This document discusses implied and realized index skews using a beta stochastic volatility model. Empirical evidence shows that implied and realized volatilities of stock indices follow log-normal distributions. A beta stochastic volatility model is presented that models volatility evolution based on changes in the index price. The model's parameters, volatility beta and residual volatility, are estimated using historical index returns and volatility data. Implied parameters from option prices generally overestimate realized values. Risk-neutral skews incorporate an additional premium due to investor risk aversion that the model quantifies using a relationship from financial studies literature. A Merton jump diffusion model is fit to the empirical data to further examine skews.
Financial Markets with Stochastic Volatilities - markov modellingguest8901f4
The document summarizes the research of Anatoliy Swishchuk on stochastic volatility models and their applications in financial mathematics. Specifically, it discusses:
1. Random evolutions (REs), which are abstract dynamical systems with random components that can model stochastic processes.
2. Applications of REs, including modeling traffic, storage, risk, and biological processes. In finance, REs can model markets with stochastic volatility.
3. Pricing of derivatives like variance swaps, volatility swaps, and swing options under stochastic volatility models like Heston. Numerical examples are provided based on S&P60 Canada index data.
Trading and managing volatility surface risks_Axpo Risk Management Workshop_2...Stian J. Frøiland
- The document discusses trading and managing volatility surface risks. It summarizes the classical Black-Scholes-Merton options pricing model and its assumptions.
- The classical BSM model assumptions do not reflect market reality as volatility is non-stationary and stochastic rather than constant. To compensate, traders use multiple local BSM models with different implied volatilities.
- Alternative models like SABR better incorporate features like stochastic volatility and non-constant drift to model the volatility surface. The SABR model provides a framework to price options and manage risks like vega, delta, vanna and volga.
The document discusses Value at Risk (VaR), a metric used to measure and manage financial risk. It provides an introduction to VaR and outlines several key concepts, including: reasons for VaR's widespread adoption; calculating VaR for single and multiple assets; assumptions underlying VaR calculations; and approaches to estimating VaR for linear and non-linear derivatives. It also covers converting daily VaR to other time periods, factors affecting portfolio risk, and stress testing as a complement to VaR analysis.
Value at Risk (VaR) is a risk measurement technique used to estimate potential losses that could occur from market risk over a specified time period. The document discusses the need for VaR, how it is defined and calculated using historical simulation, its uses, strengths and weaknesses. It emphasizes that VaR should not be used alone and other risk measures like tail measures and stress testing are also important.
Value at Risk (VaR) is a statistical technique used to measure potential portfolio losses over a specified time period and confidence level. It was originally used to measure market risk but has been extended to other risk types like credit and operational risk. VaR calculates the maximum dollar amount a portfolio could lose with a given level of confidence, usually 95%. Lower correlations between assets in a portfolio reduce overall risk. VaR is computed using weights, volatilities, and correlations of assets in a portfolio along with the confidence level and time horizon.
The document discusses forward volatility agreements (FVAs). It defines an FVA as a volatility swap contract where the buyer and seller agree to exchange a straddle option at a future date based on a specified volatility level. The key motivation for trading FVAs is that it allows investors to speculate on future volatility levels. The document provides details on pricing and hedging FVAs, including using volatility gadgets and forward start straddle options to isolate exposure to future local volatility.
This document discusses Value at Risk (VaR) and related concepts over multiple learning outcomes (LOs). It introduces VaR and explains why it was widely adopted as a risk measure. It also defines how to calculate VaR for single and multiple assets, and how to convert between time periods. The document discusses assumptions of VaR calculations and reasons for using continuously compounded returns. It also addresses factors that affect portfolio risk and how to calculate VaR for linear and non-linear derivatives. Finally, it introduces cash flow at risk (CFaR) and how VaR and CFaR can be used to evaluate projects and allocate risk.
Value at Risk (VAR) summarizes the worst potential loss over a target period at a given confidence level, accounting for risks across an institution. VAR is calculated using statistical techniques to estimate losses that may occur but are unlikely to be exceeded. It is used to measure market, credit, operational and enterprise-wide risk and determine capital requirements to withstand unexpected losses.
Financial Markets with Stochastic Volatilities - markov modellingguest8901f4
The document summarizes the research of Anatoliy Swishchuk on stochastic volatility models and their applications in financial mathematics. Specifically, it discusses:
1. Random evolutions (REs), which are abstract dynamical systems with random components that can model stochastic processes.
2. Applications of REs, including modeling traffic, storage, risk, and biological processes. In finance, REs can model markets with stochastic volatility.
3. Pricing of derivatives like variance swaps, volatility swaps, and swing options under stochastic volatility models like Heston. Numerical examples are provided based on S&P60 Canada index data.
Trading and managing volatility surface risks_Axpo Risk Management Workshop_2...Stian J. Frøiland
- The document discusses trading and managing volatility surface risks. It summarizes the classical Black-Scholes-Merton options pricing model and its assumptions.
- The classical BSM model assumptions do not reflect market reality as volatility is non-stationary and stochastic rather than constant. To compensate, traders use multiple local BSM models with different implied volatilities.
- Alternative models like SABR better incorporate features like stochastic volatility and non-constant drift to model the volatility surface. The SABR model provides a framework to price options and manage risks like vega, delta, vanna and volga.
The document discusses Value at Risk (VaR), a metric used to measure and manage financial risk. It provides an introduction to VaR and outlines several key concepts, including: reasons for VaR's widespread adoption; calculating VaR for single and multiple assets; assumptions underlying VaR calculations; and approaches to estimating VaR for linear and non-linear derivatives. It also covers converting daily VaR to other time periods, factors affecting portfolio risk, and stress testing as a complement to VaR analysis.
Value at Risk (VaR) is a risk measurement technique used to estimate potential losses that could occur from market risk over a specified time period. The document discusses the need for VaR, how it is defined and calculated using historical simulation, its uses, strengths and weaknesses. It emphasizes that VaR should not be used alone and other risk measures like tail measures and stress testing are also important.
Value at Risk (VaR) is a statistical technique used to measure potential portfolio losses over a specified time period and confidence level. It was originally used to measure market risk but has been extended to other risk types like credit and operational risk. VaR calculates the maximum dollar amount a portfolio could lose with a given level of confidence, usually 95%. Lower correlations between assets in a portfolio reduce overall risk. VaR is computed using weights, volatilities, and correlations of assets in a portfolio along with the confidence level and time horizon.
The document discusses forward volatility agreements (FVAs). It defines an FVA as a volatility swap contract where the buyer and seller agree to exchange a straddle option at a future date based on a specified volatility level. The key motivation for trading FVAs is that it allows investors to speculate on future volatility levels. The document provides details on pricing and hedging FVAs, including using volatility gadgets and forward start straddle options to isolate exposure to future local volatility.
This document discusses Value at Risk (VaR) and related concepts over multiple learning outcomes (LOs). It introduces VaR and explains why it was widely adopted as a risk measure. It also defines how to calculate VaR for single and multiple assets, and how to convert between time periods. The document discusses assumptions of VaR calculations and reasons for using continuously compounded returns. It also addresses factors that affect portfolio risk and how to calculate VaR for linear and non-linear derivatives. Finally, it introduces cash flow at risk (CFaR) and how VaR and CFaR can be used to evaluate projects and allocate risk.
Value at Risk (VAR) summarizes the worst potential loss over a target period at a given confidence level, accounting for risks across an institution. VAR is calculated using statistical techniques to estimate losses that may occur but are unlikely to be exceeded. It is used to measure market, credit, operational and enterprise-wide risk and determine capital requirements to withstand unexpected losses.
Short Variance Swap Strategies on the S&P 500 Index Profitable, Yet RiskyRYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Convertible Bonds and Call Overwrites - 2007RYAN RENICKER
The document evaluates Best Buy's 2.25% convertible bonds due 2022 with call overwrites as a risk-adjusted trade on Best Buy stock. It analyzes the trade over 3 time horizons (3 months, 5 months, 15 months) with varying degrees of call option overwrites. Selling calls at higher implied volatilities allows investors to monetize rich option premium. The trade provides upside potential if the stock rises while limiting downside through the call premium collected and bond floor. Tables show estimated returns for the convertible bond under different stock price scenarios and call overwrite strategies.
Options on the VIX and Mean Reversion in Implied Volatility Skews RYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Style-Oriented Option Investing - Value vs. Growth?RYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable E...Volatility
1) Analyze the dependence between returns and volatility in conventional stochastic volatility (SV) models
2) Introduce the beta SV model by Karasinski-Sepp, "Beta Stochastic Volatility Model", Risk, October 2012
3) Illustrate intuitive and robust calibration of the beta SV model to historical and implied data
4) Mix local and stochastic volatility in the beta SV model to produce different volatility regimes and equity delta
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
Implied volatility represents the volatility that makes the theoretical value of an option equal to its market price. It is typically expressed as an annual percentage that represents how much a stock's price could move up or down in one standard deviation. The document explains how to convert implied annual volatility into expected price movements over different time periods like days or weeks by taking the square root of the fraction of days relative to a year. For example, a stock with 35% annual implied volatility would be expected to move up or down around 2.2% within one day, 4.93% within five days, and 9.86% within 20 days. The document demonstrates how to use these expected movements to assess risk for options positions.
The variance swap market has grown exponentially over the past decade and is among the most liquid equity derivatives contracts. Variance swaps provide exposure to volatility through the difference between the implied and realized variance of an underlying asset. Historically, the implied volatility of indices has been higher than realized volatility, allowing those taking short volatility positions to profit. Standard and Poor's has developed indices to benchmark volatility arbitrage strategies, such as the S&P 500 Volatility Arbitrage Index which measures the performance of a variance swap on the S&P 500.
Enhanced Call Overwriting*
Systematically overwriting the S&P 500 with 1-month at-the-money calls, rebalanced on a monthly basis at expiration, outperformed the S&P 500 Index during our sample period (1996 – 2005). This “base case” overwriting strategy also generated superior risk-adjusted returns versus the index.
Overwriting portfolios with out-of-the-money calls tends to outperform at-the-money overwriting during market rallies, but provides less protection during market downturns. However, out-of-the money overwriting also results in relatively higher return variability and inferior risk-adjusted performance.
During the sample period, overwriting the S&P 500 with short-dated options, rebalanced more frequently, outperformed overwriting with longer-dated options, rebalanced less frequently. We discuss possible explanations for these performance differences.
We find that going long the market during periods of heightened short-term anxiety, inferred from the presence of relatively high S&P 500 1-month at-the-money implied volatility, has, on average, been a winning strategy. To a slightly lesser extent, having relatively less exposure to the market during periods of complacency – or relatively low implied market implied volatility – was also beneficial.
We create an “enhanced” overwriting strategy – whereby investors systematically overwrite the S&P 500 or Nasdaq 100 with disproportionately fewer (more) calls against the indices when risk expectations are relatively high (low).
Our enhanced overwriting portfolios handily outperformed the base case overwrite portfolios and the respective underlying indices, on an absolute and risk-adjusted basis. For example, the average annual return for the S&P 500 enhanced overwriting portfolio from 1997 – 2005 was 7.9%, versus 6.6% for the base case overwrite portfolio and 5.5% for the S&P 500 Index.
Overwriting with fewer calls when implied volatility is rich, and more calls when implied volatility is cheap, could improve the absolute and risk-adjusted performance of index-oriented overwriting portfolios.
