Understanding Options using the Binomial Tree Model

Option trading in India is picking up. Check out the article to understand how to take good trades while writing and buying options.

· 6 min read
Understanding Options using the Binomial Tree Model

Recently, options trading has grown in popularity both domestically and globally. The advantages of options to portfolio cost and risk management are clear reasons for their surge in trading volume. Whether an investor wants to buy or sell options, understanding what makes up an option’s premium is crucial in trading options.

What are Options?

Options are financial derivatives - contracts whose value is based on an underlying financial asset (like a security) or set of assets (like an index). When you buy an option, you are buying the right to buy or sell the underlying asset. You also have to exercise your option by a certain date or it expires.

How do they work?

To purchase options, you pay the seller of the option a fee, known as a “premium.”

When you buy a call option, you hope the price of the stock associated with it will increase in the near future.

When you buy a put option, you hope the price of the stock associated with it will decrease in the near future.

There are approximately 175 Indian companies on the F&O stock list stipulated by the Securities & Exchange Board of India (SEBI). To know how you can take advantage of options, let us first understand how options are valued using the following example -

Reliance Industries Limited (RIL)

Reliance Industries Limited (RIL) stock is currently trading at Rs. 2114 as on 17thAugust 2020. We are expecting the price of the stock to go up and want to buy it at a lower pre-determined price. Therefore, we will buy a call option that will give us the “right to buy”. There are many call options contracts available for this stock, however, for our convenience we shall compute the premium for a contract expiring on the 24thSeptember, 2020 with a Strike Price/Exercise Price (Sx) of Rs. 2100.

Strike Price/ Exercise Price (Sx)– The price at which you would like to buy the stock
Risk Free Rate– The risk-free rate of return is the rate of return where there is no risk of default. We have considered the 10 yr. Indian Government Bond yield here.
Current Stock Price (So)= Rs. 2114 (17thAugust 2020)
Expiry Date of the Contract– 24thSeptember 2020
Time/Duration of the Contract, T= 40 days; T (in years) = 40/365 = 0.1095
Risk Free Rate (10 yr. Indian Government Bond yield)– 5.97%
Risk Free Rate (Continuously Compounded Rate)– 5.79 %

Calculate the Annual Volatility/ Standard Deviation of Stock Returns
Using the historical stock price data of the last 365 days, we calculate the standard deviation of the returns.
Annual Volatility = σ = 2.8841 % = 0.028841

Calculate the Up Factor and Down Factor
We then calculate the up and down factor for the stock price, using the formula in the table,
UpFactor = e^ (σ √T) = e^(0.028841 x √0.1095) = 1.009589
Down Factor = 1/U = 1/1.009589 = 0.990502

Up Factor

U

e^(σ √T)

1.009589

Down Factor

D

1/U

0.990502

Multiply the Up and Down Factor by the Stock Price

We now multiply the Up Factor with the current stock price to obtain the future stock price in case of an up move, and multiply the Down Factor with the current stock price to obtain the future stock price in case of a down move. Stock price in case of an up move:
So x U = 2114 x 1.009589 = Rs. 2134.2711

Stock price in case of a down move:
So x D = 2114 x 0.990502 = Rs. 2093.9212

4.    Calculate the Risk Neutral Probabilities

Finally, we calculate the risk neutral probabilities -
∏(u) refers to the probability that the stock will go up,

∏(u) = e^rT – D/ (U-D)

= e^ (0.0579 x 0.1095) - 0.990502/ (1.009589- 0.990502)

= 0.830837

∏(d) refers to the probability that the stock will go down

∏(d) = 1- ∏(u)

= 1 - 0.830837

= 0.169163

Risk Neutral Probability

 

Probability (Up)

∏(u)

e^rT -D/(U-D)

0.830837

Probability (Down)

∏(d)

1- ∏(u)

0.169163

 

 

 

 

 

After doing the above calculations, we can now calculate the option premium for the stock. Since we are computing the value of a call option, we have the right to buy the stock at Rs. 2100. In case of an up move, the stock will go up to Rs.2134.27and we will be able to buy the stock at Rs. 2100. If we exercise the option, we will also be able to sell the stock at Rs.2134.27and make a profit of Rs.34.27 (Rs.2134.27– Rs. 2100)

