Publications

Conference Paper: Realized and implied index skews, jumps, and the failure of the minimumvariance hedging
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ABSTRACT: We develop an accurate approximation of the moment generating function for the lognormal stochastic volatility (SV) model. We show, by comparison to Monte Carlo simulations, that our approximation proves to be very precise for valuation of vanilla options on the equity and the quadratic variance using Fourier inversion techniques. We extend our approximation for the lognormal SV model with simultaneous jumps in the logprice and the volatility. To develop intuition for model parameters and their calibration, we introduce the concept of the volatility beta, which measures the sensitivity of the volatility to changes in the price, and the idiosyncratic volatility, which introduces independent changes in the volatility. For practical contribution, we introduce the concept of volatility skewbeta which serves as an empirical adjustment for the option delta. We show how to calibrate the model and compute its empirical delta under the statistical measure so the model can reproduce any dynamics of implied volatility. The calibrated model minimizes realized volatility of deltahedging P\&Ls and reduces transaction costs, especially so for nonvanilla options. We extend the model to multiasset case and show that it produces a steep correlation skew. We present empirical investigation using implied and realized volatilities of four major stock indices (S\&P 500, FTSE 100, Nikkei 225, and EURO STOXX 50) and two volatility indices to validate the assumption about lognormality of both implied and realized volatilities.Global Derivatives Trading & Risk Management 2014; 05/2014 
Conference Paper: Consistently Modeling Joint Dynamics of Volatility and Underlying To Enable Effective Hedging
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ABSTRACT: 1) Analyze the dependence between returns and volatility in conventional stochastic volatility (SV) models 2) Introduce the beta SV model by KarasinskiSepp, "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 deltaGlobal Derivatives Trading & Risk Management 2013; 04/2013 
Article: Beta Stochastic Volatility Model
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ABSTRACT: We introduce the beta stochastic volatility model and discuss empirical features of this model and its calibration. This model is appealing because, first, its parameters can be easily understood and calibrated and, second, it produces steeper forward skews, compared to traditional stochastic volatility models. 
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ABSTRACT: We consider the deltahedging strategy for a vanilla option under the discrete hedging and transaction costs. Assuming that the option is deltahedged using the BlackScholesMerton model with an implied lognormal volatility, we analyze the profitandloss (P&L) of the deltahedging strategy given that the actual underlying dynamics are driven by one of four alternative models: lognormal diffusion, jumpdiffusion, stochastic volatility and stochastic volatility with jumps. For all of the four cases, we derive approximations for the expected P&L, expected transaction costs, and P&L volatility assuming hedging at fixed times.Using these results, we formulate the problem of finding the optimal hedging frequency that maximizes the Sharpe ratio of the deltahedging strategy. We also show how to apply our results to spot and deltabased hedging strategies. Finally, we provide illustrations.SSRN Electronic Journal 05/2012; DOI:10.2139/ssrn.1865998 
Conference Paper: Achieving Consistent Modeling Of VIX and Equities Derivatives
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ABSTRACT: 1) Discuss model complexity and calibration 2) Emphasize intuitive and robust calibration of sophisticated volatility models avoiding nonlinear calibrations 3) Present local stochastic volatility models with jumps to achieve joint calibration to VIX options and (shortterm) S&P500 options 4) Present two factor stochastic volatility model to fit both the shortterm and longterm S\&P500 option skewsGlobal Derivatives Trading & Risk Management 2012; 04/2012 
Article: Filling the Gaps
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ABSTRACT: The calibration of local volatility models to market data is one of the most fundamental problems of financial engineering. Under the restrictive assumption that the entire implied volatility surface is known, this problem can be solved by virtue of the socalled Dupire equation. In reality, however, the number of available data points is very limited and construction of a nonarbitrageable implied volatility surface is difficult, if not impossible, since it requires both interpolation and extrapolation of the market data. Thus, it is more natural to build the local volatility surface directly. In this article we present a generic semianalytical approach to calibrating a parametric local volatility surface to the market data in the realistic case when this data is sparse. This approach also allows one to build a nonarbitrageable implied volatility surface. The power of the method is illustrated by considering layered local volatility and generating local and implied volatility surfaces for options on SX5E. 
