Calculation of Premiums of Put/Call Options

[gargi.adhikari]

Hi,
Kind of intuitively understand that there is a correlation between how Call & Put Options are priced and the amount by which they are "In the Money" or "Out The Money" ..but was wondering and trying to find if there is a concrete formula or a way to calculate or price the Put and Call Options based on their "Moneyness"..? Thanks in advance for your insights on this.

 

[jairamjana]

Ok.. I won't get into Binomial models(risk neutral) or the black scholes.. But We could understand a simpler method for this which involves expiration date of the call/put option (T) , the Current stock price .. S(0) , the risk free rate/discount rate (r) and finally the strike price (K) ..

Now assuming the European option, you will exercise only at expiration date (T) at price of Strike Price if it's ITM right.. We can discount this K into Present Value form taking r as discount factor, Derivatives are usually denoted in continuous time so e is used .. This value K*e^(-r*T) should be subtracted by Current Price S(0) .. So value of C= S(0)-K*e^(-r*T) ..
In this way we factor both time value and intrinsic value and this is a decent price.. Except this is not realistic as we failed to consider volatility of the stock/underlying and also the probability that the it will be in the money at time of expiration.. That's where black scholes and its variants come in which factors all these too..

Hope that's clear...

 

[ShaktiRathore]

Hi,
Yes as jairam above pointed the the value of call is simply S(0)-K*e^(-r*T) can be written as S(0)-K+K-K*e^(-r*T)=( S(0)-K)+K(1-e^(-r*T))=Moneyness+K(1-e^(-r*T)) where Moneyness=( S(0)-K) the present Intrinsic value of the option.Thus we get the value of the option in terms of ( S(0)-K) the Moneyness. The formula ( S(0)-K)+K(1-e^(-r*T)) does gives an approximate value of the option which can be negative also its just that we get an idea of the option value. But the problem with these formula is that we fail to consider the time value that is the volatility component of the underlying if we introduce the volatility components N(d1) and N(d2) then they enter the value of option equation S(0)-K*e^(-r*T) , as attach N(d1) to S(0) and attach N(d2) to K*e^(-r*T) so that S(0) becomes S(0)N(d1) and K*e^(-r*T) becomes K*e^(-r*T)N(d2) so that the value of option equation becomes S(0)N(d1)-K*e^(-r*T)N(d2) which is nothing but the black scholes formula for option valuation.If you remove the volatility component then we drop N(d1) and N(d2) to get the option valuation formula as S(0)-K*e^(-r*T).
thanks

 

[jairamjana]

For more about N(d2) and N(d1)
https://forum.bionicturtle.com/threads/interpretation-of-n-d1-and-n-d2.610/

David sir tries to explain it as intuitive as possible..
You can also later check the book Options, future and other derivatives by John hull.. The industry standard and basic book on this subject.. Part of the book is used in many of the topics in part 1 FRM..

 

[gargi.adhikari]

Thank you all so much for this in depth breakdown on this topic! Gratitude