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How Option Price Is Calculated:


Option trading is favorite among the traders due to its lower values , requirements of low margin and easy calculation of maximum possible risk.Its also an easy way for stock brokers also.But if you ask me my opinion is that option trading is far difficult from futures trading.It happens most of the times that future prices moves up in our favorable position but option prices do not in the same proportion.There is a common question among the new traders that how this option price is calculated.

In 1973,Fischer Blacks & Myron Scholes  derived a formula and presented a paper named as “The Pricing Of Options And  Corporate Liabilities” which was then become known as “Blacks-Scholes Model Of Option Pricing“.

This formula is a mathematical model  option calculation containing certain derivative investment instrument.

Model Assumptions:

This model is derived based on following assumptions:

1.Underlying asset does not pay the dividends as well there is no hedging opprtunity.

2.Underlying asset is volatile and follow the Brawnian movment.

3.Its possible to buy or short sell with any fractional amount.

4.Its possible to borrow or lend cash at a known constant Risk free interest rate.This interest rate may be considered as interest rate offered by government securuties.

5.Market is frictionless.

Model Parameters:

  • C:Price of call option.
  • P:Price of put option.
  • x:Price of underlying stock.
  • K:Strike Price of the option.
  • r:Annualised risk free interest rates,continuously compounded.
  • t:time[0,T]
  • \sigma =% Volatility.
  • V:Price of derivative as a function of x and t.

The partial differential model equation derived was:

\partial V/\partial t+1/2\sigma ^2x^2\partial ^2V/\partial x^2+rx\partial V/\partial x-rV=0

If this equation is solved for boundary and terminal conditions then roots are the prices of call option and put option.

Call Option:

C=N(d1)x-N(d2)Ke^{-r(T-t)}

Put Option:

P=Ke^{-r(T-t)}-xN(-d1).

Where

d1=\frac{ln(x/k)+(r+\sigma ^2/2)(T-t)}{\sigma \sqrt{T-t}}

d2=\frac{ln(x/k)+(r-\sigma ^2/2)(T-t)}{\sigma \sqrt{T-t}}

N represents the standard normal distribution with mean =0 and standard deviation=1.

This model is used india to calculate the option prices.Like all models,this model have also certain limitaions and not accurate under all market conditions.It has also extended for stocks paying dividends.There are also other option pricing models like Binomial options model,Monte-Carlo options model etc and are being used for suitable market conditions.

How change in one of the parameter changes the option prices with all other parameters remaining constant:

Trade date Expiry Date Spot Price Strike Price Volatility Risk Free Rate Call Option Price
20/11/2011 30/11/2011 4850 4800 20% 6% 93.20
28/11/2011 30/11/2011 4850 4800 5% 6% 58.18

In above example,all parameters except % volatility are constant.Call option price is an output value and all remaining are inputs.

Vice versa, we can calculate one of the input parameter like % volatility,risk free interest rate ,if option price is known.


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4 Responses

  1. Hema Raghavendra says:

    Good information. It could have even better if you provide one simple example along with the formula

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