Follow ups from the first Model&Method Risk seminars are exciting. It is all about the combination of valuation and risk management, better the parallelization in the work-flow with the objective of an early warning system. You cannot really think about the model risk of a valuation afterwards?!
Pricing with PDEs - What pitfalls you have to take care of I
After Andreas Binder gave insight to you that pricing financial instruments using trees can end with disaster I will show you in a multi-part blog that numerical methods for partial differential equations (PDEs) give you reliable tools for fast and stable pricing. How to choose the appropriate technique and what pitfalls you have to take care of will be discussed in detail.
UnRisk-Q High-Value at Low-Price
Recently, we took UnRisk-Q to the quant community. We made this by configuring and packaging our latest technologies of which our UnRisk product family is made. We decided to transit this benefit to the quant-finance developers. Price examples:
Shhh....Preparing Major Releases over the Summer
We have revolved and refined our short term development plans. Over the course of the next weeks, independent whether the weather becomes hot or not, we will conclude major developments and experiments on UnRisk's computational kernels as well as front-ends.
If UnRisk was a Tall Building
It most probably was the Lake Point Tower by the lake shore in Chicago. Based on a conceptual design by Ludwig Mies van der Rohe , built by Schipporeit&Heinrich.Because of its height and location on the shore of Lake Michigan, the residential skyscraper had to be designed to withstand the high winds. At the center of the building is a triangular core, which contains 9 elevators and 3 stairways. This core also holds all the vertical weight of the building. Radiating from the core are three arms, which form an asymmetrical Y-shaped floor plan.
Due to the angles between the wings the units of LPT feature some stunning panoramic views over the city, but so that the apartments would not face each other. The design also offers less surface area exposed to direct wind loads. Consequently the building is more stable and generally safer.
It does much more than expected from a first view.
Not in the Toothbrush Business
Traveling back from Zurich, Andreas read about the trends in the financial markets, see FT and Convertibles, while I tried to get insight from other businesses. In the English issue of Wired the cover story was on "Inside Google".
Financial Times Germany and Convertible Bonds
Flying home from Zurich today, I came across an article in Financial Times Deutschland saying in the headline “Convertible bonds bring good luck”.
Model Risk Seminar in Zurich
15-Jul-09, Zurich. Again, we had a sprited discussions with quant experts, who use already techniques, which are not so common in financial circles. For example the application of finite element (FE) techniques to solve complex models in their PDE form.
It Must be a Metrics? Risk ... ?
If you want to assess anything, you want to do this unbiased, thus you search for a metrics. And if it was building tick-boxes.
Are You ... ? You don't Look ... ?
As often I have a little time I read Emanuel Derman's Blog (as Paul Wilmott's and a few others). ED is one of the most influencing quantitative finance experts, author of My Life as a Quant. I like his special humor.
If UnRisk was a Ballroom Dance
It most probably was the Foxchatrot demo performed by Luca & Lorraine Barrichi.
Watch The Dance
Watch The Dance
They prove that it does not need heavy make-up, bells and whistles to dance in the most elegant and smooth way, reduced to the essence of dancing.
Their foxchatrot combines the best of several worlds: foxtrot steps and a jazzy version of Bali Hai.
At UnRisk we are also passionate about combinations from several worlds: Take Mathematica as a language, advanced numerical techniques proven in reactor modelling and treat your customers as if you had to dance with them.
Their foxchatrot combines the best of several worlds: foxtrot steps and a jazzy version of Bali Hai.
At UnRisk we are also passionate about combinations from several worlds: Take Mathematica as a language, advanced numerical techniques proven in reactor modelling and treat your customers as if you had to dance with them.
To Tree or Not To Tree
Binomial trees are perfect. Well, the Cox-Ross-Rubinstein One-Level Model is perfect to introduce the concept of a risk-neutral measure, which is (and was also not for me) not easy to understand for beginners. When there are no dividends, constant interest rates and constant volatilities, then N-level binomial trees recombine.
Recombining N-level trees need little storage and are easy to implement, even for instruments with early exercise rights. If done properly (from the algorithmic point of view), they are quite efficient.
