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 vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
This document summarizes a presentation on using particle filtering and leverage stochastic volatility models for volatility arbitrage trading. The presentation covered modeling volatility, dynamic asset pricing models, realized and implied volatility, leverage stochastic volatility models, particle filtering as a sequential Bayesian filtering technique, volatility trading strategies, variance and volatility swaps, using particle filtering to predict volatility on the S&P 500, backtesting with the VIX index, and conclusions.
This document discusses Monte Carlo simulation techniques for pricing derivatives. It begins with an overview of Monte Carlo simulation and its use in derivative pricing models. It then covers different types of Monte Carlo path evolution (element-wise, path-wise, etc.). The document discusses implementing Monte Carlo simulation, including generating Wiener paths using methods like Euler discretization and Brownian bridges. It provides an example applying these techniques to price an option on the geometric average rate and shows convergence with different levels of path stratification. The key steps in Monte Carlo simulation for derivative pricing are outlined as simulating asset paths, computing payoffs, and estimating the option value and associated standard error.
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
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
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
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.
Realized and implied index skews, jumps, and the failure of the minimum-varia...Volatility
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.
Pricing vulnerable European options when the option’s payoff can increase the risk of financial distressPeter Klein, Michael InglisJournal of Banking & Finance
This document summarizes a presentation on using particle filtering and leverage stochastic volatility models for volatility arbitrage trading. The presentation covered modeling volatility, dynamic asset pricing models, realized and implied volatility, leverage stochastic volatility models, particle filtering as a sequential Bayesian filtering technique, volatility trading strategies, variance and volatility swaps, using particle filtering to predict volatility on the S&P 500, backtesting with the VIX index, and conclusions.
This document discusses Monte Carlo simulation techniques for pricing derivatives. It begins with an overview of Monte Carlo simulation and its use in derivative pricing models. It then covers different types of Monte Carlo path evolution (element-wise, path-wise, etc.). The document discusses implementing Monte Carlo simulation, including generating Wiener paths using methods like Euler discretization and Brownian bridges. It provides an example applying these techniques to price an option on the geometric average rate and shows convergence with different levels of path stratification. The key steps in Monte Carlo simulation for derivative pricing are outlined as simulating asset paths, computing payoffs, and estimating the option value and associated standard error.
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
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
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
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.
Realized and implied index skews, jumps, and the failure of the minimum-varia...Volatility
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.
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.
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
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.
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
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
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.
- 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.
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.
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
The document discusses key concepts in financial theory including:
1. The postulate of yield of money over time, and capitalization and discounting operations which allow moving money forward or backward in time.
2. Economic laws like the arbitrage principle, law of one price, law of linearity of amounts, and law of monotonicity of amounts which are necessary for an efficient market without arbitrage opportunities.
3. Financial risks including market risk, credit risk, and liquidity risk.
4. Financial engineering and how derivatives instruments derive their value from underlying assets or indices.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
This document provides an overview of various credit default models, including:
- Merton's structural model, which uses Black-Scholes option pricing theory to estimate probability of default.
- Extensions to Merton's model, including the KMV model which maps "distance to default" to historical default rates.
- Ratings-based models that use credit rating migration probabilities provided by rating agencies.
- Multivariate factor models that model default dependence between firms using common factors like the economy.
The document discusses key aspects and assumptions of these different modeling approaches.
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.
The document discusses using Fourier and Laplace transforms to calculate expectations needed for pricing options, default probabilities, and other financial quantities. It notes that while the Fourier/Laplace transform method is fast, numerically evaluating the resulting integrals is challenging. The document proposes using "quasi-parabolic deformations" to analytically continue characteristic functions to wider regions of the complex plane, allowing more flexible integration contours that improve accuracy and speed. Examples where characteristic functions can be calculated explicitly are given for common financial models like Lévy processes and stochastic volatility models.
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.
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
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.
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
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
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.
- 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.
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.
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
The document discusses key concepts in financial theory including:
1. The postulate of yield of money over time, and capitalization and discounting operations which allow moving money forward or backward in time.
2. Economic laws like the arbitrage principle, law of one price, law of linearity of amounts, and law of monotonicity of amounts which are necessary for an efficient market without arbitrage opportunities.
3. Financial risks including market risk, credit risk, and liquidity risk.
4. Financial engineering and how derivatives instruments derive their value from underlying assets or indices.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
This document provides an overview of various credit default models, including:
- Merton's structural model, which uses Black-Scholes option pricing theory to estimate probability of default.
- Extensions to Merton's model, including the KMV model which maps "distance to default" to historical default rates.
- Ratings-based models that use credit rating migration probabilities provided by rating agencies.
- Multivariate factor models that model default dependence between firms using common factors like the economy.
The document discusses key aspects and assumptions of these different modeling approaches.
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.
The document discusses using Fourier and Laplace transforms to calculate expectations needed for pricing options, default probabilities, and other financial quantities. It notes that while the Fourier/Laplace transform method is fast, numerically evaluating the resulting integrals is challenging. The document proposes using "quasi-parabolic deformations" to analytically continue characteristic functions to wider regions of the complex plane, allowing more flexible integration contours that improve accuracy and speed. Examples where characteristic functions can be calculated explicitly are given for common financial models like Lévy processes and stochastic volatility models.
