Overcoming the Assumptions in the Black-Scholes Model

As everyone knows - there is no free lunch, or the cheese is free only in the mouse trap. So to say, we need to talk about the deficiences in the analysis by Black-Scholes and Merton. There were few assumptions put into the basis of the Black-Scholes model and equation.

Those are:

A1) There are no market imperfections, e.g., taxes, transactions costs, shortsales constraints, and trading is continuous and frictionless.

Yes, this is the GREAT assumption.

A2) There is unlimited riskless borrowing and lending at the continuously compounded rate of return r; hence a $\$1$ investment in such an asset over the time interval $\QTR{Large}{\tau }$ grows to MATH Alternatively, if MATH is the date $t$ price of a discount bond maturing at date $T$ with face value $\$1,$ then for $t\in \lbrack 0,T]$ the bond price dynamics are given by

A3) The stock price dynamics is given by a geometric Brownian motion, with the solution to the following Ito stochastic differential equation on $t\in \lbrack 0,T]$

where $B\left( t\right) $ is a standard Brownian motion, and at least one investor observes $\sigma $ without an error.

A4) There is no arbitrage.


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