Heston Question: Logarithmic Coordinates

Stylz · December 5th, 2005, 7:00 pm

I am reading the very instructive paper "Not-so-complex logarithms in the Heston model" by Kahl and Jackel. This paper has been guiding my implementation. I have a question on something on page 6. They make a comment that for long-dated options the numerical computation will fail due to overflow on GN, and that this problem can be fixed by "neglecting the subtraction of 1 in steps 6 and 9 and the whole procedure of estimating the term ln(G(u)) should be done in logarithmic coordinates). Can someone lend some advice on exactly what this means?Rgds

Fermion · December 5th, 2005, 8:14 pm

I don't know anything about the paper you refer to, but G-1 and log(G) are approximately equal for G near 1. But away from 1 they will have different behavior that may create difficulties in a numerical scheme for one function but not the other. In particular, for large G, a quantity that behaves as G-1 will overflow quicker than one which behaves as log(G).No idea if this is what was intended, but it stands out from what you wrote.

Cuchulainn · December 5th, 2005, 10:06 pm

> Not-so-complex logarithms in the Heston modelCould you explain this pls.

PaperCut · December 5th, 2005, 11:50 pm

Logarithmic coordinates: a mapping where angular coordinates are drawn on the vertical axis and the Log of radius coordinates are on the horizontal one...cf:definition

boschian · December 6th, 2005, 7:35 am

I posed the same question to Kahl direclty who replied:QuoteFor large \tau and increasing u the term e=c(u)*exp(d(u)\tau) can lead to a numerical overflow. As we can neglect the subtraction of -1 for large |e| we can calculate the logarithm of the numerator simply as:Re(Log(e)) = log(|c(u)|) + Re(d(u))\tauIm(Log(e)) = arg(c(u)) + Im(d(u))\tau

Stylz · December 6th, 2005, 12:41 pm

Thanks to all for your replies.Boschian, I wanted to make sure I exactly understand the reply you posted below. Does this mean that the only steps in Algorithm 1 I need to modify are Steps 6, 8, and 9? In other words in Step 9 I need to find a new equation for lnG = ln(GN/GD) based on the equations you provided.Let me know ... thanks again and best regards.

GraemeHamham · July 11th, 2006, 8:36 am

I am currently trying to implement the Kahl & Jackel continuity correction, but I am little puzzled.I am being correct in assuming that the phase number produced by formula (24) will always be 0as t_c will always between - PIE & PIE?Therfore only formula (28) has the potential to be non-zeroApologies, if I am being silly

PiotrW · January 10th, 2007, 2:27 pm

GraemeHamham,try out algorithm embedded in this paper of Kahl (page 9, Algorithm 1).That is true the rotation count of the denominator of G (in Kahl "Not-so-complex ...") is always 0, but note that as far as nominator of G is concerned as \tau goes to infinity (d also has its part), the rotation count of nominator can be larger than 0, isn't it?Hope it heps,PiotrW

wim · January 10th, 2007, 6:01 pm

For your information - in our paper The Little Heston Trap you find a very simple way to solve this problem of branch cuts in the Heston model.