The best-known algorithm is probably the one that divides [0,T] recursively into N intervals using a kind of bisection method. But BB has also an exact solution and it also has a defining SDE that can be solved using FDM on non-constant meshes if needed.My question is; is there a special reason for using the first solution and not the other two?

Do you have references/papers/books for the other 2 methods ?? I am familiar with the bisection type algorithm for BB

QuoteOriginally posted by: CompPDEDo you have references/papers/books for the other 2 methods ?? I am familiar with the bisection type algorithm for BBOksensdal, page 75 has the SDE and an integral solution.Kloeden 1997 (Computer book) page 59 has an explicit solutionBasically, in some cases the standard discretisation performs better than the BB and my main question is why? In that case, I would prefer a sledge-hammer accurate solution with m-core than having to integrate a BB in the algorithms.regards http://www1.cs.columbia.edu/~ap/html/BB.pdf

Hello Cuchulainn,my experience is that the standard discretization works better than the BB, if you have a derivative payoff which is hardly sensitive to the last simulated fixing, but rather to some earlier fixings.A simple vanilla call depends only on the last simulated fixing, so the BB technique will work well in pricing this vanilla option. An intuitive reasoning is that if we have a simple BS framework, it is stupid to generate intermediary fixings. Using this way we would loose the efficiency of the quasi random integration. Generate instead the Wiener path immediately at the last fixing! This is what BB does. In spite of this, if the derivative payoff does not depend on the last simulated fixing (or just hardly), but it depends strongly on few intermediary fixing, then it is better to generate the Wiener path first for these intermediary fixings, and only after that for the other fixings incorporating the last fixing too.The example of Papageorgiou is exactly such a derivative. The payoff depends on each fixing equally, thus the BB, where the simulation of the Wiener is started at the last fixing and then the simulation is jumping, does not offer a consistent advantage. Especially, if the derivative payoff is more sensitive to earlier fixings rather than to later fixings it is better to use the standard discretization than the BB.However, my experience is that such a product as of Papageorgiou is rare on the market. Most of the products, even if they are path-dependent, depend mainly on the final fixing. For instance a barrier option is a classical path-dependent option, but if the barrier is too far then the option works like a vanilla. Pricing such a derivative the BB technique will work very well.I developed a multidimensional optimization for the BB where I define the path building steps based not only on time, but also on total volatilities through time and in case of multiple assets based on the asset correlations. Moreover, as I wrote above I start the generation of the Wiener path for the payoff fixings, and then I continue with the simulation of the model requested fixings. In general, this technique works well and quickly for me and I use it for local volatility type models, where I have to simulate many model requested fixings, but with my payoff I want to focus only on a few fixings.Hopefully I did not answer something that was not your question.Peter

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: CompPDEDo you have references/papers/books for the other 2 methods ?? I am familiar with the bisection type algorithm for BBOksensdal, page 75 has the SDE and an integral solution.Kloeden 1997 (Computer book) page 59 has an explicit solutionBasically, in some cases the standard discretisation performs better than the BB and my main question is why? In that case, I would prefer a sledge-hammer accurate solution with m-core than having to integrate a BB in the algorithms.regards http://www1.cs.columbia.edu/~ap/html/BB.pdfwhat problem are you trying to solve? the question is not meaningful unless you state that.

QuoteOriginally posted by: mjQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: CompPDEDo you have references/papers/books for the other 2 methods ?? I am familiar with the bisection type algorithm for BBOksensdal, page 75 has the SDE and an integral solution.Kloeden 1997 (Computer book) page 59 has an explicit solutionBasically, in some cases the standard discretisation performs better than the BB and my main question is why? In that case, I would prefer a sledge-hammer accurate solution with m-core than having to integrate a BB in the algorithms.regards http://www1.cs.columbia.edu/~ap/html/BB.pdfwhat problem are you trying to solve? the question is not meaningful unless you state that.I am examining it to see if it is better than standard discretisations (which show oscillating 'saw-tooth' behaviour) with MC. Some applications of BB allow monotonic convergence (see the examples in the Kucherenko Wilmott article), instead of oscillating above and below the true solution. The examples were with QMC and MC.