"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven In...Quantopian
Commonality in idiosyncratic volatility cannot be completely explained by time-varying volatility. After removing the effects of time-varying volatility, idiosyncratic volatility innovations are still positively correlated. This result suggests correlated volatility shocks contribute to the comovement in idiosyncratic volatility.
Motivated by this fact, we propose the Dynamic Factor Correlation (DFC) model, which fits the data well and captures the cross-sectional correlations in idiosyncratic volatility innovations. We decompose the common factor in idiosyncratic volatility (CIV) of Herskovic et al. (2016) into the volatility innovation factor (VIN) and time-varying volatility factor (TVV). Whereas VIN is associated with strong variation in average returns, TVV is only weakly priced in the cross section
A strategy that takes a long position in the portfolio with the lowest VIN and TVV betas, and a short position in the portfolio with the highest VIN and TVV betas earns average returns of 8.0% per year.
The document discusses modeling volatility for European carbon markets using stochastic volatility (SV) models. It outlines estimating SV model parameters from market data, re-projecting conditional volatility, and using the re-projected volatility to price options and calculate implied volatilities. The modeling approach involves projecting historical returns, estimating an SV model, and then re-projecting conditional volatility and pricing options based on the estimated model. Parameters are estimated for both the NASDAQ OMX and Intercontinental Exchange carbon markets and model diagnostics are presented.
The document discusses modeling volatility for European carbon markets using stochastic volatility (SV) models. It outlines estimating SV model parameters from market data, simulating conditional volatility distributions, and using these to price options and evaluate market pricing errors. The modeling approach involves projecting returns from an SV model, estimating parameters, and then re-projecting to obtain conditional volatility forecasts for option pricing. Estimated model parameters and implied volatilities from major European carbon exchanges are presented and compared.
Algorithms Behind Term Structure Models II Hull-White ModelJeff Brooks
This document summarizes the Hull-White trinomial tree model for pricing interest rate derivatives. It describes:
1) The Hull-White model uses a trinomial tree to model the short-term interest rate as following either a Vasicek or Black-Karasinski process over discrete time steps.
2) Negative interest rates are avoided by manipulating the tree's branching probabilities or geometry. Three alternative branching processes are introduced to control rates.
3) The tree is constructed in two steps - first a "level tree" approximates the short rate process, then the tree probabilities are set to match the process expectations and variance.
The Vasicek model is one of the earliest stochastic models for modeling the term structure of interest rates. It represents the movement of interest rates as a function of market risk, time, and the equilibrium value the rate tends to revert to. This document discusses parameter estimation techniques for the Vasicek one-factor model using least squares regression and maximum likelihood estimation on historical interest rate data. It also covers simulating the term structure and pricing zero-coupon bonds under the Vasicek model. The two-factor Vasicek model is introduced as an extension of the one-factor model.
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Seminar talk at École des Ponts ParisTech about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model". - Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
1. Dimensional analysis and the concept of similitude allow experiments using scale models to be used to study full-scale systems. Dimensional analysis uses Buckingham pi theorem to determine the minimum number of dimensionless groups needed to describe a phenomenon in terms of the variables involved.
2. For a model to accurately simulate a prototype system, the dimensionless pi groups that describe the phenomenon must be equal between the model and prototype. This establishes the modeling laws or similarity requirements that a model must satisfy.
3. Common dimensionless groups in fluid mechanics include the Reynolds number, Froude number, Strouhal number, and Weber number. These groups arise frequently in analyzing experimental data from fluid mechanics problems.
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time...SYRTO Project
Spillover Dynamics for Systemic Risk Measurement Using Spatial Financial Time Series Models. Andre Lucas. Amsterdam - June, 25 2015. European Financial Management Association 2015 Annual Meetings.
L-8 VECM Formulation, Hypothesis Testing, and Forecasting - KH.pptxRiyadhJack
This lecture discusses vector error correction models (VECM), comparing the single-equation Engle-Granger approach to the multivariate Johansen approach. It outlines the VECM formulation and discusses hypothesis testing and properties including cointegration, weak exogeneity, and Granger causality. An example analyzing the relationship between short-term and long-term interest rates is used to illustrate the methodology.
This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides examples of time series data and introduces the AR(1) model. The document describes an algorithm for calculating a bootstrap confidence interval for forecasting from an AR(1) model. It then discusses a simulation study comparing empirical coverage rates of bootstrap confidence intervals under different parameters. Finally, it applies the bootstrap method to forecasting Gross National Product growth, comparing the results to a parametric approach.
This document discusses using bootstrap methods to create confidence intervals for time series forecasts. It provides background on time series models and the autoregressive (AR) process. It then presents an algorithm for calculating a bootstrap confidence interval for forecasts from an AR(1) model. A simulation study compares coverage rates for bootstrap confidence intervals under different parameters. Finally, the method is applied to US Gross National Product data to forecast and construct confidence intervals.
This document describes specification tests that can be used after estimating dynamic panel data models using the generalized method of moments (GMM) estimator. It presents GMM estimators for first-order autoregressive models with individual fixed effects that exploit moment restrictions from assuming serially uncorrelated errors. Monte Carlo simulations are used to evaluate the small-sample performance of tests of serial correlation based on GMM residuals, Sargan tests, and Hausman tests. The tests are also applied to estimated employment equations using an unbalanced panel of UK firms.
