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Paul
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What is the binomial model and how does it work?

November 14th, 2004, 10:12 am

thanks, wstguruP
 
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exotiq
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What is the binomial model and how does it work?

November 23rd, 2004, 1:50 am

The binomial model in finance is the special case of a binary tree presented by John Cox, Steve Ross, and Mark Rubinstein in 1979 where nodes represent possible future prices in a recombining manner so that there are n possible prices at time n rather than 2^n if the tree did not recombine. It's major assumption is a discrete version of the main Black-Scholes assumption, which is that you do not know whether the asset price will move up or down in the next time period, but you know know by how far it will move (volatility), and so with only two points, you can delta hedge a derivative. Then assuming you can borrow or lend to trade the asset at some interest rate r (or trade it forward), the tree can be constructed so that the moves center around the forward price and the probability of moving up or down are the same. The beauty of "binomial" is that means the probabilities of each move, expanded to the term of a European option with n timesteps can be represented as the coefficients of the binomial (0.5 + 0.5)^n, and so pricing an option is simply a matter of multiplying the value of the option by the probabilities of each of those prices and discounting back along the paths to the present. Although this elegant version requires the volatility to be constant throughout and payoff to be European, trees are remarkably flexible at handling American, exotic, and path dependent payoffs for many different types of options in a way very intuitive for even non-quants.Since one value for volatility generally does not match real option prices in the market, Rubinstein presented in 1994 a method of implied binomial trees similar to the local volatility models that binomial trees are a special numerical case for.Trees and the binomial model also appear in the work of Jarrow and Rudd, and later in the interest rate model of Black, Derman, and Toy.
 
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Errrb
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What is the binomial model and how does it work?

December 21st, 2004, 1:10 am

Binomial/Multinomial tree with constant probabilities can be viewed as a method to approximate gaussian integrals of the following form where n(x) is a pdf of gaussian distribution and f(x) is a payoff function, for example for a plain vanilla call option. American or early exercise optionality can be modeled using binomial/multinomial trees.In this scheme it is implementation of the more general dynamic programming algorithm, known as a Belman equation.
 
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Cuchulainn
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What is the binomial model and how does it work?

December 21st, 2004, 11:22 am

Why is the binomial method so popular despite its accuracy, perfiemancce limitations and difficulty with barrier options, for example?
 
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Errrb
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What is the binomial model and how does it work?

December 21st, 2004, 11:41 am

It is popular, because it is simple. It also has clear graphical representation and despite its limitations in terms of accuracy etc. it brings useful intuition about more complicated problems.
 
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Fermion
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What is the binomial model and how does it work?

December 23rd, 2004, 5:02 pm

For me, the critically important point about binomial trees is how they can be generalized. However complex the wider probability distribution for macroscopic times, one can hope to reproduce it in a fairly transparent way by assuming a normal (or lognormal, depending on the variable defining the nodes) distribution for each microscopic node with (locally variable) instantaneous drift rate and volatility. Understanding the structure of those local moments, is then the key to understanding the macroscopic probability distribution. If one is audacious enough to believe one knows the higher instantaneous moments as well, then the tree can be generalized to use that information too.In the more general case, one typically has enough freedom in the way one chooses nodes to also improve speed and/or reduce error and make an appropriate compromise. Edit: To be clearer: the essence of the generalization lies in the method of recombination (preventing node explosion). When the volatility is not constant, the regular binomial recombination condition breaks down. It is the variety of ways in which recombination can be generalized (to permit locally variable moments) that gives the power to reproduce arbitrary pdfs and optimize speed and/or error.
Last edited by Fermion on December 22nd, 2004, 11:00 pm, edited 1 time in total.
 
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exotiq
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What is the binomial model and how does it work?

December 23rd, 2004, 6:09 pm

QuoteOriginally posted by: CuchulainnWhy is the binomial method so popular despite its accuracy, perfiemancce limitations and difficulty with barrier options, for example?As a former trader once told me, trees are popular because they are how traders think. Most traders do not think in terms of PDEs, martingales, or probability integrals, but they visualize a stock as being able to go either up or down tomorrow by a certain amount, and they just want to know how many shares they have to hold to be hedged either way. In my experience, trees do not perform anywhere near as badly as many numerical analysts seem to suggest, and are an important sanity check for one-factor models. Trees have parity with PDEs and can be constructed in almost form desired, as Fermion mentioned.
 