This goes against the conventional tendency for investors to sell calls against their positions when implied volatility is high.
*Renicker, Ryan and Devapriya Mallick., “Enhanced Call Overwriting.”, Lehman,Brothers Global Equity Research Nov 17, 2005.
The Lehman Brothers Volatility Screening ToolRYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
ImperialMathFinance: Finance and Stochastics Seminarelviszhang
This paper presents a general stochastic volatility model that nests both Heston and Sabr models. It develops a correlation control variate Monte Carlo valuation method that provides significant improvements in valuation efficiency over existing simulation methods, especially for pricing barrier options. By providing an effective simulation method for this general model, the paper enables the potential to calibrate the model to both vanilla and path-dependent instruments to better fit market prices.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Volatility
This document discusses efficient numerical PDE methods to solve calibration and pricing problems in local stochastic volatility models. It begins with an overview of volatility modelling, including local stochastic volatility models that combine local volatility, jumps, and stochastic volatility. It then discusses calibrating both parametric and non-parametric local volatility models using PDE methods. The document provides examples of modelling stochastic volatility factors using implied volatility data and estimating jump parameters from historical returns. It also discusses calibrating local volatility models to vanilla option prices while including jumps and stochastic volatility.
Volatility trading strategies seek to profit from changes in a asset's volatility. Volatility measures how much the price of an asset fluctuates over time. There are several types of volatility strategies including volatility dispersion trading which buys options on index components and sells options on the overall index, volatility spreads which use option combinations to profit from different implied volatilities, and gamma trading which aims to benefit from unexpected events causing large price moves. Volatility is important for options as their pricing depends on assumptions about future volatility.
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Achieving Consistent Modeling Of VIX and Equities DerivativesVolatility
1) Discuss model complexity and calibration
2) Emphasize intuitive and robust calibration of sophisticated volatility models avoiding non-linear calibrations
3) Present local stochastic volatility models with jumps to achieve joint calibration to VIX options and (short-term) S&P500 options
4) Present two factor stochastic volatility model to fit both the short-term and long-term S&P500 option skews
This document provides an overview of key concepts related to the Capital Asset Pricing Model (CAPM). It discusses how to calculate expected returns, variances, and covariances of individual securities and portfolios. It explains how diversification reduces unsystematic risk but not systematic market risk. The efficient frontier and capital market line are introduced. The CAPM relationship between risk and expected return is presented, relating an asset's beta to its expected return.
Express measurement of market volatility using ergodicity conceptJack Sarkissian
Don't we want to base our trading decisions on current market conditions? Then why should we rely on time averages only because they are simple to comprehend? We can get current market volatility a lot faster by applying the ERGODICITY concept to financial markets. Ensemble averaging allows to measure market volatility quickly, based on only two points in time and is as relevant to volatility measurement as the traditional measures.
Short Variance Swap Strategies on the S&P 500 Index Profitable, Yet RiskyRYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Convertible Bonds and Call Overwrites - 2007RYAN RENICKER
The document evaluates Best Buy's 2.25% convertible bonds due 2022 with call overwrites as a risk-adjusted trade on Best Buy stock. It analyzes the trade over 3 time horizons (3 months, 5 months, 15 months) with varying degrees of call option overwrites. Selling calls at higher implied volatilities allows investors to monetize rich option premium. The trade provides upside potential if the stock rises while limiting downside through the call premium collected and bond floor. Tables show estimated returns for the convertible bond under different stock price scenarios and call overwrite strategies.
Options on the VIX and Mean Reversion in Implied Volatility Skews RYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Style-Oriented Option Investing - Value vs. Growth?RYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable E...Volatility
1) Analyze the dependence between returns and volatility in conventional stochastic volatility (SV) models
2) Introduce the beta SV model by Karasinski-Sepp, "Beta Stochastic Volatility Model", Risk, October 2012
3) Illustrate intuitive and robust calibration of the beta SV model to historical and implied data
4) Mix local and stochastic volatility in the beta SV model to produce different volatility regimes and equity delta
Stochastic Local Volatility Models: Theory and ImplementationVolatility
1) Hedging and volatility
2) Review of volatility models
3) Local volatility models with jumps and stochastic volatility
4) Calibration using Kolmogorov equations
5) PDE based methods in one dimension
5) PDE based methods in two dimensions
7) Illustrations
Implied volatility represents the volatility that makes the theoretical value of an option equal to its market price. It is typically expressed as an annual percentage that represents how much a stock's price could move up or down in one standard deviation. The document explains how to convert implied annual volatility into expected price movements over different time periods like days or weeks by taking the square root of the fraction of days relative to a year. For example, a stock with 35% annual implied volatility would be expected to move up or down around 2.2% within one day, 4.93% within five days, and 9.86% within 20 days. The document demonstrates how to use these expected movements to assess risk for options positions.
The variance swap market has grown exponentially over the past decade and is among the most liquid equity derivatives contracts. Variance swaps provide exposure to volatility through the difference between the implied and realized variance of an underlying asset. Historically, the implied volatility of indices has been higher than realized volatility, allowing those taking short volatility positions to profit. Standard and Poor's has developed indices to benchmark volatility arbitrage strategies, such as the S&P 500 Volatility Arbitrage Index which measures the performance of a variance swap on the S&P 500.
Enhanced Call Overwriting*
Systematically overwriting the S&P 500 with 1-month at-the-money calls, rebalanced on a monthly basis at expiration, outperformed the S&P 500 Index during our sample period (1996 – 2005). This “base case” overwriting strategy also generated superior risk-adjusted returns versus the index.
Overwriting portfolios with out-of-the-money calls tends to outperform at-the-money overwriting during market rallies, but provides less protection during market downturns. However, out-of-the money overwriting also results in relatively higher return variability and inferior risk-adjusted performance.
During the sample period, overwriting the S&P 500 with short-dated options, rebalanced more frequently, outperformed overwriting with longer-dated options, rebalanced less frequently. We discuss possible explanations for these performance differences.
We find that going long the market during periods of heightened short-term anxiety, inferred from the presence of relatively high S&P 500 1-month at-the-money implied volatility, has, on average, been a winning strategy. To a slightly lesser extent, having relatively less exposure to the market during periods of complacency – or relatively low implied market implied volatility – was also beneficial.
We create an “enhanced” overwriting strategy – whereby investors systematically overwrite the S&P 500 or Nasdaq 100 with disproportionately fewer (more) calls against the indices when risk expectations are relatively high (low).
Our enhanced overwriting portfolios handily outperformed the base case overwrite portfolios and the respective underlying indices, on an absolute and risk-adjusted basis. For example, the average annual return for the S&P 500 enhanced overwriting portfolio from 1997 – 2005 was 7.9%, versus 6.6% for the base case overwrite portfolio and 5.5% for the S&P 500 Index.
Overwriting with fewer calls when implied volatility is rich, and more calls when implied volatility is cheap, could improve the absolute and risk-adjusted performance of index-oriented overwriting portfolios.
This goes against the conventional tendency for investors to sell calls against their positions when implied volatility is high.
*Renicker, Ryan and Devapriya Mallick., “Enhanced Call Overwriting.”, Lehman,Brothers Global Equity Research Nov 17, 2005.
The Lehman Brothers Volatility Screening ToolRYAN RENICKER
Actionable trade ideas for stock market investors and traders seeking alpha by overlaying their portfolios with options, other derivatives, ETFs, and disciplined and applied Game Theory for hedge fund managers and other active fund managers worldwide. Ryan Renicker, CFA
ImperialMathFinance: Finance and Stochastics Seminarelviszhang
This paper presents a general stochastic volatility model that nests both Heston and Sabr models. It develops a correlation control variate Monte Carlo valuation method that provides significant improvements in valuation efficiency over existing simulation methods, especially for pricing barrier options. By providing an effective simulation method for this general model, the paper enables the potential to calibrate the model to both vanilla and path-dependent instruments to better fit market prices.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Volatility
This document discusses efficient numerical PDE methods to solve calibration and pricing problems in local stochastic volatility models. It begins with an overview of volatility modelling, including local stochastic volatility models that combine local volatility, jumps, and stochastic volatility. It then discusses calibrating both parametric and non-parametric local volatility models using PDE methods. The document provides examples of modelling stochastic volatility factors using implied volatility data and estimating jump parameters from historical returns. It also discusses calibrating local volatility models to vanilla option prices while including jumps and stochastic volatility.
Volatility trading strategies seek to profit from changes in a asset's volatility. Volatility measures how much the price of an asset fluctuates over time. There are several types of volatility strategies including volatility dispersion trading which buys options on index components and sells options on the overall index, volatility spreads which use option combinations to profit from different implied volatilities, and gamma trading which aims to benefit from unexpected events causing large price moves. Volatility is important for options as their pricing depends on assumptions about future volatility.
An Approximate Distribution of Delta-Hedging Errors in a Jump-Diffusion Model...Volatility
1) Analyse the distribution of the profit&loss (P&L) of delta-hedging strategy for vanilla options in Black-Scholes-Merton (BSM) model and an extension of the Merton jump-diffusion (JDM) model assuming discrete trading and transaction costs
2) Examine the connection between the realized variance and the realized P&L
3) Find approximate solutions for the P&L volatility and the expected total transaction costs
4) Apply the mean-variance analysis to find the trade-off between the costs and P&L variance given hedger's risk tolerance
5) Consider hedging strategies to minimize the jump risk
Achieving Consistent Modeling Of VIX and Equities DerivativesVolatility
1) Discuss model complexity and calibration
2) Emphasize intuitive and robust calibration of sophisticated volatility models avoiding non-linear calibrations
3) Present local stochastic volatility models with jumps to achieve joint calibration to VIX options and (short-term) S&P500 options
4) Present two factor stochastic volatility model to fit both the short-term and long-term S&P500 option skews
This document provides an overview of key concepts related to the Capital Asset Pricing Model (CAPM). It discusses how to calculate expected returns, variances, and covariances of individual securities and portfolios. It explains how diversification reduces unsystematic risk but not systematic market risk. The efficient frontier and capital market line are introduced. The CAPM relationship between risk and expected return is presented, relating an asset's beta to its expected return.
Express measurement of market volatility using ergodicity conceptJack Sarkissian
Don't we want to base our trading decisions on current market conditions? Then why should we rely on time averages only because they are simple to comprehend? We can get current market volatility a lot faster by applying the ERGODICITY concept to financial markets. Ensemble averaging allows to measure market volatility quickly, based on only two points in time and is as relevant to volatility measurement as the traditional measures.
Asset Pricing Implications of Volatility Term Structure RiskSteve Xie, Ph.D.
This document summarizes a study on the pricing of volatility term structure risk. The author finds that stocks with high sensitivities to changes in the VIX slope exhibit high average returns, with an estimated annual premium of 2.5% for bearing VIX slope risk. This risk premium cannot be explained by common factors. The author interprets the findings using a rare disaster model where economic disasters can have short or long durations, leading to different term structure shapes for the VIX. Extending the rare disaster model to include disaster duration helps explain why the VIX slope predicts future returns.
1. The document discusses risk and return, defining concepts like expected return, risk, standard deviation, beta, and models like the Capital Asset Pricing Model (CAPM).
2. It provides examples of how to calculate expected return, standard deviation, and beta for both discrete and continuous probability distributions.
