Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

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Multiperiod Binomial Tree Models

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Note: These pages are prepared with MathJax. MathJax is an open source JavaScript display engine for mathematics that works in all browsers. See http://mathjax.org for details on supported browsers, accessibility, copy-and-paste, and other features.

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### Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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### Section Starter Question

Suppose that you owned a 3-month option, and that you tracked the value of the underlying security at the end of each month. Suppose you were forced to sell the option at the end of two months. How would you ﬁnd a fair price for the option at that time? What simple modeling assumptions would you make?

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### Key Concepts

1. A multiperiod binomial derivative model can be valued by dynamic programming — computing the replicating portfolio and corresponding portfolio values back one period at a time from the claim values to the starting time.

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### Vocabulary

1. The multiperiod binomial model for pricing derivatives of a risky security is also called the Cox-Ross-Rubenstein model or CRR model for short, after those who introduced it in 1979.

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### Mathematical Ideas

#### The Binomial Tree model

The multiperiod binomial model has $N$ time intervals created by $N+1$ trading times ${t}_{0}=0,{t}_{1},\dots ,{t}_{N}=T$. The spacing between time intervals is $\Delta {t}_{i}={t}_{i}-{t}_{i-1}$, and typically the spacing is equal, although it is not necessary. The time intervals can be any convenient time length appropriate for the model, e.g. months, days, minutes, even seconds. Later, we will take them to be relatively short compared to $T$.

We model a limited market where a trader can buy or short-sell a risky security (for instance a stock) and lend or borrow money at a riskless rate $r$. For simplicity we assume $r$ is constant over $\left[0,T\right]$. This assumption of constant $r$ is not necessary, taking $r$ to be ${r}_{i}$ on $\left[{t}_{i},{t}_{i-1}\right]$ only makes calculations messier.

${S}_{n}$ denotes the price of the risky security at time ${t}_{n}$ for $n=0,1,\dots N$. This price changes according to the rule

${S}_{n+1}={S}_{n}{H}_{n+1},0\le n\le N-1$

where ${H}_{n+1}$ is a Bernoulli (two-valued) random variable such that

Again for simplicity we assume $U$ and $D$ are constant over $\left[0,T\right]$. This assumption of constant $r$ is not necessary, for example, taking $U$ to be ${U}_{i}$ for $i=0,1,\dots ,N$ only makes calculations messier. A binomial tree is a way to visualize the multiperiod binomial model, as in Figure 1.

A pair of integers $\left(n,j\right)$, with $n=0,\dots N$ and $j=0,\dots ,n$ identiﬁes each node in the tree. We use the convention that node $\left(n,j\right)$ leads to nodes $\left(n+1,j\right)$ and $\left(n+1,j+1\right)$ at the next trading time, with the “up” change corresponding to $\left(n+1,j+1\right)$ and the “down” change corresponding to $\left(n+1,j\right)$. The index $j$ counts the number of up changes to that time, so $n-j$ is the number of down changes. Several paths lead to node $\left(n,j\right)$, in fact $\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)$ of them. The price of the risky underlying asset at trading time ${t}_{n}$ is then $S{U}^{j}{D}^{n-j}$. The probability of going from price $S$ to price $S{U}^{j}{D}^{n-j}$ is

${p}_{n,j}=\left(\genfrac{}{}{0.0pt}{}{n}{j}\right){p}^{j}{\left(1-p\right)}^{n-j}.$

To value a derivative with payout $f\left({S}_{N}\right)$, the key idea is that of dynamic programming — extending the replicating portfolio and corresponding portfolio values back one period at a time from the claim values to the starting time.

An example will make this clear. Consider a binomial tree on the times ${t}_{0}$, ${t}_{1}$, ${t}_{2}$. Assume $U=1.05$, $D=0.95$, and $exp\left(r\Delta {t}_{i}\right)=1.02$, so the eﬀective interest rate on each time interval is 2%. We take ${S}_{0}=100$. We value a European call option with strike price $K=100$. Using the formula derived in the previous section

$\pi =\frac{1.02-0.95}{1.05-0.95}=0.7$

and $1-\pi =0.3$. Then concentrating on the single period binomial branch in the large square box, the value of the option at node $\left(1,1\right)$ is $7.03 (rounded to cents). Likewise, the value of the option at node $\left(1,0\right)$ is$0. Then we work back one step and value a derivative with potential payouts $7.03 and$0 on the single period binomial branch at $\left(0,0\right)$. This uses the same arithmetic to obtain the value \$4.83 (rounded to cents) at time $0$. In the ﬁgure, the values of the security at each node are in the circles, the value of the option at each node is in the small box beside the circle.

