December 12th, 2013, 8:36 am
QuoteOriginally posted by: ppauperQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: ppauperQuoteOriginally posted by: Cuchulainn3. What do you think of this approximation to exp(-r dt)?(1 - r dt) / (1 + r dt)not much[$] exp(-r\,dt))\approx 1 - r\,dt[$][$](1 - r\,dt) / (1 + r\,dt)\approx 1 - 2r\,dt[$]Both forms are special cases of Pade(p,q) rational approximants, P(0,1) O(dt^2) and P(1,1) O(dt^3), respectively. (for PDE/FDM they correspond to fully implicit and Crank Nicolson, respectively, well established).P(1,1) is more accurate and robust than P(0,1) which is nice but if simple compounding is agreed on financially then the results that P(1,1) produces will be more accurate and at the same time "wrong"? (?). A different value will be calibrated for the SDE.This goes back to the remarks of DavidJN concerning the numerical Vs programming Vs financial interpretation of this formula. // I put it in Student because #questions > #answers (speaking for myself) you might want to put a factor of 1/2 in one....[$] exp(-r\,dt))\approx 1 - r\,dt[$][$](1 - r\,dt/2) / (1 + r\,dt/2)\approx 1 - r\,dt[$] // edit DD[$](1 -\alpha r\,dt) / (1 + (1-\alpha) r\,dt)\approx 1 - r\,dt[$]You are correct. I wrote it down too fast. In fact, exp(- z) is handily written as (2 - z)/(2 + z)// I put a '/' into your formulaVarga's classic book "Matrix Iterative Analysis" has a nice table for Pade(p,q) page 266.
Last edited by
Cuchulainn on December 11th, 2013, 11:00 pm, edited 1 time in total.