This is post #4 on the binomial option pricing model. The purpose of post #4:

Post #4: Extend the oneperiod binomial option pricing calculation to more than one period.
The work in this post is heavily relying on the work in the oneperiod binomial option pricing model discussed in the part 1 post and in the part 2 post.
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Multiperiod binomial trees
We describe how to price an option based on a multiperiod binomial tree. We use a 2period tree to anchor the discussion. Assume that the length of one period is years. Then the following 2period binomial tree is to price a year option (call or put). For example, if 0.25 years, then the following binomial tree is a basis for pricing a 6month option.
The stock prices in the above binomial tree are constructed using forward prices. At the left, is the initial stock price. Then the stock prices at the end of period 1 are:
where is the annual riskfree interest rate, is the annual continuous dividend rate and is the annualized standard deviation of the continuously compounded stock return. Multiplying by adjusts the standard deviation to make the stock return appropriate for a period of length .
The stock prices at the end of period 2 are also constructed based on the idea in (1). The formula (1) takes a starting price (e.g. ) and calculates the up move, e.g. and the down move, e.g. . The same idea in (1) can then be used to build additional periods beyond the period 2.
Because the stock prices in Figure 1 are calculated by formula (1), an up move followed by a down move leads to the same stock price as a down move followed by an up move. Thus at the end of the second period. When this happens, the resulting binomial tree is called a recombining tree. When updown move leads to a different price from a downup move, the resulting tree is called a nonrecombining tree. When stock prices are calculated using the forward prices, the resulting binomial tree is a recombining tree.
Suppose that the binomial tree in Figure 1 models a year option. We can compute the value of the option at each node at the end of period 2.
The option value is $0 if it is not advantageous for the option buyer to exercise. If it is, the option value is the difference between the stock price at expiration and the strike price. For example, for a call option, if strike price is $50 and $75, then = $25. For a put option, if strike price is $50 and if $30, then $20.
Once the option values at the end of the last period are known, we can calculate the option values for the preceding periods and at time 0.
Riskneutral pricing is an efficient algorithm for pricing an option using a binomial tree. The option value at a given node is simply the weighted average using riskneutral probabilities of the two option values in the next period discounted at the riskfree interest rate. The following is the riskneutral pricing formula:
where is the option value at a given node, and is the option value at the up move and (C^*)_d$ is the option value at the down move that follow the node at . The riskneutral probability for the up move is:
For example, in Figure 3, the option value at the node for stock price is:
Once the option values at expiration (the end of the last period in the binomial tree) are known, we can use the riskneutral pricing formula (2) to work backward to derive the option value at the first node in the tree, which is the price of the option in question.
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The process of pricing an option using a multiperiod binomial tree
The process just described can be used to price a European option based on a binomial tree of any number of periods. The process is summarized as follows:

Pricing an option using a multiperiod binomial tree
 Build a binomial tree as in Figure 1. The stock prices in this tree are relative to the forward prices as shown in formula (1).
 Calculate the option values at the end of the last period in the tree as in Figure 2. This step is based on a comparison of the strike price and the stock prices at expiration of the option.
 Starting from the option values at the end of the last period, work backward to calculate the option value at each node in each of the preceding periods. One way to calculate the option value at each node is to use the riskneutral pricing formula in (2).
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Twoperiod examples
We demonstrate how to extend the oneperiod calculation to twoperiod through the following two examples.
Example 1
Price a oneyear call option using a 2period binomial tree. The specifics of the call option and its underlying stock are:
 Initial stock price is $60.
 Strike price of the call option is $55.
 The stock is nondividend paying.
 The annual standard deviation of the stock return is 0.3.
 The annual riskfree interest rate is 4%.
The oneyear option period is divided into two periods, making one period being 6 months. Thus 0.5. This example is based on Example 4 in this previous post, which is about a 6month call option with the same specifics as given above. Thus Example 1 here is Example 4 in the previous post with an additional 6month period in the binomial tree.
Usually, in working a binomial tree problem, one tree diagram suffices. In order to make the procedure clear, we use three tree diagrams to demonstrate the three steps involved. Step 1 is to build the binomial tree. The following diagram is the result.
Step 1: build the binomial tree (Example 1)
The stock prices in the above binomial tree are based on the following movement factors and .

