Posted by
Peter Caspers-4 on
May 23, 2015; 4:11pm
URL: http://quantlib.414.s1.nabble.com/Matching-results-between-HW-tree-and-simulation-models-tp16399p16584.html
nice blog in any case. You have something like
Var( r(t) | r(s) ) = \int_s^t [ exp(-a(t-u)) \sigma(u) ]^2 du
from where you can see how the reversion and model volatility affect
the conditional variance of the short rate in a MC simulation (so
bigger \sigma is "bad", bigger a is "good", which seems somewhat
consistent with your statement). However in real life you calibrate to
real market instruments and a and \sigma will adjust to match their
volatility. So if you e.g. prescribe a "big" positive reversion value
the calibrated sigma will be "big" as well and vice versa. So in the
end, if the market volatility is low, your MC simulation will produce
estimates with lower variance, if it is high, you will get a higher
variance. Which is quite obvious looking at it this way. In a zero
volatility market you will only need one path.
Another remark is that it is often useful to change the measure (e.g.
to a T-forward measure) which allows for large jumps in a simulation.
For example if you price a bermudan swaption with MC, you would only
need the exercise times as simulation times without loosing any
accuracy.
Which leads us to the convergence in other vendor products (which are
we talking about btw ?). Let's take an example, NumeriX, which is
quite popular I think. Without having access to it currently, I would
assume that they do not use the risk neutral measure in their Hull
White MC implementation and that zero bonds come out exactly
indepedently of the number of simulation paths. Alex, is that true :-)
?
best regards
Peter
On 22 May 2015 at 20:40, gouthambs <
[hidden email]> wrote:
> Hey Rakesh
>
> The convergence of Monte Carlo simulation of Hull-White model, as it turns
> out, not only depends on the number of paths that you use, but also on the
> parameters sigma and a that you chose.
>
> I discuss the convergence of Monte Carlo simulation of the Hull-White model
> <
http://gouthamanbalaraman.com/blog/hull-white-simulation-monte-carlo-convergence.html>
> in this blog post. To summarize, good convergence will be obtained for the
> P(0,T) as long as T is smaller than sqrt(a/sigma).
> In your test case, you chose sigma=a=0.1. This means you will have good
> convergence for small number of paths as long as T is roughly a few months.
>
> When you say you have seen good convergence with other vendor products, did
> you try the same parameters a, and sigma. It would be very interesting if
> that is the case.
>
> Peter, Luigi, I would welcome your thoughts on my analysis.
>
>
> Goutham
>
>
>
>
>
>
> -----
> --
>
http://GouthamanBalaraman.com> --
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http://quantlib.10058.n7.nabble.com/Matching-results-between-HW-tree-and-simulation-models-tp16399p16581.html> Sent from the quantlib-users mailing list archive at Nabble.com.
>
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