Options University Blog

One interpretation of option pricing formula is an option pricing theory, also known as derivatives pricing theory or Black-Scholes theory, from Bachelier.  Bachelier is also the inventor of Brownian motion.  People have been seeking a complete guide to option pricing formula for a long time.  The truth is that there are several possible option pricing formula variations to consider.

Derivatives pricing theory sparked several other valuation methods, which all modeled future asset values, using the Monte Carlo method of binomial trees.  Such an options pricing formula method does not provide discount rates or realistic expected returns.  It treats all financial assets as if they have an expected return equal to their risk free rate.  The problem with that is that no investor is risk neutral, so it does not really reflect a real world outcome.  Still, though, it can produce accurate option prices, if it is used properly.

Cox and Ross were the first people to mention risk neutral valuation.  They did so in 1976 in a paper about pricing options with jump processes.  They teamed up with mark Rubinstein three years later, after they realized the importance of the technique.  Together, they published a paper, which sued risk neutral valuation to develop binomial trees technique.  Soon, other authors took up the cause and formalized the mathematics involved in risk neutral valuation as a method of equivalent martingale measures.  Now, it is the option pricing formula most commonly used for derivatives in complete markets.

The Black-Scholes technique utilizes partial differential equations, so that options trading formula is sometimes know as the differential equations approach.  They usually have simple pricing formulas, due to closed-form solutions.  The Monte Carlo method of solving equations numerically is another good example of a differential equation approach.

The risk neutral approach usually involves use of stochastic calculus with changes of measure between a risk neutral world and the real world.  Therefore, the risk neutral approach is sometimes known as the stochastic calculus approach.  It can lead to either numerical or closed-form solutions, although numerical are most common.  Sometimes it is effective in pricing derivatives that the Black-Scholes approach was incapable of solving.

Financial engineers, who hold advanced degrees in physics or mathematics, design and implement derivative option pricing formula models all the time.  These “rocket scientists” are well-equipped to handle such technical information.  However, it is then up to the general public to understand it and learn to use it properly.  Financial engineering has extended, recently, to include fixed income derivatives as well.  Normally, that requires modeling entire term structures, which can be quite complicated.  They have even been extended, sometimes, to commodities markets, which can cause risk neutral valuation to be quite a problem.

So, now that you understand option pricing formula better, you can use your new-found knowledge to make the most of option pricing formulas and techniques and turn a healthy profit!

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