As another example, consider a European put on the same security. The strike price is again $100$. All of the other parameters are the same. We work backward again through the tree to obtain the value at time $0$ as $0.944$. In the ﬁgure, the values of the security at each node are in the circles, the value of the option at each node is in the small box beside the circle.

The multiperiod binomial model for pricing derivatives of a risky security is also called the Cox-Ross-Rubenstein model or CRR model for short, after those who introduced it in 1979.

The disadvantages of the binomial model are:

1. Trading times are not really at discrete times, trading goes on continuously.
2. Securities do not change value according to a Bernoulli (two-valued) distribution on a single time step, or a binomial distribution on multiple time periods, they change over a range of values with a continuous distribution.
3. The calculations are tedious.
4. Developing a continuous theory will take detailed limit-taking considerations.

The advantages of the model are:

1. It clearly reveals the construction of the replicating portfolio.
2. It clearly reveals that the probability distribution is not centrally involved, since expectations of outcomes aren’t used to value the derivatives.
3. It is simple to calculate, although it can get tedious.
4. It reveals that we need more probability theory to get a complete understanding of path dependent probabilities of security prices.

It is possible, with considerable attention to detail, to make a limiting argument and pass from the binomial tree model of Cox, Ross and Rubenstein to the Black-Scholes pricing formula. However, this approach is not the most instructive. Instead, we will back up from derivative pricing models, and consider simpler models with only risk, that is, gambling, to get a more complete understanding of stochastic processes before returning to pricing derivatives.

Some caution is also needed when reading from other sources about the Cox-Ross-Rubenstein or Binomial Option Pricing Model. Many other sources derive the Binomial Option Pricing Model by discretizing the Black-Scholes Option Pricing Model. The discretization is diﬀerent from building the model from scratch because the parameters have special and more restricted interpretations than the simple model. More sophisticated discretization procedures from the numerical analysis of partial diﬀerential equations also lead to additional discrete option pricing models that are hard to justify by building them from scratch. The discrete models derived from the Black-Scholes model are used for simple and rapid numerical evaluation of option prices rather than for motivation.

#### Sources

This section is adapted from: “Chapter 2, Discrete Processes” in Financial Calculus by M. Baxter, A. Rennie [2] and Quantitative Modeling of Derivative Securities by M. Avellaneda and P. Laurence [1].

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### Algorithms, Scripts, Simulations

#### Algorithm

The goal is to set up and solve for the value of the European call option in a two period binomial model. The scripts will output the derivative security value. First set values of $S$, $U$, $D$, $r$, $T$, and $K$. Deﬁne the derivative security payoﬀ function. (In the given scripts, it is for a European call option.) Deﬁne the risk neutral measure $\pi$. Solve for derivative values at $\left(1,1\right)$ and $\left(1,0\right)$ with the risk neutral measure formula. Solve for the derivative value with the risk neutral measure formula linear solver. Finally, print the derivative value.

#### Scripts

R
1S <- 100
2factorUp <- 1.05
3factorDown <- 0.95
4B <- 1
5effR <- 1.02
6deltati <- 1
7K <- 100
8
9f <- function(x, strike) {
10    # European call option
11    max(x - strike, 0)
12}
13
14riskNeutralMeas <- function(fUp, fDown, exprdt) {
15    # risk neutral measure pi
16    (exprdt - fDown)/(fUp - fDown)
17}
18
19piRNM <- riskNeutralMeas(factorUp, factorDown, effR)
20
21v11 <- (1/effR) * (piRNM * f(S * factorUp * factorUp, K) + (1 - piRNM) * f(S * factorUp *
22    factorDown, K))
23v10 <- (1/effR) * (piRNM * f(S * factorUp * factorDown, K) + (1 - piRNM) * f(S *
24    factorDown * factorDown, K))
25
26value <- (1/effR) * (piRNM * v11 + (1 - piRNM) * v10)
27
28cat("value:", value, "\n")