A readable piece od Mathematica code for a European call would be
BinomialEuropeanCall[S_, K_, r_, sigma_, T_, n_] :=
Module[{dt, a, up, down, P, Q, BinomTree, value, level},
dt = T/n;
a = Exp[r*dt];
up = Exp[sigma*Sqrt[dt]]*a; down = Exp[-sigma*Sqrt[dt]]*a;
P = (a - down)/(up - down); Q = 1 - P;
P = P*Exp[-r*dt]; Q = Q*Exp[-r*dt];
BinomTree = Table[Max[S*down^node*up^(n - node) - K, 0], {node, 0, n}];
Do[BinomTree =
Table[{P, Q}.{BinomTree[[node]], BinomTree[[node + 1]]},
{node, 1, level}]; {level, n, 1, -1}];
value = BinomTree[[1]];
Clear[BinomTree];
value]
Nevertheless, there must be some disadvantages of binomial trees, otherwise my blog colleague Michael Aichinger would not have any reason to tell you about, say, Finite Elements and other really clever methods.
Even or odd?
One disadvantage of naïve implementations of binomial trees is that the option value depends quite heavily (and oscillatory) on the number of levels used as the following figure exhibits:
For vanilla instruments, these oscillations could be repaired by calculating the mean of two neighbouring values. But when discontinuous conditions are of importance (like in knock-out options), things get even worse. The following figure shows the value of an up-and out call option calculated by binomial trees and its binomial delta.
Recombining N-level trees need little storage and are easy to implement, even for instruments with early exercise rights. If done properly (from the algorithmic point of view), they are quite efficient.
A readable piece od Mathematica code for a European call would be
BinomialEuropeanCall[S_, K_, r_, sigma_, T_, n_] :=
Module[{dt, a, up, down, P, Q, BinomTree, value, level},
dt = T/n;
a = Exp[r*dt];
up = Exp[sigma*Sqrt[dt]]*a; down = Exp[-sigma*Sqrt[dt]]*a;
P = (a - down)/(up - down); Q = 1 - P;
P = P*Exp[-r*dt]; Q = Q*Exp[-r*dt];
BinomTree = Table[Max[S*down^node*up^(n - node) - K, 0], {node, 0, n}];
Do[BinomTree =
Table[{P, Q}.{BinomTree[[node]], BinomTree[[node + 1]]},
{node, 1, level}]; {level, n, 1, -1}];
value = BinomTree[[1]];
Clear[BinomTree];
value]
Nevertheless, there must be some disadvantages of binomial trees, otherwise my blog colleague Michael Aichinger would not have any reason to tell you about, say, Finite Elements and other really clever methods.
Even or odd?
One disadvantage of naïve implementations of binomial trees is that the option value depends quite heavily (and oscillatory) on the number of levels used as the following figure exhibits:
For vanilla instruments, these oscillations could be repaired by calculating the mean of two neighbouring values. But when discontinuous conditions are of importance (like in knock-out options), things get even worse. The following figure shows the value of an up-and out call option calculated by binomial trees and its binomial delta.
The true (Black-Scholes) delta should become negative above 115 or so. But the calculated binomial delta is positive in most of the points and negative only at the sharp peaks downwards. So, if one delta-hedges, he/she might end up with a situation of increased instead of decreased market risk. Well done, trees!I will return to trinomial trees in my next blog entry.
We take UnRisk-Q to the Quant Community
Our PRICING ENGINE has been introduced 2001. Hundreds of front-office and risk practitioners enjoy immediate results, from instant deal types in the Excel front-end and the robustness from high-end numerical schemes.
If UnRisk was a Car
it most probably was a Fiat 500 Abarth. The name comes from the world famous tuning firm that in the 1950s and 1960s took ordinary Fiats and turned them into racing cars with a formidable reputation. It has exceptional performance, 135 bhp but safety ist maximized by the latest control technologies. It is driven by the need to improve a car's performance for comfort and security.
We, at UnRisk, have tuned proven models by high-end numerical schemes and boost performance by clever principal component application, GPU co-processing and Grid-computing. To get a speed-up of 10.000, you do not need a supercomputer. A minimalist Windows-based infrastructure is perfect. A solution, easy to set up and deploy, that is validated. UnRisk FACTORY.
Equal Valuation and Risk Management?
Valuation and risk management, two sides of the quant business, must be treated with equal sophistication, with equal respect, and with equal suspicion. And there must be closer interaction between them. At every stage of valuation and model development you must be asking questions about risk and robustness. It is dangerous to come up with some fancy model and only afterwards start asking questions about model error. Anyone who has ever calibrated a model knows that the methods used to mitigate model risk almost come as an afterthought, and are totally inconsistent with the original model.
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