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 Trees1

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 period2. 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.

1 Cursory knowledge of binomial trees is assumed for this paper, but for an introduction see Cox, Ross, and Rubinstein (1979). For more advanced treatment see Duffie (2001).2 It is possible to construct binomial trees where after each period both branches of the tree result in higher security prices. This paper will only consider the case where the security moves up or down after each period, ie Sup > So > Sdown.

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 Probabilities3.

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.

3 For extensive coverage of Risk Neutral Valuation see CRR 1979.

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 volatilityq = Up-Transitional Probability

1- p = Down-Transitional ProbabilitySu = Security Price after Up MovementSd = Security Price after Down MovementS0 = Initial Security Price

Given:

Derivation:

Use log not lnWith 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 independently4. 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

4 See (Hull, 2002) or (CCR, 1979)

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.

Figure 2 Figure 2.15

5 Figure 2 and 2.1 were produced using Matlab 6.5

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

ud

==1.0141/1.014 or 0.9862

ud

==1.00521/1.0052 or 0.9948

ud==

1.00111/1.0011 or 0.9989

q=1.0203

q=0.6952

q=

0.5398

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 LimitGiven:

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)

LeterΔt = 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 priced6. 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 6 The price or value of an option is the selling cost before premiums and commissions an option’s price is also called its premium that are charged to a customer who wishes to purchase the option. Run-on sentence7 The node numbering system is simple. Starting at the initial time, t0, each node is numbered according to its time and height from the bottom. So, the leftmost node (the current spot price) is numbered (0, 0), the two nodes at time 1, t1, are numbered, from the bottom-up, (1, 0) and (1, 1).

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 CallS0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

106.9936.99

101.4731.68

96.23 96.2326.66 26.23

91.26 91.2621.91 21.47

86.54 86.54 86.5417.41 16.98 16.54

82.07 82.07 82.0713.34 12.72 12.29

77.83 77.83 77.83 77.839.87 9.08 8.27 7.83

73.81 73.81 73.81 73.817.07 6.23 5.26 4.03

70.00 70.00 70.00 70.00 70.004.93 4.13 3.22 2.07 0.00

66.38 66.38 66.38 66.382.67 1.92 1.07 0.00

62.96 62.96 62.96 62.961.13 0.55 0.00 0.00

59.70 59.70 59.700.28 0.00 0.00

56.62 56.62 56.620.00 0.00 0.00

53.70 53.700.00 0.00

50.92 50.920.00 0.00

48.290.00

45.800.00

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 CallS0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

36.99 106.9931.68 101.47

26.66 26.23 96.2321.91 21.47 91.26

17.41 16.98 16.54 86.5413.34 12.72 12.29 82.07

9.87 9.08 8.27 7.83 77.837.07 6.23 5.26 4.03 73.81

4.93 4.13 3.22 2.07 0.00 70.002.67 1.92 1.07 0.00 66.38

1.13 0.55 0.00 0.00 62.960.28 0.00 0.00 59.70

0.00 0.00 0.00 56.620.00 0.00 53.70

0.00 0.00 50.920.00 48.29

0.00 45.80A 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.

The first column to the left of the securities represents the boundary conditions (or the payoff of the option at expiration), which are equal to Max (ST – X, 0) for a call and Max (X - ST, 0) for a put. This technique will flow through and make trinomial option pricing much simpler. A recipe for binomial option pricing is given below in table 4.

Table 4 - Recipe for Simplified Binomial Tree Option Calculations1. Obtain last year’s one-year historical volatility σ for the underlying security.2. Obtain the current risk free rate (one year government bond yield), and denote it r. 3. Calculate Δt = T / N where T is the time to expiration in years of the option and N

is the number of periods in the binomial tree. Eighty is a good rule of thumb for N.

4. Calculate , d = 1/u5. Calculate q = ( erΔt – d ) / ( u– d )6. If q is greater then one double N and repeat steps 3 to 5.7. Place S0 in the middle of the far right column of the binomial tree (the apex).

