The Black-Scholes: Equations Pricing Model

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The Black-Scholes model is perhaps the most famous and widely used options pricing model in the world. The model was first founded by Fischer Black and Myron Scholes in their 1973 and was further developed by Robert Merton who expanded the mathematical understanding of the options price model. In order to honor their contributions in finance, they were awarded the 1997 Nobel Price in Economics. People have been trading stock options for a long period of time, and the Black-Scholes model has finally set up an analytical framework for how to value an option. It is no exaggeration to say that the Black-Scholes model brings convenience and efficiency to the European stock market.
The Black-Scholes model is used to calculate the theoretical estimate
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, the strike price of the option.
, the annualized risk-free interest rate, continuously compounded (the force of interest).
, the drift rate of , annualized.
, the standard deviation of the stock 's returns, which is the square root of the quadratic variation of the stock 's log price process.
, a time in years; generally use: now=0, expiry=T.
, the value of a portfolio. , the standard normal cumulative distribution function,
.
, the standard normal probability density function, ("Black–Scholes Model").
Black–Scholes Equation
The Black–Scholes equation is a partial differential equation, which describes the price of the option over time. The equation is:

The Black-Scholes PDE can be transformed into the heat equation with following change of variables: , ,
Placing the partial derivatives into the Black-Scholes PDE, we could get: ,
We can clearly see that the first part of the equation is actually the heat equation. More changes of variables are required in order to eliminate the last two terms on the right-hand side of the equation: , , .
Computing the partials of v in terms of x and t, we get: with initial condition: where , Thus, (“Converting the
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The European Call Option is composed of what the gain from the stock minus the pay on the stock. Since N(d) is a probability, it is always going to be greater than zero and less than 1. We now know that is the stock price being weighted by some probability, and is the present value of the exercise price being weighted by some probability. Assuming we have high probability(N), the higher the stock price relative to the exercise price, the higher the stock price would be and we are thus more likely to exercise the option. The volatility() is also a deep focus of people who operate with options. From the equation d1, we can tell that if goes up, d1 goes up as well. Conversely, if goes up, d2 will go down. The change in d1 and d2 will have direct effect on increasing and decreasing , which means one will pay less and get more from the stock. Therefore, an increase in volatility() will gives stocker more value if exercise the

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