# The Volatility Premium and Black-Scholes Pricing

The implied volatility is generally equal to or significantly greater than the forecasted volatility; for instance, the BSM implied volatility is, in general, an upward biased estimator. Indeed, by selling implied volatility a risk premium is provided because of the many expected and unexpected events that may occur. Moreover, market microstructure posits that implied volatility should be biased high because market makers profit from the bid-ask spread in the options by slightly raising their quotes (i.e., going slight long volatility exposure particularly on the downside). However, this absolutely doesn’t mean that it is always possible to profit by selling implied volatility

Volatility cannot be instantaneously measured; indeed, as an average, it needs to be estimated over time. Therefore, sample length and frequency of sampling should be chosen considering long-term volatility as the objective of estimation:

- sample length is important because too little data increases noise due to augmented sampling error, but too much data relies on information no longer relevant to the current state of the market
- high sampling frequencies introduce market microstructure’s issues (like the bid/ask bounce, relevant for prices but not so for long-term volatility). However, high-frequency returns are relevant to estimate market impact but low-frequency volatility is a better predictor of high-frequency volatility. Thus, for long-term forecasting high-frequency data is generally not a good approach

Various measurement methods can be employed to characterize long-term volatility:

- Close-to-Close Estimator has well-understood sampling properties, easy to correct bias and to convert to a form involving typical daily moves; however, it is an inefficient use of data and converges very slowly.
- Parkinson Estimator employs the daily range to include additional information but it is really only appropriate for measuring the volatility of a GBM process because it cannot handle trends and jumps. Moreover, it systematically underestimates volatility.
- Garman-Klass Estimator is up to eight times more efficient than close-to-close estimator but is even more biased than the Parkinson estimator.
- Rogers-Satchell Estimator allows for the presence of trends but cannot deal with jumps.
- Yang-Zhang Estimator is the most efficient and specifically designed to have minimum estimation error and handles both drift and jumps. The performance degrades to that of the close-to-close estimator when the process is dominated by jumps.
- First Exit Time Estimator uses fundamentally different information to traditional time-series methods, it is a natural estimator for online use, converges relatively quickly, and it is not affected by the presence of noisy data or jumps. However, it requires the use of high-frequency data.