Thursday, 8th October 2009
Commencing at 6.30pm
Venue: 7city Learning, 4 Chiswell Street, 3rd floor, London
Venue: 7city Learning, 4 Chiswell Street, 3rd floor, London
Speaker: Andreas Binder, head of the UnRisk maker Mathconsult
Unrest?
This is not a review about "Unrest" a CD from "Henry Cow", an avant-garde-music group of the late 60ies founded at Cambridge University by multi-instrumentalists Fred Frith and Tim Hodgkinson, joined by some of my favorite musicians Chris Cutler, Lindsay Cooper, Dagmar Krause, ..
Because We Are Nice,
don't think we are stupid.
Our customers know it well, but prospective users often cannot believe that our licensing/pricing does not hide anything in the-small-print.
Algorithmic Architecture
In If UnRisk was a Tall Building I thought of similarities of designing-for-usability-stability-robustness in architecture and software. When Lake Point Tower uses the triangle core which holds all weight, UnRisk uses a numerically optimized C++ kernel to hold all heavy computations. Its universal interface is represented in Mathematica. See The UnRisk Langage a language about derivatives, structures and other financial objects and their valuation.
To tree or not to tree: trinomials
We have seen in a former blog article that binomial trees may lead to oscillating solutions and even to heding parameters of the wrong sign arising from these oscillations. Therefore, trinomial trees were invented and are still widely used for short rate models like Hull-White or Black-Karasinski.
As I am not a well-known tree-addict, the reader may expect some good reasons against trinomials.
The basic branching of trinomials looks like this
At each node there are three possible branches with probabilities p1, p2, p3 who should add up to 1. In contrast to the binomial model, the trinomial model is not a complete one leading to a unique price only from no-arbitrage arguments. Therefore, there is a certain arbitrariness to assign the probabilities to the different branches. The advantage of this arbitrariness is that it is fairly easy to obtain recombining trinomial trees.
When we think of a (mean-reverting) Hull-White model, the nodes and the probabilities are typically chosen in such a way that the expected value of the short rate (at the next time step) and its variance fit the analytic values available for these. If a node is too far away from the mean reverting level, this has the consequence that the one of the branches would be assigned with a negative probability leading necessarily to severe oscillations as the explicit scheme is not stable any more.
Therefore down-branching and up-branching are used for trinomials in mean reverting models.
So what?
Well, down-branching and up-branching cuts off certain parts of the calculation domain and therefore leads to wrong values even for the most easy instrument, a zero coupon bond. Yes, certainly you can fiddle around and stretch your calculation tree in such a way that you meet the discount factors again. But if you do this, you never ever can use analytic formulae for bonds, options or CMS rates any more. It would be much much easier if you had a consistent and stable numerical scheme which does not change your model.
And yes, there is one: Finite elements with proper treatment of convection (which is the reason for the stability problem) lead to much much better results.
As I am not a well-known tree-addict, the reader may expect some good reasons against trinomials.
The basic branching of trinomials looks like this
At each node there are three possible branches with probabilities p1, p2, p3 who should add up to 1. In contrast to the binomial model, the trinomial model is not a complete one leading to a unique price only from no-arbitrage arguments. Therefore, there is a certain arbitrariness to assign the probabilities to the different branches. The advantage of this arbitrariness is that it is fairly easy to obtain recombining trinomial trees.
When we think of a (mean-reverting) Hull-White model, the nodes and the probabilities are typically chosen in such a way that the expected value of the short rate (at the next time step) and its variance fit the analytic values available for these. If a node is too far away from the mean reverting level, this has the consequence that the one of the branches would be assigned with a negative probability leading necessarily to severe oscillations as the explicit scheme is not stable any more.
Therefore down-branching and up-branching are used for trinomials in mean reverting models.
So what?
Well, down-branching and up-branching cuts off certain parts of the calculation domain and therefore leads to wrong values even for the most easy instrument, a zero coupon bond. Yes, certainly you can fiddle around and stretch your calculation tree in such a way that you meet the discount factors again. But if you do this, you never ever can use analytic formulae for bonds, options or CMS rates any more. It would be much much easier if you had a consistent and stable numerical scheme which does not change your model.
And yes, there is one: Finite elements with proper treatment of convection (which is the reason for the stability problem) lead to much much better results.
Hidden In Plain Sight?
The UnRisk Language: Barrier Options
UnRisk code: click to enlarge
Let's assume we want to price a European up & out call option with the following properties:- The
spot price of the underlying equity is 100 EUR and the continuous dividends shall are 1%
- The lifetime of the option is 1 year
- The strike price is 100 EUR
- The barrier is 120 EUR
As long as the barrier has not been reached the option stays alive.
As long as the barrier has not been reached the option stays alive.
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