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A Brownian Motion is stochastic process which has a stationary independent increments along wit continuous sample paths. The increments seem to have normal distribution for given period of time, and this is what makes these increments useful in finance; as many financial experts believe that the stocks returns are normally distributed.
Financial experts use Brownian Motion to build stock price models that include lognormal process, which is considered to be the exponential of Brownian Motion. Models based on Brownian Motion are very popular because they allow continuous hedging opinions to be used when determining pricing.
It was thanks to Albert Einstein who published three famous papers in 1905 explaining the statistical mechanics of Brownian Motion. This helped in producing the heat transfer equation, which was used by Black and Scholes to derive the famous option pricing formula. Ultimately it was the drift term in the Brownian Motion structure that allowed the financial option formula to be derived and solving this must yearned mystery forever.
Paul Samuelson in 1965 worked on the valuation of warrants that led to a formula like that of Black and Scholes. The only difference was that Samuelson used expected return of the stock for drift term rather than the riskless return, which was later on used by Black and Scholes; and it is anyone’s guess what would have happened to the famous option pricing formula.
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