Variance Swaps Research Articles

Practical Applications of The Bitcoin VIX and Its Variance Risk Premium

In <b>The Bitcoin VIX and Its Variance Risk Premium</b>, published in the Spring 2021 issue of <b><i>The Journal of Alternative Investments</i></b>, <b>Carol Alexander</b> and <b>Arben Imeraj</b> (both of the <b>University of Sussex</b>) introduce the bitcoin volatility index. CryptoCompare now streams this index every 15 seconds, under the ticker BVIN. Alexander and Imeraj are the first to investigate the bitcoin variance risk premiums and the behavior of the term structure of fair-value variance swap rates. The authors collect price data on bitcoin derivatives traded on the Deribit exchange via its application programming interface. They construct a family of indexes for different maturities using the same methodology used by CBOE’s equity volatility index, the VIX. They describe the methodology, noting that it accounts for information in volatility skews but assumes no jumps in prices. They also compare the indexes with those created with an alternative technique that does not rely on the no-jump assumption. In addition, they explore the diversification potential of bitcoin variance through correlation matrixes with other assets’ volatility indexes, realized volatilities, and other variance risk premiums. <b>TOPICS:</b>Currency, mutual funds/passive investing/indexing, statistical methods, performance measurement

Model risk in the over-the-counter market

We propose a methodology to measure the parameter estimation risk and model specification risk of pricing models, as well as model selection risk of model classes, based on realized payoffs, for products in the over-the-counter market. Lévy jump models and affine jump-diffusion models are applied in estimating the fair variance strikes of variance swaps and forward starting option prices. Our results show that both parameter estimation risk and model specification risk are significant for variance swaps, while model specification risk is dominant when pricing forward starting options. We also find that the size of the model selection risk is substantial for both products.

On pricing variance swaps in discretely-sampled with High Volatility model

In this paper, the valuation of discretely sampled variance swaps is investigated in a financial asset price model with an increase in volatility. More precisely, we consider a stochastic differential equation model with an additional parameter that augments volatility. This is to cover the impact of financial crunches on the prices of a given asset. Under these settings, the calculation of the annualized delivery price of a variance swap is not sure in a closed-form. Following the literature, the delivery price can be written as a finite sum of conditional expectations. We focus our attention on the computations of these expectations and we obtain some interesting results. This leads to a semi- analytical solution to the variance swaps pricing problems. Some illustrations showing the goodness of our model are provided.

Discrete variance swap in a rough volatility economy

AbstractThe discrete variance swap is one of the most popular volatility derivatives traded on the over‐the‐counter market. This paper discusses its valuation in a rough volatility economy and the impact of roughness on the term structure of discrete variance swap prices. A semianalytic solution is obtained through stochastic convolution. Our numerical experiments show that the roughness of volatility has a significant impact on the concavity of the variance swap term structure.

Central bank tone and currency risk premia

I analyze how the tone of central bank press conferences impacts risk premia in the currency market. Tone is measured as the difference between the number of hawkish and dovish phrases mafide during a press conference, as a fraction of total phrases. A one standard deviation increase in the hawkishness of a press conference is associated with a 1.5% decline in the variance risk premium, a 4.9% decline in the subsequent variance swap return, and a 0.2% increase in option implied risk aversion. The impact of tone comes primarily from the questions & answers, or the unscripted portion of the press conference.

Analytically pricing volatility swaps and volatility options with discrete sampling: Nonlinear payoff volatility derivatives

This paper presents the first analytical pricing formulas for volatility swaps and volatility options with discrete sampling under the Black-Scholes model with time varying risk-free interest rate. Despite numerous analytical works on the pricing of variance swaps with discrete sampling under different models of asset prices, an analytical pricing formula for volatility swaps as well as volatility options had not been well addressed until now. The main challenge in pricing volatility swaps and volatility options is that payoff functions contain a square root operator, making their expectations nonlinear. By utilizing properties of noncentral chi random variables, we can compute expectations of payoff functions analytically and obtain formulas for pricing volatility swaps and volatility options, including variance swaps and variance options. Furthermore, we investigate the accuracy of the well-known convexity correction formula. Most interestingly, we extend our results to the Black-Scholes model with time varying parameters and to the Heston stochastic volatility model in which the variance process is assumed to follow the extended Cox-Ingersoll-Ross process by constructing simple closed-form approximate formulas for pricing volatility swaps and demonstrate the accuracy and efficiency of this approach by comparing the approximated prices against those obtained with Monte Carlo simulations.

