Model-Free Implied Volatility: From Surface to Index

Abstract

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. Keywords: Model-free implied volatility index; volatility forecasting; volatility surface; variance swaps. 433 Int. J. Theor. Appl. Finan. 2011.14:433-463. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 01/05/15. For personal use only. 434 M. Fukasawa et al.

Variance Swap Replication: Discrete or Continuous?

“…Fukasawa et al propose an alternative approach in Fukasawa et al (2011). While their technique has several merits with regards to the consistency of their interpolation methodology, it leads to prices very close to the one obtained by the more direct implementation of the Carr-Madan formula described above.…”

“…In Fukasawa et al (2011), Fukasawa et al describe a practical continuous replication for the variance swap based on market quotes. While their technique is elegant and their interpolation methodology is sound, they do not explore the issue of discrete replication or truncation.…”

Markov-Regime Switches in Oil Markets: The Fear Factor Dynamics

“…The development of comparable volatility indices for other markets, such as the VXJ index from the Nikkei 225 option prices by Nishina et al (2006) for the Japanese markets and from the Kospi 200 options for the Korean markets by Maghrebi et al (2007) provided additional evidence about the usefulness of the model-free implied volatility index a gauge of investors' fear and market sentiment. An alternative version of model-free volatility is proposed by Fukasawa et al (2011) based on the approximation of the expected quadratic variations of asset prices in relation to options prices. 2 Part of the empirical literature also focuses on the stochastic behavior of volatility indices in relation to asset bubbles, financial crises, and macroeconomic shocks.…”

Implicit Expectiles and Measures of Implied Volatility

“…where O(K i ) is the price of an OTM call or put with strike K i , F is the forward price, K 0 is the highest quoted strike lower than F . As shown by Demeterfi et al (1999), this quantity is a discretization of the risk neutral expectation of the realized variance, and it can be written as VIX is thus equal to the standard deviation of the risk neutral distribution, and is computed by means of all OTM available call and put prices; Fukasawa et al (2011) showed that it is possible to compute VIX also by integrating suitably modified Black-Scholes implied volatilities.…”