Volatility Forecasting : Model-Free Implied Volatility

Volatility in the financial market is an important variable, which in asset pricing, investment, risk management and policy-making process plays an important role. Methods for predicting volatility are mainly divided into two categories, one is the historical information method, based on the historical information to predict the future volatility; the other is the implied volatility method, calculating the expectation of the future volatility based on the market price of the option. We propose a model-free implied volatility method to measure the volatility. The model-free implied volatility does not depend on the option pricing model, and extracts information from all the option contracts. We provide empirical evidence from the S&P 500 index option that the model-free implied volatility is more accurate than GARCH model in predicting the future volatility. Introduction During the past thirty years, volatility forecasting is becoming more and more important in financial engineering. Many works have been found in the theoretical and practical fields. Engle [1] proposed the first ARCH model in 1982, Bollerslev [2] and Taylor [3] added the old conditional variance into the new estimation of conditional variance, which is the generalized autoregressive conditional heteroskedasticity model (GARCH). Model-free implied volatility originated from the variance swap theory. Dupire [4] and Neuberger [5], Demeterfi, Derman, Kamal and zou (DDKZ) [6,7], Britten-Jones and Neuberger [8] expanded it further. When the underlying asset price jumps exist, Britten-Jones and Neuberger did not state explicitly. Since the jump is important in financial asset prices, Jiang and Tian [9] proved the conclusions from Britten-Jones and Neuberger remain valid when the price jumps exist, thus ensuring the generalizability of this method. Jiang and Tian [10] also demonstrated that DDKZ’s variance fair value and Britten-Jones and Neuberger’s yields squares are the same. Besides, they came to a conclusion that the model-free implied volatility has more information content than the Black-Scholes implied volatility. In this paper, the theories of the model-free implied volatility and the time series model are used in the American S&P500 index option market. We compare model-free implied volatility method with GARCH model from the empirical aspect. Model-free implied volatility The squared volatility can be expressed as the integration of call option forward prices [8]: [∫ ( ) ] ∫ ( ) ( ) (1) Where, denotes the forward price of the underlying asset at time t. ( ) denotes the forward price of the call option at time 0, and K is the strike price. Through the prices of current option contracts, we can get the long-term expectation for the market volatility. The Chicago Board Options Exchange began to use this new approach to calculate S&P500 volatility index (VIX) in 2003. International Conference on Education, Management, Commerce and Society (EMCS 2015) © 2015. The authors Published by Atlantis Press 498 In Eq. 1, strike prices range from 0 to infinite, and expiration dates of different options are the same. But in the real market, strike prices are discrete and finite, then Eq. 1 can be simplified to Eq. 2: [∫ ( ) ] ∑ ( ) ( ) (2) Where , , , . The above process will bring two types of errors: truncation errors and discretization errors. Truncation errors derive from the limited range of strike prices. Strike prices are within a certain range of underlying asset prices. Jiang and Tian [9] discovered that if cut-off points ( or ) are far from , truncation errors are small. Truncation errors are negligible if or . Otherwise, volatilities of cut-off points should be used to replace those out of the strike price interval ( ), which means volatilities outside the interval are constant. Discretization errors derive from discrete strike prices. The interval between each strike price may vary, but will not tend to 0. When is smaller, the discrete errors is smaller. Jiang and Tian [9] found the discretization error can be ignored when , where is the realized volatility of the underlying assets within the remaining maturity of the option. If , the cubic spline interpolation method should be used to add the missing option prices. Empirical Research Since S&P500 index option is very active last ten years, we choose all the call options (10 days to the expiration date) from January 2006 to January 2014. Before the calculation of model-free implied volatility, errors should be analyzed at first. The realized volatility varies each day, also does the forward price . As Fig. 1 shows, the horizontal axis represents observation sequence, and the vertical axis represents the multiple: for every highest strike price each day, ; for every lowest one , . Most of them exceed , which means truncation errors are small. Figure. 1 Multiple for strike price (unit: ) In the sample, the strike price interval is 5 points. Fig. 2 shows that discretization errors are small in most cases. The horizontal axis represents the observation sequence, and the vertical axis represents on each observation day. Almost all the intervals are less than .