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## The Ultimate Guide to Option Pricing Formula

A lot of people have been asking for a complete guide to the options pricing formula. We would try to provide a comprehensive and useful guide here. Inventor of Brownian motion, Bachelier is also the root of “option pricing theory” also called “Black-Scholes theory” or “derivative pricing theory”.

This risk-neutral approach or technique also opened the door to other valuation method options that used the Monte Carlo method of binomial trees to model future asset values. In his analysis, he does not attempt to provide so-called realistic expected returns and discount rates. Users can treat all assets of a financial nature as having expected returns equal 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, but if used properly, it produces accurate option prices.

The initial mention of the risk of neutral valuation is by Cox and Ross. It was somewhere in the middle of their paper on option pricing with jump processes, published in 1976. Three years later, realizing the importance of the technique, they teamed up with Mark Rubinstein to publish a paper using risk-neutral valuation to develop the binomial tree technique. Progressively, other authors formalized the mathematics of neutral risk as a method of equivalent martingale measures. This is the main method used for derivatives in complete markets.

Financial engineers are well-paid professionals who have advanced degrees in mathematics or physics. They are sometimes called a rocket scientist or a quant. These top financial engineers design and implement derivative pricing models.

The Black Scholes approach or technique is sometimes called the differential equation approach because it uses partial differential equations. These differential equations often have closed-loop solutions that lead to fairly 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, as it seeks to incorporate the detailed use of stochastic calculus with changes in measure between the “real world” and the “risk neutral” world. This can also lead to closed-form solutions, although numerical solutions are more common. It is relatively more flexible than the Black-Scholes approach. It is effective in some cases when used to price derivatives that the Black-Scholes approach could not solve.

Methods known in financial engineering have now been extended to fixed income derivatives; this usually requires modeling the entire term structure. In other cases they are expanded to include commodity markets, at this market risk neutral valuation becomes more of an issue.

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