Jeroen Bouma

With Experience and Education in the areas of (Quantitative) Finance and Economics, my ambition is to continuously improve in the fields of Quant Finance and Python Programming

11 minute read

Binomial trees are a popular method for pricing options and other derivatives. The Finance Toolkit includes a function for binomial trees, which can be used to price European and American options. This article provides an overview of the binomial tree module and demonstrates how to use it to price options.

The binomial tree is based on up and down movements, which are determined by the volatility of the underlying asset. The formulas are as follows:

\[u = e^{\sigma \sqrt{\Delta t}}\] \[d = e^{-\sigma \sqrt{\Delta t}}\]

Where:

  • \(u\) is the up movement
  • \(d\) is the down movement
  • \(\sigma\) is the volatility of the underlying asset
  • \(\Delta t\) is the time step

The tree is constructed by moving up and down from the current price of the underlying asset. For example, if \(u\) is equal to 1.1 and \(d\) is equal to 0.9, a tree with 3 time steps would look like this:

flowchart LR; classDef boxfont fill:#3b9cba,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; Step0["$100"]:::boxfont -- U --> Step1["$110"]:::boxfont Step0["$100"] -- U --> Step2["$90"]:::boxfont Step1["$110"] -- U --> Step4["$121"]:::boxfont Step1["$110"] -- D --> Step3["$99"]:::boxfont Step2["$90"] -- U --> Step3["$99"]:::boxfont Step2["$90"] -- D --> Step5["$81"]:::boxfont Step4["$121"] -- U --> Step7["$133"]:::boxfont Step4["$121"] -- D --> Step6["$109"]:::boxfont Step3["$99"] -- U --> Step6["$109"]:::boxfont Step3["$99"] -- D --> Step8["$89"]:::boxfont Step5["$81"] -- U --> Step8["$89"]:::boxfont Step5["$81"] -- D --> Step9["$73"]:::boxfont

The option price is then calculated at the time of expiration (the final node) with the following formula for Call and Put option respectively:

\[C = max(S - K, 0)\] \[P = max(K - S, 0)\]

Where:

  • \(C\) is the Call option price
  • \(P\) is the Put option price
  • \(S\) is the price of the underlying asset
  • \(K\) is the strike price

Therefore, when reconstructing the tree, the option price at each node is calculated using the risk-neutral pricing formula:

\[p = \frac{e^{(r - y) \cdot \Delta t} - d}{u - d}\] \[q = 1 - p\]

Where:

  • \(p\) is the probability of an up movement
  • \(q\) is the probability of a down movement
  • \(r\) is the risk-free interest rate
  • \(y\) is the dividend yield
  • \(\Delta t\) is the time step

European Options

Given that European options can not be exercised early, this lead to the following formulas for the option price at each node for a Call Option:

\[C_u = e^{(-r - y) \cdot \Delta t} \cdot (p \cdot C_u + q \cdot C_d)\] \[C_d = e^{(-r - y) \cdot \Delta t} \cdot (p \cdot C_u + q \cdot C_d)\]

And the following formulas for the option price at each node for a Put Option:

\[P_u = e^{(-r - y) \cdot \Delta t} \cdot (p \cdot P_u + q \cdot P_d)\] \[P_d = e^{(-r - y) \cdot \Delta t} \cdot (p \cdot P_u + q \cdot P_d)\]

Where:

  • \(C_u\) is the Call option price at the up node
  • \(C_d\) is the Call option price at the down node
  • \(P_u\) is the Put option price at the up node
  • \(P_d\) is the Put option price at the down node

This results in the following binomial trees for Call and Put options, assuming the risk-free rate is 2%, dividend yield is 0.5%, volatility is 25%, the time to expiration is 1 year and the strike price is $100:

Call Option

flowchart TD; classDef boxfont fill:#3b9cba,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; Step0["$18.32"]:::boxfont -- U --> Step1["$29.42"]:::boxfont Step0["$18.32"] -- D --> Step2["$6.73"]:::boxfont Step1["$29.42"] -- U --> Step3["$45.13"]:::boxfont Step1["$29.42"] -- D --> Step4["$13.11"]:::boxfont Step2["$6.73"] -- U --> Step4["$13.11"]:::boxfont Step2["$6.73"] -- D --> Step5["$10.00"]:::boxfont Step3["$45.13"] -- U --> Step6["$64.19"]:::boxfont Step3["$45.13"] -- D --> Step7["$25.53"]:::boxfont Step4["$13.11"] -- U --> Step7["$25.53"]:::boxfont Step4["$13.11"] -- D --> Step8["$0.00"]:::boxfont Step5["$10.00"] -- U --> Step8["$0.00"]:::boxfont Step5["$10.00"] -- D --> Step9["$0.00"]:::boxfont