8. Fill in each row above S0 by multiplying S0 by um where m is the number of rows above the apex of the tree.

9. Fill in each row below S0 by multiplying S0 by dm where m is the number of rows below the apex of the tree.

10. Calculate the boundary condition for every other row (starting at the top) and place them in the second row from the left using the equations: Max (ST – X, 0) for a call and Max (X - ST, 0) for a put. X is the strike price and ST is the terminal price found in the right column and same row.

11. Fill in each node of the tree by discounting the boundary conditions back to time t0 using the equation:

12. The price at node (0, 0) is the option price.

See Appendix A for a copy of the corresponding VBA code for the Simplified Binomial Tree and the European tab found on Trees.xls for the working copy.

From Binomial to Trinomial Option Pricing Trees

The trinomial option tree is an extension of the binomial tree with a more intuitive design and efficient pricing process. The transfer from binomial to trinomial is as simple as combining two steps of a binomial tree into a single step of a new trinomial tree. This is illustrated in figure 5, where thin lines correspond to the original binomial tree and thick lines to the constructed trinomial tree.

Figure 5 – Example of Reconstructed Trinomial Tree

Starting from node (0, 0) the spot security price, S0 can now go up to u2S0, down to d2S0,

or remain the same udS0 = S0. This in many ways seems more intuitive than the binomial model, as actual securities often behave in such an up, down or no-change fashion.

The recombining step is accomplished by using a two-step version of equation (6). For example, over the second period of Δt years, the stock price Su becomes either Suu or Sud (which is the same as simply S); the corresponding call option values are cuu and cud. Also, in the second time period of Δt years, the stock price Sd becomes either Sdu (S) or Sdd with call option values of cud and cdd. Applying equation (6) to these two cases gives the following results:

Substituting cu and cd into equation (6) gives:

(7)

Equation (7) allows for a direct solution for the call value given at time t from the value at time t + 2. Now instead of three discounting steps from time t + 2 to time t, only one is required. This allows for a new abbreviated tree where every second column of the binomial model can simply be ignored. The new simplified tree is called a trinomial option-pricing tree.

The new time step will be twice the length of the equivalent binomial model and denoted Δttri = 2Δtbi (the subscripts tri and bi are to differentiate the binomial and trinomial time-steps, the subscript will be dropped for convenience for the remainder of the paper); the parameters for the trinomial tree are presented in table 5.

Table 5 - The Trinomial Model Standard Parameters and FormulasParameters:

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 Return

U=Up ratio

D=Down ratio

General Formulas:Δt

=T/N or 2Δtbi

Qu

=q2

Qs

=2(q(1-q))

Qd

=(1-q)2

U=u2

D=d2

c=

(8)

Figure 6 is an example of the binomial model shown in figure 4 simplified to the trinomial design. Note that in figure 4, only the right column has the security prices and at each node of the tree only an option value remains.

Figure 6 – Simplified 4 Step Trinomial Option Pricing Tree for a European CallS0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

36.99 106.9926.66 26.23 96.23

17.41 16.98 16.54 86.549.87 9.08 8.27 7.83 77.83

4.93 4.13 3.22 2.07 0.00 70.001.13 0.55 0.00 0.00 62.96

0.00 0.00 0.00 56.620.00 0.00 50.92

0.00 45.80

A copy of this tree is included in the accompanying excel file Trees.xls on the 4 Step Trinomial Tree tab Note in figure 6 that the column to the far right once again represents the underlying security prices that apply to each row, but that the incremental change in prices is much larger than in the binomial example. Each column in figure 6 replicates every other column of figure 4 resulting in a much smaller tree. Notice that the ending call value in figure 6 is still $4.93, as was the case in the binomial example.

Like the binomial example, the trinomial European call option value of $4.93 is an approximate valuation, because a four step trinomial tree is not extensive enough to include all possible underlying security prices. But it is important to note that the call value was calculated in half as many time steps, but 1.5 times the number of calculations per step, implying that trinomial trees are more efficient as binomial. The redesigned recipe for trinomial option pricing is shown below in table 6.

Table 6 - Recipe for Simplified Trinomial Tree Option Calculations1. Obtain last year’s one-year historical volatility σ for the underlying security.2. Obtain the current risk free rate (one year government bond yield)comma and

denote it r. 3. Calculate Δt = T / N (= 2Δtbi) where T is expiration in years of the option and N

is the number of periods in the trinomial tree. Fourty is a good rule of thumb for N.