3. The CAPM model relates a security's expected return to market risk (beta) and the risk-free rate, stating that expected return equals the risk-free rate plus a risk premium based on beta.
The Sharpe model provides a simpler approach to portfolio optimization compared to the Markowitz model. It assumes the return of individual securities is linearly related to a single market index. This allows estimation of systematic and unsystematic risk for individual stocks based on their beta coefficient. An optimal portfolio is constructed by selecting stocks with the highest excess returns over the risk-free rate relative to their beta, up to the cutoff point where this ratio begins declining. The percentage invested in each stock is based on its beta and unsystematic risk. This results in a portfolio with the highest expected return for a given level of risk.
This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers
http://awesomefinance.weebly.com/
The document discusses various methods of measuring and evaluating risk:
1. Standard deviation is a statistical tool used to measure risk by quantifying the variation of returns around the mean. It allows comparison of risk across stocks.
2. Covariance and correlation measure how returns of two securities move together, allowing analysis of portfolio risk. Diversification across negatively correlated securities can reduce overall risk.
3. Beta indicates how sensitive a stock's returns are to market returns. It is estimated using regression analysis and implies the level of systematic risk. Stocks with beta greater than 1 are riskier than the market.
Ch_05 - Risk and Return Valuation Theory.pptkemboies
This chapter discusses portfolio theory and asset pricing models. It introduces concepts such as portfolio risk and return, systematic and unsystematic risk, the efficient frontier, and the capital asset pricing model (CAPM). The chapter objectives are to discuss portfolio risk and return, examine the logic of portfolio theory, show how CAPM is used to value securities, and explain the arbitrage pricing theory (APT). Key models covered include the minimum variance portfolio, capital market line, security market line, and the arbitrage pricing theory as an alternative to CAPM.
Ch_05 - Risk and Return Valuation Theory.pptkemboies
This chapter discusses portfolio theory and asset pricing models. It introduces concepts such as portfolio risk and return, systematic and unsystematic risk, the efficient frontier, and the capital asset pricing model (CAPM). CAPM holds that the expected return of an asset is determined by its sensitivity to non-diversifiable market risk (beta) and the expected market return. The chapter also covers the arbitrage pricing theory, which attributes an asset's return to multiple systematic factors rather than just one market factor.
This document proposes improvements to existing customer lifetime value models. It discusses deriving current models A and B, which discount average revenues over a subscriber's expected duration. The improvements consider estimating future cash flows and growth rates through regression analysis, accounting for other revenue streams, and incorporating the value of a subscriber's social network. The proposed model uses discounted cash flow analysis and least squares regression to forecast revenues and growth rates for each subscriber, considering revenues from mobile, TV, broadband and the revenues of subscribers within their social network. It requires subscriber revenue and call data to implement the analysis.
Innovations in technology has revolutionized financial services to an extent that large financial institutions like Goldman Sachs are claiming to be technology companies! It is no secret that technological innovations like Data science and AI are changing fundamentally how financial products are created, tested and delivered. While it is exciting to learn about technologies themselves, there is very little guidance available to companies and financial professionals should retool and gear themselves towards the upcoming revolution.
In this master class, we will discuss key innovations in Data Science and AI and connect applications of these novel fields in forecasting and optimization. Through case studies and examples, we will demonstrate why now is the time you should invest to learn about the topics that will reshape the financial services industry of the future!
Topics in Econometrics
The document discusses various methods of measuring risk and volatility in investments. It defines key terms like return, risk, standard deviation and volatility. It then explains different models used to measure volatility like EWMA, ARCH, GARCH and VaR. For EWMA, it provides the formula and explains how it is used to estimate volatility. For ARCH and GARCH models, it describes the concepts and formulas for ARCH(1), GARCH(1,1) and how they model conditional heteroskedasticity. Finally, it explains the variance-covariance and Monte Carlo methods to calculate Value at Risk (VaR).
Robust Calibration For SVI Model Arbitrage FreeTahar FERHATI
This document outlines a study on interpolation and extrapolation methods for implied volatility slices and surfaces under arbitrage-free conditions. It introduces the Stochastic Volatility Inspired (SVI) model, which provides a parametric fit for implied volatility data. The SVI model is chosen for this study due to its closed-form formula and ability to fit equity market data. The document also discusses the convergence of the Heston stochastic volatility model to the SVI model at large times.
This document discusses risk and return concepts in finance. It defines types of risk like stand-alone risk and portfolio risk. It shows how to calculate the expected return and standard deviation of individual investments and portfolios. Lower risk can be achieved through diversification since unique investment risks offset each other in a portfolio. The Capital Asset Pricing Model suggests investors should only be compensated for systematic market risk rather than risks that can be diversified away. Beta is introduced as a measure of a security's non-diversifiable market risk relative to the overall market.
This project explores using volatility as an asset class to improve portfolio optimization.
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This document provides an introduction to descriptive statistics. It discusses organizing and presenting both qualitative and quantitative data. For qualitative data, it describes frequency distribution tables, relative frequencies, percentages, and graphs like bar charts and pie charts. For quantitative data, it covers stem-and-leaf displays, frequency distributions, class widths and midpoints, relative frequencies and percentages. It also discusses histograms for presenting grouped quantitative data. Examples are provided to illustrate these concepts and techniques.
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Realized and implied index skews, jumps, and the failure of the minimum-variance hedging
1. Realized and implied index skews, jumps, and
the failure of the minimum-variance hedging
Artur Sepp
Global Risk Analytics
Bank of America Merrill Lynch, London
artur.sepp@baml.com
Global Derivatives Trading & Risk Management 2014
Amsterdam
May 13-15, 2014
1
2. Plan
1) Empirical evidence for the log-normality of implied and realized volatil-
ities of stock indices
2) Apply the beta stochastic volatility (SV) model for quantifying implied
and realized index skews
3) Origin of the premium for risk-neutral skews and its impacts on profit-
and-loss (P&L) of delta-hedging strategies
4) Optimal delta-hedging strategies to improve Sharpe ratios
5) Argue why log-normal beta SV model is better than its alternatives
2
3. References
Technical details can be found in references
Beta stochastic volatility model:
Karasinski, P., Sepp, A., (2012), “Beta stochastic volatility model,” Risk,
October, 67-73
http://ssrn.com/abstract=2150614
Sepp, A. (2013), “Consistently Modeling Joint Dynamics of Volatility and
Underlying To Enable Effective Hedging”, Global Derivatives conference
in Amsterdam 2013
http://math.ut.ee/~spartak/papers/PresentationGlobalDerivatives2013.pdf
Implied and realized skews, jumps, delta-hedging P&L:
Sepp, A., (2014), “Empirical Calibration and Minimum-Variance Delta
Under Log-Normal Stochastic Volatility Dynamics”
http://ssrn.com/abstract=2387845
Sepp, A., (2014), “Log-Normal Stochastic Volatility Model: Pricing of
Vanilla Options and Econometric Estimation”
http://ssrn.com/abstract=2522425
Optimal delta-hedging strategies:
Sepp, A., (2013), “When You Hedge Discretely: Optimization of Sharpe
Ratio for Delta-Hedging Strategy under Discrete Hedging and Transaction
Costs,” Journal of Investment Strategies 3(1), 19-59
http://ssrn.com/abstract=1865998 3
4. How to build a dynamic model for volatility?
Suppose we know nothing about stochastic volatility
We want to learn only by looking at empirical data
How do we start?
4
5. Empirical frequency of implied vol is log-normal
First, check whether stationary distribution of volatility is:
A) Normal or B) Log-normal
Compute the empirical frequency of one-month implied at-the-money
(ATM) volatility proxied by the VIX index for last 20 years
Daily observations normalized to have zero mean and unit variance
Left figure: empirical frequency of the VIX - it is definitely not normal
Right figure: the frequency of the logarithm of the VIX - it does look
like the normal density (especially for the right tail)!
3%
4%
5%
6%
7%
Frequency
Empirical frequency of
normalized VIX
Empirical
Standard Normal
0%
1%
2%
-4 -3 -2 -1 0 1 2 3 4
Frequency
VIX
3%
4%
5%
6%
7%
Frequency
Empirical frequency of
normalized logarithm of the VIX
Empirical
Standard Normal
0%
1%
2%
3%
-4 -3 -2 -1 0 1 2 3 4
Frequency
Log-VIX
5
6. Empirical frequency of realized vol is log-normal
Compute one-month realized volatility of daily returns on the S&P 500
index for each month over non-overlapping periods for last 60 years from
1954
Below is the empirical frequency of normalized historical volatility
Left figure: frequency of realized vol - it is definitely not normal
Right figure: frequency of the logarithm of realized vol - again it does
look like the normal density (especially for the right tail)
4%
6%
8%
10%
Frequency
Frequency of Historic 1m
Volatility of S&P500 returns
Empirical
Standard Normal
0%
2%
4%
-4 -3 -2 -1 0 1 2 3 4
Frequency
Vol
4%
6%
8%
10%
Frequency
Frequency of Logarithm of
Historic 1m Volatility of S&P500
Empirical
Standard Normal
0%
2%
4%
-4 -3 -2 -1 0 1 2 3 4
FrequencyLog-Vol
6
7. Dynamic model for volatility evolution should
not be based on price-volatility correlation
Now we look for a dynamic factor model for volatility (next slide)
We cannot apply model based on correlation between S&P500 returns
and changes in volatility because using correlation we can only predict
the direction of change, not the magnitude of change
For risk management of options, we need a factor model for volatility
dynamics
7
8. Factor model for volatility uses regression model for
changes in vol V (tn) predicted by returns in price S(tn)
V (tn) − V (tn−1) = β
S(tn) − S(tn−1)
S(tn−1)
+ V (tn−1) n (1)
iid normal residuals n are scaled by vol V (tn−1) due to log-normality
Volatility beta β explains about 70% of variations in volatility!