Put Option

flowchart TD; classDef boxfont fill:#3b9cba,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; Step0["$3.92"]:::boxfont -- U --> Step1["$0.78"]:::boxfont Step0["$3.92"] -- D --> Step2["$7.39"]:::boxfont Step1["$0.78"] -- U --> Step3["$0.00"]:::boxfont Step1["$0.78"] -- D --> Step4["$1.64"]:::boxfont Step2["$7.39"] -- U --> Step4["$1.64"]:::boxfont Step2["$7.39"] -- D --> Step5["$13.75"]:::boxfont Step3["$0.00"] -- U --> Step6["$0.00"]:::boxfont Step3["$0.00"] -- D --> Step7["$0.00"]:::boxfont Step4["$1.64"] -- U --> Step7["$0.00"]:::boxfont Step4["$13.75"] -- D --> Step8["$3.44"]:::boxfont Step5["$13.75"] -- U --> Step8["$3.44"]:::boxfont Step5["$13.75"] -- D --> Step9["$25.15"]:::boxfont

To interpret these results, the option price at the root node is the fair value of the option. The option price at the final node is the intrinsic value of the option. The option price at the intermediate nodes is the expected value of the option, discounted at the risk-free rate.

American Options

There is a difference between European and American options. European options can only be exercised at expiration, while American options can be exercised at any time before expiration. The binomial tree for American options is constructed in the same way as for European options, but the option price at each node is calculated using the early exercise condition. The early exercise condition is the maximum of the intrinsic value of the option and the expected value of the option, discounted at the risk-free rate, this results in the following formulas:

This lead to the following formulas for the option price at each node for a Call Option:

\[C_u = max(S - K, C_\widehat{u})\] \[C_d = max(S - K, C_\widehat{d})\]

And the following formulas for the option price at each node for a Put Option:

\[P_u = max(K - S, P_\widehat{u})\] \[P_d = max(K - S, P_\widehat{d})\]

Where:

  • \(C_u\) and \(C_\widehat{u}\) the actual and European Call option price at the up node respectively
  • \(C_d\) and \(C_\widehat{d}\) the actual and European Call option price at the down node respectively
  • \(P_u\) and \(P_\widehat{u}\) the actual and European Put option price at the up node respectively
  • \(P_d\) and \(P_\widehat{d}\) the actual and European Put option price at the down node respectively

This results in the following binomial trees for Call and Put options, assuming the risk-free rate is 2%, dividend yield is 0.5%, volatility is 25%, the time to expiration is 1 year and the strike price is $90:

Call Option

flowchart TD; classDef boxfont fill:#3b9cba,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; classDef highlightfont fill:#d67f05,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; Step0["$20.56"]:::highlightfont -- U --> Step1["$29.42"]:::boxfont Step0["$20.56"] -- D --> Step2["$11.50"]:::highlightfont Step1["$29.42"] -- U --> Step3["$45.13"]:::boxfont Step1["$29.42"] -- D --> Step4["$13.11"]:::boxfont Step2["$11.50"] -- U --> Step4["$13.11"]:::boxfont Step2["$11.50"] -- D --> Step5["$10.00"]:::highlightfont Step3["$45.13"] -- U --> Step6["$64.19"]:::boxfont Step3["$45.13"] -- D --> Step7["$25.53"]:::boxfont Step4["$13.11"] -- U --> Step7["$25.53"]:::boxfont Step4["$13.11"] -- D --> Step8["$0.00"]:::boxfont Step5["$10.00"] -- U --> Step8["$0.00"]:::boxfont Step5["$10.00"] -- D --> Step9["$0.00"]:::boxfont

Put Option

flowchart TD; classDef boxfont fill:#3b9cba,stroke-width:0px,fill-opacity:0.7,color:white,radius:20px; Step0["$3.92"]:::boxfont -- U --> Step1["$0.78"]:::boxfont Step0["$3.92"] -- D --> Step2["$7.39"]:::boxfont Step1["$0.78"] -- U --> Step3["$0.00"]:::boxfont Step1["$0.78"] -- D --> Step4["$1.64"]:::boxfont Step2["$7.39"] -- U --> Step4["$1.64"]:::boxfont Step2["$7.39"] -- D --> Step5["$13.75"]:::boxfont Step3["$0.00"] -- U --> Step6["$0.00"]:::boxfont Step3["$0.00"] -- D --> Step7["$0.00"]:::boxfont Step4["$1.64"] -- U --> Step7["$0.00"]:::boxfont Step4["$13.75"] -- D --> Step8["$3.44"]:::boxfont Step5["$13.75"] -- U --> Step8["$3.44"]:::boxfont Step5["$13.75"] -- D --> Step9["$25.15"]:::boxfont