Term Structure of Risk Factor Premiums Used for Pricing Asset: Emerging vs. Developed Markets

ABSTRACT The aim of this empirical study was to estimate and compare the term structure of risk factor premiums in developed and emerging markets. Most studies use dividend and variance swap data, but as that information is not available for all markets, we use wavelet decomposition of the observed return to calculate sensitivity to risk factors and obtain a term structure for risk factor premiums. The results show that only the market risk factor (for both types of markets) and the conservative minus aggressive factor (only for developed markets) show a term structure for risk premiums.

Pricing of volatility derivatives in a Heston–CIR model with Markov-modulated jump diffusion

This paper concerns the pricing of volatility and variance swaps with discrete sampling times using a hybrid Heston–CIR model with Markov-modulated jump–diffusion. We extend the regime-switching Heston stochastic volatility model by further considering the Cox–Ingersoll–Ross (CIR) stochastic interest rate with jump diffusion. The market parameters, including the jump–diffusion part, are modulated by a continuous-time Markov chain that represents different market states. The proposed hybrid model can capture the stochastic property of volatility rate, interest rate as well as the short-term and long-term effects on the financial market caused by unexpected events and structural changes of the macroeconomic environment respectively. The fair strike prices of variance swaps and volatility swaps based on four different formulae for the realized variance (volatility) are obtained by utilizing the characteristic function method. The accuracy and efficiency of our method are validated through the semi-Monte-Carlo simulation. Finally, numerical and sensitivity analysis are conducted to show the impact of the change of each market parameter. Overall, pricing results were greatly enhanced with a more realistic financial model, and more driving forces are considered to investigate the formation of the discrete variance and volatility swap prices more comprehensively.

A New Nonparametric Estimate of the Risk-Neutral Density with Applications to Variance Swaps

Estimates of risk-neutral densities of future asset returns have been commonly used for pricing new financial derivatives, detecting profitable opportunities, and measuring central bank policy impacts. We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S&amp;amp;P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the normal inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the Chicago Board Options Exchange website.

Liquidity Guided Machine Learning: The Case of the Volatility Risk Premium

The financial industry has eagerly adopted machine learning algorithms to improve on traditional predictive models. In this paper we caution against blindly applying such techniques. We compare forecasting ability of machine learning methods in evaluating future payoffs on synthetic variance swaps. Standard machine learning methods tend to identify contracts which are illiquid, and hard to trade. The most successful strategies turn out to be those where we pair machine learning with institutional and market/traders inputs and insights. We show that liquidity guided pre-selection of inputs to machine learning results in trading strategies with improved pay-offs to the writers of variance swap contract replicating portfolio.

Variance Swap Pricing under Markov-Modulated Jump-Diffusion Model

This paper investigates the pricing of discretely sampled variance swaps under a Markov regime-switching jump-diffusion model. The jump diffusion, as well as other parameters of the underlying stock’s dynamics, is modulated by a Markov chain representing different states of the market. A semi-closed-form pricing formula is derived by applying the generalized Fourier transform method. The counterpart pricing formula for a variance swap with continuous sampling times is also derived and compared with the discrete price to show the improvement of accuracy in our solution. Moreover, a semi-Monte-Carlo simulation is also presented in comparison with the two semi-closed-form pricing formulas. Finally, the effect of incorporating jump and regime switching on the strike price is investigated via numerical analysis.

PRICING VARIANCE SWAPS UNDER DOUBLE HESTON STOCHASTIC VOLATILITY MODEL WITH STOCHASTIC INTEREST RATE

In this paper, we discuss the problem of pricing discretely sampled variance swaps under a hybrid stochastic model. Our modeling framework is a combination with a double Heston stochastic volatility model and a Cox–Ingersoll–Ross stochastic interest rate process. Due to the application of the T-forward measure with the stochastic interest process, we can only obtain an efficient semi-closed form of pricing formula for variance swaps instead of a closed-form solution based on the derivation of characteristic functions. The practicality of this hybrid model is demonstrated by numerical simulations.