Left figure: scatter plot of daily changes in the VIX vs returns on S&P
500 for past 14 years and estimated regression model
Right: time series of empirical residuals n of regression model (1)
Residual volatility does not exhibit any systemic patterns
Regression model is stable across different estimation periods
y = -1.08x
R² = 67%
0%
5%
10%
15%
20%
-10% -5% 0% 5% 10%
ChangeinVIX
Change in VIX vs Return on S&P500
-20%
-15%
-10%
-5%-10% -5% 0% 5% 10%
Return % on S&P 500
-10%
0%
10%
20%
30% Time Series of Residual Volatility
-30%
-20%
Dec-99
Dec-00
Jan-02
Jan-03
Jan-04
Jan-05
Jan-06
Jan-07
Jan-08
Jan-09
Jan-10
Jan-11
Jan-12
Jan-13
Volatility beta β: expected change in ATM vol predicted by price return
For return of −1%: expected change in vol = −1.08 × (−1%) = 1.08%8
9. More evidence on log-normal dynamics of vol: indepen-
dence of regression parameters on level of ATM vol
Estimate empirically the elasticity α of volatility by:
1) computing volatility beta and residual vol-of-vol for each month using
daily returns within this month
2) test if the logarithm of these variables depends on the log of the VIX
in that month using regression model
Left figure: test ˆβ(V ) = βV α by regression model: ln ˆβ(V ) = α ln V + c
Right: test ˆε(V ) = εV 1+α by regression model: ln |ˆε(V )| = (1+α) ln V +c
The estimated value of elasticity α is small and statistically insignificant
Indeed the realized volatility is close to log-normal
y = 0.15x + 0.14
R² = 2%
-0.5
0.0
0.5
1.0
1.5
ln(|VIXbeta|)
ln(VIX beta) vs ln(Average VIX)
-1.5
-1.0
-0.5
-2.5 -2.0 -1.5 -1.0 -0.5 0.0
ln(|VIXbeta|)
ln(Average VIX)
y = 0.14x - 0.45
R² = 4%
-0.5
0.0
0.5
ln(|VIXresidualvol)
ln(VIX residualvol) vs ln(AverageVIX)
-1.5
-1.0
-2.5 -2.0 -1.5 -1.0 -0.5 0.0
ln(|VIXresidualvol)
ln(Average VIX)
9
10. Empirical estimation of volatility elasticity α:
volatility dynamics is log-normal
(maximum likelihood estimation - see my paper on log-normal volatility)
Figure: 95% confidence bounds for estimated value of elasticity α using
realized (RV) and implied (IV) volatilities for 4 major stock indices
-1.0
-0.5
0.0
0.5
1.0
Alpha
95% confidence bounds for
estimated elasticity alpha
-1.0
VIX,Reg
VSTOXX,Reg
VIX,ML
VSTOXX,ML
IV,S&P500
IV,FTSE100
IV,NIKKEI
IV,STOXX50
RV,S&P500
RV,FTSE100
RV,NIKKEI
RV,STOXX50
Estimation results confirm evidence for log-normality of volatility:
[i] In majority of cases (7 out of 12), bounds for ˆα contain zero
[ii] One outlier ˆα = −0.4 (realized volatility of Nikkei index)
[iii] Remaining are symmetric: two with ˆα ≈ 0.2 and two with ˆα ≈ −0.2
To conclude - alternative SV models are safely rejected:
1) Heston and Stein-Stein SV models with α = −1
2) 3/2 SV model with α = 1
Also, excellent econometric study by Christoffersen-Jacobs-Mimouni (2010),
Review of Financial Studies: log-normal SV outperforms its alternatives10
11. Beta stochastic volatility model (Karasinski-Sepp 2012):
is obtained by summarizing our empirical findings for
dynamics of index price S(t) and volatility V (t):
dS(t) = V (t)S(t)dW(0)(t)
dV (t) = β
dS(t)
S(t)
+ εV (t)dW(1)(t) + κ(θ − V (t))dt
(2)
V (t) is either returns vol or short-term ATM implied vol
W(0)(t) and W(1)(t) are independent Brownian motions
β is volatility beta - sensitivity of volatility to changes in price
ε is residual vol-of-vol - standard deviation of residual changes in vol
Mean-reversion rate κ and mean θ are added for stationarity of volatility
A closer inspection shows that these dynamics are similar to other log-
normal based SV models widely used in industry:
A) in interest rates - SABR model
B) in equities - a version of log-normal based aka exp-OU SV models
We arrived to beta SV model (2) only by looking at empirical data
for realized&implied vols and using factor model for vol dynamics
11
12. Implied interpretation of volatility beta and residual vol-
of-vol from Black-Scholes-Merton (BSM) volatilities, σBSM(z) as func-
tions of log-strike z = ln(K/S), inferred form option prices
Compute vol skew SKEW and convexity CONV for small maturities:
SKEW = [σBSM(5%) − σBSM(−5%)] / (2 × 5%)
CONV = [σBSM(5%) + σBSM(−5%) − 2σBSM(0)] / 5%2
Volatility beta β[I] implied by skew:
β[I]
= 2 × SKEW
Residual vol-of-vol ε[I] implied by convexity:
ε[I]
= 3 × σBSM(0) × CONV + 2 × (SKEW)2
As model parameters, volatility beta (left figure) and idiosyncratic vol-
of-vol (right figure) have orthogonal impact on BSM implied vols
15%
25%
35%
BSMimpliedvols
Impact of volatility beta on BSM vol
Base vols with beta = -1
Down vols with beta = -0.5
Up vols with beta = -1.5
5%
15%
0.7 0.78 0.86 0.94 1.02 1.1
BSMimpliedvols
Strike
15%
25%
35%
BSMimpliedvols
Impact of residual vol-vol on BSM vol
Base vols with ResidVol=1.0
Down vols with ResidVol=0.5
Up vols with ResidVol=1.5
5%
15%
0.70 0.78 0.86 0.94 1.02 1.10
BSMimpliedvols
Strike
12
13. Topic II: Implied and realized skew using beta SV model
Use time series from April 2007 to December 2013 for one-month ATM
vols and the S&P500 index with estimation window of one month
Figure 1): Implied and realized one
month volatilities
ATM volatility tends to trade at
a small premium to realized
Figure 2): One-month average of
implied and realized volatility beta
Implied volatility beta consis-
tently over-estimates realized one
Figure 3): Average of implied and
realized residual vol-of-vol
Implied residual vol-of-vol signifi-
cantly over-estimates realized
Absolute (Abs) and relative (Rel)
spreads between implieds&realizeds
Spreads Vol Beta VolVol
Abs, Mean 0.51% -0.27 0.78
Abs, Stdev 6.2% 0.21 0.16
Rel, Mean 7% 21% 57%
Rel, Stdev 24% 17% 11%
20%
30%
40%
50%
60%
70%
80% 1m ATM Implied Volatility
1m Realized Volatility
0%
10%
20%
Feb-07
Sep-07
Apr-08
Nov-08
Jun-09
Jan-10
Aug-10
Mar-11
Oct-11
May-12
Dec-12
Jul-13
-0.5
0.0
Feb-07
Sep-07
Apr-08
Nov-08
Jun-09
Jan-10
Aug-10
Mar-11
Oct-11
May-12
Dec-12
Jul-13
Implied Volatility Beta
Realized Volatility Beta
-2.0
-1.5
-1.0
0.5
0.8
1.1
1.4
1.7 Implied Residual Vol-of-Vol
Realized Residual Vol-of-Vol
0.2
0.5
Feb-07
Sep-07
Apr-08
Nov-08
Jun-09
Jan-10
Aug-10
Mar-11
Oct-11
May-12
Dec-12
Jul-13
13
14. Explanation of the skew premium in a quantitative way
In a very interesting study, Bakshi-Kapadia-Madan (2003), Review of
Financial Studies, find relationship between risk-neutral and physical skew
using investor’s risk-aversion
Fat tails (not necessarily skewed) of returns distribution under phys-
ical measure P along with risk-aversion lead to increased negative
skeweness under the risk neutral-measure Q
Quantitatively:
SKEWENESSQ = SKEWENESSP − γ × KURTOSISP × VOLATILITYP
SKEWENESSQ is risk-neutral skeweness of price returns
SKEWENESSP is physical skeweness of price returns
KURTOSISP is kurtosis as measure of fat tails of physical distribution
VOLATILITYP is volatility of returns under physical distribution
γ > 0 is risk-aversion parameter of investors
To conclude: the risk-neutral premium arises because risk-averse
investors assign higher value to insurance puts
Important: Volatility skew is proportional to skeweness of returns
14
15. Apply Merton Jump-Diffusion (JD) with normal jumps
Figure 1: Use last 14 years of daily returns on S&P 500 index to estimate
skeweness and kurtosis of returns - see column ”Empirical P”
Table 1: Use γ = 22.0 (estimated from time series of implied vols by
inverting BKM formula) and apply BKM to obtain SKEWENESSQ = −2
Figure&Table 2: Fit Merton JD to first four moments of physical and
risk-neutral distribution (jump frequency is set to one jump per month)
From calibration: JumpMean is 0 under empirical P and -5% under Q
Empirical P Q
Stdev 21% 21%
Skeweness 0 -2
Kurtosis 8 8
Merton JD params P Q
Jump Mean 0% -5%
Jump Volatility 4% 0%
Diffusion vol 17% 13%
Jump Frequency 12 12
Figure 3: Value one month options - implied volatility from Merton JD
under Q is skewed, while implied volatility under P is symmetric
4%
6%
8%
10%
Frequency
Frequency of S&P500 daily returns
Empirical
Frequency
Normal Density
0%
2%
4%
-9% -7% -5% -4% -2% 0% 2% 4% 6% 7%
Frequency
Daily return
4%
6%
8%
10%
Frequency
Frequency of S&P500 daily returns
Empirical
Frequency
Physical Merton
under P
Risk-Neutral
0%
2%
4%
-9% -7% -5% -4% -2% 0% 2% 4% 6% 7%
Frequency
Daily return
Risk-Neutral
Merton under Q
25%
30%
ImpliedVol
Implied volatility skew for one
month options on S&P500
Physical Merton under P
Risk-Neutral Merton under Q
15%
20%
0.75 0.85 0.95 1.05 1.15 1.25
ImpliedVol
Strike
15
16. To summarize our developments so far:
1) Log-normal beta SV model is consistent with empirical distribu-
tion for realized and implied vols
2) Beta SV model is applied to quantify realized and implied skews
and the spread between them, which turns out to be significant
Any option position is mark-to-market so no point of arguing about
market prices
However, hedging strategy is discretionary and can be the ”edge”
By computing the delta-hedge: should we use implied or realized
skews?
