The binomial option pricing model – part 4

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

    Post #4: Extend the one-period binomial option pricing calculation to more than one period.

The work in this post is heavily relying on the work in the one-period binomial option pricing model discussed in the part 1 post and in the part 2 post.

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Multi-period binomial trees

We describe how to price an option based on a multi-period binomial tree. We use a 2-period tree to anchor the discussion. Assume that the length of one period is h years. Then the following 2-period binomial tree is to price a 2h-year option (call or put). For example, if h= 0.25 years, then the following binomial tree is a basis for pricing a 6-month option.

    \text{ }
    Figure 1 – 2-period binomial tree
    binomial tree - 2 period
    \text{ }

The stock prices in the above binomial tree are constructed using forward prices. At the left, S is the initial stock price. Then the stock prices at the end of period 1 are:

    \text{ }
    \displaystyle S_d=S \ e^{(r-\delta) h - \sigma \sqrt{h}} \ \ \ \ \ \ \ \ \ \displaystyle S_u=S \ e^{(r-\delta) h + \sigma \sqrt{h}} \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
    \text{ }

where r is the annual risk-free interest rate, \delta is the annual continuous dividend rate and \sigma is the annualized standard deviation of the continuously compounded stock return. Multiplying \sigma by \sqrt{h} adjusts the standard deviation to make the stock return appropriate for a period of length h.

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. S_u) and calculates the up move, e.g. S_{uu} and the down move, e.g. S_{dd}. 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 S_{ud}=S_{du} at the end of the second period. When this happens, the resulting binomial tree is called a recombining tree. When up-down move leads to a different price from a down-up 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 2h-year option. We can compute the value of the option at each node at the end of period 2.

    \text{ }
    Figure 2 – 2-period binomial tree with option values
    option value tree - 2 period
    \text{ }

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 S_{uu}= $75, then C_{uu}= = $25. For a put option, if strike price is $50 and if S_{dd}= $30, then C_{dd}= $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.

    \text{ }
    Figure 3 – 2-period binomial tree with option values
    option value tree - 2 period filled
    \text{ }

Risk-neutral 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 risk-neutral probabilities of the two option values in the next period discounted at the risk-free interest rate. The following is the risk-neutral pricing formula:

    \displaystyle C^*=e^{-r h} \ [p^* \times (C^*)_u + (1-p^*) \times (C^*)_d] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

where C^* is the option value at a given node, and (C^*)_u is the option value at the up move and (C^*)_d$ is the option value at the down move that follow the node at C^*. The risk-neutral probability for the up move is:

    \displaystyle p^*=\frac{e^{(r-\delta) h}-d}{u-d}

For example, in Figure 3, the option value C_u at the node for stock price S_u is:

    \displaystyle C_u=e^{-r h} \ [p^* \times C_{uu} + (1-p^*) \times C_{ud}]

Once the option values at expiration (the end of the last period in the binomial tree) are known, we can use the risk-neutral pricing formula (2) to work backward to derive the option value C 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 multi-period 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 multi-period binomial tree

  1. 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).
  2. 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.
  3. 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 risk-neutral pricing formula in (2).

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Two-period examples

We demonstrate how to extend the one-period calculation to two-period through the following two examples.

Example 1
Price a one-year call option using a 2-period 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 non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 4%.

The one-year option period is divided into two periods, making one period being 6 months. Thus h= 0.5. This example is based on Example 4 in this previous post, which is about a 6-month call option with the same specifics as given above. Thus Example 1 here is Example 4 in the previous post with an additional 6-month 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.

    \text{ }

    Step 1: build the binomial tree (Example 1)
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 95.45058 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 75.67718 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 62.44865 \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 49.51187 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 40.85710  \end{array}

    \text{ }

The stock prices in the above binomial tree are based on the following movement factors u and d.

