Introduction to Binomial Trees

Introduction

Binomial option pricing trees are the groundwork for numerical option pricing. However, binomial models can easily become very large, cumbersome and inefficient. This paper will introduce trinomial pricing trees as an alternative to binomial trees and demonstrate that they can be constructed to produce results equivalent to binomial option pricing trees, but with several advantages: they are smaller, easier to construct, faster to compute and more intuitive than binomial trees.

This paper will develop the trinomial tree from binomial first principles, first with European puts and calls, and then with American puts and calls with dividends. Theoretical design will then be transformed into practical applications with a series of small excel VBA applications in the workbook Tree.xls, which is included as an appendix to this paper.

The option pricing procedures found in Trees.xls will be the foundation of the final section of the paper. Several quantitative tests will be performed on the trinomial trees to compare their run times and valuations to those of a binomial tree. Then both trees will be compared to the standard in option valuation: the Black-Scholes model.

Before introducing the trinomial tree model of option pricing, this paper will first explain several key points of the basic binomial tree including: a brief introduction of up-and-down-ratios, transitional probabilities and volatility measures.

Do you need to mention the client time stuff that you added in??

Introduction to Binomial Trees[1]

The binomial model of stock price movements is a discrete time model and begins with specification of the binomial tree. The starting time of the tree is called the initial time, or initial date, while the ending time is called the terminal time, or terminal date. The construction of the tree begins with dividing the time to expiration into several time steps of duration t, where t = T/n, T equals the time to expiration, and n is the number of time steps in the tree. The model is then constructed to specify the possible security prices at the end of each time period, and to what new prices the security can move to from one time period to the next.

The tree is called a binomial tree, because the security’s price will either move up or down at the end of each time period[2]. How much the stock moves up or down is determined by its volatility. Moreover, the tree specifies the probability that the price will move up or down at each node. An example of a basic four time period binomial tree is illustrated below as figure 1.

Figure 1 - Basic Four Period CRR Binomial Tree.

Figure 1 is a recombining tree. A recombining tree has the general property that at any time, an up move followed by a down move has exactly the same effect on the price as a down move followed by an up move. A recombining binomial tree has the property that any up move, followed by a down move has exactly the same effect as a down move, followed by an up move. A recombining binomial tree is a convenient property for expansion into trinomial trees, and as such, all tree examples presented in this paper will be recombining.

Up-and-Down-ratios of Binomial Trees

Each point in an option pricing tree where lines cross is a called a node of the tree. Each node represents a possible future price of the stock being modeled. For each node of a binomial tree, there are two possibilities for the next movement of the stock price: the underlying security’s price will either go up or go down.

Suppose the price of a security is initially S0 and moves to Su (where Su > S0), run-on sentence this up movement or up-ratio is denoted as u = Su/S0. An up movement is a change in a price. An up ratio is just a number. It does not make sense to say “this up movement or up ratio.” This u is the amount by which S0 is multiplied, if the price rises. Alternatively, if the price falls, the amount d = Sd/S0, called the down-ratio, is the factor by which the price is multiplied. It is an important feature of the binomial model that u and d are the same at every node in the tree.

Note: S0 does not need to equal S0 (ud). For example, if S0 = $100, u = 1.1 and = 0.9, S0 (ud) = $99. This means the binomial tree is not recombining, and an up followed by a down move does not return the security price to its initial value. Because recombining trees offers computational advantages for binomial and trinomial option pricing, a step in which the recombining property is enforced is required. That step is called centering the tree.

Centering the Binomial Tree

In a recombining tree, if the up-ratio is u, then the down-ratio is required to be 1/u, so that an up move followed by a down move takes the underlying price back to where it started. For example, if the ratio is 1.1, then the down-ratio will be 1/1.1 = 0.909. If the price starts at $100 and moves up and back down, it first goes up to $110, and then falls back down to $100.