This question is analyzed in the third topic of my talk:
Part I - Quantitative analysis of impact of realized and implied skews on
delta-hedging P&L
Part II - Monte-Carlo simulations for empirical analysis
16
17. Statistically significant spread between realized and im-
plied skews β[R] − β[I] leads to dependence on realized
price returns and invalidates the minimum-variance hedge
Minimum-variance delta ∆ is applied to hedge against changes in price
and price-induced changes in volatility
Given hedging portfolio Π for option U on S
Π(t, S, V ) = U(t, S, V ) − ∆ × S
∆ is computed by minimizing variance of Π using SV beta dynamics (2)
under risk-neutral measure Q (classic approach) with implied vol betaβ[I]
∆ = US + β[I] × UV /S
where US and UV model delta and vega
To see dependence on return δS due to spread between implied vol beta
β[I] and realized β[R]: given δS apply beta SV for change in vol δV under
physical measure P:
δV = β[R] × δS + ε[R] × V
√
δt
By Taylor expansion of realized P&L:
δΠ(t, S, V ) = β[R] − β[I] × UV × δS + ε[R] × UV × V
√
δt + O(dt)
ε(R) is random non-hedgable part from residual vol-of-vol
O(dt) part includes quadratic terms (δS)2, (δV )2, (δS)(δV )
17
18. Volatility skew-beta is important for computing correct
option delta
Figure 1) Apply regression model
(1) for time series of ATM vols for
maturities T = {1m, 3m, 6m, 12m, 24m}
(m=month) to estimate regression
volatility beta βREGRES(T) using
S&P500 returns:
δσATM(T) = βREGRES(T) × δS
Volatility beta for SV dynamics is in-
stantaneous beta for very small T
Regression vol beta decays in log-T
due to mean-reversion: long-dated
ATM vols are less sensitive in abso-
lute values to price-returns
Figure 2) Implied vol skew for ma-
turity T has similar decay in log-T
Figure 3) Volatility skew-beta is
regression beta divided by skew
Skew-Beta(T) ∝ βREGRES(T)/SKEW(T)
It is nearly maturity-homogeneous
y = 0.19*ln(x) - 0.37
R² = 99%
-0.6
-0.4
-0.2
0.0 Regression Volatility Beta(T)
-1.0
-0.8
-0.6
0.08 0.25 0.50 1.00 2
Maturity T
Regression Vol Beta(T)
Decay of Vol Beta in ln(T)
y = 0.16*ln(x) - 0.29
R² = 99%
-0.4
-0.2
0.0 Implied Volatility Skew (T)
-0.8
-0.6
1m 3m 6m 1y 2y
Maturity, T
Vol Skew (T)
Decay of Skew in ln(T)
y = 0.06*ln(x) + 1.41
R² = 80%1.0
1.5
2.0 Volatility Skew-Beta(T)
Vol Skew-Beta (T)
0.0
0.5
1m 3m 6m 1y 2y
Maturity T
Vol Skew-Beta (T)
Decay of Vol Skew-Beta in ln(T)
18
19. Technical supplement to compute model implied skew-
beta (omitted during the talk)
Using backward pricers and PDE:
1) Compute the term structure of ATM volatility σATM(S0; T) and skew
SKEW(S0; T), with strike width α%, implied by model parameters
2) Bump the spot price down by α%, S1 = (1 − α%)S0, and apply corre-
sponding bumping rule for model state variables
For the beta SV:
V1 → V0 + βα , θ → θ +
β
2κ
α (3)
3) Compute new term structure of ATM vols σATM(S1; T)
4) Compute model implied skew-beta
Skew-Beta(T) = −
σATM(S1; T) − σATM(S0; T)
α × SKEW(S0; T)
(4)
Using Monte-Carlo pricers:
1) Specify number of paths and simulate set of independent Brownians
2) Compute paths starting from {S0, V0}
2A) Evaluate term structure of ATM volatility σATM(K = S0; T) and
skew using σ(K = S1; T), both using Brownians in 1)
3) Compute paths starting from {S1, V1} with S1 = (1 − α%)S0 and V (1)
bumped as in Eq (3), using Brownians in 1)
4) Evaluate ATM vols σATM(K = S1; T) and skew-beta by Eq (4) 19
20. Volatility and Skew contribution to P&L - important for
volatility positions with daily mark-to-market!
Mark BSM implied vol σBSM(K) in %-strike K relative to price S(0):
σBSM(K; S) = σATM(S) + SKEW × Z(K; S)
Z(K; S) is log-moneyness relative to current price S:
Z(K; S) = ln (K × S(0)/S)
SKEW < 0 is inferred from spread between call and put implied vols
In practice, this form is augmented with extras for convexity and tails
Any SV model implies quadratic form for implied vols near ATM strikes
(Lewis 2000, Bergomi-Guyon 2012) so my approach for vol P&L is generic
Volatility P&L arises from change in spot price S → S {1 + δS}:
δσBSM(K; S) ≡ σBSM (K; S {1 + δS}) − σBSM(K; S)
= δσATM(S) + SKEW × δZ(K; S)
First contributor to P&L: change in ATM vol δσATM(S):
δσATM(S) = σATM (S {1 + δS}) − σATM(S)
Second contributor to P&L: change in log-moneyness relative to skew:
δZ(K; S) = − ln(1 + δS) ≈ −δS
20
21. Example of volatility and skew P&L with regression beta
(omitted during the talk)
σATM(S(0)) = 15%, δS = −1.0%, SKEW = −0.5, βREGRESS = −1.0
It is very important how we keep log-moneyness Z(K; S):
1) For strikes re-based to new ATM level (forward-based strikes):
S → S{1 + δS} and log-moneyness does not change δZ(K; S) = 0
P&L arises from change in ATM vol predicted by price return com-
puted using βREGRESS:
δσBSM(K) = βREGRESS × δS = −1.0 × −1% = 1%
2) For strikes fixed at old ATM level (vanilla strikes with fixed S(0))
Thus log-moneyness changes by δZ(K; S) ≈ −δS = 1%
P&L is change in ATM vol adjusted for change in money-ness:
δσBSM(K) = βREGRESS ×δS+SKEW×δS = 1%+(−0.5)×(1%) = 0.50%
0.5%
0.8%
1.0%
Change in vols, strikes fixed to ATM 0
Change in vols, strikes re-based to ATM 1
0.0%
0.3%
0.5%
90% 95% 100% 105%
Strike K%
18%
21%
BSMvol(K)
BSM vol 0, strikes fixed to ATM 0
BSM vol 1, strikes fixed to ATM 0
BSM vol 1, strikes re-based to ATM 1
12%
15%
90% 95% 100% 105%
BSMvol(K)
Strike K%
21
22. Changes in skew are not correlated to changes in price
and ATM vols - important for correct predict of vol and skew P&L
Empirical observations yet again confirm log-normality dynamics!
(Using S&P500 data from January 2007 to December 2013)
Figure 1: weekly changes in 100% − 95% skew vs price returns for
maturity of one month (left) and one year (right)
Regression slope = 0.13 (1m) & 0.03 (1y); R2 = 0% (1m) & 1% (1y)
-0.1
0
0.1
0.2
0.3
ChangeinSkew
Change in 1m skew vs Price Return
y = 0.13x - 0.00
R² = 0%
-0.3
-0.2
-0.1
-15% -5% 5% 15%
ChangeinSkew
Price return
y = 0.03x - 0.00
-0.02
0
0.02
0.04
ChangeinSkew
Change in 1y skew vs Price Return
y = 0.03x - 0.00
R² = 1%
-0.06
-0.04
-0.02
-15% -5% 5% 15%
ChangeinSkew
Price return
Figure 2: weekly changes in 100% − 95% skew vs changes in ATM
vols for maturity of one month (left) and one year (right)
Regression slope = −0.15 (1m) & −0.06 (1y); R2 = 0% (1m) & 2% (1y)
-0.1
0
0.1
0.2
0.3
ChangeinSkew
Change in 1m skew vs 1m ATM vol
y = -0.15x - 0.00
R² = 0%-0.3
-0.2
-0.1
-15% -5% 5% 15%
ChangeinSkew
Change in ATM vol
-0.02
0
0.02
0.04
ChangeinSkew
Change in 1y skew vs 1y ATM vol
y = -0.09x - 0.00
R² = 2%
-0.06
-0.04
-0.02
-15% -5% 5% 15%
ChangeinSkew
Change in ATM vol
22
23. Volatility skew-beta combines the skew and volatility
P&L together
Given price return δS:
S → S {1 + δS}
Volatility P&L is computed by:
1) For strikes re-based to new ATM level
Log-moneyness does not change, δZ(K; S) = 0
P&L follows change in ATM vol predicted by regression beta and vol
skew-beta:
δσBSM(K) ≡ δσATM(S) = βREGRESS × δS
= SKEWBETA × SKEW × δS
2) For strikes fixed at old ATM level
Log-moneyness changes by δZ(K; S) ≈ −δS
P&L is change in ATM vol adjusted for skew P&L:
δσBSM(K) ≡ δσATM(S) − SKEW × δS
= [SKEWBETA − 1] × SKEW × δS
Positive change in ATM vol from negative return is reduced by
skew
23
24. Volatility skew-beta under minimum-variance approach
is applied to compute min-var delta ∆ for hedging against
changes in price and price-induced changes in implied vol
A) We adjust option delta for change in implied vol at fixed strikes
B) The adjustment is proportional to option vega at this strike:
∆(K, T) = ∆BSM(K, T) + [SKEWBETA(T) − 1] × SKEW(T) × VBSM(K, T)/
∆BSM(K, T) is BSM delta for strike K and maturity T
VBSM(K, T) is BSM vega, both evaluated at volatility skew
I classify volatility regimes using vol skew-beta for delta-adjustments:
∆(K, T) =
∆BSM(K, T) + SKEW(T) × VBSM(K, T)/S, Sticky local
∆BSM(K, T), Sticky strike
∆BSM(K, T) − SKEW(T) × VBSM(K, T)/S, Sticky delta
∆BSM(K, T) + 1
2SKEW(T) × VBSM(K, T)/S, Empirical S&P50
”Shadow” delta is obtained using ratio O (may be different from 1/2):
∆(K, T) = ∆BSM(K, T) + O × SKEW(T) × VBSM(K, T)/S
which is traders’ ad-hoc adjustment of option delta
24
25. Does ”shadow” delta create a vision or an illusion?
We need a quantitative model to describe different volatility
regimes and produce correct option delta !
25
26. Volatility skew-beta and vol regimes (also see Bergomi 2009):
SkewBeta =
2, Sticky local regime: minimum-variance delta in SV and LV
1, Sticky strike regime: BSM delta evaluated at implied skew
0, Sticky delta regime: model delta in space-homogeneous SV
Empirical estimates for skew-beta and its lower and upper bounds are
found by regression model (see my paper)
In beta SV model, with empirical estimate of vol beta and adding jumps/risk-
aversion to match skew premium, we fit empirical vol skew-beta:
1) S&P 500: empirical skew-beta of about 1.5
2) STOXX 50: strong skew-beta close to 2
3) NIKKEI: weak skew-beta is about 0.5
As result: beta SV model with jumps can produce the correct delta!
1.00
1.50
2.00
2.50
Vol Skew-Beta for S&P500
SVJ Skew-Beta with empirical beta
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
1.00
1.50
2.00
2.50
Vol Skew-Beta for STOXX 50
SVJ Skew-Beta with empirical beta
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
1.00
1.50
2.00
2.50
Vol Skew-Beta for NIKKEI
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
26
27. Second part of topic III: Monte Carlo analysis of delta-
hedging P&L
Now let’s have some fun and do some number crunching!
We are going to simulate the market dynamics and compare hedging
performance under different specifications of delta
In next few slides I briefly discuss the methodology
Details are provided for the interested for self-studying
Details are important to understand how to improve the performance of
delta-hedging strategies
Application to actual market data produces equivalent conclusions
In my talk, I will only discuss final results and conclusions
27
28. Apply beta SV for dynamics under physical measure P:
1) Index price S(t),
2) Volatility of returns Vret(t):
3) Short-term implied volatility Vimp(t):
dS(t) = Vret(t)S(t)dW(0)(t)
dVret(t) = κ[P] θ[P] − Vret(t) dt + β[P]Vret(t)dW(0)(t) + ε[P]Vret(t)dW(1)(t)
dVimp(t) = κ[I] θ[I] − Vimp(t) dt + β[I]Vimp(t)dW(0)(t) + ε[I]Vret(t)dW(1)(t)
4) At-the-money (ATM) implied vol Vatm(t) is obtained by computing
model implied ATM vol for maturity T using model dynamics for Vimp(t)
Important: Model parameters are estimated from time series by
maximum likelihood methods - as a rule, parameters for returns vol
[P] and for implied vol [I] are different
Here, apply the same parameters for clarity
Physical for Returns dVret(t), [P] Vol dVimp(t), [I]
V.(0) 16% 16.75%
θ[.] 16% 16.75%
κ[.] 3.0 3.0
ε[.] 0.5 0.5
β[.] -1.0 -1.0
28
29. Volatility and skew premiums are produced using BSM implied
volatility, σBSM(K), as function of % strike K relative to S(0):
σBSM(K) = Vatm(t) + SKEW × ln (K × S(0)/S(t)) (5)
SKEW = −0.5 is vol implied skew specified exogenously by
strike % BSM vol σBSM(K) σBSM(K) − Vret(0)
99% 17.25% 1.25%
100% 16.75% 0.75%
101% 16.25% 0.25%
Market Skew -0.50
Important - option delta is computed using two models:
1) Beta SV model with market implied beta β[I] = -1.1
2) Beta SV model with empirical beta β[I] = -1.0 and jumps (risk-
aversion) to price-in excessive skew −1.1 − 1.0 = −0.1 (discussed later)
Both SV models fit to market skew exactly!