This document provides an overview of models for the short rate, including equilibrium and no-arbitrage models. Equilibrium models like Vasicek and CIR assume the short rate follows a stochastic process with mean reversion. No-arbitrage models like Ho-Lee and Hull-White use today's term structure as an input rather than an output. The Hull-White model extends Vasicek by making the reversion level a function of time.
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.
This document presents a comparison of Gaussian and non-Gaussian stochastic volatility models for modeling financial asset returns. It estimates parameters for these models using a hidden Markov model approach on index fund daily return data from 2006 to 2016. The results show that non-Gaussian models generally perform better in terms of goodness-of-fit measures. Specifically, indexes for stocks, emerging markets and the Pacific performed better with a non-Gaussian assumption, while a bond index was nearly normally distributed. The document also discusses model specifications and concludes it would be interesting to relax independence assumptions between error terms.
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Similar to ImperialMathFinance: Finance and Stochastics Seminar (20)
ImperialMathFinance: Finance and Stochastics Seminar
1. Option valuation in a general stochastic volatility
model
Kai Zhang and Nick Webber
University of Warwick
10 November 2010
Zhang and Webber: General stochastic volatility 1 / 48
2. Abstract
Stochastic volatility models are frequently used in the markets to model the implied
volatility surface. These models have several failings. Firstly, although improvements
on a basic Black-Scholes model, they nevertheless fail to fit the entire surface
adequately. Secondly, the improvements they offer are usually at the cost of greatly
reduced tractability. Thirdly, these models still fail to fit to market prices of
non-vanilla securities.
This paper addresses the second of these three issues. A general stochastic volatility
model is described, nesting both the Heston models and Sabr-related models. A
control variate Monte Carlo valuation method for this model is presented that, when
it can be applied, is shown to be a significant improvement over existing simulation
methods; when applied to barrier option pricing, it out-performs importance
sampling methods.
By providing a plausible simulation method for this general model, the paper opens
the possibility of exploring calibration to non-vanilla, as well as vanilla, instruments.
Zhang and Webber: General stochastic volatility 2 / 48
3. Pricing in financial markets
Need to price and hedge financial instruments.
Issues in effective hedging:
Need to
1. Recover prices of hedging instruments;
2. Model evolution of prices of hedging instruments closely,
ie, match hedge deltas and gammas as closely as possibile.
Do (1): Fit to implied volatility surface
Do (2): Fit to instruments whose values are path-dependent
Zhang and Webber: General stochastic volatility 3 / 48
4. Calibration to vanillas
Calibration to market prices of vanilla options.
1. Necessary.
Must correctly price one’s hedging instruments.
2. Insufficient.
Can calibrate to vanillas but still misprice other instruments.
Vanillas are priced off asset density at their maturity time.
Do not depend upon sample path of asset process.
Barrier options, Americans/Bermudan, average rate options, etc,
Depend upon (full) sample path.
Zhang and Webber: General stochastic volatility 4 / 48
5. Fitting to the implied volatility surface
Various attempts:
Jump-diffusion models (eg Merton, Bates, etc);
Lévy models (eg VG, NIG, CGMY, etc);
Stochastic volatility (SV) models (eg Heston, Sabr, etc).
Natural to investigate SV models:
Naively attractive as implied volatility is clearly stochastic;
Allows for (relatively) persistent smiles;
Generates volatility clustering
(and other empirically observed times series properties).
Zhang and Webber: General stochastic volatility 5 / 48
6. Contribution
1. Present a general SV model,
nesting Heston, Sabr-clone, et cetera.
2. Construct a correlation-control variate for the model.
3. Apply the model to pricing:
Average rate options, focus on these
Barrier options.
4. Explore variations in barrier option pricing, not discussed
consistent with the implied volatility surface.
Zhang and Webber: General stochastic volatility 6 / 48
7. A general stochastic volatility model
Let
(S)t 0 be an asset price process,
(V)t 0 be a volatility process,
(1)
with SDEs
dSt = rStdt + σf (Vt) S
β
t dWS
t , (2)
dVt = α(µ Vt)dt + ηV
γ
t dWV
t , (3)
dWS
t dWV
t = ρdt, (4)
where f (v) = vξ and
ξ, β 0, determine the structure of volatility for S,
γ 0, determines the process for V,
γ > 0, when ξ /2 Z,
σ > 0, included for generality,
α, µ 2 R, are not required to be positive.
(5)
When ξ /2 Z, Vt is required to remain positive.
When ξ 2 Z, Vt is permitted to become negative.
Zhang and Webber: General stochastic volatility 7 / 48
8. Characteristics of the processes
β and the process for St:
β = 0, SV absolute diffusion process
β 2 (0, 1) , SV CEV process
β = 1 geometric SV process
ξ and the processes for St and Vt:
ξ = 0, St is non-stochastic GBM,
ξ = 1
2 , Vt is a variance process,
ξ = 1 Vt is a volatility process.
When ξ = 1, do not require Vt to remain positive.
When β < 1
2 impose an absorbing boundary at 0.
Zhang and Webber: General stochastic volatility 8 / 48
10. Not nested
More general specifications:
(β, f (v) , g (v)): Jourdain & Sbai (2010) (including (1, 2, 0)),
(h (S) , ξ, γ): Bourgade & Croissant (2005).
Log-specifications: β, ˆξ, γ = (1, 1, 0), with f (v) = exp v
ˆξ :
Scott (1987), Chesney and Scott (1989),
Melino and Turnbull (1990).
