Option pricing

 Buying an option gives you a right to do something. Buying a call option gives you a right to buy the underlying asset(ex.stock) at a specific price (called strike price) at a specific time. In European call option , you can exercise your option only on the last day of option expiry. American option can be exercised at any point of time before or on expiry. The region specific name in categorization is just a misnomer, as both types are traded on both sides of Atlantic. By Writing a call option, you are at the other end of the trade. You receive the option premium initially and give the privilege of exercising the option to the buyer. Options were in use since centuries ago. But the  traders lacked a formula, which could take some parameters as inputs and give the price of the option. 

 In 1973 , Black and Scholes published the famous option formula, Another economist, Merton improved it. When other economists could not quantify the risk of holding the option, these three persons, eliminated the risks in a portfolio by a technique, what is known as delta hedging.  They showed , if you have a portfolio of call options and underlying asset (stocks), it is possible to hedge your position by going long (buying) on stock and writing the call option at a calculated quantity. This will protect you from up or down movement of both prices for a short period.

 As the option and stock price moves to new values ,you need to calculate and hedge again. If you continuously hedge , you can eliminate all the risks in the portfolio. Then the portfolio becomes riskfree.It should give the returns what riskless asset ,namely US bond , gives. If the portfolio gives more returns people will not buy US bonds, they will simply construct portfolio like this and get higher riskfree returns.(Additional Returns with no additional risk between two assets is termed as arbitrage opportunity. Arbitrage opportunities are  like a 100 Rs note on a busy street. Quickly consumed in the market and no longer available). So they equated the riskfree portfolio returns to riskless bond returns and obtained theoretical prices for call options. 

 Advanced Math is involved in understanding the model. Something like, Geometric Brownian Motion(GBM) , Ito Calculus (Standard calculus , which is used to find rate of growth etc will work only for smooth curves. Patterns of Stock prices which follow GBM, has sharp curves which are continuous but nowhere differentiable. Stock prices are an example of Fractals too)etc. Black died of Throat cancer in 95. Scholes and Merton received Nobel prize in 97. Though Nobel is only for living persons, the committee recognized Black's contribution in its statement. 

 Scholes and Merton were pulled into a Hedge fund called Long Term Capital Management(LTCM).In 1994 it started investing operations , which would heavily depend on mathematical models and leverage to make higher returns than market returns. Investments poured in for the company. It gave 41% on the year 95, 42% return on 96,17% on 97 .In 1998, as Russia defaulted on its bond obligations, LTCM lost almost everything and was about to bankrupt. It was 'Too big to fail' at that time. Fed brokered a deal with leading banks to inject around 3.5 Billion dollars to avoid a total financial system collapse(A precursor to 2008).