Why do we have Log Returns in continuous time finance?
May 10, 2009
In a continuous time framework, when we slice time into smaller and smaller intervals, why does the return of an asset get expressed as natural logarithm of this periodís price divided by the last periodís price? In other words, if is this periodís price (say, today) and is last periodís price (say, yesterday) then why do we express the return of the asset as ? This is one question that initially bothers all those who try to grapple with Finance101.
It is a very fundamental question and all theories of continuous time finance is predicated on this notion. Even though the above expression does not make intuitive sense traders and quants use this measure all the time in their calculation. Actually, it comes from applying integral calculus to finance.
Asset return should be measured, as indeed it is by thousands of fund managers and investors around the world, as today price less yesterdayís price divided by yesterdayís price. In other words, if is this periodís asset price (where the time period is discrete like one year, one month, one week, one day or even one minute) and is last periodís asset price then (assuming no dividend payments) the asset return should be equal to:
This is the measure we are familiar with. Now, if we have many periods, say, N, in one time interval (i.e. 12 months in an interval of one year) then the total return of the entire interval is simply the sum of all the individual periodís returns, i.e.
However, what happens if we start to slice time into smaller and smaller intervals. Say, we start slicing time (one year) into minutes, seconds, nanoseconds and so on until we get to the mathematical definition of an infinitesimally small interval of time. We are now talking about the limit when delta t (the smallest measurable unit of time) goes to zero. Mathematically speaking, we say . In the limit, the above expression for return will reduce to:
In the limiting case if , then and the summation sign will get replaced by an integral. Therefore, the expression for the asset return (dropping the subscript for time) becomes:
And the limits of the integral are chosen in line with the time limits, i.e. at start when the asset price is and at the end of the time interval (maturity), i.e. at , the asset price is .From integral calculus, we know that and therefore, we have:
Therefore, in a time interval [0, T], if we assume a continuous time process, i.e. time is sliced as infinitesimally small intervals, the asset return will be expressed as the natural logarithm of the final price over the initial price.
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