Stratégies Options - Volatility : A First Attempt

"The volatility is the most important and elusive quantity in the theory of derivatives"(P.W.)

The question when investing on financial markets remains to manage the risk. How to invest with the smaller risk.

Risk can be define in several ways, but it can be represented as the propension an asset price has to vary and by how much. A easy way is to look at the returns. How do they move ?

I - How is it derived ?

a - Return from day t-1 to day t
Returns are the yield an asset produced by its own price variation.
It's often derived as

r(t) = Ln( P(t)/P(t-1) )

where
Ln(x) is the natural logarithm of the variable x ( Wikipedia : Natural Logarithm)
P(t) is the price at date t
P(t-1) is the price at date t-1

If R is the total return, then

R = r(1) + r(2)....r(t-2) + r(t-1) + r(t)

Hence,
R = r(1) + r(2) +....r(t-2) + r(t-1) + r(t)
R = Ln( P(1)/P(0) ) + Ln( P(2)/P(1) ) +... Ln( P(t-1)/P(t-2) ) + Ln( P(t)/P(t-1) )

Because Ln(a)+Ln(b)=Ln(a*b)

R = Ln [ (P(1)/P(0).(P(2)/P(1))....(P(t-1)/P(t-2)).(P(t)/P(t-1) ]
R = Ln( P(t)/P(0) )

b - Why the natural logarithm ?
Natural logarithm is a mathematical function that makes easy to bridge the gap between adding and multiplying.

If r is the return from time t to time t+h,

r = ( P(t+h) - P(t) ) / P(t)
r = ( P(t+h) / P(t) ) - 1

If r is small, we know that
r ≈ Ln( 1 + r )

That leads to,

( P(t+h) / P(t) ) = r + 1

and

Ln( P(t+h) / P(t) ) = Ln( 1 + r ) ≈ r

r ≈ Ln( P(t+h) / P(t) )

NB : Using n closing prices leads to (n-1) returns

II - Academic Volatility

Volatility is often derived as a standard deviation around the mean of n returns.

σ_daily = √ [ ∑ ( r (t) - r_m )² / ( n - 1 ) ]

Where r(t) is the return from t-1 to t and r_m is the mean over the period [0,t]
See the "n-1" ;-))

3 - Trader's Volatility

The formula above is meaningless if daily returns equalize the mean (r(t) = r_m), volatility is then equal to zero. Of course, it's rare to see an underlying asset that continuously increases or decreases by an amount that is exactly its mean, but this is to show how this way to derive can be misleading.
No needs for mean for small time scale ;-))

σ_daily = √ [ ∑ ( r (t) )² / ( n - 1 ) ]

Where r = Ln( P(t+h) / P(t) )
Even easier to derive.

Annualized Volatility
Formulae above would return outcomes for one day.
To grasp how the movement would be on a yearly basis, it's needed to annualize it.

Annualized Volatility = √ (365 . σ²_daily)

Annualized Volatility = √ (365 . [ ∑ ( r (t) )² / ( n - 1 ) ] )

Next : Volatility : Let's Price It !
Previous :Simple Definition Of What An Option Is

OPTIONS 101 - INDEX
OPTIONS 101 - CHAPTER I
OPTIONS 101 - CHAPTER II
OPTIONS 101 - CHAPTER III
OPTIONS 101 - CHAPTER IV
OPTIONS 101 - CHAPTER V
OPTIONS 101 - CHAPTER VI

Strategies-options.com