Given that the American option can be exercised at any time before expiration, the actual fair value of the option is higher than that of the European option when the option is in the money. This is because the option holder can exercise the option early and receive the intrinsic value of the option. As highlighted in the color orange in the binomial tree, the actual fair value of American the option is higher than that of the European option.

Using the Finance Toolkit

The Finance Toolkit provides a function to calculate the binomial tree for both European and American options. You can install the Finance Toolkit by using the following command:

pip install financetoolkit -U

Let’s start off with the actual model function, this function requires you to pass all inputs. Later on I’ll show you how to work with actual company data.

from financetoolkit.options import binomial_trees_model

option_pay_off = binomial_trees_model.get_option_payoffs(
    stock_price=100,
    strike_price=90,
    years=1,
    timesteps=3,
    volatility=0.25,
    dividend_yield=0.005,
    risk_free_rate=0.02,
    put_option=False, # True for Put Option
    american_option=False # True for American Option
)

This returns the folowing DataFrames for the European Call and Put options and the American Call and Put options respectively, which match up with binomial trees as mentioned above:

European Call Option

  0 1 2 3
UUU 18.3183 29.4228 45.1286 64.1896
UUD nan 6.73261 13.1098 25.5274
UDD nan nan 0 0
DDD nan nan nan 0


European Put Option

  0 1 2 3
UUU 3.92378 0.781139 0 0
UUD nan 7.39281 1.63935 0
UDD nan nan 13.7482 3.44045
DDD nan nan nan 25.1448

American Call Option

  0 1 2 3
UUU 20.5888 29.4228 45.1286 64.1896
UUD nan 11.4975 13.1098 25.5274
UDD nan nan 10 0
DDD nan nan nan 0


American Put Option

  0 1 2 3
UUU 3.92378 0.781139 0 0
UUD nan 7.39281 1.63935 0
UDD nan nan 13.7482 3.44045
DDD nan nan nan 25.1448

Now let’s work with actual company data, for this example I’ll use the stock price of Apple Inc. (AAPL) and Microsoft Corporation (MSFT). Here, all required inputs will be collected automatically such as the risk-free rate, dividend yield and volatility.

from financetoolkit import Toolkit

companies = Toolkit(["AAPL", "MSFT"], api_key="FINANCIAL_MODELING_PREP_KEY")

# Obtain the Binominal Model with Time to Expiration of 16 Days
companies.options.get_binomial_model(show_input_info=True, time_to_expiration=16/365)

This returns insights into each component of the binomial tree model through the show_input_info=True parameter, which is useful for understanding the model’s inputs and outputs. The output will look like this (with the 8th of February 2024 as starting date):

Based on the period 2013-02-11 to 2024-02-07 the following parameters were used:
Stock Price: AAPL (189.41), MSFT (414.05), Benchmark (498.1)
Volatility: AAPL (28.26%), MSFT (26.99%), Benchmark (17.46%)
Dividend Yield: AAPL (0.49%), MSFT (0.74%)
Up Movement: AAPL (101.89%), MSFT (101.8%)
Down Movement: AAPL (98.15%), MSFT (98.23%)
Risk Neutral Probability: AAPL (49.96%), MSFT (49.97%)
Risk Free Rate: 4.11%

The model itself calculates the binominal tree for a range of Strike Prices based on the stock price of the company. By default it takes all Strike Prices between 75% and 125% of the stock price in steps of 5. The output for Apple with a Strike Price of $190 will look like this:

