I'm not sure if there is an answer to this or not. I'm doing some analysis on binomial and trinomial trees, and trying to understand how many steps I need. As the time to expiry increases, presumably I need more and more steps to have the same percentage error. But what would be more accurate, a 10 year option with 10,000 steps, or a 1 year option with 1,000 steps. (If it makes a difference, these are barrier options I'm looking at).

A 10-year option with 10,000 steps, assuming we're just talking about the error from discretization. Obviously, the longer term security has larger error from things like parameter uncertainty and drift.To a first approximation, the percentage error introduced by discretization depends only on the number of steps, not their size. So 1,000 steps is as good or bad for a 1 day security as a 100 year security.However, with barrier options, the situation is different. Now the size of the step matters because it gives you a more accurate idea of whether the barrier is reached. This error can be reduced somewhat by probabilistic barrier crossing. If the barrier is at $50 from above and you move in $0.25 steps, you might treat a price of $50.25 as having a 25% chance of touching the barrier, $50.00 as having 50% and $49.75 as having 100%. Of course, you can get more sophisticated about the probabilities. Another technique is to use smaller price steps near the barrier. Still, in most cases over ten years, the barrier microstructure matters less than the gross movement.In your case, however, the 10,000 steps is clearly better. Each step will be 10^-0.5 times the size of the 1,000 one-year steps so it will be more accurate for the barrier, and the greater number of steps means a smaller percentage error from the macro movements.

there was a paper a year or two ago in finance and stochastics where the asymptotics of the errors for binomial trees were computed.

Try Leisen D.P., Reimer M., (1996); Binomial Models for Option Valuation – Examining and Improving Convergence, Applied Mathematical Finanace 3, 319–346.

Thanks for the answers - I'll look into those.

QuoteOriginally posted by: gjlipmanI'm not sure if there is an answer to this or not. I'm doing some analysis on binomial and trinomial trees, and trying to understand how many steps I need. As the time to expiry increases, presumably I need more and more steps to have the same percentage error. But what would be more accurate, a 10 year option with 10,000 steps, or a 1 year option with 1,000 steps. (If it makes a difference, these are barrier options I'm looking at).You are now asking a question that has to do with a branch of maths called Numerical Analysis. The binomial method is notorious for a number of reasons (I am not going to get into it). Anyways ... BN- conditionally stable: time step must be chosen small otherwise 'ziggy-zaggy'- it is only first order accurate- not good for barrier options- With small steps and many calculation you get ROUNDOFF errors, so results cannot be trusted.A better solution is to use finite differences. It is a well-established.BTW almost everyone I meet says they never use BM (maybe they are closet binomialsts (I plagiarised this remark from DaveAngel)).My humble opinion is give BM a break and move on to more robust methods. But there again, I am missing the point.Edit: the more calculations you do the more round off you get. This is a well-known problem (numerical analysis 201). People have some up with Interval Analyis to resolve this problem. http://creativelimits.net/research/interval/

if you're interested in valuing barrier options then best to go for either tian's trinomial tree or the QUAD numerical integration approach (duck, newton, widdicks et al paper in JFE) - can improve with richardson extrapolation too

> best to go for What is best? Accuracy, robustness, etc.The trinomial method is an EXPLICIT FDM schems, not something we always want. Each method has advantage/disadvantages and we should quantify.Instead of putting the cart before the horse, we should say something like: "I want to price an option in X nanoseconds to an accuracy of Y with a reliabilty factor of 99.5%" Otherwise it's just pears and apples.

QuoteOriginally posted by: AaronA 10-year option with 10,000 steps, assuming we're just talking about the error from discretization. Obviously, the longer term security has larger error from things like parameter uncertainty and drift.To a first approximation, the percentage error introduced by discretization depends only on the number of steps, not their size. So 1,000 steps is as good or bad for a 1 day security as a 100 year security.Aaron,A couple of questions about your posting above:1. Assuming one wanted to, how would one go about crunching the tree for 10,000 steps? Wouldn't the calculation of the binomial coefficient overflow at anything over ~ 1000?2. If the error of the tree is only dependent on the number of steps and not their size regardless of what the time to expiry of the option is, then I take it that bigger is better? And we want to compute the maximum number of steps possible? Is that correct? Is there an optimal method of choosing steps?Thanks.

There's no problem with 10,000 steps. It's true that if you tried to compute the 2^(-10,000) it would round to zero even with double precision. But you never do that computation and, in any case, a zero wouldn't hurt your result. It's also true that you don't have to bother with anything beyond, say, five standard deviations. If you're getting overflows, then you're doing something wrong.In general, when approximating something continuous, the more steps, the more accuracy. However, once you get the error from discretization down to a tolerable level, further increases in step size may not help, and could lead to other problems (certainly, slow runs and possibly numerical instabilities). So a good general method for choosing the number of steps is to compute the discretization error, choose a number of steps such that it is less than other sources of error (for example, errors less than one basis point of price probably don't matter in most applications). If that number of steps is too large for convenient processing, you might want to use less, but it's usually not a good idea to use more, even if the computer time is insignificant.If you are constrained, there are usually clever things you can do to increase the effective number of steps. Even if you're not constrained, some of those things can be useful.