Jump-diffusion models:
Bates (1998).
General Lévy models:
VG, NIG, CGMY, etc.
Stochastic interest rate and higher factor models:
van Haastrecht, Lord, Pelsser & Schrager (2009), etc.
Zhang and Webber: General stochastic volatility 10 / 48
11. Valuation issues
Ackward to get prices out.
PDE and lattice methods:
Two-factor PDEs are tricky;
Difficult to get accurate prices quickly.
Monte Carlo methods:
Issues with bias and convergence.
State dependent volatility is likely to be problematical, eg CIR.
No strong solution to SDE,
Zero is accessible (in equity calibrated models),
Exact simulation is possible (but expensive. Scott (1996).)
Zhang and Webber: General stochastic volatility 11 / 48
12. Issues with Heston
Valuation:
European options: direct numerical integration possible.
Path-dependent options: can use only Monte Carlo.
Simulation:
Exact simulation is possible but too expensive.
(Broadie and Kaya (2006), Glasserman and Kim (2008)).
OK for long-step Monte Carlo, infeasible for short step.
Approximate solutions are poor
(particulary when zero accessible for vt).
Need fast simulation methods.
Zhang and Webber: General stochastic volatility 12 / 48
13. Using Monte Carlo
Evolve from time t to time t + ∆t.
Short step: ∆t small.
Usually required if:
1. No exact solution to SDE.
2. Option is continuously monitored.
Long step: ∆t equal to time between reset dates.
Usually possible if:
1. Option is discretely monitored.
2. Exact solution to SDE, or a good approximation, is known.
Zhang and Webber: General stochastic volatility 13 / 48
14. Control variates and Monte Carlo
Plain Monte Carlo:
Generate M sample paths.
Get discounted payoff cj along each sample path, j = 1, . . . , M.
Plain MC option value ˆc is
ˆc =
1
M
M
∑
j=1
cj. (6)
Control variate Monte Carlo:
Along each sample path also generate CV value dj st E dj = 0.
Let β = cov cj, dj / var dj then
CV corrected option value ˆcCV is
ˆcCV =
1
M
M
∑
j=1
cj βdj . (7)
Can extend to have multiple CVs.
Zhang and Webber: General stochastic volatility 14 / 48
15. Efficiency gain
Suppose ˆc computed in time τ with standard error σ,
Suppose ˆcCV computed in time τCV with standard error σCV.
Efficiency gain E is
E =
τσ2
τCVσ2
CV
. (8)
Proportional reduction in time by CV method
to get same standard error as plain method.
If ρ = corr cj, dj then
E =
τ
τCV
1
1 ρ2
. (9)
If τCV/τ 5 and ρ = 0.99 then E 10; if ρ = 0.999 then E 100.
τCV/τ large? “Speed-up” is a better term than “variance reduction”.
Zhang and Webber: General stochastic volatility 15 / 48
16. Auxiliary model CVs
Auxiliary instument.
Suppose that have an option p “similar” to c such that
1. Along sample paths, corr cj, pj is close to 1;
2. An explict solution p is known.
Then dj = pj p is an auxiliary instrument CV.
Auxiliary model.
Write M for the pricing model.
Suppose have an auxiliary model Ma “similar” to M, such that
1. Same set of Wiener sample paths can be used for each model;
2. c has an explicit solution ca in Ma.
Write ca
j for discounted payoff on path j under Ma
(so that ca ˆca = 1
M ∑M
j=1 ca
j .)
Then dj = ca
j ca is an auxiliary model CV.
Zhang and Webber: General stochastic volatility 16 / 48
17. The SV model: Auxiliary model and instrument CV
Find suitable:
Auxiliary model Ma
j (effectively sample path dependent),
Auxiliary instrument p, so that p has an explicit value pe
j in Ma
j .
Ma
j is conditioned on a realisation of a volatility sample path:
Evolve Vt to get sample path ˜Vt,
then evolve St as if Vt were piece-wise constant with values ˜Vt.
(Works since c = E cj = E E cj j Vt .)
Along each sample path compute
cj in model M,
pj and pe
j in model Ma
j .
Correlation CV is dj = pj pe
j .
CV corrected option value ˆcCV is, as usual,
ˆcCV =
1
M
M
∑
j=1
cj βdj . (10)
Zhang and Webber: General stochastic volatility 17 / 48
18. Preliminary re-write
Set ˜ρ =
p
1 ρ2.
Write processes in the form
dSt = rStdt + V
ξ
t S
β
t ρdWV
t + ˜ρdW2
t , (11)
dVt = α(µ Vt)dt + ηV
γ
t dWV
t , (12)
dWV
t dW2
t = 0, (13)
where
WS
t = ρWV
t + ˜ρW2
t and
σ has been absorbed into Vt.
Zhang and Webber: General stochastic volatility 18 / 48
19. The SV model: Auxiliary model, 1
First step. Transform SDE of St to make volatility independent of St.
Two cases: β = 1 and β 2 (0, 1).
β = 1 case. Set Yt = ln St, then
dYt = r
1
2
V
2ξ
t dt + V
ξ
t dWS
t (14)
β 2 (0, 1) case. Set Yt = 1
1 β S
1 β
t , then
dYt = r(1 β)Yt
β
2(1 β)Yt
V
2ξ
t dt
+V
ξ
t ρdW1
t + ˜ρdW2
t , (15)
Zhang and Webber: General stochastic volatility 19 / 48
20. The SV model: Auxiliary model, 2
Second step. Discretize the transformed process.