[i] Premium of implied vol to realized vol is:
16.75% − 16% = 0.75% (in line with empirical spread)
[ii] Premium of implied and empirical beta is:
β[I] − β[R] = -1.1 − ( -1.0 ) = -0.1 (empirical is about −0.2)
As we saw using Madan-Merton fits, physical dynamics don’t need to
have asymmetric jumps to produce skew premium - now, skew premium
arises from excess kurtosis produced by empirical SV model for returns29
30. Consistency with market skew does not guar-
antee fit to empirical dynamics
Both hedging models are consistent with market implied skew
However, we observe discrepancy:
SV model with market implied beta,called Minimum variance hedge
Implies vol skew-beta about 2.0 , which is inconsistent with empirical
dynamics
SV model with jumps and empirical beta, called Empirical hedge:
Implies vol skew-beta about 1.6 , which is consistent with empirical
dynamics
Important - no re-calibration along a MC path is applied:
Both hedging models are initially consistent with the market skew - as
price S(t) and vol Vimp(t) change, both models remain very close to
market skew
Log-normality assumption - independence of implied&realized skew
from volatility - comes into play
30
31. Specification for trading in delta-hedged positions:
1) Straddle - short ATM put and call
Figure 1: P&L profile with Delta= 0 is function of realized return squared
Important: P&L/delta of straddle are not sensitive to realized/implied
skew - Benefits from small realized variance of price returns
2) Risk-reversal - short put with strike 99% and long call with strike
101% of forward
Figure 2: P&L profile with Delta= −0.8 is function of realized return
Important: P&L/delta of risk-reversal are very sensitive to real-
ized/implied skew - Benefits from small realized covariance of changes
in price and ATM vol
-5.0%
-2.5%
0.0%
2.5%
5.0%
PayOff+PV-DeltaHedge with Delta=0
PayOff
-10.0%
-7.5%
-5.0%
-10%-8%-6%-4%-2% 0% 2% 4% 6% 8%10%
Straddle P&L vs Price return
-2.5%
0.0%
2.5%
5.0%
7.5%
10.0%
PayOff+PV-DeltaHedge with delta=-0.8
PayOff
-10.0%
-7.5%
-5.0%
-2.5%
-10%-8%-6%-4%-2% 0% 2% 4% 6% 8%10%
Risk-Reversal P&L vsPrice return
31
32. Specification for notionals of delta-hedged positions
Notionals are normalized by CashGamma=(1/2) × (S2)×OptionGamma
Notionals for straddle:
PutNotional(tn) = CallNotional(tn) = −
0.5
ATM CashGamma(tn)
Notionals for risk-reversal:
PutNotional(tn) = −
0.5 × (Vatm(tn))2T
2% × {Put Vega(tn)}
CallNotional(tn) = +
0.5 × (Vatm(tn))2T
2% × {Call Vega(tn)}
where 2% comes from strike width 2% = 101% − 99%
Important: for Straddle, cash-gamma is 1.0
For Risk-reversal, the vanna (vega of delta) is 1.0
32
33. Monte-Carlo analysis: P&L accrual
Daily re-balancing at times tn, n = 1, ..., N
At the end of each day, we roll into new position so straddle is at-the-
money and risk-reversal has the same strike width
Realized P&L is P&L on hedges minus P&L on options position:
P&L =
N
n=1
{∆(tn−1) S(tn) − S(tn−1)
− Π (T − dt, S(tn), Vatm(tn)) − Π T, S(tn−1), Vatm(tn−1) }
Π (T, S(tn), Vatm(tn)) is options position computed using BSM formula and
implied volatility skew (5) with Vatm(tn), T = 1/12, dt = 1/252
Transaction costs are 2bp (k = 0.0002) per delta-rebalancing:
TC = k |∆(t0)| S(t0) + k
N
n=1
|∆(tn) − ∆(tn−1)| S(tn)
where ∆(tn) is combined delta for newly rolled position
Important: P&L across different days and paths is maturity-time
and strike-space homogeneous - robust for statistical inference!
33
34. Monte-Carlo analysis - final notes
Trade notional is 100,000,000$
Realized P&L and explanatory variables are reported in thousands of $
Option maturity: one month
Daily re-hedging with total for each path: N = 21
P&L is annualized by multiplying by 12
Draw 2,000 paths and compute realized P&L and price return, variance,
volatility beta for changes in price and ATM vol, etc
Price and volatility paths are the same for straddle and risk-reversal
and different hedging strategies
A) Analyze realized delta-hedging P&L (Profit and Loss) by
[i] Realized P&L and its volatility, transaction costs
[ii] Sharpe ratios
B) P&L Explain using regression model with explanatory variables
What factors (realized variance, covariance, etc) contribute to P&L
34
35. 1. Analysis of realized P&L for straddle
Figure left - realized P&L with no accounting for transaction costs
Right - realized P&L with transaction costs
Approximately, straddle P&L is spread between implied&realized vols2:
P&L = Γ × (Vatm)2 − (Vret)2
= 100, 000 × (16.75%)2 − (16.00%)2 = 246
where Γ is cash-gamma notional in thousands $
Realized P&L little depends on the delta hedging strategy
Important is that asset drift is zero, otherwise P&L-s for different hedging
strategies have directional exposure to realized asset drift
244 243
100
200
300 Straddle P&L, zero trans costs
0
Minimum var Empirical beta
161 161
100
200
300 Straddle P&L after trans costs
161 161
0
Minimum var Empirical beta
35
36. 2. Analysis of realized P&L for risk-reversal
Figure: left - realized P&L with no accounting for transaction costs
Right - realized P&L with transaction costs
Approximately, risk-reversal P&L is spread between implied and realized
co-variance of price and vol returns:
P&L = V × −SKEW × (Vatm)2 + (Vret)2 + β[R] × (Vret)2
= 100, 211 × 0.5 × (16.75%)2 + (16.00%)2 − 0.88 × (16.00%)2 = 431
where V is vanna notional in thousands $
Again, realized P&L little depends on the delta hedging strategy when
asset drift is zero
423 423
100
200
300
400
500 Risk-Reversal P&L, zero trans costs
0
100
Minimum var Empirical beta
190 192100
200
300
400
500 Risk-Reversal P&L after trans costs
190 192
0
100
Minimum var Empirical beta
36
37. 3. Analysis of transaction costs
Transaction costs are 2bp per traded delta notional or 1$ per 5, 000$
Left figure: realized transaction costs
1) Risk-reversal has higher transaction costs due to larger delta notional
2) Minimum variance hedge and empirical hedge imply about equal trans-
action costs for straddle
3) Minimum variance hedge implies higher transaction costs for
risk-reversal because of over-hedging the put side
Right figure: volatility of transaction costs
Volatility is about uniform and very small compared to mean costs
233 231100
200
300 Realized Transaction costs
83 82
0
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for
risk-reversal
5 5
2
4
6
Volatility of Transaction costs
2 2
0
2
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for
risk-reversal
37
38. 4. Volatility of Realized P&L
Left figure: P&L volatility without accounting for transaction costs
Empirical hedge implies lower P&L volatility for:
[i] Risk-reversal (about 20%)
[ii] Straddle (about 2 − 3%)
Because Minimum Variance delta over-hedges for put side and make delta
more volatile
Right figure: volatility of realized P&L accounting for costs
1) Transaction costs increase P&L slightly by about 1 − 2%
2) Contrast with reduction of realized P&L by about 50%
328 320
100
200
300
400 P&L Volatility, zero transaction costs
122 102
0
100
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for
risk-reversal
331 323
100
200
300
400 P&L Volatility, after transaction costs
122 102
0
100
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for
risk-reversal
38
39. 5. Sharpe ratios of realized P&L-s
Left figure: Sharpe ratios for delta-hedging P&L without account-
ing for transaction costs
Right figure: Sharpe ratios for P&L accounting for costs
1) For straddle, both Minimum Variance and Empirical hedges imply
about the Sharpe ratio
2) For risk-reversal, Minimum Var hedge implies smaller Sharpe
than Empirical hedge (by about 20%) because of higher P&L volatility
and transaction costs
3.46
4.14
1.00
2.00
3.00
4.00
Sharpe ratio, zero tranaction costs
0.74 0.76
0.00
1.00
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for risk-
reversal
1.56
1.88
0.50
1.00
1.50
2.00
Sharpe ratio, after transaction cost
0.49 0.50
0.00
0.50
Min var for
straddle
Empirical
beta for
straddle
Min var for
risk-reversal
Empirical
beta for risk-
reversal
39
40. P&L Attribution to risk factors is applied to understand
what factors contribute to P&L by using regression
P&L = α + s1X1 + s2X2 + s3X3 + s4X4 + s5X5 + s6X6 (6)
α (”Alpha”) is theta related P&L - P&L we would realize if nothing would
move
X1 (”Var”) is returns variance: X1 = S(tn)
S(tn−1)
− 1
2
X2 (”VolChange”) is change in ATM vol: X2 = Vatm(tn) − Vatm(tn−1)
X3 (”Covar”) is covariance: X3 = S(tn)
S(tn−1)
− 1 Vatm(tn) − Vatm(tn−1)
X4 (”VarVol”) is variance of vol changes: X4 = Vatm(tn) − Vatm(tn−1) 2
X5 (”Return3”) is cubic return: X5 = S(tn)
S(tn−1)
− 1
3
X6 (”Return”) is realized return: X6 = S(tn)
S(tn−1)
− 1
Summation runs from n = 1 to n = N, N = 21
R2 indicates how well the realized variables explain realized P&L (not
accounting for transaction costs) - we should aim for R2 = 90%
Some explanatory variables are correlated so it is robust to test reduced
regressions
40
41. P&L explain for straddle by realized variance of returns:
Empirical hedge has stronger explanatory power
Is needed to confirm theoretical P&L explain by MC simulations
For P&L of straddle hedged at implied vol, first-order approximation:
V 2
atm −
n
S(tn)
S(tn−1)
− 1
2
First term is alpha or ”carry” - approximate alpha is
α = Γ × V 2
atm = 100, 000 × 0.16752 = 2806
Second term is short risk to realized variance - key variable for P&L
Theoretical slope should be −Γ = −100, 000
Figure: explanatory power using only realized variance is weak because
of impact of other variables and skew (for multiple variables, R2 ≈ 90%)
P&L = -48,768*Var + 1,559
R² = 30%
0
4,000
P&L
Straddle P&L by Min-Var Hedge
-8,000
-4,000
0.00 0.05 0.10 0.15
P&L
Realized Variance
P&L = -55,132*Var + 1,730
R² = 40%
0
4,000
P&L
Straddle P&L by Empirical hedge
-8,000
-4,000
0.00 0.05 0.10 0.15
P&L
Realized Variance
41
42. P&L explain for risk-reversal by realized vol beta:
Empirical hedge implies that realized vol beta is clear
driver behind P&L of risk-reversal with R2 = 50%
For P&L of risk-reversal hedged at implied vol skew, approximation:
−SKEW × V 2
atm +
n
S(tn)
S(tn−1)
− 1
2
+
n
S(tn)
S(tn−1)
− 1 (Vatm(tn) − Vatm(tn−1))
In terms of returns vol Vret and implied vol beta βR:
−SKEW × V 2
atm + V 2
ret + β[R]
× V 2
ret
First term is ”carry” or alpha
Second term is risk to realized beta between returns and vol - key variable
In our example: α = 0.5 × V × {(16.75%)2 + (16.00%)2} = 2, 682
Slope= V × (16.00%)2 = 2, 560
P&L = 2129*Beta + 2570
R² = 37%
0
1500
3000Risk-Reversal P&L
by Min-Var Hedge
-1500
0
-2.0 -1.5 -1.0 -0.5
P&L
Realized Volatility Beta
P&L = 2050*Beta + 2490
R² = 49%
0
1500
3000Risk-Reversal P&L
by Empirical Hedge
-1500
0
-2.0 -1.5 -1.0 -0.5
P&L
Realized Volatility Beta
42
43. Important: vol beta (for skew) is comparable to Black-
Scholes-Merton (BSM) implied volatility (for one strike)
1) Volatility and vol beta are meaningful and intuitive model pa-
rameters which can be inferred from both implied and historical data
Implied vol σ[I] is inferred from option market price
Realized vol σ[R] is volatility of price returns
Implied vol beta β[I] is inferred from market skew (β[I] ≈ 2 × SKEW)
Realized vol beta β[R] is change in implied ATM volatility predicted by
price returns: β[R] = dS(t)dVatm(t) /(σ[R])2
2) Both serve as directs input for computation of hedges
3) Both allow for P&L explain of vanilla options in terms of implied
and realized model parameters:
Implied/realized volatility- P&L of delta-hedged straddle:
σ[I] 2
− σ[R] 2
Implied/realized volatility beta- P&L of short delta-hedged risk-reversal
(more noisy because of contribution from σ[R]):
−β[I] ×
1
2
σ[I] 2
+ σ[R] 2
+ β[R] × σ[R] 2
43
44. Conclusion: existing practical approaches for hedging
improvement are not fully satisfactory - we need proper
model for dynamic delta-hedging!
