# Valuation of options using Binomial Model with Example

Introduction:

In the previous blog, we saw how Binomial model works. In this blog, we will illustrate the same with examples for a PUT option and a CALL option. Please see attached Options Valuation using Binomial Model Excel Sheet which has the valuation of PUT/CALL options using Binomial Model. Throughout this blog, we will point to the numbers in Valuation Excel Sheet. Downloading and having it handy side by side would help, as you go through this blog.

Inputs for Binomial Model:

Stock price as on valuation date: \$500.00

Exercise price of option which is set forth as on valuation date (this is the price at which option buyer enjoy their rights): \$510

Expiry or maturity of the option: 1 Year

Risk free rate of interest in terms of continuous compounding: 4% per annum

Annualized volatility of underlying stock: 10%

Calculation of factors:

Binomial model is based on the assumption that stock price can only take two possible values at the end of each period. Those values depend upon the upward and downward movement factor. These movement factors are calculated using annualized volatility of stock.

We calculated upward movement factor which is exponential of annualized volatility of stock and scaled with square root of time which is duration of each period in binomial tree setup. In our model, duration of each period is 1 year. Please refer cell number “C12” in tab – “Assumption Sheet”.

Downward movement factor is inverse of upward movement factor. Please refer cell number “C13” in tab – “Assumption Sheet”.

In our example, we have calculated upward movement factor as ~1.1052 and downward movement factor as ~0.9048.

Risk free rate factor is the exponential of the continuous compounded risk-free rate which is yield on treasury security. In our example, risk-free rate factor is ~1.0408. Please refer cell number “C14” in tab – “Assumption Sheet”.

Calculation of Risk-Neutral Probability:

Risk neutral probability is calculated using upward movement, downward movement and risk-free rate factors.

We have calculated upward risk neutral probability as 0.6787. Please refer cell number “C17” in tab – “Assumption Sheet”.

Formula for Upward Risk – Neutral Probability = We have calculated downward risk neutral probability as 0.3213. Please refer cell number “C18” in tab – “Assumption Sheet”.

Formula for Downward Risk – Neutral Probability= 1 – q

Call Option Value:

Binomial model based on the assumption that stock price can only take two possible values at the end of each period. So, we get two expected stock prices. Expected Stock prices are \$552.5855 and \$452.4187. Please refer cell number “E6” and “E8” in tab – “Call Option Valuation”.

We calculated pay-off for option buyer at each node. Please refer cell number “F6” and “F8” in tab – “Call Option Valuation”.

We calculated expected payoff using risk neutral probability. Please refer cell number “E11” in tab – “Call Option Valuation”.

Finally, value of the call option is the present value of expected payoff discounted at risk free rate which is ~\$27.77. Please refer cell number “E14” in tab – “Call Option Valuation”.

Put Option Value:

Binomial model based on the assumption that stock price can only take two possible values at the end of each period. So, we get two expected stock prices. Expected Stock prices are \$552.5855 and \$452.4187. Please refer cell number “E6” and “E8” in tab – “Put Option Valuation”.

We calculated pay-off for option buyer at each node. Please refer cell number “F6” and “F8” in tab – “Put Option Valuation”.

We calculated expected payoff using risk neutral probability. Please refer cell number “E11” in tab – “Put Option Valuation”.

Finally, value of the put option is the present value of expected payoff discounted at risk free rate which is ~\$17.7735. Please refer cell number “E14” in tab – “Put Option Valuation”.

Conclusion:

We saw how to calculate the value of PUT and CALL option using binomial model with an example, as illustrated in the above mentioned Valuation Spreadsheet. Hope this was useful to understand how binomial model is used in valuation of PUT and CALL options.