Discretize the process for Vt, with an approximation ˜Vt, as you like.
Discretize the process for Yt, conditional on ˜Vt,
with an approximation ˜Yt, so that increments are normal.
Set ˜Yi = ˜Yti
, ˜Vi = ˜Vti
, then have µY ˜Yi, ˜Vi and σY ˜Yi, ˜Vi such that
˜Yi+1 = ˜Yi + µY ˜Yi, ˜Vi + σY ˜Yi, ˜Vi εY
i (16)
for εY
i N (0, 1) normal iid.
The discretization of Yt determines the auxiliary model Ma
j .
˜Yi has normal increments (conditional on Vt)?
More likely to get explicit solutions in Ma
j .
Zhang and Webber: General stochastic volatility 20 / 48
21. The SV model: Auxiliary model, 3
Need some definitions. Set
Ii =
Z ti+1
ti
V
2ξ
s ds, (17)
Ji =
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds. (18)
Models determine the form of Ii and Ji, eg:
Model Ii Ji
Heston
R ti+1
ti
Vsds, α (µ∆t Ii) ,
Garch
R ti+1
ti
Vsds,
R ti+1
ti
V
1
2
s αµ α + 1
4 η2 Vs ds,
Sabr
R ti+1
ti
V2
s ds, αµ∆t α
R ti+1
ti
Vsds
J&S
R ti+1
ti
V2
s ds, 1
4 η2∆t +
R ti+1
ti
V
1
2
s α (µ Vs) ds
Zhang and Webber: General stochastic volatility 21 / 48
22. The SV model: Auxiliary model, 4
β = 1 case. Integrating 14, obtain
Yti+1
= Yti
+ r∆t
1
2
Z ti+1
ti
V
2ξ
s ds + ˜ρ
Z ti+1
ti
V
ξ
s dW2
s
+
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds
9
=
;
,
(19)
so
Yti+1
= Yti
+ r∆t +
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Ji
9
=
;
1
2
Ii + ˜ρ
p
IiεY
i . (20)
Hence
µY ˜Yi, ˜Vi = r∆t +
ρ
η
8
<
:
˜V
ξ γ+1
ti+1
˜V
ξ γ+1
ti
ξ γ + 1
Ji
9
=
;
1
2
Ii, (21)
σY ˜Yi, ˜Vi = ˜ρ
p
Ii. (22)
Zhang and Webber: General stochastic volatility 22 / 48
23. The SV model: Auxiliary model, 5
β 2 (0, 1) case. Integrating 15,
Yti+1
= Yti
+ r(1 β)
Z ti+1
ti
Ysds
β
2(1 β)
Z ti+1
ti
V
2ξ
s
Ys
ds + ˜ρ
Z ti+1
ti
V
ξ
s dW2
s
+
ρ
η
8
<
:
V
ξ γ+1
ti+1
V
ξ γ+1
ti
ξ γ + 1
Z ti+1
ti
V
ξ γ
s α(µ Vs) +
1
2
(ξ γ)η2
V
2γ 1
s ds
9
=
;
.
(23)
Freezing Ys (eg at initial value Y0) and integrating,
˜Yi+1 = ˜Yi + r(1 β)Y0∆t
β
2(1 β)Y0
Ii +
ρ
η
˜V
ξ γ+1
i+1
˜V
ξ γ+1
i
ξ γ + 1
Ji
!
+˜ρ
p
IiεY
i , (24)
so
µY ˜Yi, ˜Vi = r(1 β)Y0∆t
β
2(1 β)Y0
Ii +
ρ
η
˜V
ξ γ+1
i+1
˜V
ξ γ+1
i
ξ γ + 1
Ji
!
,(25)
σY ˜Yi, ˜Vi = ˜ρ
p
Ii. (26)
Zhang and Webber: General stochastic volatility 23 / 48
24. The SV model: Auxiliary instrument, 1
Example: arithmetic average rate option.
K Reset dates at times T1 < < TK = T, final maturity date,
Tk Tk 1 = ∆T constant, i = 1, , K.
Discretization times 0 = t0 < t1 < < tN = TK,
ti ti 1 = ∆t constant, i = 1, , N.
Assume that ∆T = δ∆t, so Tk = tik
for some index ik.
Index vector κ = (i1, , iK) is indexes of reset dates.
Write ˜Sj = ˜S0
j , . . . , ˜SN
j , ˜S0
j = S0, for a sample path of St.
Discounted payoff aong ˜Sj is
cj = e rT
AA
j X
+
(27)
where X is strike, AA
j is the arithmetic average along ˜Sj,
AA
j =
1
K ∑
k2κ
˜Sk. (28)
Zhang and Webber: General stochastic volatility 24 / 48
25. Auxiliary instrument, 2
Auxiliary instruments:
β = 1 case. Option on discrete geometric average, AG,
AG
= ∏
k2κ
˜Sk
! 1
K
. (29)
Discounted payoff is pj = e rT AG
j X
+
.
β 2 (0, 1) case. Option on discrete β-average, Aβ,
Aβ
=
1
K ∑
k2κ
˜S
1 β
k
! 1
1 β
. (30)
Discounted payoff is pj = e rT A
β
j X
+
.
Need to compute pe
j in each case.