    \displaystyle u=e^{(0.04-0) 0.5 + 0.3 \sqrt{0.5}}= 1.261286251

    \displaystyle d=e^{(0.04-0) 0.5 - 0.3 \sqrt{0.5}}= 0.825197907

The following details the calculations for the stock prices:

    \displaystyle S_u=60u= 60 (1.261286251) = $75.67717506
    \displaystyle S_d=60d= 60 (0.825197907) = $49.51187441

    \displaystyle S_{uu}=u S_u 1.261286251 (75.67717506) = $95.45058041
    \displaystyle S_{du}=S_{ud}=d S_u 0.825197907 (75.67717506) = $62.44864645
    \displaystyle S_{dd}=d S_d 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.

    \text{ }

    Step 2: add option values at expiration (Example 1)
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 95.45058 \\   \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\$ 40.45058 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 75.67718 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 62.44865 \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{ud}=\$ 7.44865 \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 49.51187 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 40.85710  \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\$ 0 \end{array}
    \text{ }

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 risk-neutral pricing described in (2). The following diagram shows the results.

    \text{ }

    Step 3: work backward to obtain option price (Example 1)
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 95.45058 \\   \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\$ 40.45058 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 75.67718 & \text{ } & \text{ } \\  \text{ } & \text{ } & C_u=\$ 21.76625 & \text{ } & \text{ } \\  \text{ } & \text{ } & \Delta=1.0 & \text{ } & \text{ } \\  \text{ } & \text{ } & B=- \$ 53.91093 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 62.44865 \\  C=\$ 11.30954 & \text{ } & \text{ } & \text{ } & C_{ud}=\$ 7.44865 \\  \Delta=0.70710 & \text{ } & \text{ } & \text{ } & \text{ } \\  B=- \$ 31.11633 & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 49.51187 & \text{ } & \text{ } \\   \text{ } & \text{ } & C_d=\$ 3.26482 & \text{ } & \text{ } \\   \text{ } & \text{ } & \Delta=0.34498 & \text{ } & \text{ } \\   \text{ } & \text{ } & B=- \$ 13.81577 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 40.85710  \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\$ 0 \end{array}
    \text{ }

As mentioned above, to calculate the option values, we use risk-neutral probabilities:

    \displaystyle u=e^{(0.04-0) 0.5 + 0.3 \sqrt{0.5}}= 1.261286251

    \displaystyle d=e^{(0.04-0) 0.5 - 0.3 \sqrt{0.5}}= 0.825197907

    \displaystyle p^*=\frac{e^{(0.04-0)0.5}-0.825197907}{1.261286251-0.825197907}= 0.447164974

    \displaystyle 1-p^*= 0.552835026

The following computes the option values:

    C_u=e^{-0.04 (0.5)} [ p* (40.45058041)+(1-p^*)(7.44864645)]= $21.76624801

    C_d=e^{-0.04 (0.5)} [ p* (7.44864645)+(1-p^*)(0)]= $3.264820056

    C=e^{-0.04 (0.5)} [ p* (21.76624801)+(1-p^*)(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 C_{u} is computed as follows:

    \displaystyle \Delta=e^{0(0.5)} \ \frac{C_{uu}-C_{ud}}{S_{uu}-S_{ud}}=\frac{40.45058041-7.44864645}{95.45058041-62.44864645}= 1.0

    \displaystyle \begin{aligned} B&=e^{-0.04(0.5)} \ \frac{u C_{ud}-d C_{uu}}{u-d} \\&=e^{-0.02} \ \frac{1.261286251 (7.44864645)-0.825197907 (40.45058041)}{1.261286251-0.825197907} \\&=- \$ 53.91093  \end{aligned}

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Example 2
This is Example 5 in this previous post. Example 5 in that post is a 3-month put option. We now price the same 3-month put option using a 2-period binomial tree. Thus the 3-month 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 non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 5%.

We carry out the same three steps as in Example 1. The following diagram captures the results of all three steps.

