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Natural Log e = 2.71828.....


The natural log was first introduced by the 16th century mathematician Leonhard Euler who noticed the following pattern:

e = 1 + 1/1! +1/2! + 1/3! 1/4! + 1/5! + 1/6! + ..... = 2.71828...

This pattern reflects continuous growth and can be more intuitively recognized in the calculation for interest (the growth of money). For example, suppose you are considering depositing $5,000 at the bank earning 10% ( this I know is unrealistic in today's world, but the differences in compounding interest are clearer with higher rates, but more importantly, over long periods of time), and the bank tells you they will compound the interest annually. This means that after one year you will have:

$5,000 (1 + .1/1) ^ 1 = $5,500

You don't like this so you go to another bank and they are willing to compound the interest quarterly (they will at the end of the quarter credit you the interest and then calculate next quarter's interest on that number). At the end of one year at this bank you will have:

$5,000 (1 + .1/4) ^ 4 = $5,519

This isn't much difference so you go to another bank and they tell you they will compound your interest daily:

$5,000 (1 + .1/365) ^ 365 = $5,525.78

Finally another bank tells you they will compound the interest continuously. This means that we let the number of periods in which to compound the interest go to infinity:

$5,000 x e ^ .1 = $5, 525.85

As I said, this difference is not that great, but the difference between compounding a sum annually versus continuously over thirty years can be dramatic. The point is that e is a natural number that connotates continuous (exponential growth).

An extremely useful tool is the inverse operation called the natural logarithm that undoes the exponential:

ln (e ^ x) = x

If ln (5) = 1.609437912, then e ^ 1.609437912 = 5. This tells us at what rate we have to grow some number exponentially to get another number. These things are essential in engineering, finance, and yes, to traders.

A trader, unlike the common perception that she is someone who buys and sells stocks back and forth, is (should be) in reality a pseudo- arbitrager. For example, I as a trader provide liquidity to the system because I am agnostic as to direction; my focus is to buy cheap options and sell expensive ones. If a call buyer wants to bet that the stock is going up, I will sell him that call and hedge the delta if he is over paying for it. I use a myriad of mathematical relationships to determine correlations and risks. For example, I use e in the black-scholes model to normalize price movements: a stock going up $1 from 50 is the same to me as a stock going up $2 from 100.

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