Variance Swaps

We investigate the popular formula of Chriss and Morokoff for the fair value of the log contract (Gatheral (2006, Chap. 11), Carr and Lee (2009, Sect. 5)) and show that the formula holds under the sole assumption that the log-spot is integrable. This extends the recent result of Fukasawa (2026).

The theory of pricing to acceptability developed for incomplete markets is applied to marking one's own default risk. It is observed in agreement with Heckman (2004), that assets and liabilities are not to be valued in …nancial reporting at the same magnitude. Liabilities are to be marked at the ask prices of two price economies that are above the asset mark at the bid prices for such economies. Applying cones of acceptability de…ned by concave distortions it is observed that counterintuitive pro…tability resulting from credit deterioration is then mitigated. We argue that the di¤erence between the liability mark at two price ask and the asset mark at the two price bid be taken as an upfront expense deposited in a special account called the ODOR account for Own Default Operating Reserve. Procedures are described for pricing coupon bonds separately as assets and liabilities. These procedures employ the default time distribution embedded in the CDS market.

Markets passively accept a convex cone of cash ‡ows that contains the the nonnegative cash ‡ows. Di¤erent markets are de…ned by di¤erent cones and conditions are established to exclude the possibility of arbitrage between markets. Operationally these cones are de…ned by positive expectation under a concave distortion of the distribution function of the cash ‡ow delivered to market. Firms access risky assets and risky liabilities and regulatory bodies ensure that su¢ cient capital is put at stake to make the risk of excess loss acceptable to taxpayers. Firms approach equity markets for funding and can come into existence only if they can raise su¢ cient equity capital. Firms that are allowed to exist approach debt markets for favorable funding opportunities. The costs of debt limit the amount of debt. Firms with lognormally distributed and correlated assets and liabilities are analysed for their required capital, their optimal debt levels, the value of the option to put losses back to the taxpayer, the costs of debt and equity, and the level of …nally reported equity in the balance sheet. The relationship between these entities and the risk characteristics of a …rm are analysed and reported in detail. We thank Martijn Pistorius for comments on an earlier version of the paper.

Portfolios are selected in non-Gaussian contexts to maximize a Cherny and Madan index of acceptability. Analytical gradients are developed for the purpose of optimizing portfolio searches on the unit sphere. It is shown that though an acceptability index is not a preference ordering, many utilities will concur with acceptability maximization. A stylized economy illustrates the advantages from the perspective of acceptability of nonlinear securities and options. In sample results for the year 2008 indicate that maximizing the acceptability index can lead to portfolios that second order stochastically dominate their Gaussian counterparts. Backtests over the period 1997 to 2008 re ‡ect gains to maximizing acceptability over holding a maximal Sharpe ratio portfolio.

In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, w...

Observing that pure discount curves are now based on a variety of tenors giving rise to tenor speci…c zero coupon bond prices, the question is raised on how to construct tenor speci…c prices for all …nancial contracts. Noting that in conic …nance one has the law of two prices, bid and ask, that are nonlinear functions of the random variables being priced, we model dynamically consistent sequences of such prices using the theory of nonlinear expectations. The latter theory is closely connected to solutions of backward stochastic di¤erence equations. The drivers for these stochastic di¤erence equations are here constructed using concave distortions that implement risk charges for local tenor speci…c risks. It is then observed that tenor speci…c prices given by the mid quotes of bid and ask converge to the risk neutral price as the tenor is decreased and liquidity increased when risk charges are scaled by the tenor. Square root tenor scaling can halt the convergence to risk neutral pri...

The theory of two price markets of Cherny and Madan (2010) yields closed forms for bid and ask prices. De…ning pro…ts as the di¤erence between the mid quote and the risk neutral expectation and capital as difference between the ask and the bid price one obtains precise expressions for these entities and thereby also returns. New expressions are developed for the bid and ask prices in terms of the sensitivity of the inverse distribution function to the quantile level. The latter turns out to be a measure of risk exposure at the quantile level. The theory is illustrated on unhedged exposures in the Black Merton Scholes model, followed by variance swaps and call options for variance gamma underliers. It is argued that markets should economize capital and furthermore the maximization of expected utility may involve an uneconomic use of capital. We further observe that stock positions should be revised downwards from zero delta in left skewed markets in response to the target gamma when minimizing capital committments. This paper is the private opinion of the authors and does not necessarily re ‡ect policy or research of Morgan Stanley. This paper is being provided to you solely for your review and consideration for future publication. It should be treated as con…dential material of the submitter and not shared with anyone outside of your organization. No rights to publish or publicly distribute are granted at this time.

