 View : 438 Click : 0 April 5, 2014

# Options Basics: Understanding Maths Behind Options

April 5, 2014

Most of the traders lose money by trading in options because they start trading in options without understanding basics of options.  Options are not just about buying put and calls there are lot of things that works behind option pricing. You need to understand those crucial things for success in options trading. Here I am discussion some mathematical factors that decides option pricing.

### Intrinsic and Extrinsic option value

In the money(ITM) options have intrinsic value. If Nifty is currently trading at 6440 then 6400 ITM call option has intrinsic value of 40 . The 6400 put option has 0 intrinsic value ( technically it should be negative but we take what we get which is 0 )

If Nifty is at  6440 then 6500 call option will have intrinsic value zero  and 6500 put option will have intrinsic value 60.

Hope you understood what is intrinsic value. It is simply the difference between current stock or index price and the strike price of the option.  So ITM options has positive intrinsic value and OTM has zero intrinsic value.

OTM call have extrinsic value which is actually a non existent value. If Nifty is at 6400 then 6500 call option has 100 points of extrinsic value. It is non existent value because its just the value of hope, anyone buying 6500 call option when Nifty is at 6400 is only buying on hope that Nifty can cross 6500.

Some important parameters that effects option value

1. Theta θ : which is time decay

2. Delta Δ : the amount by which the option price changes for a change in the underlying

3. Gamma Γ : the rate of change of delta to the underlying

4. Vega ν : change in price of option with change in volatility (uncertainty / fear factor)

5. Rho Ρ : change in price of option to changes in Rate of interest or the cost of money ( not a big factor for us)

### Theta θ Time Decay

The premium of options keep on decreasing as we move towards the expiry, this decrease with time is called as time decay and is also known as premium decay. The greater the certainty about an option’s expiry value, the lower the time value. Conversely, the greater the uncertainty about an option’s expiry value, the greater the time value. Time decay is more in OTM options than ITM options.

There is no exact formula to calculate time decay or theta value it is calculated theoretically.  A rule-of-thumb is that 1/3rd of the premium will decay in 1/2 the time left for expiry. Based on this theory we can calculate time decay.

Assume Nifty currently trading at 6470 and 6400 strike price call option is at 133. The intrinsic value is 70 here so actual premium is 133- 70 = 63

If there are 20 days left for expiry, then in the next 10 days premium would  decay by  1/3 of 63 = 21

After 10 days 6400 call option would have a value of intrinsic value(70) +( 63-21 ) = 122 , assuming nifty remains at same 6470 level.

OTM call has no intrinsic value, the premium value is everything. If Nifty 6500 call option is at 90 then its value will decay to

90-(1/3)*90 =  60 , assuming nifty remain at same level

Delta Δ : the amount by which the option price changes for a change in the underlying asset

Delta measures the sensitivity of of an option’s price to a change in the price of the underlying stock. Deltas value ranges from 0 to 1.00 call options have positive delta value while put options’ deltas are negative. If Nifty moves up by 100 points and if the delta value is 0.5 then ITM call option will increase by 50% or 50 points while ITM put option will decrease by 50% or 50 points.  Many broker’s offer trading software that provides delta value.  The exact delta value is calculated by options pricing model such as the Black-Scholes Model. The typical values for delta are given below you can use them they also works fine.

1. At The Money options have options delta value of 0.5.

2. Nearest In The Money Options have options delta value of close to 0.75.

3. Next Deeper In The Money Options have options delta value of close to 0.9.

4. Deep In The Money Options have options delta value of close to 1.

5. Nearest Out Of The Money Options have options delta value of close to 0.25.

6. Next further out of the money options have options delta value of close to 0.1.

7. Far Out Of The Money Options have options delta value of close to 0.

Gamma Γ : the rate of change of delta to the underlying

You can observe that delta for ATM option is 0.5 and as it gets more ITM it moves towards 1 and as it gets more OTM it moves towards 0.

Vega ν : change in price of option with change in volatility (uncertainty / fear factor)

Volatility measures the amount of uncertainty. As implied volatility (IV) increases or decreases option prices increases or decreases even when there is no significant move in the  stock or index.  The effect of IV can generally be seen during infosys results when both put and call options price increases simultaneously as there is very high uncertainty on whether the result will be good or bad. Both the put and call options value decreases simultaneously after the results are out because uncertainty is gone. This special effect of IV is also visible during various events like budget, election results etc.  If IV is high movement in option price will be more and movement will be less if IV is low.

I am not going to discuss how to calculate IV as NSE display it for every derivative equity. You can find the IV column that displays the vega factor of options chain at NSE website.

Rho – change due to Rate of Interest changes

This is the cost of money factor as you are blocking money to hold an option. For all practical purposes, rate of interest does not change in the short term and can be ignored for trading purposes. Hope this clears the mathematical terms that works behind option pricing. In the coming posts I will discuss some more detailed aspects of options.

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