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Cuchulainn
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What is the binomial model and how does it work?

January 3rd, 2005, 7:56 am

> trees do not perform anywhere near as badly as many numerical analysts seem to suggestI have just developed templated C++ classes for binomial and trinomial trees and will hopefully soon come up with objective results, performance wise when compared to FDM.Any ideas anyone on a common benchmark example?On the other, the problems with trees ARE well known (barrier options, saw behaviour).Trees are explcit FDM, so the time step k has to be small enough
 
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Cuchulainn
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What is the binomial model and how does it work?

January 3rd, 2005, 12:44 pm

Fermion How would one apply binomial method to Merton jump model or even Levy processes? has this been done anywhere?
 
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exotiq
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What is the binomial model and how does it work?

January 3rd, 2005, 3:12 pm

QuoteOriginally posted by: CuchulainnFermion How would one apply binomial method to Merton jump model or even Levy processes? has this been done anywhere?You generally can't use binomial methods for jump-diffusion or arbitrary Levy processes, since with only two possible moves, you can model either two possible diffusive steps (like Black diffusion in the original CRR tree), or drift vs jump steps to model intesity of pure jump processes (I use these for credit derivatives sometimes).General trees/lattices/meshes/grids can handle almost any process you like. A simple extention of a binomial tree having a jump step coming off as a third node is probably the most straightforward way to handle jump diffusion. For example, from a node in the binomial tree with the usual +1%/-1% steps, you add a -10% step to model the jump, and solve for its probability. One less conventional but usefully clear trick about n-nomial trees is that they can measure arbitrary processes characterized by n-1 moments, by solving the linear system Ap = m where A = (for n = 4) [1,1,1,1;dd,d,u,uu;dd^2,d^2,u^2,uu^2;dd^3,d^3,u^3,uu^3] (that is, row vectors of powers of the possible returns), and m are the moments of those returns. It is not too hard to see that this is just solving for the distribution from the definition of the distributions' moments, and that it is easy to ensure all these probabilities are positive.
 
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Cuchulainn
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What is the binomial model and how does it work?

January 4th, 2005, 8:31 am

I can imagine that very clever an ingenious methods can and have been found fir the binomial method. However, PDE/FDM methods stand on the shoulders of giants like Gauss, Fourier, Laplace, Hilbert, Courant, van Neumann and many others and have been used in real dificult problems.The inuition behind the up/down of binomial method is great. I explained it to my 1-year old son last night (I'm serious) . Once he understood it he immediately discretised the paper into rectangular meshes he ws writing on. It looked like a rectangular FDM mesh.IMO generating a mesh of S values is more or less the same as generating an FDM mesh
 
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Cuchulainn
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What is the binomial model and how does it work?

January 4th, 2005, 8:33 am

OOPS!!!> I explained it to my 1-year old son last night I meant to say 10 (ten) years, that would be going a bit too far!
 
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Cuchulainn
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What is the binomial model and how does it work?

January 4th, 2005, 8:58 am

This jpg contains:top part: a random Walk middle part: standard binomial (my work)bottom: extrapolation to a FDM mesh (not my work)
 
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Cuchulainn
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What is the binomial model and how does it work?

January 7th, 2005, 3:16 pm

have developed a binomial option calculator using C++ and templates. I have taken a standard European option with T = 1 (1 year expiry date). The steps are:1. I create the tree in mempry (n subdvisions => n! nodes approx.), use a structure like list<vector<double> > This is forward induction.2. Then using backward induction we calcukate bith call and put prices on same tree from 1Results on a Pentium 4, 256 Mb, 1.8 GHz N == number of subdivisions, R = response time in secondsN R2000 : 23000 : 45000 : 11How does this compare in other languages? Matlab, VBA etc.
 
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Fermion
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What is the binomial model and how does it work?

January 21st, 2005, 8:51 pm

For benchmarking numerical methods, European options (or anything else with an exact closed form solution) provide a simple opportunity, because we can compare speeds for a given accuracy. That is, take S=100, T= 1, r=0, vol = 25%, for example, and compute the time required to generate the BS price within (say) 1 cent, for a range of strikes. The algorithm for deciding termination must not depend on the actual price. (That is, the method must have an algorithm for deciding when sufficient accuracy has been reached without knowing what the BS price is.) Without such a method, the accuracy will not be known in other cases.I posted an iteration algorithm for using trees to price to a given accuracy in the thread "binomial model against analyical approximation" in the General Forum.