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

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Rating

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

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Question of the Day

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 determine 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

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 the figure:

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

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

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. 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 $4.82 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.

Advantages and Disadvantages of the model

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 more complete theory is going to take some detailed and serious 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 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 which 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 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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Problems to Work for Understanding

1. Consider a two-time-stage example. Each time stage is a year. A stock starts at 50. In each year, the stock can go up by 10% or down by 3%. The continuously compounded interest rate on a $1 bond is constant at 6% each year. Find the price of a call option with exercise price 50, with exercise date at the end of the second year. Also, find the replicating portfolio at each node.
2. Consider a three-time-stage example. The first time interval is a month, then the second time interval is two months, finally, the third time interval is a month again. A stock starts at 50. In the first interval, the stock can go up by 10% or down by 3%, in the second interval the stock can go up by 5% or down by 5%, finally in the third time interval, the stock can go up by 6% or down by 3%. The continuously compounded interest rate on a $1 bond is 2% in the fist period, 3% in the second period, and 4% in the third period. Find the price of a call option with exercise price 50, with exercise date at the end of the 4 months. Also, find the replicating portfolio at each node.
3. A European cash-or-nothing binary option pays a fixed amount of money if it expires with the underlying stock value above the strike price. The binary option pays nothing if it expires with the underlying stock value equal to or less than the strike price. A stock currently has price $100 and goes up or down by 20% in each time period. What is the value of such a cash-or-nothing binary option with payoff $100 at expiration 2 time units in the future and strike price $100? Assume a simple interest rate of 10% in each time period.
4. A long strangle option pays $max\left(K_1-S,0,S-K_2\right)$ if it expires when the underlying stock value is $S$. The parameters $K_1$ and $K_2$ are the lower strike price and the upper strike price, and $K_1<K_2$. A stock currently has price $100 and goes up or down by 20% in each time period. What is the value of such a long strangle option with lower strike $90$ and upper strike $110$ at expiration 2 time units in the future? Assume a simple interest rate of 10% in each time period.
5. A long straddle option pays $|S-K|$ if it expires when the underlying stock value is $S$. The option is a portfolio composed of a call and a put on the same security with $K$ as the strike price for both. A stock currently has price $100 and goes up or down by 10% in each time period. What is the value of such a long straddle option with strike price $K=110$ at expiration 2 time units in the future? Assume a simple interest rate of 5% in each time period.

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Reading Suggestion:

References

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Outside Readings and Links:

1. Peter Hoadley, Options Strategy Analysis Tools.. A useful link on basics of the Black Scholes option pricing model. It contains terminology, calculator, animated graphs, and Excel addins (a free trial version) for making a spreadsheet model. Submitted by Yogesh Makkar, September 9,2003.
2. Binomial Tree Option Pricing by Simon Shaw, Brunel University. The web page contains an applet that implements the Binomial Tree Option Pricing technique, and gives a short outline of the mathematical theory behind the method. Submitted by Chun Fan, September 10, 2003.

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