A) Hedge all vega exposure
B) Recalibration for computing delta-risks (most common):
⊗ Project change in implied volatility using empirical backbone
(For example, by applying empirical volatility skew-beta)
⊗ Re-calibrate valuation model to bumped volatility surface
⊗ Re-valuate and compute delta by finite-differences
However runs into problems:
1) A) - vega-hedging is (very) expensive and unprofitable unless
implied skew and vol-of-vol are sold at large premiums to future realizeds
2) B) - re-calibration works poorly for path-dependent and multi-
asset products and it makes P&L explain very noisy
Recall applying regression for P&L explain of straddle and risk-reversal
3) any mix of A) and B) becomes very tedious for CVA computations
Important: the choice between local vol (LV) or stoch vol (SV) is irrel-
evant when hedging using minimum variance hedge at implied vol skew -
any combination of LV and SV produces almost the same deltas! 44
45. Beta SV model with jumps is fitted to empirical&implied
dynamics for computing correct delta (Sepp 2014):
dS(t)
S(t)
= (µ(t) − λ(eη − 1)) dt + V (t)dW(0)(t) + (eη − 1) dN(t)
dV (t) = κ(θ − V (t))dt + βV (t)dW(0)(t) + εV (t)dW(1)(t) + βη dN(t)
1) Consistent with empirical dynamics of implied ATM volatility by
specifying empirical volatility beta β
2) Has jumps, as degree of risk-aversion, to make model fit to both
empirical dynamics and risk-neutral skew premium
Only one parameter with simple calibration! - explained in a bit
Jumps/risk-aversion under risk-neutral measure Q produced by:
Poisson process N(t) with intensity λ:
negative&positive jumps in returns&vols with constant size η < 0&βη > 0
3) Easy-to-implement (with no extra parameters) extension to multi-
asset dynamics using common jumps - produces basket correlation skew
4) Beta SVJ model is robust to produce optimal hedges for path-
dependent and multi-asset trades and CVA
45
46. Third to last topic: closed-form solution for log-normal
Beta SV
Mean-reverting log-normal SV models are not analytically tractable
I derive a very accurate exp-affine approximation for moment generating
function (details in my paper)
Idea comes from information theory: apply Kullback-Leibler relative en-
tropy for unknown PDF p(x) and test PDF q(x) with moment constraints:
xkp(x)dx = xkq(x)dx, k = 1, 2, ...
Now let’s think in terms of moment function:
[i] MGF for Beta SV model with normal driver for SV (as in Stein-Stein
SV model) has exact solution, which has exp-affine form
[ii] Correction for log-normal SV has an exp-affine form
15%
20%
25%
30%
35% Implied vol for 1y S&P500
options, beta SV, NO JUMPS
Analytic for Normal SV
5%
10%
15%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Strike
Analytic for Normal SV
Closed-form for Log-normal SV
Monte-Carlo for Log-normal SV
15%
20%
25%
30%
35% Implied vol for 1y S&P500
options, beta SV, WITH JUMPS
5%
10%
15%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Strike
Analytic for Normal SV
Closed-form for Log-normal SV
Monte-Carlo for Log-normal SV
46
47. Proof that closed-form MFG for log-normal model pro-
duces theoretically consistent probability density
1) Derive solutions for excepted values, variances, and covariances of the
log-price and quadratic variance (QV) by solving PDE directly
2) Prove that moments derived using approximate MGF equal to theo-
retical moments derived in 1)
Using closed-form MFG for log-normal model, we apply standard valuation
methods for affine SV models based on Lipton-Lewis formula
Implementation of closed-form moment function (MGF), MC, and PDE
pricers produce values of vanilla options on equity and quadratic variance
that are equal within numerical accuracy of these methods
15%
20%
25%
30%
35% Implied vol for 1y S&P500
options, beta SV, NO JUMPS
Closed-form MGF
5%
10%
15%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Strike
Closed-form MGF
Monte-Carlo
PDE, numerical solver
15%
20%
25%
30%
35% Implied vol for 1y S&P500
options, beta SV, WITH JUMPS
Closed-form MGF
5%
10%
15%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Strike
Closed-form MGF
Monte-Carlo
PDE, numerical solver
47
48. Second to last topic - optimal hedging under discrete
trading and transaction costs
To adopt to practical option trading:
a) change ”kid” to ”trader”
b) change ”cantaloupes” to ”millions of delta over an infinitesimal time
period dt”
As we saw in simulation of P&L, we need quantitative framework that
incorporates discrete hedging and optimizes trade-off between:
the reward - higher P&L and lower transaction costs
the risk - higher P&L volatility
48
49. Illustration of trading in implied&realized vol with strad-
dle: unique optimal hedging frequency can be found!
Figure 1) Forecast expected upside:
the spread between implied and real-
ized vol for given maturity T
This is independent of valua-
tion&hedging model and hedging
frequency
Figure 2) Forecast P&L volatility
and transaction costs
These depend on valuation&hedging
model and hedging frequency
Part of P&L volatility is not hedge-
able due to vol-of-vol and jumps -
Not optimal to hedge too fre-
quently
Figure 3) Obtain Sharpe ratio as ra-
tio of forecast P&L after costs and
P&L volatility
1%
2%
3%
Expected/Forcasted P&L(N)
0%
1%
10 160 310 460 610 760 910
N - hedging frequency
1%
2%
3% Expected/Forcasted P&L
Volatility and Costs (N)
P&L Volatility
Transaction costs
0%
1%
10 160 310 460 610 760 910
N - hedging frequency
0.6
0.8
1.0
1.2
1.4 Expected/Forcasted
Sharpe Ratio (N)
0.0
0.2
0.4
0.6
10 160 310 460 610 760 910
N - hedging frequency
49
50. Solution for optimal Sharpe ratio with dynamics under
physical measure driven by Diffusion and SV with jumps
Sharpe(N) =
Expected P&L − TransactionCosts(N)
P&L Volatility(N)
N is hedging frequency - for details see my paper on optimal delta-hedging
Using this solution we can analyze:
Figure 1) What maturity is optimal to trade given the forecast spread
between implieds and realizeds
(longer maturities have higher spreads but their P&L is more volatile
because of higher risk to ATM vol changes)
Figure 2) What is optimal hedging frequency for each maturity
Translate into approximations of optimal bands for price and delta triggers
Naturally, results are sensitive to assumed price dynamics
Under SV with jumps: lower Sharp ratio and less frequent hedging
1.0
1.3
1.6
Optimal Sharpe ratio
Diffusion
Stochastic volatility with jumps
0.4
0.7
1m
4m
7m
10m
13m
16m
19m
22m
25m
28m
31m
34m
37m
Option maturity in months
4
6
8
10
12 Optimal hedging frequency in days
Diffusion
Stochastic volatility with jumps
0
2
4
1m
4m
7m
10m
13m
16m
19m
22m
25m
28m
31m
34m
37m
Option maturity in months
50
51. Last topic: why the beta stochastic vol model with
jumps is better than its alternatives (for stock indices)
The most important feature for dynamic hedging model:
1) Ability to produce different volatility regimes as observed in the
market and to imply empirically consistent delta
Recall definition of volatility skew-beta: change in term structure of ATM
volatility, σATM(T), predicted by price return times SKEW(T)
We saw that vol skew-beta is very important to account for correct P&L
arising from change in BSM implied vols
Skew-consistent SV and LV models imply skew-beta of 2
Empirical vol skew-beta: S&P 500 ≈ 1.5; STOXX50 ≈ 1.8; Nikkei≈ 0.5
1.00
1.50
2.00
2.50
Vol Skew-Beta for S&P500
SVJ Skew-Beta with empirical beta
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
1.00
1.50
2.00
2.50
Vol Skew-Beta for STOXX 50
SVJ Skew-Beta with empirical beta
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
1.00
1.50
2.00
2.50
Vol Skew-Beta for NIKKEI
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
51
52. Why the beta SV with jumps is better than its alterna-
tives
Extra arguments to look at apart from implied volatility skew-beta
2) Fit to empirical distribution of implied and realized volatilities
3) Interpretation of model parameters in terms of impact on model implied
BSM vols
4) P&L explain for delta-hedging strategies of vanilla options in terms of
implied and realized model parameters
5) Stability of model parameters
Calibration to vanilla options is not a problem in practical applications -
it is easy to achieve by introducing a (small) local vol part
Calibration problem is solved by Dupire (1994) for diffusions, Andersen-
Andreasen (2000) for jump-diffusions, Lipton (2002) for SV with jumps
52
53. I. Non-parametric local volatility model - textbook im-
plementation of Dupire local volatility using discrete set
of option prices and interpolation
It is a good competitor to start my comparison because it fails on all
counts
As for interpretation and stability of model parameters - see quiz on the
next slide
53
54. Interpretation and stability of non-parametric LV
Quiz: one of two figures below is
[A] local vol surface fitted to discrete prices of S&P500 index options
[B] Eckert IV map projection of a ski mountain where I was skiing last
winter
Which is which ?