QuoteBasically, in some cases the standard discretisation performs better than the BB and my main question is why? ...I am examining it to see if it is better than standard discretisations (which show oscillating 'saw-tooth' behaviour) with MC. Some applications of BB allow monotonic convergence (see the examples in the Kucherenko Wilmott article), instead of oscillating above and below the true solution. The examples were with QMC and MC.The former point is one of the main issues of current QMC research: distributing the descriptive power available for a sequence along the most important directions of the integrand at will (as opposed to just improving discrepancy "in general", which is too much of a conservative goal). Sobol/Faure have good distribution properties in the first lower dimensions, and a BB change of variables alignes something like an european asian so that most variation is there (and orthogonal sections are almost constant so that sampling them with worse behaved projections is less of an issue). The fact is macroscopic for Sobol with unit initialization, and better initializations or scramblings do not really solve the issue, the consequences of such property just get shifted farther away from the eye... (an analogy would be i.e. improving the constant but not the order of convergence of an algo).Other QMC sequences dont have this ordering of dimensions so that projection quality can be constant on each slice (with equal index spacings) by construction (at a certain price, as always; and that's just one of the reasons why they're not always welcomed by practitioners).When you use BB on a digital a-la Papagorgeou on the other hand the "bad" directions of something like Sobol are not aligned with the unimportant directions of the resulting integrand anymore. Ofcourse this is a special case built specifically to break down Sobol+BB: in fact if we take each term as being equally important (in an ANOVA decomposition) and still we dont spread evenly the descriptive budget of the samples (here QMC sampling axes and path subintervals are "aligned" from the start, that's the trick), then some of it will be "lost" for no gain, and thus the result poor... (incidentally this also tells you why Sobol here is not the best to use even with the standard discretization hope that the mix of the two factors doesnt make my explanation too obscure though)There's no silver bullet, dont blame BB or Sobol as the result depends on the interplay of three different factors: integrand, transformation and QMC sequence. So in this case you either find a better transform to change the integrand or build up an integration rule specifically for your "fixed" function (ofcourse the former option is much more practical often even for a QMC expert).Also, the important thing is the order of convergence, not just the oscillatory behaviour, as the latter again will depend on the interplay of samples and integrand and it shouldnt be hard to come up with functions having oscillating QMC solutions too, it's simply uncommon. Dont base your code on that i mean, i.e. for error estimation or related issues like halting. (one shouldnt use sample variance either, with MC or RQMC, but that's preaching in vain, I know..)

QuoteOriginally posted by: CuchulainnI am examining it to see if it is better than standard discretisations (which show oscillating 'saw-tooth' behaviour) with MC. Some applications of BB allow monotonic convergence (see the examples in the Kucherenko Wilmott article), instead of oscillating above and below the true solution. The examples were with QMC and MC.Using BB should make zero difference with MC; the joint distribution of the random variables is identical. With QMC it can improve convergence if you do the right coordinate change to make the low dimensions do more work. (as quartz said at greater length.)

QuoteUsing BB should make zero difference with MC; the joint distribution of the random variables is identical. I appoximate the underlying SDE using numerical method and decreasing deltaT and/or increasing NSIM does not always lead to monotone convergence. Some writers claim that BB is better for problems with long-term simulations, I will need to check it (we want to switch between std Wiener and BB in the code).QuoteWith QMC it can improve convergence if you do the right coordinate change to make the low dimensions do more work. (as quartz said at greater length.)It's nice getting the rationale, sometimes, and is quite helpful. And dare I mention it, what about Karhunen-Loeve?

On an aside issue related to this specific topic I am building a C++ generic framework that can be customised to accomodate different kinds of requirements. So, for example we could switch between path evolvers, Increment generators, SDE, and RNGs, for example.In this way I can test the accuracy of the different methods. If we could define standard interfaces that would be nice; then you can plug in. What is new the use of policy classes using templates.