Dr. Alex Castaldo
Article Review: "The Model-Free Implied Volatility and Its Information Content"
George J. Jiang and Y.S. Tian
The Review of Financial Studies, vol 18 #4, Winter 2005, pages 1305-1342
The familiar Implied Vol (IV) that we all know and love is a Model-Based IV. The model being used is of course (in the vast majority of cases) the Black Scholes model. Suppose an option is trading in the market place for a price P_mkt. We plug in different values of volatility into the Black Scholes formula until it produces a price P_model that matches the observed price (i.e P_mkt = P_model). That volatility, by definition, is the implied volatility (at that maturity and strike). This concept goes back to the beginnings of BS option theory.
The old VIX (VXO) was a model based IV. Because an option that is exactly ATM and exactly 1 month to maturity is usually not available, VXO was computed by interpolating the BS IV's of several options close to the desired maturity and strike. A little more complicated but still a model based IV.
In the past 5 or so years, a completely different concept has been discovered, called Model-Free Implied Volatility. The new VIX is a model-free IV and its calculation does not involve the Black Scholes formula at all. It is a completely different calculation (described on the CME web site). How/why does it work? Up to now the only derivation I had seen relied on some obscure passages in a 1998 paper by Emanuel Derman, which was not about IV per se at all (it was about variance swaps and their relation to hedging of a theoretical construct called a 'log contract'). The spec_list had a contest for a clear and straightfoward derivation of the formula, but I never gave out the prize because I was not happy with any of the entries. This new paper by Jiang and Tiang in the RFS (the most high powered of the academic finance journals) starts to clarify some of the mysteries behind model free IV.
Of course you cannot get something for nothing: the model free IV requires more inputs than the model based IV. The price of a single option is not enough to recover a model free IV, you need the prices of all options having the desired maturity. The more strikes available the better (theoretically an infinite number of strikes are required, but in practice there are enough strikes available on the exchange, both above and below the current price, to allow a very decent approximation).
The key result of Jiang/Tian is the following: the expected value of the return variance between now and the maturity date T is equal to the following expression involving the current price of all call options that expire at T:
2 * Integral_from_0_to Infinity (( C_K - max(0,F_0-K)) / K^2) dK
Jiang and Tian derive this expression and show that it holds under a wide variety of assumptions (including diffusion+jumps+stochastic volatility). C_K is the price of a call with strike K. F_0 is the current price of S&P futures. K is the strike price of the option, and the integration takes place with respect to K (that is why you need all (an infinite number!) of options).
The authors then discuss the truncation and discretization errors that you make when you don't have an infinite number of options prices available and show that they are manageable.
In the empirical part of the paper the authors look at CBOE prices of options on the S&P from June 1988 to December 1994. They compute three kinds of vol: the historical vol, the model based (BS) vol and the model free vol, to see how well they forecast the future vol. Their conclusions are:
Alessandro Castaldo, CFA, is a researcher and trader for Manchester Trading. Dr. Castaldo wrote his PhD dissertation on stock market volatility at the City University of New York, and taught courses in finance and options to undergraduates at Baruch College (CUNY) from 1998-2001. He has been associated with Circle T Partners, LP, a $400 million equity hedge fund; and Willowbridge Associates, a $1 billion-plus commodities trading adviser, where his responsibilities included the ongoing refinement of a market-neutral statistically based ("stat-arb") stock selection model. Dr. Castaldo holds a B.S. in electrical engineering/computer science and an M.S. in management from the Massachusetts Institute of Technology, and worked as a software engineer at SEI Corporation/TMI Systems, Software Research Corp. and Systems Constructs Inc. before entering the finance profession.
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