In this work we consider the pricing of a special class of volatility derivatives, the so-called variance swaps. The fair value of a variance swap is equal to the expected value of the realized variance of the underlying of the swap during the lifetime of the contract. It is shown how this expected value can be computed by means of an exotic option with logarithmic pay-off. We show how to statically replicate this pay-off in terms of a basket of synthetic vanilla call and put options. We apply this construction to the TNLP4 ticker of BOVESPA and synthetize a basket with pure exposure to volatility using actual market prices.

In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, w...

We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.

It is well documented that a model for the underlying asset price process that seeks to capture the behaviour of the market prices of vanilla options needs to exhibit both diffusion and jump features. In this paper we assume that the asset price process S is Markov with càdlàg paths and propose a scheme for computing the law of the realized variance of the log returns accrued while the asset was trading in a prespecified corridor. We thus obtain an algorithm for pricing and hedging volatility derivatives and derivatives on the corridor-realized variance in such a market. The class of models under consideration is large, as it encompasses jump-diffusion and Lévy processes. We prove the weak convergence of the scheme and describe in detail the implementation of the algorithm in the characteristic cases where S is a CEV process (continuous trajectories), a variance gamma process (jumps with independent increments) or an infinite activity jump-diffusion (discontinuous trajectories with ...

We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.

Volatility swaps and volatility options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are often priced by continuously sampled approximations to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under a general stochastic volatility model. We first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to price discrete volatility derivatives through either semi-analytical pricing formulae (up to an inverse Fourier transform), or an efficient Fourier-cosine series method. Numerical experiments show that our approximation is more accurate in comparison to the approximations in the literature. We remark that although discretely sampled variance swaps and options are usually more expensive than their continuously sampled counterparts, discretely sampled volatility swaps are more prone to be cheaper than the continuously sampled counterparts. An analysis is then provided to explain why this is the case in general for realistic contract specifications and reasonable model parameters.

We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market's expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility.

We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston (1993) two-factor stochastic volatility model embedded in the framework proposed by Little and Pant (2001). In comparison with all the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include: (a) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (b) numerical values can be very efficiently computed from the newly found analytic formula.

This paper is an extension to a recent paper by Zhu and Lian (2011) [1], in which a closedform exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston (1993)'s [2] two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved.

We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market's expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility.

I rony, according to the Oxford English Dictionary, is (a) a figure of speech in which the intended meaning is the opposite of that expressed by the words used (b) a condition of affairs or events of a character opposite to what was, or might naturally be, expected (In French, ironie du sort). In the following lines, I will propose a rereading of quantitative finance where irony, as opposed to theory, emerges as a leitmotiv, perhaps even a main guide. Variance swaps will provide me with a literal rehearsal of this ironic point, as I will show that the element of irony is inscribed in the movement motivating their existence, indeed in their very contractual terms. The irony in quantitative finance, or specifically, in derivative pricing theory, is captured by the following observation. While the typical derivative paper is expressed in words and formulas aiming at the theoretical value of the derivative, it is really intended for derivative trading, which is the domain farthest away ...

We develop a simplified analytical approach for pricing discretely-sampled variance swaps with the realized variance, defined in terms of the squared log return of the underlying price. The closed-form formula obtained for Heston’s two-factor stochastic volatility model is in a much simpler form than those proposed in literature. Most interestingly, we discuss the validity of our solution as well as some other previous solutions in different forms in the parameter space. We demonstrate that market practitioners need to be cautious, making sure that their model parameters extracted from market data are in the right parameter subspace, when any of these analytical pricing formulae is adopted to calculate the fair delivery price of a discretely-sampled variance swap.

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston (1993) two-factor stochastic volatility model embedded in the framework proposed by Little and Pant (2001). In comparison with all the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include: (a) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (b) numerical values can be very efficiently computed from the newly found analytic formula.

Pricing variance swaps under stochastic volatility with discretely-sampled realized variance has been a hot subject pursued recently; quite a few papers have already been published (Zhu and Lian (2009, 2011, [11,4]); Swishchuk and Li (2011) [5]). In this paper, we present a simplified approach to price discretely-sampled variance swaps. Compared with the approach presented by Zhu and Lian (2011) [4], an important feature of our approach is that there is no need for the introduction of a new state variable and the utilization of the generalized Fourier transform. This has significantly simplified the solution procedure and will thus enable researchers to view this type of problems from a different angle.

This paper is an extension to a recent paper by Zhu and Lian (2011) [1], in which a closedform exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston (1993)'s [2] two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved.

Derivative securities are frequently priced within the Black-Scholes methodology. Theoretically this entails maintaining a hedge consisting of the underlying asset and cash which needs to be rebalanced continuously. In practice, traders would only rebalance such hedges on a discrete basis. We examine the effects of discrete rebalancing of derivative hedges written on the FTSE/JSE TOP40 index.

