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The Ultimate Guide to Option Pricing Formula

Mar 22, 2008
A lot of people have sought a complete guide to option pricing formula. We would attempt to provide here a comprehensive useful guide. The inventor of Brownian motion, Bachelier also is the root of the "Option pricing theory" also called "Black-Scholes theory" or "derivatives pricing theory".

This risk neutral approach or technique also opened a door to other options of valuation methods that used the Monte Carlo method of binominal trees to model future asset values. It does not attempt to provide so called realistic expected returns and discount rates in its analysis. Users are able to treat all assets of a financial nature as having expected returns that are equaled to the risk free rate. All cash flows can be discounted at the risk free rate. No investor can be risk neutral, so the risk neutral technique is not a true reflection of the real world, still if correctly used it produces correct option prices.

Initial mention of risk neutral valuation was by Cox and Ross. It lay somewhere in the midst of their paper on pricing options with jump processes, released 1976. Three years later, realizing the importance of the technique they teamed up with Mark Rubinstein to publish a paper that uses risk neutral valuation to develop the technique of binomial trees. Progressively other authors formalized the mathematics of risk neutral as a method of equivalent martingale measures. This is the main method used for derivatives in complete markets.

Financial engineers are well paid professionals holding advanced degrees in mathematics or physics. There are sometimes referred to as rocket scientist or quants. These top financial engineers design and implement derivatives pricing models.

The Black Scholes approach or technique is sometimes called the differential equations approach because they employ partial differential equations. These differential equations often have closed-form solutions which lead to quite simple pricing formulas. Examples include the original Black Scholes formula or the Monte Carlo method used to solve equations numerically.

The risk neutral approach is also called the stochastic calculus approach, because it tends to involve detailed use of stochastic calculus with changes of measure between a "real world" and a "risk neutral" world. It could also lead to closed form solutions, although numerical solutions are more usual. It is relatively more flexible than the Black-Scholes approach. At some instances it is effective when used to price derivatives that the Black-Scholes approach could not solve.
Methods known for financial engineering have now been extended to fixed income derivatives; this normally requires the modeling of entire term structures. They have at other instances been extended to include commodities markets, at this markets risk neutral valuation becomes quite more of a problem.
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