4. Calculate , D = 1/U

5. Calculate q = ( erΔtbi – d ) / ( u– d )6. If q is greater then one, double N and repeat steps 3 to 5.7. Calculate Qu = q2, Qd = (1-q)2, Qs = 2(q(1-q)) = 1 - Qu - Qd

8. Place S0 in the middle of the far right column of the trinomial tree (the apex).9. Fill in each row above S0 by multiplying S0 by Um comma where m is the number

of rows above the apex of the tree.10. Fill in each row below S0 by multiplying S0 by Dm comma where m is the number

of rows below the apex of the tree.11. Calculate the boundary condition for every row (starting at the top) and place

them in the second row from the left using the equations: Max (ST – X, 0) for a call and Max (X - ST, 0) for a put. X is the strike price and ST is the terminal price found in the right column and same row.

12. Fill in each node of the tree by discounting the boundary conditions back to time t0 using the equation: c =

13. The price at node (0, 0) is the option price.

See Appendix B for a copy of the corresponding VBA code for the Simplified Trinomial Tree and the European tab found on Tress.xls for the working copy.

Adjusting the Trinomial Tree for American Options

Adjusting the trinomial tree to take into account American features is relatively straightforward with the trinomial tree, requiring only that each node of the tree be checked to see if early exercise is optimal (Hull, 2002). This can be accomplished by working backward through the tree and choosing the maximum of early exercise (St-X, for a call, X-St for a put), or the option’s current value (ct for a call or pt for a put). In other words, replacing step 12 from table 6 above with the following:

c = (9)

Thus far, all examples have been illustrated using a call option, but put options are just as simple and require only two simple adjustments to the boundary conditions and the above discounting equation (9). A put option has the boundary condition Max (X - ST, 0) and the discounting equation .

The put option will be used for the remainder of this paper because it will demonstrate cases where early exercise of the American option is optimal. To make an American call option display rational early exercise, negative interest rates or dividends need to be introduced. The American call option has the property of being “Better Alive then Dead or (BAD)” under most typical valuation cases.

Figure 7 shows the calculation of a European put option price. The same put is then priced to include the American feature in figure 8, where the numbers in italics represent where early exercise would be optimal. The difference between the European and American option value is called the premium, or adder, for the American feature. Figure 7 and 8 display the differences between a Non-Dividend paying European and American trinomial tree.

Note that the American feature may not be fully valued when there are too few steps because there is not enough detail in the tree to identify where early exercise is optimal.

Figure 7 – Simplified 4 Step Trinomial Option Pricing Tree for a European PutS0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

0.00 106.990.00 0.00 96.23

0.00 0.00 0.00 86.540.74 0.38 0.00 0.00 77.83

3.20 2.83 2.35 1.64 0.00 70.006.87 6.72 6.61 7.04 62.96

12.51 12.94 13.38 56.6218.64 19.08 50.92

24.20 45.80

A copy of this tree is included in the accompanying excel file Trees.xls on the 4 Step Trinomial Tree tab

Figure 8 – Simplified 4 Step Trinomial Option Pricing Tree for a American PutS0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs

0.00 106.990.00 0.00 96.23

0.00 0.00 0.00 86.540.76 0.38 0.00 0.00 77.83

3.36 2.96 2.45 1.64 0.00 70.007.26 7.04 7.04 7.04 62.96

13.38 13.38 13.38 56.6219.08 19.08 50.92

24.20 45.80

A copy of this tree is included in the accompanying excel file Trees.xls on the 4 Step Trinomial Tree tab See Appendix C for a copy of the corresponding VBA code for the American Trinomial Tree, and the American tab found on Trees.xls for the working copy. An American enabled version of the binomial model is also included in the Appendix D, and the working code on the tab is labeled American.

Adjusting the American Trinomial Tree for Dividends

There are many ways to adjust the trinomial tree for dividends and all are similar to the binomial methods of adjustment. This paper will present the Constant Yield Method adapted from “Option Pricing: A Simplified Approach” (CRR 1979) and the Constant Dollar Method, where we assume that all dividends are known cash amounts.