Biases in standard variance swap rates can induce substantial deviations below market rates. Defining realised variance as the sum of squared price (not log-price) changes yields an 'arithmetic' variance swap with no such biases. Its fair value has advantages over the standard variance swap rate: no discrete-monitoring or jump biases; and the same value applies for any monitoring frequency, even irregular monitoring and to any underlying, including those taking zero or negative values. We derive the fair-value for the arithmetic variance swap and compare with the standard variance swap rate by: analysing errors introduced by interpolation and integration techniques; numerical experiments for approximation accuracy; and using 23 years of FTSE 100 options data to explore the empirical properties of arithmetic variance (and higher-moment) swaps. The FTSE 100 variance risk has a strong negative correlation with the implied third moment, which can be captured using a higher-moment arithmetic swap.

The concept of the gamma of a financed return as the highest level of stress that a return distribution can withstand is introduced. Stress is measured by positive expectation under a concave distortion of the return distribution accessed. Four distortions introduced in Cherny and Madan (2008) are employed in studying the distribution of returns available in the hedge fund universe. It is shown that the skewness, peakedness and tailweightedness of the standardized investment return significantly affects the Sharpe ratios required to reach a target gamma level.

The theory of two price markets of Cherny and Madan (2010) yields closed forms for bid and ask prices. De…ning pro…ts as the di¤erence between the mid quote and the risk neutral expectation and capital as difference between the ask and the bid price one obtains precise expressions for these entities and thereby also returns. New expressions are developed for the bid and ask prices in terms of the sensitivity of the inverse distribution function to the quantile level. The latter turns out to be a measure of risk exposure at the quantile level. The theory is illustrated on unhedged exposures in the Black Merton Scholes model, followed by variance swaps and call options for variance gamma underliers. It is argued that markets should economize capital and furthermore the maximization of expected utility may involve an uneconomic use of capital. We further observe that stock positions should be revised downwards from zero delta in left skewed markets in response to the target gamma when minimizing capital committments. This paper is the private opinion of the authors and does not necessarily re ‡ect policy or research of Morgan Stanley. This paper is being provided to you solely for your review and consideration for future publication. It should be treated as con…dential material of the submitter and not shared with anyone outside of your organization. No rights to publish or publicly distribute are granted at this time.

We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market's expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility.

Volatility measures variability, or dispersion about a central tendency --- it is simply a measure of the degree of price movement in a stock, futures contract or any other market. Volatility also has many subtleties that make it challenging to analyze and implement. The following questions immediately come to mind: is volatility a simple intuitive concept or is it complex in nature, what causes volatility, how do we estimate volatility and can it be managed? The management of volatility is currently a topical issue. Due to this, many institutional and individual investors have shown an increased interest in volatility as an investment vehicle. Variance swaps or variance futures (also called variance contracts) are equity derivative instruments offering pure exposure to daily realised future variance.

This paper provides the first estimation strategy for the Wishart Affine Stochastic Correlation (WASC) model. We provide elements to show that the utilization of empirical characteristic function-based estimates is advisable: this function is exponential affine in the WASC case. We use a GMM estimation strategy with a continuum of moment conditions based on the characteristic function. We present the estimation results obtained using a dataset of equity indexes. The WASC model captures most of the known stylized facts associated with financial markets, including the leverage and asymmetric correlation effects.

This paper provides the first estimation strategy for the Wishart Affine Stochastic Correlation (WASC) model. We provide elements to show that the utilization of empirical characteristic function-based estimates is advisable: this function is exponential affine in the WASC case. We use a GMM estimation strategy with a continuum of moment conditions based on the characteristic function. We present the estimation results obtained using a dataset of equity indexes. The WASC model captures most of the known stylized facts associated with financial markets, including the leverage and asymmetric correlation effects.

Derivative exchanges and the trading of derivatives have existed for a long time. CBOE 1 started trading call options way back in 1973. In South Africa, an organised trading in index future contracts started with the establishment of Safex 2 during 1988. Trading in single name futures (also known as single stock futures) started in 1997. The liquidity in the ALSI and DTOP index futures as well as the liquidity in most of the FTSE/JSE TOP 40 single name futures has grown tremendously over the past 5 years. A liquid and vibrant onscreen market makes it now possible to use these futures as alternative hedging instruments.

In 2007, the SAVI was launched as an index designed to measure the market's expectation of the 3-month market volatility. The SAVI soon became the benchmark for measuring the market sentiment, and in this light can be thought of as a market "fear" index. Two years later, in 2009 the Johannesburg Stock Exchange updated the SAVI to reect a new way of measuring the expected volatility, one that is consistent with the theoretical framework, risk-management and the way traders trade options. The new SAVI is calculated as the at-the-money volatility adjusted for the volatility skew as determined by the actively traded options in
the market. The aim of this note is to introduce the new SAVI calculation method, and briefly discuss the benets of the new SAVI.