Verified
About the Black–Scholes equation all the facts are given here:
• The equation follows by certain supposition and from a financial and mathematical argument which involves hedging.
• The equation is homogeneous and linear (as we say ‘here is no right-hand side,’ that is no non-V terms) therefore you can value a portfolio of derivatives with summing the values of the single contracts.
• This is a partial differential equation since it has more than one independent variable where asset (S) and time (t).
• This is of parabolic type, implies that one of the variables, t, just has a first-derivative term, and the other asset has a second-derivative term.
• This is of backward type, implies that you specify a last condition representing the option payoff at expiry and after that solve backwards in time to find the option value at this instant. You can tell it’s backward with looking at the sign of the time (t)-derivative term and the second asset (S)-derivative term, while on the same side of the equivalents sign they are both identical sign. When they were of opposite signs then this would be a forward equation.
• The equation is an illustration of a diffusion equation or heat equation. These equations have been approximately for nearly two hundred years and have been utilized to model all sorts of physical phenomenon.
• The equation needs importance of two parameters, the risk-free interest rate and the asset volatility. This interest rate is easy sufficient to measure and the option value is not so sensitive to this anyway. However, the volatility is other matter, quite harder to forecast accurately.
• Because the major uncertainty in the equation is the volatility one sometimes thinks of the equation less as a valuation tool and more like a way of understanding the connection between volatility and options.
• The equation is simple to solve numerically, along with finite-difference or Monte Carlo methods, for illustration.
• The equation can be generalized to permit for dividends, another payoff, jumping stock prices and stochastic volatility, etc.
And then there is the Black–Scholes formula that is solutions of the equation in particular cases, such as for puts and calls.