In this three part series, we introduced the Option Greeks in the first post. In the second post, we discussed the practical application of option Greeks with respect to options trading.
In this concluding post, we will understand the usage of an option calculator. An option calculator is a tool which helps you calculate the Greeks, i.e. the delta, gamma, theta, vega, and rho of an option. Along with the calculation of the option Greeks, the option calculator can also be used to calculate the theoretical price of an option (also called fair value of an option’s premium) and the implied volatility of the underlying.
The option calculator uses a mathematical formula called the Black-Scholes options pricing formula, also popularly called the ‘Black-Scholes Option Pricing Model’. This is probably the most revered valuation model in Economics, so much so that its publishers (Robert C. Metron and Myron Scholes) received a Nobel Prize in Economics in 1997.
Briefly, the framework for the pricing model works like this:
- We feed the model with a bunch of inputs
- Inputs include: Spot price, Interest rate, dividend, and the number of days to expiry. Along with these mandatory inputs, we also input either the price of the option or the implied volatility of the underlying, but not both .
- The pricing model churns out the required mathematical calculation and gives a bunch of outputs
- The output gives us the value of option Greeks. Along with the option Greeks, we also get one of the following:
- The Implied volatility of the underlying, provided one of the input is the option price or
- The theoretical value of option’s premium, provided the input is the implied volatility of the underlying
The illustration below gives the schema of a typical options calculator:
Let us inspect the input side:
- Spot Price – This is the price at which the underlying is trading. Note, we can even replace the spot price by the futures price. We use the futures price when the option contract is based on futures as its underlying. Usually, commodity and in some cases currency options are based on futures. For equity option contacts, always use the spot price.
- Interest Rate – This is the risk-free rate prevailing in the economy. Use the RBI 91 day Treasury bill rate for this purpose. As of September 2014, the prevailing rate is 8.6038% per annum.
- Dividend – This is the dividend expected per share in the stock, provided the stock goes ex-dividend within the expiry period. For example, today is September 11 and you wish to calculate the option Greeks for the ICICI Bank option contract. Assume ICICI Bank is going ex-dividend on September 18 with a dividend of Rs. 4. The expiry for September series is September 25. In this situation you need to give an input of Rs. 4.
- Number of days to expiry – This the number of calendar days left to expiry.
- Volatility – This is where it gets a little confusing, so I suggest you pay extra attention. As mentioned earlier, along with option Greeks you can use the option calculator to calculate either the implied volatility of the underlying or the theoretical option price but not both at the same time
- If you wish to calculate the theoretical option price as one of the desired outputs, then volatility has to be one of the inputs. For Nifty option contracts, use the India VIX index value. Alternatively, if you have a view on volatility from today to expiry, you can input that as well. You can do the same thing for stocks.
- Option Price, also called the ‘Actual Market Value’ – If you wish to calculate the implied volatility of the underlying you need to input actual market value data. The actual market data is simply the price at which the option is trading in the market.
Once these inputs are fed to Black-Scholes option pricing model, the model churns out the math to give us the required output. The logic on which Black-Scholes model works is quant heavy involving concepts of stochastic calculus. For a quick introduction on the working of a Black-Scholes model, I’d encourage you to watch this video .
We get the following values on the output side:
Along with the Greeks, the output includes either the implied volatility of the underlying or the theoretical option price.
Option Calculator on Zerodha Trader (ZT)
Keeping the above framework in perspective, let us explore the Option Calculator on Zerodha Trader (ZT). To invoke the option calculator, click Tools –> Option Calculator as shown below. Or you can simply place your cursor on an option scrip and use the shortcut key Shift+O.
This is how the calculator appears on the terminal:
The calculator can be broken down into three sections as shown in the image below:
The top section highlighted in blue is used to select the option contract, this is fairly straightforward.
The left section highlighted in red is the input
field. Let us look into this.
We begin by selecting either the ‘Underlying’ or the ‘Futures’ price. I’d suggest you select ‘underlying’ as the default option. Once the underlying has been selected, you need to manually enter the value of the underlying in the ‘Spot Price (in Rupees)’ field.