    \text{ }

    Example 2: binomial tree for pricing put option
    \text{ }
    \displaystyle \begin{array}{lllll} \displaystyle   \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\$ 50.07448 \\   \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\$ 0 \\   \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & S_u=\$ 44.75466 & \text{ } & \text{ } \\  \text{ } & \text{ } & C_u=\$ 2.35281 & \text{ } & \text{ } \\  \text{ } & \text{ } & \Delta=-0.46983 & \text{ } & \text{ } \\  \text{ } & \text{ } & B=- \$ 23.37970 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  S= \$ 40 & \text{ } & \text{ } & \text{ } & S_{ud}=\$ 40.50314 \\  C=\$ 5.56462 & \text{ } & \text{ } & \text{ } & C_{ud}=\$ 4.49686 \\  \Delta=-0.72087 & \text{ } & \text{ } & \text{ } & \text{ } \\  B=\$ 34.39932 & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\   \text{ } & \text{ } & S_d=\$ 36.20016 & \text{ } & \text{ } \\   \text{ } & \text{ } & C_d=\$ 8.51947 & \text{ } & \text{ } \\   \text{ } & \text{ } & \Delta=-1 & \text{ } & \text{ } \\   \text{ } & \text{ } & B=- \$ 44.71963 & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\  \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\$ 32.76128  \\  \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\$ 12.23872 \end{array}
    \text{ }

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 2-period binomial tree here is $5.56462. It is not uncommon for binomial option prices to fluctuate when the number of periods n is small. When n is large, the binomial price will stabilize.

Note that the option period is 3-month long (a quarter of a year). Thus one period is h= 0.25/2 = 0.125 of a year. To build the binomial tree, the following shows the calculation for the stock prices S_d and S_{du}.

    \displaystyle u=e^{(0.05-0) 0.125 + 0.3 \sqrt{0.125}}=e^{0.1625}= 1.118866386

    \displaystyle d=e^{(0.05-0) 0.125 - 0.3 \sqrt{0.125}}=e^{-0.1375}= 0.905003908

    \displaystyle S_d=40d= 40 (0.905003908) = $36.20015632

    \displaystyle S_{ud}=S_{du}=S_d \ d= 36.20015632 (1.118866386) = $40.50313807

As in Example 1, we perform risk-neutral pricing. The following shows the calculation of the option values.

    \displaystyle p^*=\frac{e^{(0.05-0)0.25}-0.87153435}{1.176448318-0.87153435}= 0.462570155

    \displaystyle 1-p^*= 0.537429845

    C_u=e^{-0.05 (0.125)} [ p* (0)+(1-p^*)(4.496861938)]= $2.352809258

    C_d=e^{-0.05 (0.125)} [ p* (4.496861938)+(1-p^*)(12.23871707)]= $8.519470762

    C=e^{-0.05 (0.125)} [ p* (2.352809258)+(1-p^*)(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 C_{d} is computed as follows:

    \displaystyle \Delta=e^{0(0.125)} \ \frac{C_{du}-C_{dd}}{S_{du}-S_{dd}}=\frac{4.496861938-12.23871707}{40.50313806-32.76128293}= -1.0

    \displaystyle \begin{aligned} B&=e^{-0.05(0.125)} \ \frac{u C_{dd}-d C_{du}}{u-d} \\&=e^{-0.00625} \ \frac{1.118866386 (12.23871707)-0.905003908 (4.496861938)}{1.118866386-0.905003908} \\&=- \$ 44.71962708      \end{aligned}

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Binomial trees with more than two periods

Since one or two-period 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 one-period 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 three-period 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 6-month call option. In this example, we price the same call options using a 3-period 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 non-dividend paying.
  • The annual standard deviation of the stock return is \sigma= 0.3.
  • The annual risk-free interest rate is r= 4%.

Now one period is 2-month long. Hence in the calculation h= 2/12 = 0.16667. The results of the 3-period 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 \Delta and the amount of borrowing B) at each node.

