Saturday, 15 December 2007

Implied Volatility, Historical Volatility and Volatility Smile

There are so many things I want to share in my blog as I learnt and I'm trying to prioritize what should I blog first. If you have follow through my blog on Black-Scholes formula to derive the theoretical option price using Microsoft Excel, you will noticed that at the end of the day, we are using the Goal Seek function to find implied volatility of the call option. If you have been trading warrants and/or options, chances are, you may come across historical volatility and implied volatility before. What is historical volatility and implied volatility then? And what is volatility smile?

If you understand what are warrants and options, you should know that there are seven factors that influence an option and warrant price. These factors are as follow:

  • The type of option or warrant (call or put)
  • The price of the underlying asset
  • The exercise price or strike price of the option or warrant
  • The expiration date
  • The Risk-free interest rate which is normally taken as the 3-months T-Bills rate
  • Dividends and stock splits
  • Volatility - Implied and Historical

When you trade stocks, you must be aware of volatility. Volatility is a measure of how a security’s price is moving. Volatility is recognized as a measure of risk. If a stock price fluctuates all over the place in wild swings, then you had find it uncomfortable because you would not have a clue what it was going to do next, and it would feel risky. On the other hand, if a stock price remains static all the time, then you might get a bit bored and feel that it might lack liquidity. Hence, higher volatility is predicted by wider and faster price fluctuations. This means greater risk. The greater the volatility, the more expensive options and warrants premiums become as there is higher chance that a currently out or at the money option or warrant may become in the money. However, the reverse is true as well.

Volatility is calculated by measuring the standard deviation of the closing prices, and then expressed as annualized percentage figure. Volatility is not directional. Vega measures an option’s or warrant’s sensitivity to the stock’s volatility. This volatility is known as the historical or statistical volatility. On the price charts, Bollinger Bands can provide a visual representation of volatility.

I shall give an example on how to interpret historical volatility. Suppose a stock is trading at $10 with a 30d historical volatility of 10%, then, based on the theories of statistics, there is a 68% probability of the time that the stock will be trading within the range of $10 +/- $10 x 10%, i.e. 1 standard deviation away from the mean. Similarly, there will be a 95% probability of the time that the stock will be trading within the range of $10 +/- $10 x 1.96 x 10%, i.e. 1.96 or approximately 2 standard deviations away from the mean. I purposely choose a 30d historical volatility so I can assume the distribution is normal, there are some books which actually used 20d historical volatility.

Recall from above the seven factors influencing the option and warrant price. Six of the variables are known with certainty. The only variable that is now known with certainty is the expected or implied volatility of the stock going forward.

Though there is a saying of history repeats itself, historical volatility does not necessary predict where the price of the underlying will move towards in future. This is what the weak efficient market hypothesis is trying to prove. As such, there are several mathematical models for calculating the theoretical value of an option or warrant. The Black-Scholes is one such model. To be more exact, the Black-Scholes option pricing model is really used for American-style options (I did not mention warrant here as warrants in Singapore are all European-style or rather a more exact term should be Asian-style, especially for stock warrants) and the Black’s option pricing model for European-style option and warrant. In general, implied volatility increases when the market is bearish and decreases when the market is bullish. This is due to the common belief that bearish markets are more risky than bullish markets.

Lastly, the volatility smile is a common graphical shape that results from plotting the strike price and implied volatility of a group of options with the same expiration date. A picture speaks a thousand words. The image below will provide a better illustration why is it known as a volatility smile. :)


16 comments:

Anonymous said...

One thing needs to be discussed is the risk neutral valuation when getting BS option price. When solve implied from option market quote we attribute cause only on vol but not interest rate. But in reality, option price should be explained both from historical vol as well as expected return ( interest rate on BS ). The difference of the expected return is another reason to create difference between historical vol and implied vol. In our current prictice we use only vol to explain but that is not correct.

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Anonymous said...

Lovely sharp post. Never thought that it was this easy. Extolment to you!.

livevol said...

Hi Neavens!!
You are really a great blogger. I think your CFA study would gives you a lot of things to learn. My work is much related to stock market and that's why i want to learn the hidden factors behind the implied volatility and the rest that you thing is to be important.
But thanks anyway for this awesome stuff.
Implied volatility

Anonymous said...

1)Volatility is computed by taking the standard deviation of RETURNS, not PRICES.
2)The volatility smile picture is incorrect. The left of center should read "In the money calls and Out of the money puts". The right of center should read "In the money puts and Out of the money calls". You reversed them.
Thanks

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