[Show abstract][Hide abstract] ABSTRACT: We develop an accurate approximation of the moment generating function for the log-normal 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 log-normal SV model with simultaneous jumps in the log-price 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 skew-beta 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 delta-hedging P\&L-s and reduces transaction costs, especially so for non-vanilla options. We extend the model to multi-asset 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 log-normality of both implied and realized volatilities.
Global Derivatives Trading & Risk Management 2014; 05/2014
[Show abstract][Hide abstract] ABSTRACT: 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
Global Derivatives Trading & Risk Management 2013; 04/2013
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: We consider the delta-hedging strategy for a vanilla option under the discrete hedging and transaction costs. Assuming that the option is delta-hedged using the Black-Scholes-Merton model with an implied log-normal volatility, we analyze the profit-and-loss (P&L) of the delta-hedging strategy given that the actual underlying dynamics are driven by one of four alternative models: log-normal diffusion, jump-diffusion, 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 delta-hedging strategy. We also show how to apply our results to spot- and delta-based hedging strategies. Finally, we provide illustrations.
[Show abstract][Hide abstract] ABSTRACT: 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
Global Derivatives Trading & Risk Management 2012; 04/2012
[Show abstract][Hide abstract] 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 so-called Dupire equation. In reality, however, the number of available data points is very limited and construction of a non-arbitrageable 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 semi-analytical 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 non-arbitrageable 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.
[Show abstract][Hide abstract] 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 non-parametric local volatility volatility models with jumps and stochastic volatility using PDE methods
5) Numerical methods for PDEs
6) Illustrations using SPX and VIX data
Global Derivatives Trading & Risk Management 2011; 04/2011
[Show abstract][Hide abstract] 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 longer-term options on the realized variance, this approximation underestimates option values for short-term maturities (with maturities up to three months). We propose a method of mixing of the discrete variance in a log-normal 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 semi-analytical Fourier transform methods developed by Sepp (2008) for pricing shorter-term options on the realized variance.
Journal of Computational Finance 01/2011; 16(2). · 0.30 Impact Factor
[Show abstract][Hide abstract] 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 dimensions
Financial Engineering Workshop, University of Leicester; 12/2010
[Show abstract][Hide abstract] ABSTRACT: We analyse the profit-and-loss (P&L) of delta-hedging 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 risk-reversal, and a butterfly. Using our results, we derive the break-even realized skew and convexity that equate the average realized P&L of the risk-reversal and the butterfly, respectively, to zero. Furthermore, we analyse the impact of the volatility skew on the delta-hedging of these option strategies. We present some empirical results using implied volatilities of options on the S&P 500 index.
[Show abstract][Hide abstract] ABSTRACT: 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
Financial Engineering Workshop, Cass Business School; 05/2010
[Show abstract][Hide abstract] ABSTRACT: We introduce a jump-diffusion 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 delta-hedging strategy for vanilla options under the diffusion model (DM) and the proposed jump-diffusion model (JDM) assuming discrete trading intervals and transaction costs, and derive an approximation for the probability density function (PDF) of the profit-and-loss (P&L) of the delta-hedging strategy under the both models. We find that, under the log-normal model by Black-Scholes-Merton, the actual PDF of the P&L can be well approximated by the chi-squared 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 mean-variance 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.
[Show abstract][Hide abstract] ABSTRACT: We present a multi-dimensional jump-diffusion 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 non-risky counterparty. We identify features having the biggest impact on credit value adjustment: namely, default correlation and spread volatility.
Journal of Credit Risk 01/2009; 5(2):123-146. · 0.35 Impact Factor
[Show abstract][Hide abstract] 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 Stein-Stein 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 sign-definite in SSM and is non-negative in HM. It is well-known that both models produce closed-form expressions for the prices of vanilla options via the Lewis-Lipton 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 non-negative. 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 closed-form 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 Fourier-transformed 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(34). DOI:10.1088/1751-8113/41/34/344012 · 1.58 Impact Factor
[Show abstract][Hide abstract] 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 forward-start 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 log-normal approximation to the density of the realized variance in the Heston model and obtain accurate approximate solution for volatility options with longer maturities.
Journal of Computational Finance 04/2008; 11(4). · 0.30 Impact Factor
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] 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 jump-to-default model. For the sec-ond purpose, we concentrate on factor models which de-scribe default events within a basket of obligors. In contrast to previous studies of factor credit models, we do not re-strict ourselves to tractable, but not necessarily financially motivated, affine dynamics of the common factor and in-dividual default intensities. Instead, we model the defaults using the logit survival function which depends on an ap-propriately 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 jump-diffusion stochastic process. To solve the calibration and pricing problem, we develop robust partial integro-differential equation (PIDE) based numerical solution methods for the forward and back-ward 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(2):211-227. DOI:10.4310/SII.2008.v1.n2.a1 · 2.93 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 1) Heston stochastic volatility model with the term-structure of ATM volatility and the jump-to-default: 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, implications
[Show abstract][Hide abstract] 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 risk-managing. In this article we apply the Heston stochastic volatility model, which is by now a widely accepted pricing model in many markets, to derive closed-form solutions for pricing and risk-managing of conditional variance swaps.