Replicating Americans a la Jaeckel - Rebonato

AVt · Post by **AVt** » Sun Mar 13, 2005 5:23 pm

This approach allows to approximate american options by a portfolio

of europeans which care for 'user-defined' smiles. The portfolio is

constructed backwards in time by a recursive numerical procedure.

Trying to follow their paper I have troubles to get similar numbers

as they can be guessed from their graphics (and actually if using

more steps the methods fails for me). So i may have an error ...

In the first step i get 111.39073413968 as critical strike S_star_n-1

and weight 0.37185102253954 for the 'optiolina'.

For the next step my point seems to be Eq 20, p.14. The argument of

the expectation operator equals the sum of 2 europeans, which is (19).

The second of those is just the weighted pay-off function for call

with strike S_star_n-1, while the first is an european call with a

strike K which expires at time T.

Now the expectation at time t = t_n-2 of an european call expiring

at T is the BS value for the call with life time T-t (which holds not

only by starting backwards from the pay-off function, but is true in

general - no? They use discounted expectation, cf p. 11).

So (20) stands for BSCall(s,K,2*tau) + h_n-1*BSCall(s,S_star_n-1,tau)

where tau = t_n-2 - t_n-1 = t_n-1 - t_n is taken to be equidistant

(so the weighted call expires in tau years and the other in 2*tau).

For that i get a critical strike S_star_n-2 = 116.28308442250 and a

weight h_n-2 = 0.17724854493624.

Now 'looking' at figure 3 ff their critical point is below 116 (while

mine is above) and their price is ~ 10.5 (while mine is 11.28 ) by

'visual inspection'.

This is done for their data: spot=100,strike=105,expiry=3,rate=0.045,

yield=0.065,vola=0.1 and a time stepping of 1 year, ie n= 3.

Now if i choose 1/2 year as steps (n=6) already in the second step

the constructed portfolio (with 1 'optiolina') does not fall below

the intrinsic value spot-K and the procedure fails.

Has anybody checked the paper or can point me to my errors Dead ?

FDAXHunter · Post by **FDAXHunter** » Sun Mar 13, 2005 9:21 pm

Okay using your your inputs I get the exact same numbers that you did.

S*1 = 111.3907

Weight 1 = 0.3718

S*2 = 116.2830

Weight 2 = 0.17724

Now, what I want to point out to you is that the gentlemen Jäckel and Rebonato actually used [b]4.25%[/b] as their rate and [b]NOT[/b] 4.5% as you did! (However, they later reverted to another implied volatility of 11.35% in the numerical section... go figure).

In which case we get:

S*1 = 110.9823

Weight 1 = 0.3931

S*2 = 115.44

Weight 2 = 0.2011

Now, I tried the same thing by decreasing the step size to 0.5 and get the same problem. Can't get the portfolio with one optiolina to fall below intrinsic Dead .

I will try to work on that tomorrow again, if I have time. I would like to solve that too.

FDAXHunter · Post by **FDAXHunter** » Tue Mar 15, 2005 10:20 am

Played around with this some more. Still can't get it to work, also not for a 4 year option with dT=1 year. Phuck knows, I'm obviously too stupid to see the problem.

AVt · Post by **AVt** » Tue Mar 15, 2005 8:21 pm

Thx ... sigh ... it also does not work for puts ... can not find the authors emails

[img]/beta/User%20Files/66/X27.gif[/img]

edited to add: the only (reasonable?) idea i had was the portfolio is too expensive

and to lower costs one can sell calls above the critical values (as that portfolio is

assumed to be eliminated above). But i have no systematic approach ... And the

weightings are used to make the deltas jump to 1, they do not achieve it smoothly ..

FDAXHunter · Post by **FDAXHunter** » Thu Mar 17, 2005 9:59 am

You can reach the author at riccardorebonatorboscom Let me know what you find out.

AVt · Post by **AVt** » Wed Mar 23, 2005 2:52 pm

the actual email has the prefix "no.answers.for.unknowns." ... as always :-) so what ...

Does somebody have american put prices (with dividend yield = 0) of high quality

as reference values? By that i mean at least 6 - 8 reliable (proved?) digits of

exactness (as far as i have understood the theme a binomial method gives 3 to 4).

mj · Post by mj » Wed Apr 20, 2005 8:46 am

you'd do better to talk to peter jaeckel -- it was his method

Nonius · Post by **Nonius** » Wed Apr 20, 2005 8:53 am

Isn't he totally evil, like Carlos the Jackal?

JumpStart · Post by **JumpStart** » Thu Apr 21, 2005 4:08 am

here's what i get using binomial, so not high quality. couldn't quite understand the paper... [url=/User%20Files/584/1.zip]1.zip[/url]

mj · Post by mj » Fri Apr 22, 2005 2:04 pm

maybe you guys could write your understanding of the algorithm and i'll check whether it corresponds to mine?