The Quantitative Finance Library

Volatility Interpolation We present an effcient algorithm for interpolation and extrapolation of a discrete set of European option prices into a an arbitrage consistent full double continuum in expiry and strike of option prices. The method is based on an application of the fully implicit finite difference method and related to the local variance gamma model of Carr (2008). In a numerical example we show how the model can fitted to all quoted prices in the SX5E option market (12 expiries, each with roughtly 10 strikes) in 0.05 seconds of CPU time. , J.Andreasen, B.N.Huge (2010)

Options on Realized Variance and Convex Orders Realized variance option and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk neutral densities are said to be increasing in the convex order. For L?vy processes such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability then the resulting risk neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally we consider modelling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a L?vy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order. , P.Carr, H.Geman, M.Yor, D.P.Madan (2010)

A Consistent Pricing Model for Index Options and Volatility Derivatives We propose and study a flexible modeling framework for the joint dynamics of an index and a set of forward variance swap rates written on this index, allowing options on forward variance swaps and options on the underlying index to be priced consistently. Our model reproduces various empirically observed properties of variance swap dynamics and allows for jumps in volatility and returns. An affine specification using L?vy processes as building blocks leads to analytically tractable pricing formulas for options on variance swaps as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index. The calibration of the model is done in two steps, first by matching VIX option prices and then by matching prices of options on the underlying. , R.Cont, T.Kokholm (2009)

Multi-asset Stochastic Local Variance Contracts Variance swaps now trade actively over-the-counter (OTC) on both stocks and stock indices. Also trading OTC are variations on variance swaps which localize the payoff in time, in the underlying asset price, or both. Given that the price of the underlying asset evolves continuously over time, it is well known that there exists a semi-robust hedge for these localized variance contracts. Remarkably, the hedge succeeds even though the stochastic process describing the instantaneous variance is never specified. In this paper, we present a generalization of these results to the case of two or more underlying assets. , P.Carr, P.Laurence (2009)

Volatility Derivatives Volatility derivatives are a class of derivative securities where the payoff explicitly depends on some measure of the volatility of an underlying asset. Prominent examples of these derivatives include variance swaps and VIX futures and options. We provide an overview of the current market for these derivatives. We also survey the early literature on the subject. Finally, we provide relatively simple proofs of some fundamental results related to variance swaps and volatility swaps. , P.Carr, R.Lee (2009)

Hedging (Co)Variance Risk with Variance Swaps In this paper we introduce a new criterion in order to measure the variance and covariance risks in financial markets. Unlike past literature, we quantify the (co)variance risk by comparing the spread between the initial wealths required to obtain the same final utility in an incomplete and completed market case. We provide explicit solutions for both cases in a stochastic correlation framework where the market is completed by introducing volatility products, namely Variance Swaps. Using real data on major indexes, we find that this criterion provides a better measure of the market risks with respect to the (misleading) traditional approach based on the hedging demand. , J.Da Fonseca, M.Grasselli, F.Ielpo (2008)

Pricing and Hedging Volatility Derivatives This paper studies the pricing and hedging of variance swaps and other volatility derivatives, including volatility swaps and variance options, in the Heston stochastic volatility model. Pricing and hedging results are derived using partial differential equation techniques. We formulate an optimization problem to determine the number of options required to best hedge a variance swap. We propose a method to dynamically hedge volatility derivatives using variance swaps and a finite number of European call and put options. , M. Broadie, A. Jain (2008)

Volatility and Dividends This article revisits a simple and robust method on how to incorpo- rate the handling of cash dividends and credit risk into equity derivatives pricing and risk management. We will show that the method presented is the only such method which is consistent with the assumption of cash dividends and simple credit risk, and we will demonstrate the impact on the pricing of various derivatives: plain Europeans, American options, Barriers and ?nally variance swaps and related derivatives. This article is mainly based on results discussed in the first chapter of in Equity Hybrid Derivatives", Overhaus et al, Wiley 2006 [10] and the IQPC presentation Options On Variance: Pricing And Hedging" given in 2006 [2]. , H.Buehler (2008)

Consistent Modeling of SPX and VIX options This article revisits a simple and robust method on how to incorpo- rate the handling of cash dividends and credit risk into equity derivatives pricing and risk management. We will show that the method presented is the only such method which is consistent with the assumption of cash dividends and simple credit risk, and we will demonstrate the impact on the pricing of various derivatives: plain Europeans, American options, Barriers and ?nally variance swaps and related derivatives. This article is mainly based on results discussed in the first chapter of in Equity Hybrid Derivatives", Overhaus et al, Wiley 2006 [10] and the IQPC presentation Options On Variance: Pricing And Hedging" given in 2006 [2]. , J.Gatheral (2008)