Zhang and Webber: General stochastic volatility 25 / 48
26. Auxiliary instrument, 3.
Computing pe
j , β = 1 case.
Set g = 1
K ∑k2κ
˜Yk so that AG = exp (g). From 20,
Yti
= Y0 + rti
1
2
i 1
∑
j=0
Ij +
ρ
η
0
@
V
ξ γ+1
ti
V
ξ γ+1
0
ξ γ + 1
i 1
∑
j=0
Jj
1
A + ˜ρ
i 1
∑
j=0
εj
q
Ij, εj N(0, 1).
(31)
Set ¯T = 1
K ∑k2κ tk and
ν =
1
η (ξ γ + 1)
1
K ∑
k2κ
˜V
ξ γ+1
k
˜V
ξ γ+1
0 . (32)
Then
g = ˜Y0 + r ¯T
1
2
H1 + ε
p
H2, ε N(0, 1), (33)
where
H1 =
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
˜Ii +
2ρ
η
˜Ji 2ρν, (34)
H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii. (35)
Zhang and Webber: General stochastic volatility 26 / 48
27. Auxiliary instrument, 4
Hence g is normally distributed and
pe
j = e rT
E
h
(eg
X)+
j V, I, J
i
(36)
= er( ¯T T)
cBS (K, ¯T, S0, y, r, ¯σ) , (37)
where cBS(K, ¯T, S0, y, r, ¯σ) is Black-Scholes European call value with
strike K, maturity time ¯T,
on an asset with initial value S0, volatility ¯σ, and dividend yield y,
when the riskless rate is r, and
¯T =
1
K ∑
k2κ
tk, ¯σ2
=
H2
¯T
, y =
1
2 ¯T
(H1 H2). (38)
Zhang and Webber: General stochastic volatility 27 / 48
28. Auxiliary instrument, 5.
Computing pe
j , β 2 (0, 1) case.
Set g = 1 β
K ∑k2κ
˜Yk so that Aβ = g
1
1 β . From 24, g is normal,
g = (1 β) m + (1 β) sε, ε N(0, 1), (39)
where
m = ˜Y0 + r(1 β) ˜Y0
¯T + ρν
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
β
2(1 β) ˜Y0
˜Ii +
ρ
η
˜Ji , (40)
s2
= H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii. (41)
Set g+ = (g)+
, so that (g+)
1
1 β is well defined.
When g is normal can compute
pe
j = e rT
E g+
1
1 β
X
+
j V, I, J . (42)
Zhang and Webber: General stochastic volatility 28 / 48
29. Auxiliary instrument, 6
Write λ = 1 β and set b = Kλ λm /λs. Have
E g+
1
1 β
K
+
= E
h
g
1
λ j g > Kλ
i
P
h
g > Kλ
i
. (43)
P g > Kλ = n (b) is known; get series expansion for E
h
g
1
λ j g > Kλ
i
.
Let Mi = E Zi j Z > b where Z N (0, 1) is normal. Then
E
h
g
1
λ j g > Kλ
i
= (λm)
1
λ +
∞
∑
i=1
1
i!
(λm)
1
λ i
si
Mi
i 1
∏
j=0
(1 jλ). (44)
Can compute Mi rapidly, with only a single evaluation of N (b).
Let φ = n(b)
1 N(b)
then (Dhrymes (2005))
M0 = 1, (45)
M1 = φ, (46)
Mi = bi 1
φ + (i 1)Mi 2, (47)
Can truncate 44 at a level Nmax where Nmax = 10 is small.
Zhang and Webber: General stochastic volatility 29 / 48
30. Special case: Auxiliary model, zero-correlation CV, 1
Obtain Ma,0
j auxiliary model from Ma
j auxiliary model by setting ρ = 0.
Why?
Is cheaper to compute,
May yield higher value of corr cj, dj .
Verified empirically:
Zero correlation CV often performs better
than standard correlation CV.
Zero correlation CV:
Can also be used in the f (V) = eV case.
Zhang and Webber: General stochastic volatility 30 / 48
31. Special case: Auxiliary model, zero-correlation CV, 2
β = 1 case. In 31, 33, 34 and 35, set ρ = 0, then
˜Yi = ˜Y0 + rti
1
2
i 1
∑
j=0
Ij +
i 1
∑
j=0
εj
q
Ij, εj N(0, 1) (48)
and
g = ˜Y0 + r ¯T
1
2
h1 + ε
p
h2, ε N(0, 1), (49)
where
h1 =
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
˜Ii, (50)
h2 = H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii. (51)
Hence g is normally distributed and
pe
j = e rT
E
h
(eg
X)+
j V, I, J
i
(52)
= er( ¯T T)
cBS (K, ¯T, S0, y, r, ¯σ) , (53)
with ¯σ2 = 1
¯T
h2, y = 1
2 ¯T
(h1 h2).
Zhang and Webber: General stochastic volatility 31 / 48
32. Special case: Auxiliary model, zero-correlation CV, 3
β 2 (0, 1) case.
In 40 and 41 set ρ = 0, then
m = ˜Y0 + r(1 β) ˜Y0
¯T
1
K
K 1
∑
k=0
(K k)
ik+1 1
∑
i=ik
β
2(1 β) ˜Y0
˜Ii,
s2
= H2 =
˜ρ2
K2
K 1
∑
k=0
(K k)2
ik+1 1
∑
i=ik
˜Ii, (54)
and formula for pe
j follows from 44, as before.