Conference Paper: Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in Local Stochastic Volatility Models
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ABSTRACT: 1) Volatility modelling 2) Local stochastic volatility models: stochastic volatility, jumps, local volatility 3) Calibration of parametric local volatility models using partial differential equation (PDE) methods 4) Calibration of nonparametric local volatility volatility models with jumps and stochastic volatility using PDE methods 5) Numerical methods for PDEs 6) Illustrations using SPX and VIX dataGlobal Derivatives Trading & Risk Management 2011; 04/2011 
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ABSTRACT: We analyse the effect of the discrete sampling on the valuation of options on the realized variance in the Heston (1993) stochastic volatility model. It has been known for a while (Buehler (2006)) that, even though the quadratic variance can serve as an approximation to the discrete variance for valuing longerterm options on the realized variance, this approximation underestimates option values for shortterm maturities (with maturities up to three months). We propose a method of mixing of the discrete variance in a lognormal model and the quadratic variance in a stochastic volatility model, which allows to accurately approximate the distribution of the discrete variance in the Heston model. As a result, we can apply semianalytical Fourier transform methods developed by Sepp (2008) for pricing shorterterm options on the realized variance. 
Conference Paper: Stochastic Local Volatility Models: Theory and Implementation
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ABSTRACT: 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 6) PDE based methods in two dimensionsFinancial Engineering Workshop, University of Leicester; 12/2010 
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ABSTRACT: We analyse the profitandloss (P&L) of deltahedging strategies for vanilla options in the presence of the implied volatility skew and derive an approximation for the P&L under the quadratic parametrization of the implied volatility. We apply this approximation to study the P&L of a straddle, a riskreversal, and a butterfly. Using our results, we derive the breakeven realized skew and convexity that equate the average realized P&L of the riskreversal and the butterfly, respectively, to zero. Furthermore, we analyse the impact of the volatility skew on the deltahedging of these option strategies. We present some empirical results using implied volatilities of options on the S&P 500 index.SSRN Electronic Journal 10/2010; DOI:10.2139/ssrn.1697829 
Conference Paper: An Approximate Distribution of DeltaHedging Errors in a JumpDiffusion Model with Discrete Trading and Transaction Costs Motivation
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ABSTRACT: 1) Analyse the distribution of the profit&loss (P&L) of deltahedging strategy for vanilla options in BlackScholesMerton (BSM) model and an extension of the Merton jumpdiffusion (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 meanvariance analysis to find the tradeoff between the costs and P&L variance given hedger's risk tolerance 5) Consider hedging strategies to minimize the jump riskFinancial Engineering Workshop, Cass Business School; 05/2010 
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ABSTRACT: We introduce a jumpdiffusion model for asset returns with jumps drawn from a mixture of normal distributions and show that this model adequately fits the historical data of the S&P500 index. We consider deltahedging strategy for vanilla options under the diffusion model (DM) and the proposed jumpdiffusion model (JDM) assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profitandloss (P&L) of the deltahedging strategy under the both models. We find that, under the lognormal model by BlackScholesMerton, the actual PDF of the P&L can be well approximated by the chisquared distribution with specific parameters. We derive an approximation for the P&L volatility in the DM and JDM. We show that, under the both DM and JDM, the expected loss due to transaction costs is inversely proportional to the square root of the hedging frequency. We apply the meanvariance analysis to find the optimal hedging frequency given the hedger's risk tolerance. Since under the JDM it is impossible to reduce the P&L volatility by increasing the hedging frequency, we consider an alternative hedging strategy, following which the P&L volatility can be reduced by increasing the hedging frequency.Quantitative Finance 05/2010; DOI:10.1080/14697688.2010.494613 · 0.75 Impact Factor 
Conference Paper: Quantitative Methods for Counterparty Risk
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ABSTRACT: 1) Counterparty risk 2) Modelling aspects 3) Pricing of credit instruments 4) Analytical Methods 5) FFT based methods 6) PDE based methodsQuantitative Finance Workshop, Technical University of Helsinki; 09/2009 
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ABSTRACT: We present a multidimensional jumpdiffusion version of a structural default model and show how to use it in order to value the credit value adjustment for a credit default swap. We develop novel analytical and numerical methods for solving the corresponding boundary value problem with a special emphasis on the role of negative asset value jumps. Using recent market data, we show that under realistic assumptions credit value adjustment greatly reduces the value of a credit default swap sold by a risky counterparty compared with one sold by a nonrisky counterparty. We identify features having the biggest impact on credit value adjustment: namely, default correlation and spread volatility. 
