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Topic Author
Posts: 28
Joined: June 7th, 2012, 11:05 am

Appropriateness of Difference of Log Prices

The calculation of return as the difference in logs of consecutive price ticks is only accurate if the return is infinitesimally small (see below). If that is the case, why is the calculation so widely used in financial models, given we know that returns on equity holdings can be fat tailed and the return between two consecutive ticks may not always be infinitesimally small? I wonder if this formula is more appropriate for high-frequency trading where the holding period is in minutes or even seconds as opposed to other trading methods where the investment horizon is much longer. Would anyone please comment.Pt = Price at time tPt+1 = Price at time t+1R = Return between two consecutive ticks Pt+1 = Pt * (1 + R)or ln(Pt+1) = ln(Pt) + ln(1 + R)or ln(Pt+1) = ln(Pt) + R (assuming R is tiny)or R = ln(Pt+1/Pt)

quantmeh
Posts: 5974
Joined: April 6th, 2007, 1:39 pm

Appropriateness of Difference of Log Prices

it works for exponential growth model, regardless of the time interval.

countblessings
Topic Author
Posts: 28
Joined: June 7th, 2012, 11:05 am

Appropriateness of Difference of Log Prices

Thanks, but stock returns are not guaranteed to exhibit an exponential or even a linear growth model?

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

Appropriateness of Difference of Log Prices

One can just as easily say that R = (Pt+1)/(Pt) - 1 is only an approximation that only works for infinitesimally small returns. If I get a R1 = (P1/P0) - 1 in time period 1 and R2 = (P2/P1) - 1 in time period 2, then total return over both periods is equal to R1+R2 only if R1 and R2 are infinitesimally small. So it all depends on how you define "returns" and what properties you want your returns variable to have.The advantage of log-returns is that they are additive for sequential returns. If Pt = 100, Pt+1 = 50, and Pt+2 = 100, then the log returns are ln( Pt+1 / Pt ) = ln(50/100) = ln(1/2) = -0.69 and then ln( Pt+2 / Pt+1 ) = ln(100/50) = ln(2) = +0.69 which adds to 0 and accurately reflects the fact that Pt = Pt+2. In contrast the linear model would assert that the investor lost 50% in the first timeperiod but then gained 100% in the second which sounds great until you do the math.Having sequentially additive returns helps in discussing returns models on different timescales (quantmeh's point) and in certain mathematical operations (e.g., calculus, differential equations, etc.). The log model also has the advantage that no matter how negative or positive the return, the end price is always above zero.The larger point is that we can define returns any way we want as long as we're consistent. That provides the freedom to choose a calculation of returns that has useful numerical or mathematical properties.
Last edited by Traden4Alpha on June 7th, 2012, 10:00 pm, edited 1 time in total.

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