Zhang and Webber: General stochastic volatility 32 / 48
33. Approximating volatility integrals
Computing values for ˜Ii and ˜Ji.
Approximate the integrals by a single-step trapizium, eg,
Z ti+1
ti
V2
s ds
1
2
V2
i+1 + V2
i . (55)
Take care:
discretizations of Vt must not allow ˜Vi to become negative.
Zhang and Webber: General stochastic volatility 33 / 48
34. Numerical results
Apply the two new correlation CVs
(correlation CV (ρ 6= 0); zero correlation CV (ρ = 0))
to average rate options.
Compare with 3 “old” CVs.
GBM auxiliary CV; GBM delta CV;
European call (where explicit solution exists).
Apply to average rate options: 4, 16, 64 resets.
Maturity T = 1;
Three cases: ITM, X = 80; ATM, X = 100; OTM, X = 120;
Use M = 106 sample paths for plain MC, M = 104 for CV MC,
N = 320 times steps.
Evolving Vt: Model dependent.
Log-normal approximation, exact or Milstein.
Evolving Yt: Can use Euler (absorbed at zero when β < 1
2 ).
Zhang and Webber: General stochastic volatility 34 / 48
35. Choice of parameters
4 models: Heston, Garch, Sabr, Johnson & Shanno (J&S),
Two cases each.
Parameters (β, ξ, γ) S0 V0 r α µ η ρ
Heston Case 1: 1, 1
2 , 1
2 100 0.0175 0.025 1.5768 0.0398 0.5751 0.5711
Case 2: 1, 1
2 , 1
2 100 0.04 0.05 0.2 0.05 0.1 0.5
Garch Case 1: 1, 1
2 , 1 100 0.0175 0.025 4 0.0225 1.2 0.5
Case 2: 1, 1
2 , 1 100 0.04 0.05 2 0.09 0.8 0.5
Sabr Case 1: (0.4, 1, 1) 100 2 0.05 0 0 0.4 0.5
Case 2: (0.6, 1, 1) 100 2 0.05 0 0 0.4 0.5
J&S Case 1: 0.4, 1, 1
2 100 2 0.05 2 2 0.1 0.5
Case 2: 0.6, 1, 1
2 100 2 0.05 2 2 0.1 0.5
Zero is accessible in: Heston, Case 1; Sabr, Case 1; J&S, Case 1.
Heston, case 1, is Albrecher et al. (2007); case 2 is Webber (2010)
Garch cases: Vt parameters are Lewis (2000)
Sabr and J&S: Vt parameters chosen to give IVs of 15% and 30%.
Zhang and Webber: General stochastic volatility 35 / 48
36. Results, plain Monte Carlo
Value options with plain Monte Carlo, M = 106 sample paths.
Heston Garch Sabr J&S
Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
itm
20.91
(0.008)
[740]
21.57
(0.01)
[730]
20.76
(0.008)
[600]
21.72
(0.01)
[600]
21.48
(0.007)
[610]
22.46
(0.02)
[610]
21.49
(0.007)
[710]
22.25
(0.02)
[700]
atm
3.93
(0.005)
[740]
5.87
(0.008)
[730]
3.90
(0.005)
[610]
6.70
(0.009)
[600]
4.27
(0.005)
[600]
8.39
(0.01)
[600]
4.25
(0.005)
[710]
8.43
(0.01)
[710]
otm
0.039
(0.0006)
[740]
0.446
(0.002)
[740]
0.039
(0.0005)
[600]
0.88
(0.003)
[600]
0.012
(0.0002)
[610]
1.70
(0.005)
[610]
0.030
(0.0004)
[700]
2.00
(0.006)
[710]
In each box:
Top number is MC option value;
Middle number (round brackets) is standard error;
Bottom number (square brackets) is time in seconds.
Results are poor.
Takes 600 - 700 seconds to achieve prices accurate to 1 - 2 bp,
ie, in 6 - 7 seconds prices accurate only to 10 - 20 bp.
Zhang and Webber: General stochastic volatility 36 / 48
37. Extreme cases
In each case also look at:
Correlation extremes: ρ = +0.9; ρ = 0.9.
High volatility extreme:
Heston and Garch: Set V0 = µ = 0.25;
Sabr: Set V0 = 8 (case 1), V0 = 3 (case 2)
J&S: Same as Sabr, but also set µ = V0.
Value options with plain Monte Carlo, M = 106 sample paths.
Heston Garch Sabr J&S
Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ρ "
3.65
(0.009)
[750]
5.79
(0.009)
[740]
3.83
(0.006)
[600]
6.64
(0.01)
[600]
4.07
(0.006)
[610]
8.24
(0.01)
[610]
4.23
(0.005)
[710]
8.46
(0.01)
[710]
ρ #
3.88
(0.004)
[740]
5.89
(0.007)
[740]
3.88
(0.005)
[600]
6.70
(0.009)
[600]
4.29
(0.004)
[610]
8.40
(0.01)
[600]
4.25
(0.005)
[700]
8.44
(0.01)
[710]
σ "
11.73
(0.02)
[740]
12.44
(0.02)
[740]
11.83
(0.02)
[600]
12.33
(0.02)
[610]
12.45
(0.02)
[610]
11.84
(0.02)
[610]
12.61
(0.02)
[700]
11.99
(0.02)
[700]
Zhang and Webber: General stochastic volatility 37 / 48
38. Empirical correlations
Correlations, corr cj, dj , for K = 64 average rate options.
