## 12 December 2006

### Implied Volatilty

Volatility has more importance than just calculating statistical probabilities of price movement in options trading, it is actually one of the inputs into the Black Scholes Option Pricing Model (or BSOPM) I shall hereafter use this acronym as proxy for any of the various option pricing models. (American style options are more likely the be priced using the Cox, Ross & Rubinstein binomial model, but the difference is not great)

For future reference the six inputs into BSOPM are:

Price of the underlying Asset
Exercise price
Time till expiry
Dividend amount/date
Risk free interest rate
Volatility

In determining a price for an option all of the above must be put into the pricing equation to derive an output price. However there is a problem here. The first five inputs are absolute, as these values are known at any one point in time, but as the volatility input required is the forward volatility.

In other words we need to know the volatility of the underlying between now (the analysis date) and expiry. As this is in the future, it cannot be known. Therefore an estimate of this future volatility must be made, using tools such as statistical volatility and the traders best guess as to what trading conditions will be.

In reality, option prices are set by supply and demand, as well as the influence of arbitragers who will quickly leap on pricing anomalies.

The net result of this is that the volatility figure as input, is worked out from the other five inputs and the option price, using algebra. This is called "Implied Volatility". It is the volatility "implied" by the options tradeable price.

Most of the time, the implied volatility of the option "tends to reflect" the actual statistical volatility of the underlying. This means that they will be reasonably similar, but with some forward looking by option traders that will make for some degree of difference, as shown in the chart of (BHP) volatilities below.
The image above is a one year chart with the 30 day statistical volatility plotted in blue and an average of option implied volatilities in gold.

There are times when option traders may be expecting a sudden change in volatilities, such as when an earnings announcement or FDA decision is due. This will be reflected in implied volatilities being a great deal different to the statistical volatility as shown in the Rambus (RMBS) volatility chart below
You can see that IV was rising from the middle of March as SV was declining to quite low levels. The stock went quiet, yet option traders were expecting action, which eventually arrived with a sudden lift of SV in late April. Notice however that IV raised to much higher levels than the eventual realized volatility. This is the type of situation that can be traded, using the greeks; this is also the type of situation where the unwary have their money taken from them.

The purpose of analyzing volatilities in this way is paramount in selecting appropriate strategies and defining risk as a function of the greeks.

Many beginning and/or short term option traders imagine that these factors do not apply to them, or are merely complicating what may be kept simple. It is these traders who ultimately blame market makers for some of their bewildering losses, (Been there, done that, got the t-shirt) but as shown, it is just a factor of option pricing that must be known.

Next we'll have a look at the greeks, starting with Delta.