(μ is the population average and σ is the standard deviation)
Second, the mathematical description of Brownian motion:
The price time function P(x), the difference between the price p (T+t) at t+t and the price p (T) at t: p (T+t)-p (t) is a normal random variable with a mean value of μt and a standard deviation of. (T & gt0，t & gt0)
Main defects:
1. According to this price, there may be a negative value in theory, but it is impossible to have a negative value in practice.
2. No matter what the initial price value is, the price difference with a fixed time length has the same normal distribution, which is unreasonable.
Three, geometric Brownian motion:
Converting price difference into price fluctuation: it can avoid the defect of describing price directly with Brownian motion, that is, geometric Brownian motion.
Is a normal random variable, the average expected value of the distribution is μt, and the standard deviation is. (T & gt0，t & gt0)
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Geometric Brownian motion
The function of geometric Brownian motion is to simulate the change of stock price. Its advantage is that the value of general Brownian motion may be negative, while the value of geometric Brownian motion is never less than 0, which conforms to the characteristic that stock prices are never negative.
Representation of differential form of geometric Brownian motion. Or SDE (stochastic differential equation) form:
Where S(t) can be understood as the stock price.
Formal expression of geometric Brownian motion function;
The above formula tells us that we can form a general form of Brownian motion, then find its exponential function, and finally multiply it by S(0), that is, the initial stock price, and we can get geometric Brownian motion.
Supplement: Why is there one more term for the coefficient of T here? See Ito formula for details.
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