Black-Scholes Model
The Black-Scholes-Merton model is a model to price European-style stock options. The model assumes that the stock price follows Geometric Brownian motion (also called exponential Brownian motion)
where 
is stock's expected rate of return, 
is the volatility of the stock price, and dz is the Wiener process (also called standard Brownian motion).
Black, Scholes and Merton 1973 created a riskless portfolio of the option and stock to eliminate the Wiener process from the above equation, and obtained Black-Scholes-Merton partial differential equation
where f is the value of an option.
Final and Boundary conditions
The European call option gives the payoff S-K at t=T when S>K and is worthless otherwise, so the final condition is
The option is worthless, that is, C(0,t)=0 when S=0. And, when the asset price increases without bound, that is, 
, the exercise becomes less and less important. Hence, the boundary conditions are
The solution to the Black-Scholes-Merton partial differential equation with these final and boundary conditions is called Black-Scholes-Merton formula for European-style call option
Using put-call parity for European option
We can obtain the formula for European put option
Where
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