What happens if the price of the stock falls? There is no need exercise the option in this case as we wanted to buy the stock at a lower price. Now, since there is a 83.084 %chance that the price of the stock will go up, we can calculate the expected payoff from the option in the future by multiplying the profit with the probability - Payoff = Profit x Probability (Up) = 34.27x 0.830837= Rs. 28.472

Using this value, we can now compute the expected value of the option premium at present – Co = Payoff x e^(-rT) = 28.472x e^(- 0.0579 x 0.109589) = Rs.28.29

Therefore, value of the option premium is Rs. 28.29 What does this mean?

This means that for Rs. 28.29, we now have the right to buy a share of RIL at Rs. 2100 before 24thSeptember 2020, in case the price of the share goes up.

There are two basic components of this option premium –

1.    Intrinsic value: The intrinsic value of an option is the amount of money you would get if you exercised the option immediately. It is the difference between the underlying stock’s price and the option’s strike price. In case of RIL –

Intrinsic Value = Stock Price (Sx) – Exercise Price (So)

= 2114 – 2100

= Rs.14

Therefore, the intrinsic value of the option is Rs. 14

2.    Time Value: The time value of an option is the difference between the price of the option and its intrinsic value. The price of the call option we calculated is Rs. 28.29. Therefore, its time value is -

Time Value = Price of Option – Intrinsic Value

= 28.29 – 14

= Rs. 14.29

This means that it would cost Rs. 14.29 more to buy the option and exercise it today than it would to just buy the stock outright. This extra Rs. 14.29 is the cost of holding the option for the period of time until the option expires. Because of this time element, this cost is called the time value. As the option approaches its expiration date, the time value decreases. When the option expires, it becomes worthless.

What should you do?

Options premiums don’t always trade at the price we have calculated. The last traded price of the call option of RIL is Rs. 70. This could be because people believe that there is a good chance the price of RIL will rise during the term of the optionand they are willing to pay more than Rs. 14 to take advantage of the higher price.

You could short sell the option here and buy it closer to expiry date, however keep in mind that selling an option brings with it an obligation.If you sell a call option, you agree to sell the underlying stock at the strike price even if its price later rises.

Also, sometimes thestock may also have a high option premium because it is more volatile than other stocks.  In this case, it may be a bad idea to sell options on it. Maybe the premium is high, but because of the high risk of the owner exercising the option, the premium is not high enough to warrant taking the risk.

Whether or not you base your trades on finding stocks with the highest option premiums depends on your strategy. It stands to reason that if you’re a buyer of options, you want to find stocks with low option premiums so you can buy them cheaper. This allows you to control stocks at a lower cost. On the other hand, if you’re primarily a seller of options, you’ll want high option premiums.

Apart from granting the “right to buy”, does the option premium provide you any other benefits?

Yes, it does. The option premium of Rs. 28.29 allows you to gain an exposure on one share of RIL without having to invest Rs. 2114 in one share. If you want to buy shares of RIL at Rs. 2114 per share, it would cost you Rs. 2,11,400 to buy 100 shares. By contrast, if you bought one call option contract affording you control of 100 shares, it would cost you only Rs. 2829. Whether you've purchased 100 shares or one call option contract, you're now long 100 shares of the RIL This way you have managed to limit your risk and you now have more free capital to invest in other opportunities.

This lower cost of entry also provides another great benefit: leverage. Because an option is cheaper to buy than the equivalent amount of stock, there's greater potential for higher percentage gains on your investment - It's simply a lot easier to double your money on a Rs. 3000 investment than a Rs. 2,00,000 investment.

Disclaimer – This article is for educational purposes only. While the option premium in this article has been calculated using the Binomial Tree Model, there are other methods to calculate it as well.

This article was written by Ayesha S. for Pvot. She is a FRM level 2 candidate and an avid reader. Her primary interest areas are in the field of quantitative finance and investments.

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