    \text{ }

    Example 3 – the binomial tree and option values
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & S_{uuu}=\$ 88.39081 \\   \text{ } & \text{ } & \text{ }   & C_{uuu}=\$ 33.39081 \\        \text{ } & \text{ } & S_{uu}=\$ 77.68226   & \text{ } \\   \text{ } & \text{ } & C_{uu}=\$ 23.04770   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  S_{uud}=\$ 69.18742 \\  \text{ } & \text{ } & \text{ }   &  C_{uud}=\$ 14.19742 \\     \text{ } & S_u=\$ 68.27104  & \text{ }    & \text{ } \\   \text{ } & C_u=\$ 14.23394  & \text{ }    & \text{ } \\     S=\$ 60 &  \text{ } & S_{ud}=S_{du}=\$ 60.80536    & \text{ } \\   C=\$ 8.26318 &  \text{ } & C_{ud}=\$ 6.61560    & \text{ } \\    \text{ } & S_d=\$ 53.43878 \text{ }   &  \text{ } \\   \text{ } & C_d=\$ 3.08486 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  S_{udd}=\$ 54.15607 \\   \text{ } & \text{ } & \text{ }   &  C_{udd}=\$ 0 \\      \text{ } & \text{ } & S_{dd}=\$ 47.59506   & \text{ } \\     \text{ } & \text{ } & C_{dd}=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & S_{ddd}=\$ 42.39037 \\  \text{ } & \text{ } & \text{ } & C_{ddd}=\$ 0 \\      \end{array}

    \text{ }

    \text{ }

    Example 3 – Replicating portfolios
    \text{ }
    \displaystyle \begin{array}{llll} \displaystyle   \text{Initial Price} & \text{Period 1} & \text{Period 2}   & \text{Period 3} \\  \text{ } & \text{ } & \text{ }   &  \text{ } \\  \text{ } & \text{ } & \text{ }   & \text{N/A} \\   \text{ } & \text{ } & \text{ }   & \text{N/A} \\        \text{ } & \text{ } & \Delta=1   & \text{ } \\   \text{ } & \text{ } & B=-\$ 54.63456   & \text{ } \\      \text{ } & \text{ } & \text{ }   &  \text{N/A} \\  \text{ } & \text{ } & \text{ }   &  \text{N/A} \\     \text{ } & \Delta=0.97364  & \text{ }    & \text{ } \\   \text{ } & B=-\$ 52.23779  & \text{ }    & \text{ } \\     \Delta=0.75168 &  \text{ } & \Delta=0.94386    & \text{ } \\   B=-\$ 36.83749 &  \text{ } & B=-\$ 50.77586    & \text{ } \\    \text{ } & \Delta=0.50079 \text{ }   &  \text{ } \\   \text{ } & B=-\$ 23.67681 \text{ }   &  \text{ } \\       \text{ } & \text{ } & \text{ }   &  \text{N/A} \\   \text{ } & \text{ } & \text{ }   &  \text{N/A} \\      \text{ } & \text{ } & \Delta=0   & \text{ } \\     \text{ } & \text{ } & B=\$ 0   & \text{ } \\       \text{ } & \text{ } & \text{ } & \text{N/A} \\  \text{ } & \text{ } & \text{ } & \text{N/A} \\      \end{array}

    \text{ }

The call option price using one-period tree in Example 4 in the previous post is $9.06302. The 3-period option price using a 3-period 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:

    \displaystyle u=e^{(0.04-0) \frac{1}{6} + 0.3 \sqrt{\frac{1}{6}}}= 1.137850725

    \displaystyle d=e^{(0.04-0) \frac{1}{6} - 0.3 \sqrt{\frac{1}{6}}}= 0.890646371

    \displaystyle S_{uuu}=S_{uu} \ u= 77.68225631 (1.137850725) = $88.39081165

    \displaystyle S_{uud}=S_{uu} \ d= 77.68225631 (0.890646371) = $69.18741967

The option value at the node S_{uu} is calculated as follows using risk-neutral probabilities:

    \displaystyle p^*=\frac{e^{(0.05-0)0.25}-0.87153435}{1.176448318-0.87153435}= 0.462570155

    \displaystyle 1-p^*= 0.537429845

    C_u=e^{-0.05 (0.125)} [ p* (0)+(1-p^*)(4.496861938)]= $2.352809258

    C_d=e^{-0.05 (0.125)} [ p* (4.496861938)+(1-p^*)(12.23871707)]= $8.519470762

    C=e^{-0.05 (0.125)} [ p* (2.352809258)+(1-p^*)(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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\copyright \ \ 2015 \ \text{Dan Ma}

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