The Constant Yield Method

The Constant Yield Method assumes that the underlying security maintains a constant dividend yield on each ex-dividend date. Suppose a security that followed a trinomial tree process has a price of S0, one period before a dividend is paid (cum-dividend). When the security goes ex-dividend, the holder of the security will receive a dividend of either: uS0, dS0, or S0. The security price ex-dividend will either be u(1 – )vS0, d(1 – )vS0 or (1 – )vS0, where v = 1 if the end of the period is an ex-dividend date and v = 0 otherwise. Both and v are assumed to be known with certainty. See figure 9 below.

Figure 9 – Dividend Process of the Constant Yield Trinomial Tree

It is important to realize that uS0 <> dS0 <> S0. That is to say, dividends are not constant dollar but consist yield or proportional to S. With this treatment, the following step in the tree is just like any other, the tree will recombine and u (1 – )vS0 * d = (1 – )vS0 * u and d (1 – )vS0 * u = (1 – )vS0 * d. See figure 10.

Figure 10 - Simplified 4 Step Trinomial Option Pricing Tree for an American Put Adjusted for a Constant Dividend Yield of 1.5% Per Year

S0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs, Div = 1.5%

Dividend Timing 0 0 1 0 0

0.00 105.390.00 0.00 94.78

0.06 0.00 0.00 85.241.07 0.67 0.24 0.00 76.67

3.94 3.58 3.10 2.38 1.05 68.958.10 7.99 7.99 7.99 62.01

14.23 14.23 14.23 55.7719.84 19.84 50.16

24.89 45.11

Simplified Trinomial Put with Constant Yield Dividend

See Appendix E and F for a copy of the corresponding VBA code for the American Binomial and Trinomial Trees with Constant Dividend Yield, and the Dividends tab found on Trees.xls for the working copy.

The Constant Dollar Method

The Constant Dollar Method assumes that dividends are known cash amounts that pay Ω on each ex-dividend date. This means that no matter which path the trinomial tree takes between cum and ex-dividend dates, the payout will be exactly the same and the value of the underlying security will decrease by Ω. The security price ex-dividend will be either uS0 - v Ω dS0 - v Ω or S0 - v Ω, where v = 1 if the end of the period is an ex-dividend date

S0

u(1-δ)1S0 dividend paid: δuS0

(1-δ)1S0 dividend paid: δS0

d (1-δ)1S0 dividend paid: δdS0

cum-dividend

ex-dividend

and v = 0 otherwise. Both Ω and v are assumed to be known with certainty. See figure 11 below.

Figure 11 – Dividend Process of the Constant Dollar Trinomial Tree

The constant dollar dividend method requires the initialization of a second variable Div, which is handled in parallel to the security S. The security, S is modeled using the standard European or American trinomial approach as above.

If the option price has been calculated for t+1, and there is no dividend at t, then the option price at t is just the expected discounted value of the option price at t+1. Update the dividend varable: Divt = erΔt Divt+1. If there is a dividend payment at t then Divt = erΔt

Divt+1 + Ω. Finally, ensure that Div0 + root S0 is equal to the initial security price S0. This is done by placing S0 –Div0 at the apex, or center, of the far right-hand column of the tree during the initial step.

If the option is American, then the present value of the dividend payment must be subtracted from the current security price (St – Divt) before comparison to the strike price to check for early exercise. See figure 12.

Figure 12 - Simplified 4 Step Trinomial Option Pricing Tree for an American Put Adjusted for a Constant Dollar Dividend of $1.05 Per Year

S0 = $70, = 0.3, Exercise Price = $70, r = 0.1, T = 0.25 Yrs, Div = 1.5%

S0

uS0 - v Ω dividend paid: Ω

S0 - v Ω dividend paid: Ω

dS0 - v Ω dividend paid: Ω

cum-dividend

ex-dividend

Dividend Payments 1.04 1.04 1.05 0 0

0.00 105.410.00 0.00 94.80

0.06 0.00 0.00 85.261.07 0.67 0.24 0.00 76.68

3.93 3.57 3.10 2.37 1.04 68.968.09 7.98 7.98 7.98 62.02

14.22 14.22 14.22 55.7819.83 19.83 50.17

24.88 45.12

Simplified Trinomial Put with Constant Dollar Dividend

See Appendix G and H for a copy of the corresponding VBA code for the American Binomial and Trinomial Trees with Constant Dividend Yield, and the Dividends tab found on Trees.xls for the working copy.