1.261286251
0.825197907
The following details the calculations for the stock prices:

60 (1.261286251) = $75.67717506
60 (0.825197907) = $49.51187441
1.261286251 (75.67717506) = $95.45058041
0.825197907 (75.67717506) = $62.44864645
0.825197907 (49.51187441) = $40.85709513
Step 2 is to obtain the option values at expiration. For a European call option, the option value at expiration is the mximum of $0 or the stock price less the strike price. Simply compare the strike price of $55 with the stock prices at the end of the binomial tree. Any node with stock price above the strike price $55 has positive option value. The following tree shows the result.
Step 2: add option values at expiration (Example 1)
Step 3 is to work backward from the end of the tree to the front of the tree. For example, calculate the option value at each node in period 1 by using the option values of the associated up and down nodes in period 2. We take the approach of using riskneutral pricing described in (2). The following diagram shows the results.
Step 3: work backward to obtain option price (Example 1)
As mentioned above, to calculate the option values, we use riskneutral probabilities:

1.261286251
0.825197907
0.447164974
0.552835026
The following computes the option values:

$21.76624801
$3.264820056
$11.30954269
The diagram in Step 3 also shows the replicating at each node. For example, the replicating portfolio at the node for is computed as follows:

1.0
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Example 2
This is Example 5 in this previous post. Example 5 in that post is a 3month put option. We now price the same 3month put option using a 2period binomial tree. Thus the 3month option period is divided into two periods. The following gives the specifics of this put option:
 Initial stock price is $40.
 Strike price of the put option is $45.
 The stock is nondividend paying.
 The annual standard deviation of the stock return is 0.3.
 The annual riskfree interest rate is 5%.
We carry out the same three steps as in Example 1. The following diagram captures the results of all three steps.
Example 2: binomial tree for pricing put option
Note that the put option calculated in Example 5 in this previous post using one binomial period is
$5.3811 whereas the put option price from a 2period binomial tree here is $5.56462. It is not uncommon for binomial option prices to fluctuate when the number of periods is small. When is large, the binomial price will stabilize.
Note that the option period is 3month long (a quarter of a year). Thus one period is 0.25/2 = 0.125 of a year. To build the binomial tree, the following shows the calculation for the stock prices and .

1.118866386
0.905003908
40 (0.905003908) = $36.20015632
36.20015632 (1.118866386) = $40.50313807
As in Example 1, we perform riskneutral pricing. The following shows the calculation of the option values.
0.462570155
0.537429845
$2.352809258
$8.519470762
$5.564617421
The diagram in Example 2 also shows the replicating at each node. For example, the replicating portfolio at the node for is computed as follows:

1.0
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Binomial trees with more than two periods
Since one or twoperiod binomial trees are unlikely to be accurate model of stock price movements, option prices based on the binomial model with one or two periods are unlikely to be accurate. It is then necessary to use more periods in the binomial tree, i.e. divide the time to expiration into more periods to create more realistic model of stock price movements. Therefore realistic applications of the binomial option pricing model require the use of software.
Another point we would like to make is that using more periods in the binomial tree requires no new concepts or new methods. The same three steps described above are used – build the binomial tree, calculate the option values at expiration and work backward to derive the option price. The calculation at each node still uses the same oneperiod binomial option formulas. It is just that there are more periods to calculate. Hence realistic binomial option pricing is a job that should be done by software. To conclude this post, we present an example using a threeperiod binomial tree.
Example 3
Like Example 1 above, this example is based on Example 4 in this previous post. Example 4 in that post is to price a 6month call option. In this example, we price the same call options using a 3period binomial tree. All other specifics of the call option and the underlying stock remain the same. They are repeated here:
 Initial stock price is $60.
 Strike price of the call option is $55.
 The stock is nondividend paying.
 The annual standard deviation of the stock return is 0.3.
 The annual riskfree interest rate is 4%.
Now one period is 2month long. Hence in the calculation 2/12 = 0.16667. The results of the 3period binomial calculation are show in the following two trees. The first one displays the stock prices and the option values. The second one displays the replicating portfolios (the hedge ratio and the amount of borrowing ) at each node.
Example 3 – the binomial tree and option values
Example 3 – Replicating portfolios
The call option price using oneperiod tree in Example 4 in the previous post is $9.06302. The 3period option price using a 3period tree is $8.26318. Once again, there is no need to be alarmed. Binomial option prices can wildly fluctuate when the number of periods is small. The example here is only meant to illustrate the calculation in binomial option model.
Just to be clear on the process, the stock prices in the upper two nodes in the third period are calculated as follows:

1.137850725
0.890646371
77.68225631 (1.137850725) = $88.39081165
77.68225631 (0.890646371) = $69.18741967
The option value at the node is calculated as follows using riskneutral probabilities:
0.462570155
0.537429845
$2.352809258
$8.519470762
$5.564617421
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Practice problems
For practice problems on how to calculate price of European option using multiperiod binomial tree, go here in the practice problem companion blog.
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Tagged: Binomial Option Pricing Model, Call Option, Derivative contract, European Call Option, European Put Option, Financial Math, Forward Tree, Option, Put Option, Replicating portfolio, RiskNeutral Pricing, RiskNeutral Probability
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