Movement 2024-02-07 2024-02-08 2024-02-10 2024-02-11 2024-02-13 2024-02-15 2024-02-16 2024-02-18 2024-02-19 2024-02-21 2024-02-23
UUUUUUUUUU 4.32 6.10 8.39 11.22 14.51 18.18 22.05 26.02 30.06 34.18 38.38
UUUUUUUUUD nan 2.53 3.80 5.57 7.91 10.85 14.31 18.09 21.98 25.94 29.98
UUUUUUUUDD nan nan 1.26 2.03 3.22 4.97 7.39 10.53 14.20 18.02 21.91
UUUUUUUDDD nan nan nan 0.47 0.84 1.48 2.54 4.257 6.85 10.38 14.13
UUUUUUDDDD nan nan nan nan 0.10 0.21 0.42 0.83 1.66 3.32 6.63
UUUUUDDDDD nan nan nan nan nan 0 0 0 0 0 0
UUUUDDDDDD nan nan nan nan nan nan 0 0 0 0 0
UUUDDDDDDD nan nan nan nan nan nan nan 0 0 0 0
UUDDDDDDDD nan nan nan nan nan nan nan nan 0 0 0
UDDDDDDDDD nan nan nan nan nan nan nan nan nan 0 0
DDDDDDDDDD nan nan nan nan nan nan nan nan nan nan 0

This represents a binominal tree for a Call Option with a Strike Price of $190 for Apple Inc. (AAPL) with a time to expiration of 16 days (23th of February, 2024) and 10 time steps. The actual fair value of the option is $4.32.

With this fair value calculation in hand, it is then possible to compare this to the actual option prices by looking at the options chains of the company. This can be done by using the get_option_chains method of the options module. Here, the time to expiration is set to 2024-02-23 which reflects the same timeframe as that of the binominal tree as shown above.

companies.options.get_option_chains(expiration_date="2024-02-23")

Then, Apple is selected to return the following selection of option prices (shrunken down to view the most relevant rows). Here, you can spot the actual option price for the Call Option with a Strike Price of $190:

Strike Price Contract Symbol Currency Last Price Change Percent Change Open Interest Bid Ask Expiration Last Trade Date Implied Volatility In The Money Volume
180 AAPL240223C00180000 USD 10 0 0 947 0 0 2024-02-23 2024-02-07 0 True 263
185 AAPL240223C00185000 USD 5.75 0 0 4128 0 0 2024-02-23 2024-02-07 0 True 449
190 AAPL240223C00190000 USD 2.67 0 0 8817 0 0 2024-02-23 2024-02-07 0.0039 False 5594
195 AAPL240223C00195000 USD 0.95 0 0 7124 0 0 2024-02-23 2024-02-07 0.0313 False 4149
200 AAPL240223C00200000 USD 0.34 0 0 17056 0 0 2024-02-23 2024-02-07 0.0625 False 2927

As is visible, based on the binominal trees model, the actual fair value of the option is $4.32, while the actual option price is $2.67. This means that the option is undervalued, which could mean that the option could converge to its fair value in the future. This is a useful insight for investors who are looking to capitalize on fair value discrepancies in the options market.

As a bonus, it is also possible to use the Black Scholes model to perform these calculations. This can be done by using the get_black_scholes_model method of the options module. Here, the time to expiration is set to 2024-02-23 which reflects the same timeframe as that of the binominal tree as shown above.

companies.options.get_black_scholes_model(expiration_time_range=20)

When selecting for Apple Inc. (AAPL) the following output is returned (shrunken down to view the most relevant rows):

Strike Price 2024-02-20 2024-02-21 2024-02-22 2024-02-23 2024-02-24 2024-02-25 2024-02-26
180 10.49 10.6 10.71 10.81 10.92 11.02 11.13
185 6.73 6.87 7.02 7.16 7.29 7.43 7.56
190 3.86 4.02 4.18 4.33 4.48 4.62 4.76
195 1.96 2.1 2.24 2.37 2.5 2.63 2.76
200 0.87 0.97 1.07 1.17 1.27 1.37 1.47

As you can see, both the Binomial Trees and Black Scholes model point to a similar fair value of the option, with the Binomial Tree returning a fair value of $4.32 and the Black Scholes model returning a fair value of $4.33.

Statistical Option Arbitrage
The method of evaluating the option price via the Binomial Tree or Black Scholes model and trading on the deviations based on the actual option prices is an actual method used within the financial industry and refers to the concept of “Statistical Option Arbitrage”. For example see the article titled “Statistical Arbitrage in the Black-Scholes Framework” in the Journal of Financial and Quantitative Analysis (here) which proves the existence of statistical arbitrage opportunities in the Black-Scholes framework by considering trading strategies that consists of borrowing from the risk free rate and taking a long position in the stock until it hits a deterministic barrier level.

Did you find this article helpful? If you have any questions or suggestions, feel free to reach out to me via any of the links on the side.

Updated:

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