The Log Moment formula for implied volatility

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that when the underlying stock price martingale admits finite log-moments E[|log (S)|^q] for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile is less constrained than Lee's bound. The result is rationalised by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying martingale to admit any negative moment. In this respect, the result can derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral-Fukasawa formula expressing variance swaps in terms of the implied volatility.

Pricing Variance Swaps on Time-Changed Markov Processes

We prove that the variance swap rate (fair strike) equals the price of a co-terminal European-style contract when the underlying is an exponential Markov process, time-changed by an arbitrary conti...

The effects of transaction costs and illiquidity on the prices of volatility derivatives

The prices of assets differ when considering transaction costs and their illiquidity, thus affecting their returns. As volatility derivatives such as variance swaps, gamma swaps, corridor variance swaps and volatility swaps depend on the expected values of these returns, we develop a high-order algorithm so that differences in the fair values of the volatility derivatives under different transaction costs and illiquidity models can be precisely observed. A high-order finite-difference algorithm is used to solve a framework of partial differential equations. A local mesh refinement strategy is applied at the singularities that appear in the pricing problem of some volatility derivatives so that nonuniform five-point stencil finite differences can be implemented in space. The time-stepping is dealt with by an efficient iterative scheme coupled with a Richardson extrapolation and Carathéodory–Fejér approximations. Consequently, the highly convergent solutions are used to analyze the effect of transaction costs and illiquidity on the fair values of volatility derivatives, and the results demonstrate an increase in these values when different transaction costs and liquidity models are implemented.

Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields

Analytically pricing variance swaps in commodity derivative markets under stochastic convenience yields

Multi-asset generalized variance swaps in Barndorff-Nielsen and Shephard model

This paper proposes swaps on two important new measures of generalized variance, namely, the maximum eigenvalue and trace of the covariance matrix of the assets involved. We price these generalized variance swaps for Barndorff-Nielsen and Shephard model used in financial markets. We consider multiple assets in the portfolio for theoretical purpose and demonstrate our approach with numerical examples taking three stocks in the portfolio. The results obtained in this paper have important implications for the commodity sector where such swaps would be useful for hedging risk.

Variance Swaps Under Multiscale Stochastic Volatility of Volatility

Many hedge funds and retail investors demand volatility and variance derivatives in order to manage their exposure to volatility and volatility-of-volatility risk associated with their trading positions. The Heston model is a standard popular stochastic volatility model for pricing volatility and variance derivatives. However, it may fail to capture some important empirical features of the relevant market data due to the fact that the elasticity of volatility of volatility of the underlying price takes a special value, i.e., 1/2, whereas it has a merit of analytical tractability. We exploit a multiscale stochastic extension of volatility of volatility to obtain a better agreement with the empirical data while taking analytical advantage of the original Heston dynamics as much as possible in the context of pricing discrete variance swaps. By using an asymptotic technique with two small parameters, we derive a quasi-closed form formula for the fair strike price of variance swap and find useful pricing properties with respect to the stochastic extension parameters.

English

English

A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean

Based upon the fact that a constant long-term mean could not provide a good description of the term structure of the implied volatility and variance swap curve, as suggested by Byelkina and Levin (in: Sixth world congress of the Bachelier Finance Society, Toronto, 2010) and Forde and Jacquier (Appl Math Finance 17(3):241–259, 2010), this paper presents a new stochastic volatility model, by assuming the long-term mean of the volatility in the Heston model be stochastic. An important feature of our model is that it still preserves the essential advantage of the Heston model, i.e., the analytic tractability, because a closed-form pricing formula for European options can be derived, which could not only facilitate the risk management process but also help save plenty of time in terms of model calibration. The effect of the newly introduced stochastic long-term mean is demonstrated through the numerical comparison with the Heston model. It is also shown that the current model can overall lead to more accurate option prices than the Heston model, through a carefully designed empirical study.