The Volatility Surface

The almost standard approach of quoting option prices in terms of their Black-Scholes implied volatilities is probably best not viewed as a ringing endorsement of same. The 1987 stock market crash shattered the received wisdom that asset returns can be modeled unquestionably lognormal process. Practitioners however did not abandon Black Scholes model, instead they seemed to have re-purposed Black Scholes as as an interpolation tool to deduce volatilities consistent with market prices of options. In other words, market prices gained greater weight relative to model generated prices and practitioners sought to back out volatility using grid-search style techniques. The resultant variation of Implied Volatility ranging across option strikes and maturities is widely referred to as the volatility surface and is characteristic of many option markets post 1987. Traders would view the Volatility Surface as a key reference for determining priciness of options. A good account of this phenomenon is provided by Jim Gatheral. Robert Shiller also provides some useful intuition. The implied volatility surface is generally set up as a 3-D plot that maps out the smile/skew/smirk and term structure of volatility implied from option chain data. This surface is normally developed by taking option market data relating to prices, bids and offers and interpolating that data using Black Scholes. The VIX similarly is an estimate of ex ante market volatility but differs in respect to being a model free estimate of implied volatility. The SPX VIX, is an index of annualized volatility based on a notional 30-day options maturity for the S&P 500 calculated from a wide range of calls and puts. The volatility surface is estimated typically by brokers with a view to capturing similar insights regarding ex ante volatility or 'nervousness'. The more "nervous" markets become the more "pricey" options are. The smile/skew/smirk also reflects evidence that is now widely accepts that stock returns are not, de facto, normally distributed. Market measures of volatility relative to "moneyness" exhibit skew. This has led to the emergence of what is known as the ad hoc model developed by Dumas, Fleming, and Whaley (1998). To construct an Implied Volatility Surface, please follow the playlist.

The practitioners or ad hoc procedure involves estimating Black-Scholes (1973) implied volatilities and then smoothing across strike prices and maturities. These can be then plugged back into the Black-Scholes formula. This “ad hoc Black-Scholes” approach constitutes a benchmark of sorts for evaluating the forecast accuracy of option pricing models by virtue of its consistently solid empirical performance. Dumas, Fleming and Whaley (1998), for example, find that the ad hoc Black-Scholes (ABS) model forecasts option prices out-of-sample more accurately than Black Scholes and other approaches. Heston and Nandi (2000) point out that the ad hoc approach compares favorably with a close-form GARCH option pricing model. Christoffersen and Jacobs (2004) find that the ABS model outperforms Heston’s (1993) theoretical model both in and out of sample. The Black Scholes ad hoc approach represents a sophisticated interpolation tool used to ensure that an option is priced consistently with the market prices of other traded options.

VBA code for estimating the Implied Volatility of a Call:

VBA code for estimating the Implied Volatility of a Put:

Function BSPut(S, K, r, q, sigma, T)

Dim dOne, dTwo, NNd1, NNd2

dOne = (Log(S / K) + (r - q + 0.5 * sigma ^ 2) * T) / (sigma * Sqr(T))

dTwo = dOne - sigma * Sqr(T)

NNd1 = Application.NormSDist(-dOne)

NNd2 = Application.NormSDist(-dTwo)

BSPut = -Exp(-q * T) * S * NNd1 + Exp(-r * T) * K * NNd2

End Function

Function BSPutImVol(S, K, r, q, T, putmrkpr)

H = 5