The same conclusion is for any SV and LV model implemented naively
using direct calibration to market prices and mean-variance hedging:
1-2) Wrong hedges and empirical inconsistency
3-5) Uninterpretable and unstable model parameters
54
55. II. Industry-standard alternative (in equity derivatives)
Implied volatility@strike-into-density@price approach (my terminology)
Conceptually:
σimpl(K; T) → Pimpl(S(T) = K) (7)
where→ is Dupire LV formula in terms of implied vols at strike K&mat T
Figure 1A) Given parametric form for implied vols σimpl(K; T)
Figure 1B) Given backbone function fbackbone(δS; K, T) to map price
changes δS into changes in vols δσimpl(K; T) according to specified regime
Figure 2) → in Eq(7) serves as interpolator from implied vols in strike
space to implied densities in price space
Figure 3) LV model projects densities to option prices in ”model-independent”
way using MC or PDE methods
20%
30%
40%
50%
ImpliedVolatility
Implied Volatility (S0=1.00)
Implied Volatility (S1=0.95)
Change in IV from backbone function
0%
10%
20%
0.6 0.7 0.8 0.9 1.0 1.1 1.2
ImpliedVolatility
Strike
0.75%
1.50%
2.25%
Density
Implied Density from LV mapping (S0=1.00)
Implied Density from LV mapping (S1=0.95)
Change in Density from LV mapping
-0.75%
0.00%
0.6 0.7 0.8 0.9 1.0 1.1 1.2
Density
Spot
0.10
0.20
0.30
0.40
0.50
PV
PV Risk-Reversal 95-105% (S0)
PV Risk-Reversal 95-105% (S1)
Change in PV Risk-Reversal 95-105%
-0.20
-0.10
0.00
0.10
0.6 0.7 0.8 0.9 1.0 1.1 1.2
Spot
55
56. 1) Hedging performance for local vol approach are pri-
mary driven by parametric form for implied vols σimpl(K; T)
and empirical backbone function
2) No consistency with empirical distribution of implied and realized vol
3) & 4) Model interpretation and P&L explain are possible only in terms
of parameters of functional form for implied volatility
Key drawback of implied volatility-into-density approach:
[i] For computation of delta it requires a re-calibration of local vol and
re-valuation for any change in market data
[ii] Lacks vol-of-vol so it is inconsistent for hedging of path-dependent
options sensitive to forward vols and skews
56
57. Alternatives for local vol or σimpl(K; T) → Pimpl(S(T) = K)
approach do not produce improvements
Instead of LV to map implied vol into price density, it is also customary
to use SV or LSV models as interpolators with extra degree of freedom
Hereby hodel choice is typically motivated by availability of a ”closed-
form” solution, not empirical consistency!
Figure: SV and LSV models are not applied for hedging as dynamic
models since their model delta is wrong - with and without minimum
variance hedge - but through re-calibration to empirical backbone
1.5%
2.5%
3.5%
Change in implied vol, S1-S0=-0.05
SV model delta
SV with Min Var Hedge
Empirical backbone
-1.5%
-0.5%
0.5%
0.70 0.80 0.90 1.00 1.10 1.20
Strike
0.40
0.60
0.80
1.00Delta for 1y call option on
S&P500
SV model
0.00
0.20
0.40
0.70 0.80 0.90 1.00 1.10 1.20 Strike
SV with Min Var hedge
Sticky-Strike BSM delta
SV re-calibrated to empirical backbone
To conclude I use a quote from Richard P. Feynman:
It doesn’t matter how beautiful your theory is, it doesn’t matter
how smart you are. If it doesn’t agree with experiment, it’s wrong!
57
58. III. Arguments in favor of Beta SVJ model:
1) the model has ability to fit empirical vol skew-beta
and produce correct option delta without re-calibration
Figure: delta from SVJ model fits empirical backbone
0.40
0.60
0.80
1.00 Delta for 1y call option on
S&P500
SV model
0.00
0.20
0.40
0.70 0.80 0.90 1.00 1.10 1.20Strike
SV with Min Var hedge
SV re-calibrated to Market backbone
SVJ (without re-calibration)
2) Consistent with the empirical distributions of implied and realized
volatilities, which are very close to log-normal
3) It has clear intuition behind the key model parameters:
Volatility beta is sensitivity to changes in short-term ATM vol
Residual vol-of-vol is volatility of idiosyncratic changes in ATM vol
4) P&L explain is possible in terms of implied and realized quantities
of key model parameter - vol beta
5) Stability and calibration - next slide
58
59. Calibration of beta SV model is based on econometric
and implied approaches without large-scale non-linear
and non-intuitive calibrations
1) Parameters of SV part are estimated from time series
2) Jump/risk-aversion params are fitted to empirical vol skew-beta
Params in 1) & 2) are updated only following changes in volatility regime
3) Small mis-calibrations of the SV part and jumps are corrected
using local vol (LV) part
Contribution to skew from LV part is kept small (no more than 10-15%)
Local vol part is re-calibrated on the fly to reproduce small variations in
some parts of implied vol surface, which are caused by temporary supply-
demand factors specific to that part
It is also robust to compute bucketed vega risk in this way
In practical terms:
1) Local volatility part accounts for the noise from idiosyncratic
changes in implied volatility surface
2) Stochastic volatility and jumps serve as time- and space-homogeneou
factors for the shape of the implied volatility surface
59
60. More details on calibration of beta SV model (technical
part omitted during the talk)
1) Parameters of SV part are calibrated using maximum likelihood meth-
ods from time series of 1m implied ATM volatility (or the VIX)
[i] mean-reversion κ is estimated over longer-period, at least 5 years, -
better to keep it constant at 3.00
[ii] vol beta β and residual vol-vol ε are estimated over shorter periods,
1y, - typically β ≈ −1.00 and ε ∈ [0.60, 1.00]
2) Negative jump in return η is fitted using Merton jump model to put
options with maturity of 6 months and [80% − 100%] OTM strikes -
typically η = −30%
3) Given 1) and 2): 3A) jump intensity λ is calibrated to fit the empirical
sensitivity of implied volatility changes to price changes, aka volatility
skew-beta, - typically λ ∈ [0.03, 0.2]
Given all above: 3B) initial vol V (0) and mean vol θ calibrated to fit the
current term structure of ATM vols
Parameters in 1), 2) 3A) (relatively uniform for major stock indices) are
updated infrequently
4) Local vol part is added to fit daily variations in implied vol surface
60
61. For risk-neutral pricing, distribution of jumps does not
matter - jumps are only needed to fit skew premium
Recall illustration of emergence of Q-
skew using Bakshi-Kapadia-Madan
formula and Merton jump model:
[i] Under P, jumps are symmetric
with mean of 0% and volatility of 4%
[ii] The risk-neutral mean jump is
−5% with zero volatility
4%
6%
8%
10%
Frequency
Frequency of S&P500 daily returns
Empirical
Frequency
Physical Merton
under P
Risk-Neutral
0%
2%
4%
-9% -7% -5% -4% -2% 0% 2% 4% 6% 7%
Frequency
Daily return
Risk-Neutral
Merton under Q
Yet, jumps are needed to fit market prices & compute correct deltas
Also jumps are important to fit market prices of options on realized and
implied volatilities (VIX) - see my presentations at GD in 2011 & 2012
Practical explanation for excessive risk-neutral skew premium:
1) Risk-averse investors always ready to over-pay for insurance ir-
respectively of price changes
2) As part of index correlation skew premium, when holders of
stock portfolios buy index puts for (macro) protection
To make things robust, I assume constant jumps with simple calibration
61
62. How to explain the difference between implied
and realized dynamics using preference theory
For retail option buyer - option value is derived from his preference/utility
for specific payoffs in certain market scenarios
For institutional option seller - option value is derived from:
[i] Expected hedging costs
[ii] Smooth stream of fees and P&L
[iii] Premium for suffering losses in bad market conditions
As a result:
1) Option prices in the market are set by demand-supply equilibrium
between sellers and buyers
2) Risk-aversion parameters is a degree of demand-supply imbal-
ance
3) Implied and realized vols and, in particular, skews are different
62
63. To conclude: we can think of jumps as a mea-
sure of risk-aversion for pricing kernel!
Recently, interesting research is made and also presented at Global Deriva-
tives on how to imply the ”expected-implied” physical distribution from
options market prices and specified risk-aversion
Stephen Ross:The Recovery Theorem, GD2012, Journal of Finance 2014
Peter Carr: Can we recover?, Global Derivatives 2013
Computation of ”empirical” delta and calibration of excessive skew are
related concepts:
1) Compute option delta under ”expected-implied” physical distri-
bution using empirical vol beta
2) Fit level of risk-aversion to excessive skew premium observed in
market prices of index options
These concepts and volatility skew-beta are related to the interplay be-
tween the implied and realized risk premiums:
[i] high implied / positive realized risk premiums - sticky strike vol regime
[ii] low implied / negative realized risk premiums - sticky local vol regime
To be continued at next year Global Derivatives conference...
63
64. Summary
1) Dynamics of implied and real-
ized vols are log-normal
2) Implied vol beta significantly
overestimates realized beta
3) Vol skew-beta is important for
correct P&L - any dynamic hedging
model should fit empirical skew-beta
Risk-Aversion/Jumps parameter
is added to fit empirical skew-beta
SVJ fits empirical skew-beta≈ 1.5,
unlike Minimum Var delta≈ 2.0
4) Beta SVJ model applied for
delta-hedging risk-reversal is tool to
produce P&L from spread be-
tween implied and realized skews
Log-normal beta SVJ model:
⊗Is consistent with the empirical
dynamics of ATM volatility
⊗Produces correct option deltas
⊗Can significantly improve Shar-
pe ratios for delta-hedging P&Ls
3%
4%
5%
6%
7%
Frequency
Empirical frequency of
normalized logarithm of the VIX
Empirical
Standard Normal
0%
1%
2%
3%
-4 -3 -2 -1 0 1 2 3 4
Frequency
Log-VIX
-0.5
0.0
Feb-07
Sep-07
Apr-08
Nov-08
Jun-09
Jan-10
Aug-10
Mar-11
Oct-11
May-12
Dec-12
Jul-13
Implied Volatility Beta
Realized Volatility Beta
-2.0
-1.5
-1.0
1.00
1.50
2.00
2.50
Vol Skew-Beta for S&P500
SVJ Skew-Beta with empirical beta
0.00
0.50
1m
3m
5m
7m
9m
11m
13m
15m
17m
19m
21m
23m
T in months
SVJ Skew-Beta with empirical beta
Sticky local with Min-var delta
Empirical bounds
P&L = 2050*Beta + 2490
R² = 49%
0
1500
3000
P&L
Risk-Reversal P&L vs
Realized Vol Beta
-1500
0
-2.0 -1.5 -1.0 -0.5
P&L
Realized Volatility Beta
64
65. Disclaimer
The views represented herein are the author own views and do not neces-
sarily represent the views of Bank of America Merrill Lynch or its affiliates
65
66. References
Andersen, L., Andreasen, J., (2000), “Jump-Diffusion Processes - Volatility Smile Fitting
and Numerical Methods for Option Pricing ,” Review of Derivatives Research 4, 231-262
Bakshi, G., Kapadia, N., Madan, D., (2003), “Stock return characteristics, skew laws,
and the differential pricing of individual equity options,” Review of Financial Studies 16
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