Realized Volatility and Variance: Options via Swaps In this paper we develop strategies for pricing and hedging options on realized variance and volatility. Our strategies have the following features. * Readily available inputs: We can use vanilla options as pricing benchmarks and as hedging instruments. If variance or volatility swaps are available, then we use them as well. We do not need other inputs (such as parameters of the instantaneous volatility dynamics). * Comprehensive and readily computable outputs: We derive explicit and readily computable formulas for prices and hedge ratios for variance and volatility options, applicable at all times in the term of the option (not just inception). * Accuracy and robustness: We test our pricing and hedging strategies under skew-generating volatility dynamics. Our discrete hedging simulations at a one-year horizon show mean absolute hedging errors under 10%, and in some cases under 5%. * Easy modification to price and hedge options on implied volatility (VIX). Specifically, we price and hedge realized variance and volatility options using variance and volatility swaps. When necessary, we in turn synthesize volatility swaps from vanilla options by the Carr-Lee 4] methodology; and variance swaps from vanilla options by the standard log-contract methodology. , P.Carr, R.Lee (2007)

Moment Methods For Exotic Volatility Derivatives The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail. , C.Albanese, A.Osseiran (2007)

Variance Risk Premia We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks. 2 , P.Carr, L.Wiu (2007)

Volatility Markets Consistent modeling, hedging and practical implementation We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks. 2 , H.Buehler (2006)

Consistent Variance Curve Models We introduce a general approach to model a joint market of stock price and a term structure of variance swaps in an HJM-type framework. In such a model, strongly volatility-dependent contracts can be priced and risk-managed in terms of the observed stock and variance swap prices. To this end, we introduce equity forward variance term-structure models and derive the respective HJM-type arbitrage conditions. We then discuss ?nite-dimensional Markovian representations of the ?xed time-to-maturity forward variance swap curve and derive consistency results for both the standard case and for variance curves with values in a Hilbert space. For the latter, our representation also ensures non-negativity of the process. We then give a few examples of such variance curve functionals and discuss brie?y completeness and hedging in such models. As a further application, we show that the speed of mean-reversion in some standard stochastic volatility models should be kept constant when the model is recalibrated. , H.Buehler (2006)

Introduction to Variance Swaps The purpose of this article is to introduce the properties of variance swaps, and give insights into the hedging and valuation of these instruments from the particular lens of an option trader. • Section 1 gives general details about variance swaps and their applications. • Section 2 explains in ‘intuitive’ financial mathematics terms how variance swaps are hedged and priced. , Sebastien Bossu (2006)

Pricing Options on Realized Variance Models which hypothesize that returns are pure jump processes with independent increments have been shown to be capable of capturing the observed variation of market prices of vanilla stock options across strike and maturity. In this paper, these models are employed to derive in closed form the prices of derivatives written on future realized quadratic variation. Alternative work on pricing derivatives on quadratic variation has alternatively assumed that the underlying returns process is continuous over time. We compare the model values of derivatives on quadratic variation for the two types of models and find substantial differences. , P.Carr, D.P.Madan, M.Yor, H.Geman (2005)

Pricing Methods and Hedging Strategies for Volatility Derivatives In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive delta-hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation. , H. Windcliff, P.A. Forsyth, K.R. Vetzal (2003)

Robust Replication of Volatility Derivatives In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive delta-hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation. , P.Carr, R.Lee (2003)

Towards a Theory of Volatility Trading In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the effects of variations in the contract design, such as the differences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the difficulties associated with hedging these products. Naive delta-hedging strategies are ineffective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation. , P. Carr, D. Madan (2002)

More Than You Ever Wanted to Know About Volatility Swaps Volatility swaps are forward contracts on future realized stock volatility. Variance swaps are similar contracts on variance, the square of future volatility. Both of these instruments provide an easy way for investors to gain exposure to the future level of volatility. Unlike a stock option, whose volatility exposure is contaminated by its stock-price dependence, these swaps provide pure exposure to volatility alone. You can use these instruments to speculate on future volatility levels, to trade the spread between realized and implied volatility, or to hedge the volatility exposure of other positions or businesses. In this report we explain the properties and the theory of both variance and volatility swaps, first from an intuitive point of view and then more rigorously. The theory of variance swaps is more straightforward. We show how a variance swap can be theoretically replicated by a hedged portfolio of standard options with suitably chosen strikes, as long as stock prices evolve without jumps. The fair value of the variance swap is the cost of the replicating portfolio. We derive analytic formulas for theoretical fair value in the presence of realistic volatility skews. These formulas can be used to estimate swap values quickly as the skew changes. We then examine the modifications to these theoretical results when reality intrudes, for example when some necessary strikes are unavailable, or when stock prices undergo jumps. Finally, we briefly return to volatility swaps, and show that they can be replicated by dynamically trading the more straightforward variance swap. As a result, the value of the volatility swap depends on the volatility of volatility itself. , E.Derman, K.Demeterfi, M.Kamal (1999)