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The Volatility Smile
- Author / Creator
- Available as
- Online
Details
- Format
- Books
- Language
- English
- Contributors
- Publication
-
- 1st ed
- Newark : John Wiley & Sons, Incorporated, 2016
- ©2016
- Series
- Physical Details
-
- 1 online resource (531 pages)
- ISBNs
- 1118959175, 1119289254, 1118959183, 9781119289258, 1118959167, 9781118959176, 9781118959169, 9781118959183
- OCLC
- ocn948670280, ocn958572770
- Intro -- The Volatility Smile -- Contents -- Preface -- Acknowledgments -- About the Authors -- 1 Overview -- Introduction -- The Black-Scholes-Merton Model and Its Discontents -- A Quick Look at the Implied Volatility Smile -- No-Nonsense Financial Modeling -- About Theorems and Laws -- On Financial Engineering -- The Purpose of Models -- A Simple but Prototypical Financial Model -- Additional Advantages of Using a Model -- Styles of Modeling: What Works and What Doesn't -- 2 The Principle of Replication -- Replication -- The One Law of Quantitative Finance -- Valuation by Replication -- Styles of Replication -- The Limits of Replication -- Modeling the Risk of Underliers -- The Efficient Market Hypothesis -- Uncertainty, Risk, and Return -- The Behavior of a Share of Stock -- The Risk of Stocks -- Riskless Bonds -- The Key Question of Investing -- Some Investment Risks Can Be Avoided -- World #1: Only a Few Uncorrelated Stocks and a Riskless Bond -- World #2: An Infinite Number of Uncorrelated Stocks and a Riskless Bond -- World #3: An Infinite Number of Stocks All Simultaneously Correlated with the Entire Market, and a Riskless Bond -- Derivatives Are Not Independent Securities -- End-of-Chapter Problems -- 3 Static and Dynamic Replication -- Exact Static Replication -- Put-Call Parity -- Replicating a Collar -- Generalized Payoffs -- Approximate Static Hedge for a European Down-and-Out Call -- A Simplified Explanation of Dynamic Replication -- What Should You Pay for Convexity? -- The Distinction between Implied Volatility and Realized Volatility -- Notation for Implied Variables -- Hedging an Option Means Betting on Volatility -- End-of-Chapter Problems -- 4 Variance Swaps -- The Volatility Sensitivity of an Option -- Volatility and Variance Swaps -- Replicating Volatility Swaps
- Replicating Variance Swaps out of Options in a Black-Scholes-Merton World -- A Portfolio of Vanilla Options with 1/K2 Weights Produces a Log Payoff -- Value of a Log Contract in the Black-Scholes-Merton World -- Proof That the Fair Value of a Log Contract with S* = S0 Is the Realized Future Variance -- Replicating Variance When Volatility Is Stochastic -- Valuing the Variance -- Replication with a Finite Number of Options -- Errors in Replication -- The VIX Volatility Index -- End-of-Chapter Problems -- 5 The P& -- L of Hedged Option Strategies in a Black-Scholes-Merton World -- The Black-Scholes-Merton Equation -- The P& -- L of Hedged Trading Strategies -- The Effect of Different Hedging Strategies in the BSM World -- The P& -- L When Hedging with Realized Volatility -- Bounds on the P& -- L When Hedging at the Realized Volatility -- The P& -- L When Hedging with Implied Volatility -- End-of-Chapter Problems -- 6 The Effect of Discrete Hedging on P& -- L -- Replication Errors from Discrete Rebalancing -- A Simulation Approach -- Understanding the Hedging Error Analytically -- An Example -- Conclusion: Accurate Replication and Hedging Are Very Difficult -- End-of-Chapter Problems -- 7 The Effect of Transaction Costs on P& -- L -- The Effect of Transaction Costs -- The Simplest Rebalancing Strategy: Rebalancing at Regular Intervals -- A More Practical Rehedging Strategy: Rehedging Triggered by Changes in the Hedge Ratio -- Analytical Approximation of the Effect of Transaction Costs -- A PDE Model of Transaction Costs -- End-of-Chapter Problems -- 8 The Smile -- Smile, Term Structure, Surface, and Skew -- How to Graph the Smile -- Variable Choice Can Matter -- Delta and the Smile -- The Relationship between Delta and Strike -- Smiles in Different Option Markets -- Consequences of the Smile for Trading -- End-of-Chapter Problems
- 9 No-Arbitrage Bounds on the Smile -- No-Arbitrage Bounds on the Smile -- The Merton Inequalities for European Option Prices as a Function of Strike -- Inequalities for the Slope of the Smile -- End-of-Chapter Problems -- 10 A Survey of Smile Models -- An Overview of Smile-Consistent Models -- Local Volatility Models -- Stochastic Volatility Models -- Jump-Diffusion Models -- A Plenitude of Other Models -- Problems Caused by the Smile -- Hedging Vanilla Options -- Valuing Exotic Options -- End-of-Chapter Problem -- 11 Implied Distributions and Static Replication -- Implied Distributions -- State-Contingent Securities -- The Breeden-Litzenberger Formula -- Static Replication: Valuing Arbitrary Payoffs at a Fixed Expiration Using