Geometric Brownian motion 229Note: The probability of whatSdoes next depend only on the currentstate,t. This is calledMarkov property.In other words: In a Markov process future movements in a variabledepend only on where we are, notthe history of how we got where weare.
Over a small time interval[t,t+dt]a GBM has the followingeconomicinterpretation:stock return = mean return + volatility * normal random disturbancelarge volatilityÄlarge random fluctuationssmall volatilityÄsmall random fluctuations()()dtdtNdS SdtNdWwheredWdtdS St tttt t² , ~ , 0 ~ ,
σμσμ⇒+=Derivative securities: Options - Black-Scholes model