Before continuing, it is necessary to introduce the forward contract on a non-dividend-paying stock to be delivered at time T. A forward contract is an agreement to buy an asset at a certain price F0, at a certain future time T. For a security that pays no dividends (S) the arbitrage free forward price is:

F0 = S0erT

An arbitrage opportunity is a trading strategy that takes advantage of two different prices on the same security being mispriced relative to each other - in this case the forward price F0 and the securityspot price S0. If F0 > S0erT,then an arbitrageur can buy the asset with borrowed money, at the risk free rate and enter into a short forward contract to deliver the asset at T in exchange for the forward price F0. This set of transactions gives a zero net payoff at time 0, and the payment F0 - S0erT > 0 at time T. If F0 < S0erT,then an arbitrageur can short-sell the asset, invest the proceeds at the risk free rate and enter into a long forward contract to buy the asset at time T for F0. The riskless payoff S0erT- F0 > 0 occurs at time T.

It is possible to choose the down-ratio so that an up move followed by a down move brings the price back to the corresponding forward price of the stock. That is, ud = e2rΔt,where u is the up-ratio, d is the down-ratio, r is the constant, continuously compounding risk free interest rate, and Δt is the length of the time period. The reason the number two appears in the above expression is that an up move followed by a down move takes two time periods. Trees with such up and down ratios are said to be “grown along the forward” (Chriss, 1997), which refers to the fact that an up move followed by a down move takes the spot price to the arbitrage-free forward price at t, for delivery two periods later.

In general, the choice of where to return the price after an up-down movement is called the centering condition.

Risk-Neutral Valuation

In a risk neutral world all individuals are indifferent to risk (uncertainty) and therefore derive no extra utility for avoiding it, or accepting more of it. They require no compensation for risk, in the form of a higher-than-riskless expected rate of return on a risky asset; so the expected return rate of return on all securities is the risk-free interest rate (Hull, 2002). In a risk neutral world the expected rate of return, r solves the following equation over the time interval [0,T]:

E (ST) = S0 erT

Risk-Neutral Valuation is an important general principle in option pricing and is central to the introduction of Transitional Probabilities[3].

Binomial Transitional Probabilities

The probabilities that the asset price will move up, or respectively down, from one node to the next are called the up-transitional probability (qu), and the down transitional probability (qd). The up-transitional probability and the down transitional probability must sum to one. Hence, the down transitional probability can effectively be ignored and rewritten as (1 - qu) or simply (1 – q).

The one period binomial tree representing a security S with initial price S0, up price Su, and down price Sd now has the single period expected value:

qSu + (1 – q) Sd = quS0 + (1 – q) d S0

From the above Risk-Neutral Valuation:

quS0 + (1 – q) d S0 = S0 erΔt

This allows the S0 to be canceled out to obtain:

qu+ (1 – q) d= erΔt

q = (erΔt – d ) / ( u– d )(1)

Equation (1) for variable q is the original risk-neutral probability proposed by Cox, Ross and Rubinstein (CRR) in their 1979 paper, “Option Pricing: A Simplified Approach”. Using this equation, the expected value of S0 after two periods is S0 e2rΔt, and after n periods, is S0enrΔt. In other words, the long-term expected returns are simply powers of the one period expected gross rate of return.

Binomial Trees and Volatility

A positive random variable that changes over time, say S(t), has constant volatility if the standard deviation of log(S(t+1))-log(S(t)) does not depend on t. In this case, the volatility of S is the constant standard deviation. The assumption is that time is measured in years. If volatility were to change between every time period Δt, then both up and down ratios, as well as up and down probabilities would have to be recalculated at every node in the binomial tree. In order to overcome this challenge, local-volatility must be shown to be the consistent at every node in the binomial tree.

