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Guy Cohen

Volatility and Options

by Guy Cohen 18. November 2007 17:52
The seven factors that influence an option’s price

There are seven factors that influence an option’s price:

  1. The type of option (call or put)
  2. The price of the underlying asset
  3. The exercise price (or strike price) of the option
  4. The expiration date
  5. Volatility – Implied and Historical
  6. Risk-free interest rate
  7. Dividends and stock splits

 Vega starts with a V and stands for Volatility

When you trade stocks, you must be aware ofvolatility.  Volatility is a measure of how a security’s price ismoving.  Volatility is recognized as a measure of risk.  If a stockprice fluctuates all over the place in wild swings, then you’d find ituncomfortable because you wouldn’t have a clue what it was going to donext and it would feel risky.  If a stock price remains static all thetime, then you might get a bit bored, but you wouldn’t have to reachfor the Pepto-Bismol! 

So, higher volatility is predicated by wider,faster price fluctuations.  This translates into greater risk.  Thegreater the volatility and risk, the more expensive options premiumsbecome.

Volatility is calculated by measuring thestandard deviation of closing prices, then expressed as an annualizedpercentage figure.  Volatility is not directional.  If a stock ispriced at $100 and has volatility of 20%, then we expect the stock totrade in the range of $80-$120 for the next year. 

Vega Κ (also known as Kappa or Lambda)

Vega measures an option’s sensitivity to the stock’s volatility. This volatility is known as Historical or Statistical volatility.

Diagram: Volatility

Volatility Cause.jpg

There are two categories of volatility: Historical and Implied.

Historical Volatility
is derived from the standard deviation of the underlying asset price movement over a known period of time.
Implied Volatility   
is derived from the market price of the option itself.


Remember that there are 7 variables that affect an option's premium.  Six of these variables are known with certainty:

  1. stock price
  2. strike price
  3. type of option
  4. time to expiration
  5. interest rates
  6. dividends

The final variable can be considered not to be known with certainty and is the expected volatility of the stock going forward. 

Implied Volatility

There are several mathematical models for calculating the theoretical value of an option.  In the main they manipulate the above seven variables to arrive at the correct theoretical option price.  I stress the word theoretical because the theoretical price is not themarket price for the option.  Sometimes the figures will be the same,sometimes they’ll be different, there’s no magic rule. 

  • The thing to remember is that the Theoretical option price uses Historical Volatility (of the stock) to calculate the theoretical value of an option.  So, all the seven factors go into the pot and we emerge with a theoretical option price. 
  • The market price of an option premium has a volatility figure implied within it.  We reverse the theoretical option price model in order to find out what figure for volatility was implied. So, with a real market option where we know what price it is tradingat, we mix the 6 factors (not volatility) into the pot with the actualmarket option price to work out what the Implied Volatility figure must be to create that market price. 

Diagram: Theoretical option price

  Theoretical option price.jpg

Diagram: Implied Volatility calculated from real option prices in the market

  Implied Vol from Real.jpg

This expected volatility figure is expressed as anannualized percentage and, working back from the option premium itself,is an "implied" figure, hence Implied Volatility

A reminder: Historical Volatility is the annualized standard deviation of past price movements of the stock.  We can use Historical Volatility as a reference figure for calculating what the Fair Valueof the option should be, given the stock's Historical Volatility.  Inthe real world, option premiums frequently trade away from their fairvalues, adopting trading ranges driven more by demand and supply in thecut and thrust of market activity. 


Volatility   Based on …
Historical Underlying stock volatilityover a period of time, for example the past 20 trading days.  Expressedas a % reflecting the average annual standard deviation.
Implied: The volatility derived from the option’s traded market price using an option pricing model.  Expressed as a %.

The mechanical pricing of options involves complex mathematicalformulae, which we don’t need to explore here. There are also a numberof different methodologies available for options pricing models, eachwith their associated merits. Typically I’ll be tacitly referring tothe Black-Scholes Options Pricing Model (for stocks and American-style(early exercise) options) and Black’s option Pricing Model (for futuresand European-style (no early exercise) options).

What we need to remember is that there are sevenmajor influences for pricing an option (above). We also need toremember that Volatility is one of them. In the actual marketplace thevalue assigned to an option is determined by market forces. This cangive rise to inconsistency between the Fair Value of an option and theactual price of the option in the marketplace. The Fair Value of anoption is the mathematically based calculation of the option price,using Historical Volatility as the figure for volatility.

The inconsistency emerges when the market pricediffers from the Fair Value, which is a common occurrence. Out of allseven factors that influence the option price, the only one which couldbe subject to any form of debate is Volatility. Let’s go through theseven factors again:

Factor influencing option price              
The type of option (call or put):
This is fixed and cannot be changed, the option is either a call or a put
The price of the underlying asset: 
No room for manoeuvre here because the option price is directly correlated with the underlying asset price
Strike Price: The strike price is fixed for each option
The expiration date:
The expiration date is fixed for each option
Volatility* – Implied and Historic:

Although Historical Volatility itself is fixed(with respect to whatever time period we’re assigning to it, say, 20trading days), the choice of time frame can be somewhat arbitrary anddoesn’t necessarily fit with the time left to the option’s expiration.

