Stratégies Options - Actualization: A Basic Principle #2

I - Continuously compounded

Example above shows how easy it’s to be adapted if the expiry is not 1 year.
In fact if the expiry is in n years from now:

X' = X / ( ( 1 + r ) . ( 1 + r ) . .....( 1 + r ) )

Since once again one can place this amount during n years using a rate r, thus to have:

X' . [ ( 1 + r ) . ( 1 + r ) . ..( 1 + r ) ]

This leads to,
[ X / ( ( 1 + r ) . ( 1 + r )...( 1 + r ) ) ] . [ ( 1 + r ) . ( 1 + r ) . ..( 1 + r ) ] = X

Thus,

[ X / ( ( 1 + r )ⁿ ) ] . [ ( 1 + r ) ⁿ ] = X

And,

X' = [ X / ( ( 1 + r )ⁿ ) ] or X = X' . ( ( 1 + r )ⁿ )

II - Effects.

Let us admit that rates are proportional to their durations, i.e. that the amount perceived over 6 months is half compared with 1 year one, that the amount perceived for 3 months corresponds to half of that over 6 months… Intuitively that means that for a 6 months investment, one perceives half of what one would have earned if money had been invested 1 year long.
One would have then over a period of 1 year to place 2 times 6 months yield at the rate (r/2) the X' amount and thus

X = X' . [ ( 1 + (r/2) ) . ( 1 + (r/2 ) ) ]
X = X' . [ 1 + (r/2) ]²

So instead of placing over 6 months one places 4 times of following 3 months yield one would have :

X = X' . [ ( 1 + (r/4) ) . ( 1 + (r/4 ) ) . ( 1 + (r/4) ) . ( 1 + (r/4 ) ) ]
X = X' . [ 1 + (r/4) ]⁴

So instead of placing over 3 months one places from day to day, it leads to:

X = X' . [ ( 1 + (r/365) ) . ( 1 + (r/365 ) )..... ( 1 + (r/365) ) . ( 1 + (r/365 ) ) ]
X = X' . [ 1 + (r/365) ]³⁶⁵

However it is known that for r small (<<1)

Where exp (.) is the exponential function.

That gives us over t year(s):

X = X' . [ exp( r . t ) ]

That leads to the notion of continuously compounded rate over a period t.

To remember :
In finance, as soon as one sees exp (rt), exp (- rt) one must have the reflex:

→ exp (rt) means the future value of the placement of 1 ($,€,£…) over t year at the annualized continuous rate r
- from where, 10*exp (0.02*0.5) means the future value of the placement of 10 at rate 0.02 (2%) over 0.5 year (6 months)
- and 18.53*exp (0.1*3.25) means the future value of the placement of 18.53 at rate 0.1 (10%) over 3.25 year (3 years and 4 months)

Same way,
→ exp (- rt) means the current value of 1 ($,€,£…) in t years if r is the annualized continuous rate r for the period.
- from where, 5*exp (- 0.03*1) means the current value of 5 in 1 year if 0.03 (3%) are the rate for the period
- and, 10.28*exp (- 0.0215*0.75) means the current value of 10.28 in 0.75 year (9 months) if 0.0215 (2.15%) are the rate for the period

For the pricing purpose, every rate used is always a continuously compounded one.

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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

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