Volatility most frequently refers to the standard deviation of the change in value of a financial instrument within a specific time horizon. Conceptually, local volatility represents the amount of variation in the spot price at the particular node of the binomial tree under study. To calculate local volatility first recall that if the spot price is S(0) today and S(Δt ) one period further along the binomial tree, then the annualized, continuously compounded return rate of return on S over this period of time, denoted R(0,Δt), is defined to satisfy . The assumption is that the asset pays no dividends in the interval [0,Δt]. Consequently,

In the binomial model, S(t) changes from S0 at time 0 to Su or Sd at time Δt, where Δt is one period following 0. If, as above, the up transitional probability is q, then the expected value of the rate of return over one period is:

To calculate local volatility (σloc) of the return, take the standard deviation of expected return defined as:

Derivation of Local Volatility

Parameters:

Δt / = / Length of time periods in years
loc / = / Local volatility
q / = / Up-Transitional Probability
1- p / = / Down-Transitional Probability
Su / = / Security Price after Up Movement
Sd / = / Security Price after Down Movement
S0 / = / Initial Security Price
Given: /

Derivation:

Use log not ln

With a recombining binomial tree uS0 = Su and S0/u= Sd. Therefore, the above local volatility equation can be rewritten as:

(3)

Since none of the terms in the above equation depend on time or the asset price, the implicit assumption is that local volatility is the same at every node. This is an extremely important assumption, which allows historical volatility to be estimated and used as a proxy for volatility in the binomial tree.

Historical volatility (σ) can be defined as the standard deviation of the returns provided by a security in one year when the return is expressed using continuous compounding (Hull, 2002) or the standard deviation of log(S(t+1))-log(S(t)) as above.

The General Binomial Tree

Thus far, u, the up-ratio, has yet to be specified. Due to the sheer length of complex calculations that would be required, and with the understanding that this is a paper about binomial extensions to trinomial trees, the derivation of Cox-Ross-Rubinstein (CRR) up and down-ratios will be omitted and left for the reader to investigate independently[4]. The CRR up- and down-ratios are as follows:

(4) and (5)

Now the set of formulas for the standard binomial model is complete and presented below in table 1.

Table 1 - The Binomial Model Standard Parameters and Formulas
Parameters:
T
=
Total time in years
N
=
Number of periods
Δt
=
Length of time periods in years

=
Historical volatility, in percent per annum
r
=
Risk Free Rate
u
=
Up-ratio
d
=
Down-ratio
General Formulas:

=

Δt
=
T/N
q
=
(erΔt – d ) / ( u– d )
u
=

d
=

There are limitations to the above equations if the CRR up-ratio is used. It turns out that many extreme values of low volatility and high interest rates (and therefore high market return) results in an up-ratio probability (p) that is greater than 1, and therefore impossible. These results would never occur with reasonable tree parameters, but for interest, an extreme case is presented below. Consider a binomial tree with the parameters shown in table 2.

Table 2 - Example Binomial Parameters (σ = 0.1, r =.75, T = 1 year)
N
=
52
N
=
365
N
=
8760
Δt
=
1 week
Δt
=
1 day
Δt
=
1 hour
u
d
=
=
1.014
1/1.014 or 0.9862
u
d
=
=
1.0052
1/1.0052 or 0.9948
u
d
=
=
1.0011
1/1.0011 or 0.9989
Figure 2 / Figure 2.1[5]

The tree with the parameters specified in table 2 and Δt of one week has an up transitional probability of 1.0203, which by definition is impossible (See figure 2). But once Δt is reduced to one day, the up transitional probability falls to 0.6952 (See figure 2.1). Further, when Δt is down to one hour, the up transitional probability becomes 0.5398, which implies that q may in fact have a limit. In fact, no matter how low volatility becomes and how far interest rates rise, Δt can be chosen small enough to yield an up probability arbitrarily close to 0.5; see the derivation in table 3.

In order to calculate the limit of q as Δt approaches zero this paper will employee small-o arithmetic and the Infinitesimal Asymptotic property of the exponential function that states:

ex = 1 + x + x2/2 + o(x3) as x → 0

The above property expresses the fact that the error of the exponential function is smaller in absolute value than some constant times x3 if x is close enough to 0.