The next two input fields are ‘Actual Market Value’ and ‘Volatility %’. At this stage you need to decide what the option calculator should calculate for you.
If you want to calculate the ‘fair value of the option premium’ also called the ‘Theoretical Option Price’ then leave the ‘Actual Market Value’ field blank and proceed to enter the volatility data. As I mentioned earlier, for Nifty options use the India VIX index value for the ‘volatility %’ field.
Alternatively, if you want to calculate the volatility of the underlying leave the ‘Volatility %’ blank, but make sure you input the market price of the option in ‘Actual Market Value’.
For the ‘interest rate %’, take the 91 day T-bill rate data from the RBI website .
‘Dividends (in Rupees)’ would be for the index and the actual dividend value in case of a stock. Also, in case dividends are expected within the expiry of the contract, make sure you enter the ex-dividend date.
The last input field is the number of days left to expiry. Input the total number of calendar days here.
Note, Zerodha Trader (ZT) has two models based on which the Greeks can be calculated, i.e. Black-Scholes Pricing Model and another model called the ‘Cox-Ross-Rubinstein Binomial Method’.
The binomial method is also popularly used, however, I’d advocate the Black-Scholes model as it is more advanced and precise. It is worth mentioning that the difference in output values between the two models is not really much.
Lastly, look at the bottom section of the Output field (highlighted in green ). Just besides the ‘Calculate’ button you have two options:
- Option Price
Select Volatility if you want the option calculator to calculate the volatility for you. If you want to calculate the theoretical option price, select the ‘Option Price’.
Have a look at the image below with all the input data loaded:
Notice two things:
- Along with the Greeks, I intend to calculate the Option price (highlighted in blue). Also ‘Actual Market Value’ is left blank (highlighted in red). I’ve taken the volatility value from the India VIX index.
- The dividend field is blank since I have selected 8100 Nifty Call option ( index option), hence the value in ex-dividend date field is irrelevant.
Once the input values are loaded, click Calculate to generate the output. The following image shows the output:
The first field in the output field is the theoretical option price (also called the fair value) of the call and put option. The calculator is suggesting the fair value of 8100 call option should be 81.14 and the fair value of 8100 put option is 71.35. However, the call option value as seen on the NSE option chain is 83.85.
The difference, though not significant, mainly occurs due to factors such as wrong volatility assumptions, bid-ask spread, liquidity, transaction charges, and taxes.
Following the theoretical option price you can find the data on Greek values. As of today Nifty spot is 8085, and the closest ATM option is 8100. As we had discussed in the previous post, the ATM option should have a delta of approximately 0.5. In fact, the calculator is telling us that the delta is 0.525 for the call option and -0.475 for the put option. This is in line with our discussion on delta in the previous post. Following the delta value we find other Greek values such as Gamma, Theta, Vega, and Rho.
Also, by default the calculator calculates the Greeks of:
- Put option of the same strike, same expiry
- A simple long straddle
Option Calculator to calculate volatility
Let us now use the option calculator to calculate the volatility of the underlying. To do this, I leave the ‘Volatility %’ field blank (highlighted in blue) and select “Volatility” (highlighted in red) option.
Further, I input the “Actual Market Value” of the 8100 Call option as observed on NSE, which in this case happens to be 83.85 (see the NSE Quote image above).
After selecting this click calculate:
It turns out that the volatility of Nifty is 12.96% as opposed to 12.5175% as India VIX suggested. Well, the difference is less than 50 basis points; this should also explain why the calculator calculated the Theoretical Option Price as 81.14 as opposed to 83.84. In fact, instead of 12.5175% if we now give Volatility % input as 12.96% we will get the accurate Option price. See the image below:
Option calculators are mainly used to calculate the option Greeks, volatility of the underlying, and the theoretical option price. Sometimes small differences arise owing to variations in input assumptions. Hence for this reason, it is good to have room for the inevitable modeling errors. However, by and large, the option calculators are fairly accurate.