The discretion between the option’s market valueand its Fair Value is therefore interpreted as an anomaly of volatility(it simply cannot be any of the other six factors). Implied Volatilityis a calculated figure arising from the actual market price itself.

Risk-free interest rate:
The risk-free rate is fixed
Dividends and stock splits: 
This is fixed

*Volatility is always expressed as a percentage.

Question:  What does Historical Volatility mean?
Answer:    Historical Volatility is a reflection of how the underlying asset has moved in the past.


Consider a stock priced at $41.41 on 1 May and with July $40 strike calls and puts priced at $9.30 and $7.40 respectively.

Expiration Option premium Historical Volatility (23 days) Implied Volatility
  Call strike 40  July  $9.30  196.74%  111%
  Put strike 40  July  $7.40  196.74%  111%

If the options were priced in the market according to the Historical Volatility, the call would be worth $15.41 and the put would be priced at $13.51.Are we getting a bargain here for our options?4 Well, that would dependon whether Implied Volatility is usually at a discount or premium toHistorical Volatility with this particular stock, as well as a numberof other factors. Each stock, each underlying asset will have differentcharacteristics with regard to the relationship between Implied andHistorical Volatility of their options chains. Just like you have tofamiliarize yourself with a stock’s personality, you also have tofamiliarized yourself with its option chain’s personality and thehistorical relationship between Historical and Implied Volatility.

For now, just remember that Historical Volatilityis figure derived from the underlying asset price movement, and ImpliedVolatility is derived from the actual market premium of the optionitself. 

Based on

Underlying asset volatility over a period of time, for example, the past 20 trading days. 

Expressed as a percentage reflecting the average annual range (i.e. standard deviation)


The volatility derived from the option’s traded market price using an option pricing model. 

Expressed as a percentage and based on the perception of where market will be in the future. 

This is the volatility figure derived from the Black-Scholes Options Pricing Model.

(Thehigher the Implied Volatility, the higher the option price will be andvice versa. If Implied Volatility is substantially lower thanHistorical Volatility, there could be an argument to suggest good valuein the option price itself.)

In terms of trading, if you can recognize how Implied and HistoricalVolatility relate to each other with a specific stock, you can alsoidentify powerful ways with which to trade the options.

The following table is a typical guide to how totrade the relationship between Implied and Historical Volatility, but Iurge you to exercise caution here. Typical does not necessarilymean it’s right! The key is what the relationship has been like in thepast and whether the present is significantly different. Volatilityswings are often likened to the ‘rubber band effect’ where if therubber band is stretched too tight in one direction or too loose in theother, it will generally revert back to its most natural position mostof the time. Therefore, if Implied Volatility is generally around 70%for a stock, but for a period of time it plummets to, say 30%, could itbe possible that the options prices might be good value? Or, using thesame example, say Implied Volatility rockets up to 110%, could theoptions perhaps be overvalued? This is how the rubber band effect isbest illustrated. Over the medium to long term, Implied Volatility doestend to veer towards the Historical Volatility figure, but this willdepend upon how consistent the Historical Volatility of the underlyingasset is.

Look for 

Typical interpretation (not necessarily the right interpretation)
Implied > Historical: 
Options prices could be overvalued as a result of higher implied volatility, therefore look to sell options premiums
Historical > Implied:
Options prices could be undervalued, indicatinggood buying opportunities, particularly if you anticipate underlyingasset price movement

Diagram: Implied Volatility and the rubber band effect

Rubber Band Effect.jpg

So, in simplistic terms some traders look to buyoptions with low Implied Volatility (because the option premium will below) compared with the Historical Volatility of the underlying stock. In this way, the perception is that the options are cheap orundervalued; therefore they must represent a good trade. 

As stated above, this is a dangerous assumptionto make.  For a start, option premiums often have Implied Volatilitiesconsistently inconsistent with the Historical Volatility of theunderlying stock.  Secondly, just because an option is cheap today,doesn’t mean it’ll be expensive tomorrow.  So the rationale for thattactic is flawed.  Of far more relevance would be to look at thehistory of Implied Volatility and see if current options prices aretrading away from their own averages of Implied Volatility. 

Similarly, some traders look to sell options withpremiums reflecting high Implied Volatility (because the option premiumwill be high) compared with the Historical Volatility of the stock. Again, this is a flawed methodology in the real world of trading, evenif the logic initially looks plausible. 

Vega characteristics

Vega is identical and positive for (long) callsand puts. This reflects the fact that higher volatility increases theoption’s premium.  When Vega is positive, it generally suggests thatincreasing volatility is helping our position.  When Vega is negative,it generally suggests that increasing volatility is hurting ourposition. 


Let’s take a stock on 1 May.  The stock price is$49.40 and we’ll look at the December 50 strike calls and puts, whichare priced at $7.50 and $6.90 respectively.

Chart: Long Call Vega profile

Long Call Vega.jpg

Chart: Long Put Vega profile

Long Put Vega.jpg

As you can see, Vega is identical for both callsand puts. Notice how it increases around the money (strike price) andalso how Vega is vastly reduced where there is less time to expiration.This is because there is less time for increased volatility to make animpact on the Time Value component of the option’s value.

Diagram: Vega summary

Vega Summary.jpg