Implied Distributions -- Replication Using Standard Options -- The Heaviside and Dirac Delta Functions -- Using Static Replication to Estimate the Effect of a Skew -- The Black-Scholes-Merton Risk-Neutral Probability Density -- End-of-Chapter Problems -- 12 Weak Static Replication -- Summary of the Book So Far -- Introducing Weak Static Replication -- Some Insights into the Static Replication of Barrier Options -- A European Up-and-In Put with Barrier = Strike -- Valuing a Down-and-Out-Barrier Option under Geometric Brownian Motion with a Zero Riskless Rate and Zero Dividend Yield -- Valuing a Down-and-Out-Barrier Option under Geometric Brownian Motion with a Nonzero Riskless Rate -- The Static Hedge Suggested by the Valuation Formula -- Another Approach: Static Replication of an Up-and-Out Call -- Replication Accuracy -- The Generalized Approach -- Barrier Option Parity -- End-of-Chapter Problems -- 13 The Binomial Model and Its Extensions -- The Binomial Model for Stock Evolution -- First Solution: The Cox-Ross-Rubinstein Convention -- Another Solution: The Jarrow-Rudd Convention -- The Binomial Model for Options Valuation
- Options Valuation -- The Black-Scholes-Merton Partial Differential Equation and the Binomial Model -- Extending the Black-Scholes-Merton Model -- Base Case: Zero Dividend Yield, Zero Riskless Rate, and the Riskless Bond as the Numeraire -- Extension to Nonzero Rates -- Stock with a Continuous Known Dividend Yield -- Time-Dependent Deterministic Volatility: A Volatility Smile with Term Structure but No Skew -- End-of-Chapter Problems -- 14 Local Volatility Models -- Modeling a Stock with Variable Volatility -- Binomial Local Volatility Modeling -- The Relationship between Local Volatility and Implied Volatility -- The Rule of Two: Understanding the Relationship between Local and Implied Volatilities -- Difficulties with Binomial Trees -- Further Reading -- End-of-Chapter Problems -- 15 Consequences of Local Volatility Models -- Dupires Equation for Local Volatility -- Understanding the Equation -- A Binomial Derivation of the Dupire Equation -- The Tree -- The Calendar Spread -- The Butterfly Spread -- A More Formal Proof of the Dupire Equation -- An Exact Relationship between Local and Implied Volatilities and Its Consequences -- Implied Variance Is the Average of Local Variance over the Life of the Option When There Is No Skew -- The Rule of Two Revisited -- At Short Expirations, Implied Volatility Is a Harmonic Average of Local Volatility between the Current Stock Price and the Strike -- End-of-Chapter Problems -- 16 Local Volatility Models -- Hedge Ratios in Local Volatility Models -- The Correct Hedge Ratio of a Vanilla Option -- The Theoretical Value of Exotic Options in Local Volatility Models -- Up-and-Out Call with Strike = 100 and Barrier = 110 -- An Up-and Out Call That Has No Black-Scholes-Merton Implied Volatility -- Lookback Call Option -- End-of-Chapter Problems -- 17 Some Final Remarks on Local Volatility Models
- The Pros and Cons of Local Volatility Models -- Advantages -- The Big Question -- Disadvantages -- Testing the Local Volatility Model for Index Options -- The Impact of Different Market Regimes on the Variance of the Hedged Portfolio's P& -- L -- 18 Patterns of Volatility Change -- Heuristic Relationships between the Slope of the Skew and Its Dynamics -- The Sticky Strike Rule -- The Sticky Delta Rule -- The Local Volatility Model -- Summary of the Rules -- Stickiness in the Real World -- Toward Stochastic Volatility Models -- End-of-Chapter Problems -- 19 Introducing Stochastic Volatility Models -- Introduction to Stochastic Volatility -- Approaches to Stochastic Volatility Modeling -- A Heuristic Approach for Introducing Stochastic Volatility into the Black-Scholes-Merton Model -- The Extended Black-Scholes-Merton Model: A Stochastic Differential Equation for Volatility -- Adding Mean Reversion -- A Survey of Some Stochastic Volatility Models -- Risk-Neutral Valuation and Stochastic Volatility Models -- End-of-Chapter Problems -- 20 Approximate Solutions to Some Stochastic Volatility Models -- Extending the Local Volatility Model -- Extending the BSM Model: Valuing Options with Stochastic Volatility via the Replication Principle -- The Meaning of (S, , t) in Terms of Sharpe Ratios -- The Characteristic Solution to the Stochastic Volatility Model -- End-of-Chapter Problem -- 21 Stochastic Volatility Models -- The Zero-Correlation Smile Depends on Moneyness -- The Zero Correlation Smile Is Symmetric -- Two-State Stochastic Path Volatility: An Example -- The Smile for GBM Stochastic Volatility with Zero Correlation -- An Analytic Approximation for the Smile for GBM Stochastic Volatility with Zero Correlation -- End-of-Chapter Problem -- 22 Stochastic Volatility Models -- Mean-Reverting Volatility with Zero Correlation
- Volatility versus Path Volatility