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ABSTRACT: We use stochastic volatility models to describe the evolution of the asset price, its instantaneous volatility, and its realized volatility. In particular, we concentrate on the SteinStein model (SSM) (1991) for the stochastic asset volatility and the Heston model (HM) (1993) for the stochastic asset variance. By construction, the volatility is not signdefinite in SSM and is nonnegative in HM. It is wellknown that both models produce closedform expressions for the prices of vanilla options via the LewisLipton formula. However, the numerical pricing of exotic options by means of the Finite Difference and Monte Carlo methods is much more complex for HM than for SSM. Until now, this complexity was considered to be an acceptable price to pay for ensuring that the asset volatility is nonnegative. We argue that having negative stochastic volatility is a psychological rather than financial or mathematical problem, and advocate using SSM rather than HM in most applications. We extend SSM by adding volatility jumps and obtain a closedform expression for the density of the asset price and its realized volatility. We also show that the current method of choice for solving pricing problems with stochastic volatility (via the affine ansatz for the Fouriertransformed density function) can be traced back to the Kelvin method designed in the nineteenth century for studying wave motion problems arising in fluid dynamics. This paper is dedicated to Professor Darryl Holm on the occasion of his sixtieth birthday.Journal of Physics A Mathematical and Theoretical 08/2008; 41. DOI:10.1088/17518113/41/34/344012 · 1.69 Impact Factor 
Article: Pricing Options on Realized Variance in the Heston Model with Jumps in Returns and Volatility
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ABSTRACT: We develop analytical methodology for pricing and hedging options on the realized variance under the Heston stochastic variance model (1993) augmented with jumps in asset returns and variance. By employing generalized Fourier transform we obtain analytical solutions (up to numerical inversion of Fourier integral) for swaps on the realized volatility and variance and for options on these swaps. We also extend our framework for pricing forwardstart options on the realized variance and volatility, including options on the VIX. Our methodology allows us to consistently unify pricing and risk managing of different volatility options. We provide an example of model parameters estimation using both time series of the VIX and the VIX options data and find that the proposed model is in agreement with both historical and implied market data. Finally, we derive a lognormal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solution for volatility options with longer maturities. 
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ABSTRACT: We first discuss the positive volatility skew observed in the implied volatilities of VIX options. To model this feature, we apply the square root stochastic variance model with variance jumps for the evolution of the S&P500 index volatility. Then we develop a robust method for unified pricing and hedging of different volatility products on the implied and realized variance of the S&P500 index and show how to apply this formula for pricing the VIX futures and options. 
Article: Dynamic credit models
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ABSTRACT: We present a dynamic framework to model the default events of individual obligors and the correlation between these default events. For the first purpose, we present the concepts of the dynamic jumptodefault model. For the second purpose, we concentrate on factor models which describe default events within a basket of obligors. In contrast to previous studies of factor credit models, we do not restrict ourselves to tractable, but not necessarily financially motivated, affine dynamics of the common factor and individual default intensities. Instead, we model the defaults using the logit survival function which depends on an appropriately chosen common factor. In the static version of the model, the distribution of the common factor is discrete, while in the dynamic version of the model the evolution of the common factor is driven by a jumpdiffusion stochastic process. To solve the calibration and pricing problem, we develop robust partial integrodifferential equation (PIDE) based numerical solution methods for the forward and backward Kolmogoroff equations. We also show how to augment the pricing problem with the loss intensity rate, and apply it to price structured credit products within the dynamic model. Finally, we provide an example of calibrating both the static and dynamic models to iTraxx credit index data.Statistics and its interface 01/2008; 1:211227. DOI:10.4310/SII.2008.v1.n2.a1 · 0.46 Impact Factor 
Conference Paper: Volatility derivatives and default risk
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ABSTRACT: 1) Heston stochastic volatility model with the termstructure of ATM volatility and the jumptodefault: interaction between the realized variance and the default risk 2) Analytical and numerical solution methods for the pricing problem 3) Case study: application of the model to the General Motors data, implicationsQuant Congress London; 11/2007 
Article: Variance Swaps Under No Conditions
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ABSTRACT: Conditional variance swaps are claims on realized variance which is accumulated when the underlying asset price stays within a certain range. Being highly sensitive to movements in both asset price and its variance, they require a very reliable model for pricing and riskmanaging. In this article we apply the Heston stochastic volatility model, which is by now a widely accepted pricing model in many markets, to derive closedform solutions for pricing and riskmanaging of conditional variance swaps.

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