Derivatives on Volatility: Some Simple Solutions Based on Observables

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Abstract: Proposals to introduce derivatives whose payouts are explicitly linked to the volatility of an underlying asset have been around for some time. In response to these proposals, a few papers have tried to develop valuation formulae for volatility derivatives-derivatives that essentially help investors hedge the unpredictable volatility risk. This paper contributes to this nascent literature by developing closedform/analytical formulae for prices of options and futures on volatility as well as volatility swaps. The primary contribution of this paper is that, unlike all other models, our model is empirically viable and can be easily implemented.More specifically, our model distinguishes itself from other proposed solutions/models in the following respects: (1) Although volatility is stochastic, it is an exact function of the observed path of asset prices. This is crucial in practice because nonobservability of volatility makes it very difficult (in fact, impossible) to arrive at prices and hedge ratios of volatility derivatives in an internally consistent fashion, as it is akin to not knowing the stock price when trying to price an equity derivative.(2) The model does not require an unobserved volatility risk premium, nor is it predicated on the strong assumption of the existence of a continuum of options of all strikes and maturities as in some papers. (3) We show how it is possible to replicate (delta hedge) volatility derivatives by trading only in the underlying asset (on whose volatility the derivative exists) and a risk-free asset. This bypasses the problem of having to trade numerously many options on the underlying asset, a hedging strategy proposed in some other models. JEL classification: G12, G13

A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility

“…Grunbichler and Longstaff (1996) first developed a pricing model for volatility futures based on mean-reverting squared-root volatility process. Heston (2000) derived an analytical solution for both variance and volatility swaps based on the GARCH volatility process. Javaheri et al (2004) also discussed the valuation and calibration for variance swaps based on the GARCH(1,1) stochastic volatility model.…”

Pricing Variance Swaps Under Stochastic Volatility and Stochastic Interest Rate

“…One important finding was the contrast characteristics between volatility derivatives and options on traded assets. However, it was later noted by Heston and Nandi [67] that specification of the mean-reverting square-root process is difficult to be applied to the real market. Thus, the latter proposed a user-friendly model by working on the discrete-time GARCH…”

Modeling volatility smile: Empirical evidence from India

“…Thereafter, two important assumptions of the Black-Scholes model, namely, constant volatility and log normal distribution (Fama, 1965), were empirically challenged by a host of researchers. Enforced by wrong distributional assumptions and its implications, researchers focused on determining the alternative model defining the volatility smile aligned with non-lognormal distributional assumptions of Black-Scholes Victor, 1993a, 1993b;Derman and Kani, 1994;Duan, 1996;Backus et al, 1997;Heston and Nandi, 2000). When implied volatility (volatility obtained by inverting the Black-Scholes model is known as implied volatility) is plotted against the time to maturity and moneyness (ratio of stock price and strike price), it manifests systematic pricing bias and formed a 'smile' or 'skew' pattern (theoretically it should remain neutral for all maturity-moneyness on the same underlying assets).…”

“…The same was also analyzed and improved upon by Christoffersen and Jacobs (2004) and they named it Practitioner BlackScholes model. The Deterministic Volatility Function (DVF) approach has been extensively studied in Monte Carlo settings by Dumas et al (1998) and Pena et al (1999), Heston and Nandi (2000) and Christoffersen and Jacobs (2004). In recent times, the model has been evaluated by many.…”