## A brief discussion on the topic of volatility, as used in the valuation of options. One of the most important, and baffling, concepts of option theory is that of

Option Volatilityvolatility. Confusion can arise when valuing options (for example, when using an option model like Black-Scholes), as the termvolatilityis used in 3 different senses:Confusion between these three meanings can make the comprehension of option theory harder than it need be. Below, we briefly describe each of these types of volatility.

Historic Volatility- a measure of the past fluctuation of share prices,Implied Volatility- a theoretical volatility implied by an option price (using a particular option model), andForecast Volatility- the input required for an option model to calculate an option's fair value.

Note:In all of the discussion here, references toshare pricescan be taken equally to refer to any option-underlying security or asset.

## A measure of the previous fluctuations in share price (crudely: an indicator of the share's

Historic Volatilityup/downess). There is much discussion over the best method of calculating the historic volatility. The most usual method is the standard deviation of the log of price returns, this procedure is fairly standard and can be found in most text books.While the calculation itself is straight forward, it is accurate only within the parameters of each calculation (e.g. the specific time period: 3 months, 3 years etc.). There is great scope for analyzing the share price behavior over different time periods, and thereby calculating different historic volatilities.

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## Instead of inputting a volatility parameter into an option model (e.g. Black-Scholes) to determine an option's fair value, the calculation can be turned around, where the actual current option price is

Implied Volatilityinputand the volatility isoutput. The termimplied volatilityis obviously self-explanatory, the level of volatility that will calculate a fair value actually equal to the current trading option price.This calculation can be very useful when comparing different options. The implied volatility can be regarded as a measure of an option's "expensiveness" in the market, and is used by traders setting up combination strategies, where they have to identify relatively cheap and expensive options (even though these options have different terms).

It is perhaps useful to note that implied volatility only has any meaning in the context of a particular option model (it is not intrinsic to the option itself). So, although options have existed for a long time, implied volatility has only had any meaning since the option pricing model of Fisher Black and Myron Scholes (devised in the early 1970's) stated that the value of an option was a function of the volatility of the underlying share price.

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## When looking at an option calculator for the first time, everything seems fairly straight-forward: OK, I can input the

Forecast Volatilityshare price;strike price;interest rate(well, around, x%, it doesn't seem to make that much difference);dividend yield(ditto);maturityeasy...but,volatility, where do I find that? Unlike the input parametersinterest rateordividends, there is no obvious reference, and, importantly, the option fair valueissensitive to small changes in this volatility figure.To calculate the fair value, an option model requires the input of volatility, or, more precisely, the input of:

forecast volatility of the share price over the period to expiry of the option. The big question (theart) of option theory is how to estimate this forecast volatility. This single stage provides gainful employment for a legion of academics, analysts and traders. Anestimate of future share price fluctuations,plenty of room for subjectivity there!It must be said immediately, that

there is no one correct method of determining the volatility input. Every trader works constantly at refining and adapting their option valuation parameters. The common place to start when estimating volatility input, is to look at the historic volatility of the share price. So, for example, if share price historic volatility over the past 2 years is 25%, this figure may be used as the model's forecast volatility input (i.e. forecasting that share price volatility will be 25% over the option's life).However, as noted above, the historic volatility value will

depend greatly on the time periodchosen for the calculation. Some might argue that the most appropriate period to look at, for valuing a 3-month option, is just the last 3 months of share prices. Others will calculate historic volatilities for a range of periods and then take some kind of weighted average.It is always useful to refer to the implied volatility; but this is of only limited use when estimating a forecast volatility input. One must be careful of a certain element of circularity here ("if a fair value of x is given by the input of y, then the input of y gives a fair value of .... x."). Sometimes, the implied volatilities for other options (within the same sector or market) may also be determined and an average used for the calculation for a specific option.

One of the assumptions of the Black-Scholes model is that volatility is constant over the life of the option. This is obviously unrealistic in real life, and can cause problems, particularly when valuing longer dated options. Some of the latest research into option theory is looking at this variation in volatility and trying to refine the model accordingly.

Above all, however, the most important factor to take into account when estimating volatility is:

what are you going to do with the output calculation? For example, one may calculate an option value of $2.50, see that the option is actually trading at $2.00 in the market, and therefore deem it "undervalued". Now what? Just because an option is subjectively undervalued doesn't mean that the price is going to rise. Hence, while one never wants to buy grossly over priced options, fair value is of limited use when speculating with options, by far the over riding determining factor will be the performance of the share price. However, if looking at a combination option strategy (or hedging or arbitraging) an option model is more useful, and the forecast volatility may have to be more arelative(against other options) rather than an entirelyabsoluteestimate.[ top]

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