Break the binomial model to evaluate options

In the financial world, Black-Scholes and binomial option valuation models are the two most important concepts in modern financial theory. Both are used to value options, and each has its own advantages and disadvantages.

Some of the basic advantages of using the binomial model are:

  • Multi-period view
  • transparency
  • The ability to combine probabilities

In this article, we will explore the advantages of using a binomial model instead of the Black-Scholes model, and provide some basic steps to develop the model and explain how to use it.

Multi-period view

Binomial models provide multi-period views of underlying asset prices and option prices. In contrast to the Black-Scholes model, which provides numerical results based on inputs, the binomial model allows the calculation of assets and options over multiple periods and the range of possible results for each period (see below).

The advantage of this multi-period view is that users can visualize asset price changes in different periods and evaluate options based on decisions made at different points in time. For US options that can be exercised at any time before the expiry date, the binomial model can provide insights about when it may be desirable to exercise the option and when it should be held for a longer period of time. By looking at the binary tree of value, the trader can determine in advance when a decision about exercise may occur. If the value of the option is positive, there is a possibility of exercise, and if the value of the option is less than zero, it should be held for a longer period of time.

transparency

Closely related to the multi-period review is that the binomial model can provide transparency to the underlying value of assets and options over time. The Black-Scholes model has five inputs:

  1. Risk-free rate
  2. Strike price
  3. The current price of the asset
  4. Mature time
  5. Implied volatility of asset prices

When these data points are entered into the Black-Scholes model, the model will calculate the value of the option, but the influence of these factors will not be displayed period by period. Using the binomial model, traders can see the changes in the price of the underlying asset in different periods and the corresponding changes in the price of the option.

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Merger probability

The basic method of calculating the binomial option model is to use the same probability of success and failure in each period before the option expires. However, traders can combine different probabilities for each period based on new information acquired over time.

For example, the probability that the price of the underlying asset will rise or fall by 30% within a period of time is 50/50. However, in the second period, the possibility of an increase in the price of the underlying asset may increase to 70/30. For example, if an investor is evaluating an oil well, the investor is not sure what the value of the oil well is, but the probability of a price increase is 50/50. If oil prices rose in the first period to make oil wells more valuable, and now the market fundamentals point to continued increases in oil prices, the possibility of further price appreciation may now be 70%. The binomial model allows this flexibility; the Black-Scholes model does not.

Development model

The simplest binomial model will have two probabilities that add up to an expected return of 100%. In our example, the oil well has two possible outcomes at each point in time. A more complex version may have three or more different results, each with a probability of occurrence.

To calculate the return for each period from time zero (now), we must determine the value of the underlying asset for a period from now. In this example, we assume the following:

  • Underlying asset price (P): $500
  • Call option exercise price (K): $600
  • Risk-free interest rate during the period: 1%
  • Price changes in each period: up or down 30%
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The price of the underlying asset is US$500, and in the first period, it can be worth US$650 or US$350. This is equivalent to an increase or decrease of 30% in a period of time. Since the exercise price of the call option we hold is $600, if the underlying asset is ultimately less than $600, the value of the call option will be zero. On the other hand, if the underlying asset exceeds the strike price of $600, the value of the call option will be the difference between the price of the underlying asset and the strike price.The formula for this calculation is [max(P-K),0].

Maximum

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P

K

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Among them: P = underlying asset price K = call option exercise price begin{aligned} &max{left[left(P-Kright),0right]}\ \ &textbf{where:}\ &P=text{underlying asset price} \ &K=text{call option exercise price} \ end{aligned}

Maximum[(PK),]Where:phosphorus=Underlying asset pricePotassium=Call option strike price

Assume that the probability of rise is 50% and the probability of fall is 50%.Taking period 1 as an example, the calculation formula is

Maximum

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$

6

5

$

6

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]

* 0. 5 + max

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$

3

5

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6

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* 0. 5 = $ 5 0 * 0. 5 + $ 0 = $ 2 5 begin{aligned} &max{left[left($650-$600right),0right]}*0.5+max{left[left($ 350-$ 600right),0right]}*0.5\ & = $ 50 * 0.5 + $ 0 = $ 25\ end{alignment}

Maximum[($65$6),].5+Maximum[($35$6),].5=$5.5+$=$25

To get the current value of the call option, we need to discount the $25 in period 1 back to period 0, which is

$ 2 5 / (1 + 1%) = $ 2 4. 7 5 $25/left(1+1%right) = $24.75

$25/(1+1%)=$24.75

You can now see that if the probability changes, the expected value of the underlying asset will also change. If the probability should be changed, it can also be changed in each subsequent period, and it does not have to remain the same all the time.

The binomial model can be easily extended to multiple periods. Although the Black-Scholes model can calculate the result of extending the due date, the binomial model extends the decision point to multiple periods.

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The purpose of the binomial model

In addition to being used as a method to calculate the value of options, the binomial model can also be used for projects or investments with high uncertainty, capital budgeting and resource allocation decisions, as well as embeddings that have multiple periods or continue or abandon projects at specific points in time式options.

A simple example is a project that requires drilling for oil. The uncertainty of this type of project is whether there is any oil in the land being drilled, the amount of oil that can be drilled, whether oil is found, and the price at which the oil can be sold after extraction.

Binomial option models can help make decisions at every point in an oil drilling project. For example, suppose we decide to drill, but only when we find enough oil and the price of oil exceeds a certain amount, the oil well will be profitable. It takes a complete period to determine how much oil we can extract and the price of oil at that point in time. After the first period (for example, one year), we can decide whether to continue drilling or abandon the project based on these two data points. These decisions can be made continuously until a point where there is no drilling value is reached, at which point the well will be abandoned.

Bottom line

The binomial model allows the price of the underlying asset and option prices in multiple periods to be viewed in multiple periods, as well as the range of possible results for each period, thereby providing a more detailed view. Although both the Black-Scholes model and the binomial model can be used to value options, the binomial model has a wider range of applications, is more intuitive and easier to use.

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