred stocks
4.3.1 Pricing of warrants
4.3.2 Pricing of convertible securities
4.4 CCA and various corporate issues
4.4.1 Dividends as contingent claims
4.5 CCA for banks and sovereign debt
4.5.1 CCA for banks
4.5.2 CCA for sovereign debt
Acknowledgment
References
Index