# log normal diffusion

Discussion in 'P1.T2. Quantitative Methods (20%)' started by Mudaltiru, Jul 11, 2008.

1. ### MudaltiruProcess Assistant

Hi David,
Thanks for your daily screencasts. They help us a lot than so many pages of reading.
1. Pls. explain in plain words the log normal diffusion, jump diffusion etc.
2. I also hope that you will do a detailed reading/explanation on two free readings recommended by GARP viz. 1.'CRMPG-II' and 2. 'Basel-II:International....version:June 2006' as they are too lengthy. Still if you insist, then I will read them.

Regards
mudaltiru

2. ### David Harper CFA FRMDavid Harper CFA FRM (test)Staff Member

Hi mudaltiru,

Thanks for liking the screencasts, sincerely, that helps make the effort worthwhile.

1. For equities, in the FRM, easily the most important process is Brownian motion (GBM) which underlies the Black-Scholes. This GBM is a lognormal diffusion. This is two words and two different ideas:
* lognormal refers to the distribution of prices.
* Diffusion refers to the (stochastic) process; i.e., are prices evolving thru time smoothly (continuously) and/or do they exhibit abrupt jumps.

Regarding jump versus diffusion, it is really not so bad: a jump is handled with an additional term that may/may not add a bit of "hop" to the series. So the GBM we study can be altered into a jump/diffusion hybrid with:

"diffusion" GBM
change in stock = [expected drift][time change] + [volatility][random shock][square root of time]

change in stock = [expected drift][time change] + [volatility][random shock][square root of time] + [0 = no jump | 1 = jump]*[insert some distribution]

so now the smooth GBM has an on/off switch that adds a bit of hop to the series (and notably, can create fatter tails) and now it is a jump diffusion process.

To be practical, for the FRM, jump/diffusion won't be tested. Rather, I'd think of DIFFUSION in regard to counterparty risk: in Credit B, I have a slide illustrating how Monte Carlo sim for an interest rate swap first shows "diffusion" then "amortization" (i borrowed from Jorion handbook). This diffusion is related to diffusion in the process above. Both are spreading out. This "spreading out" effect can be seen with the random component of the GBM, which captures the diffusion:

[volatility][random shock][square root of time]
i.e., the longer the time, the greater the variance/uncertainty. This is the diffussion. But, again, re: the exam, think about counterparty exposure: this diffusion is creating greater exposure over time, at least initially.

On the lognormal, that refers to the distribution of PRICE LEVELS. You need to know this, write back if we should talk further on this.

First, PERIODIC RETURNS are normal: LN(S1/S0) is normally distributed.
that is the price distribution.

So, for equities, we say:
periodic returns are normal: ln(s1/s0) ~ N(mean,variance). This implies
period price levels are lognormal: S1 or S1/S0 ~ lognormal

2. Monday will publish the Ops (IV) notes, that will have notes on BASEL/president's group. And, yes, definitely screencasts on these (although some have already viewed last years, but I will still be re-recording for an improvement).

Thanks, David

That's fine.
Thanks David
mudaltiru