Correlation CV: correlations increase as options go OTM
ρ 6= 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 0.84 0.87 0.86 0.87 0.86 0.89 0.87 0.88
ATM 0.96 0.91 0.92 0.91 0.92 0.93 0.89 0.90
OTM 0.991 0.98 0.991 0.97 0.997 0.98 0.98 0.94
Zero correlation CV: correlations increase as options go ITM
ρ = 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 0.94 0.9990 0.997 0.9988 0.995 0.997 0.99996 0.99988
ATM 0.89 0.997 0.994 0.997 0.993 0.992 0.9994 0.9990
OTM 0.67 0.997 0.98 0.994 0.92 0.989 0.9994 0.9990
Correlations are almost always higher with zero-correlation CV.
Exception: OTM options, when zero is accessible.
Zhang and Webber: General stochastic volatility 38 / 48
39. Efficiency gains, K = 64 average rate options, 1
Zero correlation CV
ρ = 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 5.6 340 140 340 84 110 9000 3300
ATM 3.3 130 71 120 55 47 4100 1500
OTM 1.2 81 19 62 7.5 35 640 370
Correlation CV
ρ 6= 0 Heston Garch Sabr J&S
CV Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 3.1 3.7 2.9 3.1 3.0 3.5 2.6 2.7
ATM 11 5.3 4.7 4.4 4.6 5.2 3.0 3.4
OTM 48 21 45 11 200 14 21 5.6
Both CVs combined
Both Heston Garch Sabr J&S
CVs Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2
ITM 7.1 330 110 280 70 73 6600 2600
ATM 14 170 83 120 74 63 3700 1100
OTM 38 140 66 68 250 64 510 240
Zhang and Webber: General stochastic volatility 39 / 48
40. Efficiency gains, K = 64 average rate options, 2
Efficiency gain of 6000? Get same se as plain MC in 0.1 seconds.
Gains for ρ = 0 CV decrease as options go from ITM to OTM.
Gains for ρ 6= 0 CV decrease as options go from ITM to OTM.
Gains for ρ 6= 0 CV usually much less than ρ = 0 CV.
Exception: ATM/OTM in Heston and Sabr, when zero accessible.
Usually better to use both CVs together (except for J&S cases).
Using correlation CVs with other CVs
GBM: GBM auxiliary model
delta: delta CV in GBM auxiliary model
Do not compare with Heston or Sabr explicit call CV.
Gains with these CVs, for average rate call option, are not large.
Zhang and Webber: General stochastic volatility 40 / 48
46. Efficiency gains, ATM K = 64 average rate option, 8
High volatility cases:
CV combinations Heston Garch Sabr J&S
ATM, K = 64 C1 C2 C1 C2 C1 C2 C1 C2
ρ = 0 23 190 60 82 45 45 2200 930
ρ 6= 0 4.9 5.2 5.0 4.8 5.4 5.3 3.4 3.5
ρ = 0 + ρ 6= 0 34 160 58 78 63 56 1600 700
GBM 10 120 22 30 9.5 13 37 82
delta 28 27 31 36 18 20 59 63
GBM + delta 26 65 31 35 19 21 79 87
ρ = 0 23 190 58 82 44 45 2400 980
GBM + ρ 6= 0 19 140 27 34 15 19 28 56
both 30 180 64 84 66 56 1700 710
ρ = 0 29 83 35 45 24 26 950 380
delta + ρ 6= 0 49 61 43 49 40 39 53 55
both 54 77 50 54 55 51 830 330
all 66 110 47 52 65 63 1200 410
Best to use GBM + ρ = 0 + ρ 6= 0.
Except: J&S: Best is GBM + ρ = 0 (ρ = 0 is main contributor),
Heston, c1: Best to use all CVs (delta is main contributor).
Zhang and Webber: General stochastic volatility 46 / 48
47. Comparisons of efficiency gains
Relative improvement over existing methods:
Best gain including new CVs / Best gain with old CVs alone.
Relative Heston Garch Sabr J&S
performance C1 C2 C1 C2 C1 C2 C1 C2
ITM 1.7 2.4 1.7 2.7 1.9 2.7 6.3 9.4
ATM 6.9 2.5 2.3 2.9 3.1 2.8 11 10
OTM 34 10.7 14 5.8 92 8.4 26 3.6
ρ " 1.2 1.2 1.0 1.0 1.3 1.2 1.9 2.1
ρ # 3.1 1.4 1.3 1.2 1.6 1.7 3.8 6.6
σ " 2.4 1.6 2.1 2.3 3.5 3.0 30 11
.
Ordinary parameter values:
New CVs enhance performance by sizable factors.
Generally improve as options go OTM.
Extreme correlations: Perform less well but still give speed-ups
Extreme volatility: Still give very reasonable speed-ups
Zhang and Webber: General stochastic volatility 47 / 48
48. Summary
Have priced average rate options in a general SV model,
nesting Heston, Sabr, et cetera.
Have derived a pair of correlation CVs.
Have demonstrated their effectiveness compared to existing CVs.
The new CVs apply more generally to other options,
incuding barrier options.
Can attempt to calibrate to options other than vanillas.
Zhang and Webber: General stochastic volatility 48 / 48