We define and study properties of implied volatility for American perpetual put options. In particular, we show that if the market prices are derived from a local volatility model with a monotone volatility function, then the corresponding implied volatility is also monotone as a function of the strike price.

This study builds upon the existing literature on the Working curve and backwardation to explore the impact of storage regimes on the volatility measures of substitute agricultural commodity markets. We investigate the impact of commodity fundamentals (storage regime and stocks-to-use ratio), commodity-specific financial variables (options hedging pressure-long and -short), world economic activity, market-wide volatility index, seasonality, and time-to-maturity on nearby and deferred implied volatility (IV) series of selected commodity pairs of corn-soybean and winter wheat-spring wheat. Our work confirms that, in some cases, grain and oilseed IV derived from options premia respond to shocks in commodity (and substitute commodity) fundamentals which are in line with the behaviour of volatility in futures markets. Own-storage regime effects on price variability are stronger in the selected markets, while spillover effects from substitute commodity storage regimes show a modest impact on volatilities. We also find some evidence for the stocks-to-use ratio of both corn and soybean to impact both their own and each other’s IV, while options hedging pressure has some impact only on wheat IVs.

This paper proposes a novel stochastic volatility model with a flexible jump structure. This model allows both contemporaneous and independent arrival of jumps in return and volatility. Moreover, time-varying jump intensities are used to capture jump clustering. In the proposed framework, we provide a semi-analytical solution for the pricing problem of VIX futures and options. Through numerical experiments, we verify the accuracy of our pricing formula and explore the impact of the jump structure on the pricing of VIX derivatives. We find that the correct identification of the market jump structure is crucial for pricing VIX derivatives, and misspecified model setting can yield large errors in pricing.

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated.

The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

We extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner–Ellis theorem and sharp large deviations tools.

Economists and others need estimates of future cash price volatility to use in risk management evaluation and education programs. This paper evaluates the performance of alternative volatility forecasts for fed cattle, feeder cattle, and corn cash price returns. Forecasts include time series (e.g. GARCH), implied volatility from options on futures contracts, and composite specifications. The overriding finding from this research, consistent with the existing volatility forecasting literature, is that no single method of volatility forecasting provides superior accuracy across alternative data sets and horizons. However, evidence is provided suggesting that risk managers and extension educators use composite methods when both time series and implied volatilities are available.

The local volatility model is a well-known extension of the Black–Scholes constant volatility model, whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in foreign exchange option trading. In this paper, we address a question of validation of the local volatility model. Different stochastic models for the underlying asset can be calibrated to provide a good fit to the current market data, which should be recalibrated every trading date. A good fit to the current market data does not imply that the model is appropriate, and historical backtesting should be performed for validation purposes. We study delta hedging errors under the local volatility model using historical data from 2005 to 2011 for the AUD/USD implied volatility. We performed backtests for a range of option maturities and strikes using sticky delta and theoretically correct delta hedging. The results show that delta hedging errors under the standard Black–Scholes model are no worse than those of the local volatility model. Moreover, for the case of in- and at-the-money options, the hedging error for the Black–Scholes model is significantly better.

This note explores the behaviour of the implied volatility of a European call option far from maturity. Asymptotic formulae are derived with precise control over the error terms. The connection between the asymptotic implied volatility and the cumulant generating function of the logarithm of the underlying stock price is discussed in detail and illustrated by examples.

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.