Market Completeness

In order for a market to be complete, it must satisfy the “lottery ticket completeness criterion”. This lemma requires that for any adapted sequence8 of numbers, there is a portfolio strategy9 whose portfolio dividend process10 exactly matches the sequence, at all times t > 0, and at all nodes for those times. That is not lottery ticket completeness, which requires only that for every future node, you can create a portfolio strategy that pays $1 at the node, and 0 at all other future nodes.

Simply put, the trinomial tree does not satisfy the “lottery ticket completeness criterion,” and hence is not a complete model, but instead a simplification of the binomial design. This simplification unfortunately results in the loss of market completeness, but still retains the valuation power of the original design.

Testing the Trees for Valuation Convergence to Black-Scholes

Although the pricing theories behind both binomial and trinomial trees are intellectually interesting, they are all for naught if the option value calculated does not converge with the street-standard Black-Scholes (1973). Fortunately, both the binomial and trinomial models converge to within a penny of the European Black-Scholes model (BSM), as well as its American extensions that were developed by Barone-Adesi and Whaley (BAW) 8 Definition: A sequence of random variables X0, X1, X2, … , XT is adapted if for all t, the realized value of Xt

depends only on the node that is reached at t. Thus X0, X1, X2, … , XT is adapted if for each time t, and every node nε Nt, there is exactly one value, say Xtn, that Xt can possibly have if node n occurs.9 Definition: A portfolio strategy, denoted θ, is a complete specification of how much of each security to hold, at all times and in all situations. Thus for each t=0, 1, … , T, for each node n∈Nt at t, and for each security j=1, 2, …, K, θ has to tell you how much of security j to hold at time t if node n has occurred.10 Definition: A portfolio dividend process denoted δθ for portfolio strategy θ, is the payment schedule resulting from θ, or each t=0, 1, … , T, for each node nεNt at t, and for each security j=1, 2, …, K.

and Bjerksund and Stensland (BS). The convergence happens after a surprisingly small number of steps.

This paper will consider convergence to have occurred once the absolute difference between the binomial or trinomial option value and its Black-Scholes equivalent is less than one-half percent of the Black-Scholes calculated option value. When this measure is used, conversion occurs rapidly and is summarized in table 7 and figure 13.

Table 7 – Convergence Table Binomial and Trinomial Tree vs. Black-Scholes and Extensions

S0 = $70, = 0.5, Exercise Price = ($60, $70, $80) , μ = 0.2, T = 1 Yrs

X = StrikeS = Initial Price Binomial Trinomial Binomial Trinomial Binomial Trinomial Binomial Trinomial

X = S 34 18 34 18 96 48 56 54X < S 18 10 18 10 70 35 * *X > S 20 10 20 10 22 12 10 10

American Call European Put American PutEuropean CallTime Steps

* The American Put did not converge to the average of the BAW and BS as these valuations were so far apart. The binomial and trinomial models did however converge to a steady state after approximately thirty and sixty steps respectively.

The above table is included in the accompanying excel file Trees.xls on the Convergence tab

Figures 13.1, 13.2 and 13.3 illustrate how quickly sequences of trinomial prices can converge to their continuous time counterparts. When comparing the trinomial to the binomial model, convergence occurs in about half the number of time steps.

It is worth noting that the American continuous time approximations of Barone-Adesi and Whaley and Bjerksund and Stensland are only approximations. These approximations (Haug, 1997) produce two closed-form solutions which themselves do not agree. It therefore follows that these approximations may in fact be less reliable than the binomial and trinomial models presented above (Whaley, 1982).

