Volatility Derivatives Research Articles

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.

Asymptotics for volatility derivatives in multi-factor rough volatility models

We study the small-time implied volatility smile for Realised Variance options, and investigate the effect of correlation in multi-factor models on the linearity of the smile. We also develop an approximation scheme for the Realised Variance density, allowing fast and accurate pricing of Volatility Swaps. Additionally, we establish small-noise asymptotic behaviour of a general class of VIX options in the large strike regime.

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.

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.

Withdrawn: Yan, C, Zhao, B. (2018). A general jump‐diffusion process to price volatility derivatives. J Futures Markets, 39, 15–37. https://doi.org/10.1002/fut.21962

Withdrawn: Yan, C, Zhao, B. (2018). A general jump‐diffusion process to price volatility derivatives. J Futures Markets, 39, 15–37. https://doi.org/10.1002/fut.21962

Fast maximum likelihood estimation of parameters for square root and Bessel processes

Abstract Explicit formulae for maximum likelihood estimates of the parameters of square root processes and Bessel processes and first and second order approximate sufficient statistics are supplied. Applications of the estimation formulae to simulated interest rate and index time series are supplied, demonstrating the accuracy of the approximations and the extreme speed-up in estimation time. This significantly improved run time for parameter estimation has many applications where ex-ante forecasts are required frequently and immediately, such as in hedging interest rate, index and volatility derivatives based on such models, as well as modelling credit risk, mortality rates, population size and voting behaviour.

Volatility and variance swaps and options in the fractional SABR model

Appropriate capturing the nature of financial market volatility is a significant factor for the pricing of volatility derivatives. A recent study by Gatheral, Jaisson and Rosenbaum [2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949] has found that log-volatility behaves as a fractional Brownian motion with a small Hurst exponent at any reasonable time scale. Also, there are several empirical works showing that a stochastic volatility model driven by the fractional Brownian motion well approximates at-the-money volatility skew near expiration. In this paper, we choose the log-normal SABR model with fractional stochastic volatility to valuate variance and volatility swaps. We derive a closed-form exact solution for the fair strike price of the variance swap by using fractional Ito calculus, while we obtain an approximate solution for the fair strike price of the volatility swap by exploiting the shifted log-normal approximation. Also, solution formulas for the variance and volatility option prices are derived. Their accuracy is confirmed through numerical studies. Calibration to market variance swap rates demonstrates the strength of fractional SABR model compared to the Heston and SABR models.

The economics of the financial market for volatility trading

We examine the economics of the financial market for volatility trading based on an equilibrium model with three kinds of traders: dealers, asset managers, and leveraged funds. Our model reveals that the negative price of volatility is due to the high short positions of dealers, low short positions of leveraged funds, and high long positions of asset managers. It also explains well the negative variance risk premium and the negative returns of volatility derivatives. Our empirical analysis based on VIX futures position data with weekly frequency from 2006 to 2016, furthermore, supports the model's implications.

Volatility Derivatives for Children

We describe the steps required to approximately price volatility derivatives by making use of only three near the money vanilla index options per maturity. The only assumption we make is that the index process is a diffusion with a volatility process that does not depend on the index level. No pretense of mathematical rigor or sophistication is made; this note should be seen as a playful and rudimentary attempt at pricing volatility derivatives with minimal assumptions and market instruments.

Volatility and volatility-linked derivatives: estimation, modeling, and pricing

This article serves the purpose of reviewing recent developments of the estimation and modeling of volatilities for financial products as well as on the pricing and hedging of financial derivatives that are related to volatility under certain models. It also provides a reading guide for the contributed papers. Emphases are put on continuous processes in continuous time. Discrete time models and models with jumps are not included in the discussion. Presentation of the article is more intuitive and heuristic rather than mathematically sound and rigorous in nature.

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.

On Carr and Lee’s Correlation Immunization Strategy

ABSTRACTIn their seminal work Robust Replication of Volatility Derivatives, Carr and Lee show how to robustly price and replicate a variety of claims written on the quadratic variation of a risky asset under the assumption that the asset’s volatility process is independent of the Brownian motion that drives the asset’s price. Additionally, they propose a correlation immunization strategy that minimizes the pricing and hedging error that results when the correlation between the risky asset’s price and volatility is non-zero. In this paper, we show that the correlation immunization strategy is the only strategy among the class of strategies discussed in Carr and Lee's paper that results in real-valued hedging portfolios when the correlation between the asset’s price and volatility is non-zero. Additionally, we perform a number of Monte Carlo experiments to test the effectiveness of Carr and Lee’s immunization strategy. Our results indicate that the correlation immunization method is an effective means of reducing pricing and hedging errors that result from a non-zero correlation.

An efficient exponential twisting importance sampling technique for pricing financial derivatives

This paper develops an efficient Exponential Twisting Importance Sampling technique for variance reduction when it is used to price financial derivatives. A general class of exponential change of measure is well explored. This is the generalization of the proportional exponential twisting function, , discussed as an Example 4.6.2 in Glasserman’s book (2004). Then a new framework to find the optimal importance sampling density function is proposed based on the Cauchy-Schwartz Inequality and the Least Square Approach. In the special Gaussian case, the theory of our general exponential change of measure means finding the new drift vector and covariance matrix simultaneously. The method proposed by the paper has little smoothness requirements for the payoff functions and doesn’t rely on the initial values. It is illustrated that this method is high efficient for pricing financial derivatives, such as Asian options, Straddle options, Volatility derivatives and the pricing under the stochastic volatility models. Furthermore, this method can be extended to more general importance sampling densities such as non-Gaussian or multi-modal distributions.