Table 3 - Derivation of Up Probability Limit
Given:
q = (erΔt – d ) / ( u – d )
u =
u = 1/d



Problem:
Find the limiting value of q as Δt → 0
Solution:
If q = (erΔt – d ) / ( u – d )
q = (erΔt – ) / ( u – )
q = (erΔt – ) / ( – ) (i)
Let
erΔt = ey and = ex
Substitute the values of x and y into (i):
q = (ey – e-x) / (ex – e-x ) (ii)
After applying the Infinitesimal Asymptotic property of exponential functions, (ii) becomes:
(iii)
Expand (iii) and rearrange:

(iv)
Substitute the original values of x and y into each section of (iv) and apply determine the limitas
as (v)
as (vi)
as (viii)
as (ix)
as (x)
Substitute (v) thru (x) into (iv):
as
Therefore:

The preceding technique is necessary only if an impossible probability measure is encountered. In fact, the larger the time period (Δt) is, the faster computation will be.

The Basic Binomial Tree

With a complete set of binomial parameters and formulas a basic tree can be constructed and a simple European option, put or call, can be valued or priced[6]. This is accomplished by filling in each node of the standard binomial tree with a node specific security price and each branch of the tree with a corresponding probability.

To begin, the spot price S0 at node (0, 0)[7] is multiplied by d (the down-ratio) to yield dS0 and placed at node (1, 0). Then, S0 is multiplied by u (the up-ratio) to yield uS0 and placed at node (1, 1). This same process is continued at each subsequent node until the entire tree is filled in and the expiration date has been reached. This process creates an approximately log normal distribution of n + 1 stock prices (Hull, 2002) at the expiration of the option. Next, the values of the option are calculated at expiration (the boundary conditions). The option price at each expiration node is calculated as Max (ST – X, 0) for a call option and Max (X - ST, 0) for a put option, where ST is the stock price at expiration and X is the exercise price. Finally, the option values for each step preceding the expiration date are determined using the following equation:

(6)

Here, c is the current value of a call option, cu is the value of a call option in the next time increment if the stock price goes up, and finally cd is the value of a call option in the next time increment if the stock price goes down (replace c’s with p’s for put options). See figure 3 for an example of an 8-Step binomial option-pricing tree for a European call option.

Figure 3 – Example of An 8 Step Binomial Option Pricing Tree for a European Call
S0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

A copy of this tree is included in the accompanying excel file Trees.xls on the 8 Step Binomial Tree tab

Figure 4 shows an eight step binomial option-pricing model. The upper number in each pair is the stock price at that time step. The lower number is the option value at that time step. The column at the far right represents the boundary condition values of the option at expiration. The European call option value of $4.93 is an approximate valuation, because an eight step binomial tree is not extensive enough to include all possible underlying security prices. A much larger tree is required to reach a more accurate valuation - the same call option after 80 steps is valued at $5.04 and after 8,760 steps is $5.05. The use of an eight step binomial tree in this paper is simply to demonstrate the model’s construction techniques.

Simplifying the Binomial Option Pricing Tree for Faster Computation

Notice in figure 4 that the stock price on any row of the tree is always the same and in fact, every other column is identical after the addition of a new upper and lower value. Knowing this, it naturally follows to remove such redundancies. Figure 4 shows a reconstructed eight step binomial option-pricing tree.

Figure 4 – Simplified 8 Step Binomial Option Pricing Tree for a European Call
S0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

A copy of this tree is included in the accompanying excel file Trees.xls on the Simplified 8 Step Binomial tab

In the diagram of a simplified binomial tree only a single column of security prices remains in the far right column of the tree. The security price at each node is removed and only the option price remains. Further, the midpoint of the column of security prices is the original spot price, and each row above the midpoint is simply the previous row multiplied by u (the up multiplier). Likewise, each row below the midpoint is simply the pervious row multiplied by d (the down multiplier). The stock price that corresponds to any option value can be found on the same row in the far right column.