*The Business Professor*, updated January 30, 2020, last accessed October 29, 2020, https://thebusinessprofessor.com/lesson/binomial-tree-definition/.

### Binomial Tree – Definition

In the finance field, the term binomial tree refers to a graphical representation with possible intrinsic values showing that an option may take place at different periods or nodes. Under this model, the optionās value depends on the underlying financial instruments, such as bonds or stock. On the other hand, the nodeās option depends on the possibility that the underlying assetās price will either increase or decrease at any particular node.

**A Little More on What is a Binomial Tree**

A binomial tree is an essential tool for those individuals who want to price embedded options and American options. The tree is simple to model; however, there when it comes to the possible values that the underlying asset can take within one period of time. The underlying asset can only be worth under a binomial tree model when there is precisely one out of the two possible values. Unfortunately, this is not realistic, because the worth of assets can take any given number value within a various range.

Unlike other models, binomial option pricing has the capacity to handling a good number of conditions. For this reason, many individuals use this approach. The reason for this is that it is based on the underlying instrumentās description over some time and no single point. It is, therefore, used when valuing American options, which happen to be exercisable in a given interval and any time. The model is also used to do valuation of Bermudan options that are also exercisable given time instances.

**How it Works**

You can use the binomial pricing model approach to trace the option keyās underlying variables in discrete-time. You apply the binomial tree, also known as a lattice for several time steps between the expiration dates and valuation. Note that each node in the tree represents the underlyingās possible price at any given point in time.

Valuation using a binomial tree is performed iteratively, begging at each the nodes that you can reach at the expiration time, and then compute backward through lattice towards the first valuation date (first node). Note that the value calculated at each stage becomes the optionās value at that particular time.

When you do option valuation using a binomial tree, the process will take three steps, as shown below:

- Price tree generation
- Option valueās calculation at each final node
- Option valueās sequential computing at each preceding nod

**Why Practitioners Prefer Binomial Tree**

A binomial tree may be slightly slower being slower compared to the Black-Scholes formula but is more accurate, especially for longer-dated options on securities and dividend payments. It is for this reason that most practitioners prefer using the binomial modelās various versions in the options markets.

**Binomial Tree Limitation**

One major limitation of a binomial tree is that it may not be practical when it comes to some options. Note that some options have several uncertainty sources, while some have complicated features, making the binomial approach not fit for such. Monte Carlo simulation is the most preferred model when valuing these types of options. However, Monte Carlo simulation is time-consuming, making it not ideal for computing simulation with a small value of numbers.

### Reference for āBinomial Treeā

https://www.investopedia.com/terms/b/binomial_tree.asp

https://en.wikipedia.org/wiki/Binomial_options_pricing_model

https://subscription.packtpub.com/book/application_development/…/binomial-trees

www.businessdictionary.com/definition/binomial-tree.html

https://demonstrations.wolfram.com/BinomialTree/

### Academics research on āBinomial Treeā

Convergence ofĀ **binomial treeĀ **methods for European/American path-dependent options, **Jiang, L., & Dai, M. (2004). Convergence of binomial tree methods for European/American path-dependent options.Ā ***SIAM Journal on Numerical Analysis***,Ā ***42***(3), 1094-1109. **The binomial tree method, first proposed by Cox, Ross, and Rubinstein [*Journal of Financial Economics*, 7 (1979), pp. 229–263], is one of the most popular approaches to pricing options. By introducing an additional path-dependent variable, such methods can be readily extended to the valuation of path-dependent options. In this paper, using numerical analysis and the notion of viscosity solutions, we present a unifying theoretical framework to show the uniform convergence of binomial tree methods for European/American path-dependent options, including arithmetic average options, geometric average options, and lookback options.}** **The rate of convergence of theĀ

**binomial treeĀ**scheme,

**Walsh, J. B. (2003). The rate of convergence of the binomial tree scheme.Ā**

*Finance and Stochastics***,Ā**

*7***(3), 337-361.**We study the detailed convergence of the binomial tree scheme. It is known that the scheme is first order. We find the exact constants, and show it is possible to modify Richardson extrapolation to get a method of order three-halves. We see that the delta, used in hedging, converges at the same rate. We analyze this by first embedding the tree scheme in the Black-Scholes diffusion model by means of Skorokhod embedding. We remark that this technique applies to much more general cases

Implied binomial trees, **Rubinstein, M. (1994). Implied binomial trees.Ā ***The Journal of Finance***,Ā ***49***(3), 771-818. **This article develops a new method for inferring riskāneutral probabilities (or stateācontingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting riskāneutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending riskāneutral probability distributions.

Generalized binomial trees, **Jackwerth, J. C. (1996). Generalized binomial trees.Ā ***Journal of Derivatives***,Ā ***5***(2), 7-17. **In a novel approach, standard and implied binomial trees are completely specified in terms of two basic inputs: the ending nodal probability distribution and a linear weight function which governs the stochastic process resulting in that distribution. Several key economic principles, such as no interior arbitrage, are intuitively related to these basic inputs. A simple and computationally efficient three-step algorithm, common to all binomial trees, is found. Noting that the currently used linear weight function is unnecessarily restrictive, a binomial tree even more versatile is introduced, the generalized binomial tree. Applications to recovering the stochastic process implied in (European, American, or exotic) options of several times-to-expiration are developed.

On the rate of convergence of theĀ **binomial treeĀ **scheme for American options, **Liang, J., Hu, B., Jiang, L., & Bian, B. (2007). On the rate of convergence of the binomial tree scheme for American options.Ā ***Numerische Mathematik***,Ā ***107***(2), 333-352. **An American put option can be modelled as a variational inequality. With a penalization approximation to this variational inequality, the convergence rateĀ O((Īx)2/3)O((Īx)2/3)Ā of the Binomial Tree Scheme is obtained in this paper.