Volatility Derivatives and Market (Il)liquidity

We study how derivatives (with nonlinear payoffs) affect the liquidity of the underlying asset. In a rational expectations equilibrium, informed investors expect low conditional volatility and sell derivatives to the others. These derivative trades affect investors’ utility differently, possibly amplifying liquidity risk. As investors delta hedge their derivative positions, price impact in the underlying drops, suggesting improved liquidity, because informed trading is diluted. In contrast, effects on price reversal are ambiguous, depending on investors’ relative delta hedging sensitivity, i.e., the gamma of the derivatives. The model cautions of potential disconnections between illiquidity measures and liquidity risk premium due to derivatives trading.

It Takes Three to Smile

An equation relating the Black-Merton-Scholes Greeks vega, vanna, and volga is derived by making use of the generalised Hull-White formula. Given only three options the equation automatically gives a robust and accurate approximation for the volatility swap strike, the variance swap strike, and a quantity which we name the adjusted correlation. Once the aforementioned quantities have been implied from the three pillar options, an approximation for the entire volatility skew can be constructed. A potentially more interesting implication of the vega-vanna-volga relationship is that the volatility skew can be deformed into a smile in a logically consistent manner. In other words we give a prescription for converting vanilla options priced in a world with non-zero correlation between the index and volatility, to prices in a world in which the correlation would be approximately zero. As valuation of volatility derivatives is significantly easier in a zero correlation world, the translation offers a simplification of the problem of valuation of volatility derivatives in addition to the fact that all the aforementioned results can be achieved starting with only three quoted options on the index.

Variance and skew risk premiums for the volatility market: The VIX evidence

AbstractWe extract variance and skew risk premiums from volatility derivatives in a model‐free way and analyze their relationships along with volatility index and equity index returns. These risk premiums can be synthesized through option trading strategies. Using a time series of option prices on the VIX, we find that variance swap excess return can be partially explained by volatility index and equity index excess returns while these latter variables carry little information for the skew swap excess return. The results sharply contrast with those obtained for the equity index option market underlining very specific characteristics of the volatility derivative market.

Withdrawn: A general jump‐diffusion process to price volatility derivatives

This paper proposes a general but tractable approach to price volatility derivatives which includes most existing stochastic volatility models as special cases. Specifically, we use the generalized beta prime distribution to model the dynamics of the volatility underlying, which is flexible enough to capture the empirical marginal density function and yield desired implied volatility slope consistent with empirical data. Based on that, we use a Jacobi process to add a diffusion component, and further extend our model through stochastic time changes to capture jumps as well as to accommodate the term structure of volatility index futures. Our approach is supported by calibrations.

INDEX OPTIONS AND VOLATILITY DERIVATIVES IN A GAUSSIAN RANDOM FIELD RISK-NEUTRAL DENSITY MODEL

We propose a risk-neutral forward density model using Gaussian random fields to capture different aspects of market information from European options and volatility derivatives of a market index. The well-structured model is built in the framework of the Heath–Jarrow–Morton philosophy and the Musiela parametrization with a user-friendly arbitrage-free condition. It reduces to the popular geometric Brownian motion model for the spot price of the market index and can be intuitively visualized to have a better view of the market trend. In addition, we develop theorems to show how the model drives local volatility and variance swap rates. Hence, volatility futures and options can be priced taking the forward density implied by European options as the initialization input. The model can be accordingly calibrated to the market prices of these volatility derivatives. An efficient algorithm is developed for both simulating and pricing, and a numerical study is conducted using real market data.

The behaviour of energy-related volatility indices around scheduled news announcements: Implications for variance swap investments

This paper investigates the effect of scheduled news announcements on two energy-related volatility indices calculated by the Chicago Board of Options Exchange. These volatility indices are the crude oil volatility index (OVX) and the energy sector volatility index (VXXLE), which measure the market's expectation of 30-day volatility of crude oil prices and energy sector returns, respectively. We find that energy-related volatility indices tend to fall following several US macroeconomic news releases and OPEC meetings. Put differently, these news releases resolve investors' uncertainty about the future development of energy-related markets. Our results also suggest that macroeconomic releases and oil-related events do not seem to exert significantly different impacts on energy market uncertainty. We further document that the observed announcement day patterns cannot be exploited to generate significant returns using variance swaps once we incorporate transaction costs. Our findings are relevant because there are currently no other volatility derivatives traded in exchange markets, such as options or futures, that offer investors pure exposure to energy-related volatility alone as variance swaps do.

Forecasting and trading high frequency volatility on large indices

The present paper analyses the forecastability and tradability of volatility on the large S&P500 index and the liquid SPY ETF, VIX index and VXX ETN. Even though there is already a huge array of literature on forecasting high frequency volatility, most publications only evaluate the forecast in terms of statistical errors. In practice, this kind of analysis is only a minor indication of the actual economic significance of the forecast that has been developed. For this reason, in our approach, we also include a test of our forecast through trading an appropriate volatility derivative. As a method we use parametric and artificial intelligence models. We also combine these models in order to achieve a hybrid forecast. We report that the results of all three model types are of similar quality. However, we observe that artificial intelligence models are able to achieve these results with a shorter input time frame and the errors are uniformly lower comparing with the parametric one. Similarly, the chosen models do not appear to differ much while the analysis of trading efficiency is performed. Finally, we notice that Sharpe ratios tend to improve for longer forecast horizons.