# What Is Implied Volatility?

Implied volatility (IV) is one of the most important concepts in options trading. Unfortunately it’s also one of the most complex.

Therefore, let’s build up the concept slowly with an understanding firstly of historical volatility as an estimate of an option’s risk, then we’ll look at implied volatility and how this relates to options pricing and finally where a consideration of IV is important in practice.

## Historical Volatility

The volatility of a stock is how much it moves up and down, its risk in other words, expressed by its standard deviation. If you’re not familiar with this risk concept, this video explains it well.

If you’re not interested in the stats, just see standard deviation as a measure of how risky a stock is. A large multinational paper wholesale is going to be less risky than, say, a new start up yet to make a profit and hence we would expect the latter to have a larger volatility.

Historical volatility is just what this measure has been, well, historically and is used as an estimate of volatility now. Note this is just an estimate: there are many reasons for a stock’s volatility to change (eg a new product launch, an imminent US election etc).

Thankfully we can calculate historical volatility:

## How To Calculate Historical Volatility

- Download the end of day stock prices for a decent period before now (6-12 months, say)
- Calculate the mean, or average, price for the period.
- For each day calculate the difference between the stock price and
- Sum all these results (the ‘sum of the squares’)
- Divide by the number of days less one (ie find the average of these square numbers). So if you have 365 days of data, divide the sum of squares by 364.
- Calculate the square root of this average. This is the historical volatility.

## Implied Volatility & Options Pricing

Before defining implied volatility we need to discuss how an option is priced.

We’ve gone into this in more detail here, but in summary an option’s fair value (ie what you should pay for it) depends on 5 things:

- The price of the underlying stock or financial instrument
- The exercise, or strike, price of the option (and whether it’s a put or call)
- The number of days to expiry
- Interest rates
- Volatility

In particular if the first 4 factors remain the same, more volatile stocks’ options will be more expensive.

This makes sense. There is a greater chance of a stock ending up in-the-money, and hence being exercised, if is is more prone to jump around. An option seller should be compensated for this higher risk.

Using historical volatility as an estimate for volatility, as above, we can therefore calculate the fair price of any option.

## Volatility Implied By The Market

That’s great, but what about implied volatility? Well, in practice, the market only uses historical volatility as a guide to future volatility. In reality the market is constantly expressing its view on what it believes will be the volatility over the remaining life of an option.

How does it do this? Via the market price of an option.

The other factors are fixed (over a short period of time). The only way an option can rise or fall in value is if the market changes its view of the stock’s volatility.

If the market perceives that a stock has become riskier, say, then all things being equal a stock’s option price will rise (and vice versa).

The option price ‘implies’ a volatility figure in the above calculation – because the other factors are fixed and known.

Given a market price of an option – and knowing its strike price, the price of the underlying, the days to expiry and interest rates – we can reverse engineer the option pricing calculation to work out what the market’s view of the stock’s volatility actually is.

If we know an option price (which in an open market, we do) we know the market’s view on volatility.

This ‘implied’ volatility is, well, implied volatility. It’s the market’s view at a point of time of the riskiness of a stock which has been priced into the stock’s options.

## Implied Volatility In Practice

There are a few things where a consideration of IV is important:

### Vega: How A Change in IV affects option price

We’ve already stated that an increase in IV increases an option’s price. Vega, one of the options greeks, is a measure of how much it changes – how sensitive an option price is to implied volatility.

The percentage change in option price given a 1% change in IV – all other things being equal – is Vega.

### VIX Index

The CBOE runs the VIX index – an average (sort of) of the IVs in the market. It’s therefore a measure of how risky the market views stocks and is a useful overview of risk.

### Volatility smile

Options theory tends to assume that implied volatility is the same for all options for the same underlying and expiry date, whatever its strike price.

In practise, however, the market seems to value out of the money options (especially puts) at a higher IV than those at the money. This is the ‘volatility smile’, – a reference of the shape if the graph of volatility to strike price.