Comparing the Speed of the Binomial and Trinomial Models

This paper considers speed in so called “Client Time11”or the speed in seconds, or fractions of seconds, that it takes a computer to value an option once all the inputs have been delivered to a function call. In an attempt to be scientific, the appendix includes copies of all the code used in the time tests and every effort has been made to ensure that both the binomial and trinomial models have the same basic architecture. Please refer to appendices A thru F for further details. Appendices G and H include a VBA code for barrier options with up-and-out and down-and-out puts and calls to further test the robustness of the trinomial design.

Since every CPU will result in different computation times, comparisons have been simplified to the percent difference between the binomial and trinomial models. A 11 Client time is: “The time you leave a client on hold while your computer derives your desired value.” This time should be short; a decent reasonable benchmark is 15 seconds.

reading of 21% in the percentage difference column of table 8 below indicates that the trinomial model performed 21% faster, with equal inputs.

When performing the speed tests, the trinomial tree always has an input of half the time steps (n) of its binomial equivalent. This may lead to the assertion that it would naturally compute in half the time. In fact, this is generally the case and is due, for the most part, to the smaller number of time steps.

When the binomial and trinomial trees use 1,000 and 500 time steps they are traveling through 501,501 and 251,001 nodes respectively12, so it makes sense that computation time should be less. Again, note that the trees are equivalent, meaning they have the same maximum and minimum values of possible ending security prices and final option valuations (refer back to figures 4 and 6). This means that even though the steps are halved, the range of possible terminal prices remains the same.

Examining table 8, European trinomial trees can be solved 21% faster then an equivalent binomial tree. This is certainly impressive, but it is when the American option is added in, that the true computation savings are realized. American option with dividends or barriers solve an average of 50% faster when the trinomial model is employed - a significant advantage over the binomial model. Table 8 – Computation Time: Binomial vs. Trinomial Trees

CallsBinomial Value ($)

Trinomial Value ($)

Binomial Compution

Time (seconds)

Trinomial Compution Time

(seconds)Percentage Difference

European $19.87 $19.87 0.1900 0.1500 21%American $19.87 $19.87 9.1330 4.5260 50%Dividends $17.39 $17.39 9.2630 4.5160 51%PutsEuropean $7.18 $7.18 0.1900 0.1500 21%American $8.91 $8.90 9.1830 4.5070 51%Dividends $9.65 $9.64 9.1830 4.4960 51%Barrier OptionsCall Up and Out (H = 80) $12.47 $11.70 4.8470 2.4040 50%Put Up and Out (H = 80) $7.28 $7.28 4.8470 2.4130 50%Call Down and Out (H = 60) $12.68 $13.76 4.8970 2.4040 51%Put Down and Out (H = 60) $6.42 $6.42 4.7970 2.3740 51%

Number of time steps ( n ) (1000 Binomial, 500 Trinomial)Volatility ( ) 50% Strike price ( X ) 70Time to maturity ( T ) 1.00 Asset price ( S ) 70Expected Return (μ) 20% Div Yield (Div) 5%

Computations done on a Sony Viao Laptop Celeron CPU 1.80 GHz, 256 MB of Ram

Conclusions

12 The number of nodes on a binomial tree = (n+2) * (n+1) / 2 and the trinomial tree = (2N + 2) * (n + 1) / 2

This paper illustrates a method of constructing simplified trinomial option pricing trees from binomial first principles. Binomial pricing trees are often used to value options; however, they can become very large and cumbersome. Computation time can be improved by moving to a trinomial design, where the equivalent range of security prices can be model in fewer computation steps, while still producing corresponding results.

This paper displays methods for valuing put and call options, European and American options, and techniques for dealing with dividends that do not interfere with the design of the simplified trees. Tests of successful convergence to the Black-Scholes formula were given and showed that the trinomial model can be used alongside the proven Black-Scholes as a reliable approach to option valuation.The Black-Scholes model has not been “proven”. In particular, asset prices are definitely not log-normal. Their logarithms have thicker tails, and higher densities at the mean, than any normal distribution.

Finally, “Client Time” speed tests were performed and the trinomial model was found to be up to fifty percent faster than the binomial model for American style options, which is clearly a significant result.

In summary, trinomial tree option valuation is a superior alternative to the binomial model displaying equal valuation power with faster computational time.