Stock price probabilities after 364 days when each day the variance is 0.25:
### (a)
pnorm(100, mean = 100, sd = sqrt(0.25*364), lower.tail = F)## [1] 0.5### (b)
pnorm(110, mean = 100, sd = sqrt(0.25*364), lower.tail = F)## [1] 0.1472537### (c)
pnorm(120, mean = 100, sd = sqrt(0.25*364), lower.tail = F)## [1] 0.01801584Moment Generating Function of binomial to calculate mean and variance:
\[ g(t) = \sum_{i=1}^{\infty} e^{tx_i} p(x_i) = (pe^t + q)^n \] \[ g'(0) = n(pe^0 + q)^{n-1}pe^0 = np \] \[ g''(0) = n(n - 1)p^2 + np \] \[ \mu = np \] \[ \sigma^2 = g''(0) - g'(0)^2 = n(n - 1)p^2 + np - (np)^2 = np(1-p) \]
Moment Generating Function of exponential to calculate mean and variance:
\[ g(t) =\int_{-\infty}^{\infty} e^{tx}f(x)dx = \int_{0}^{\infty} e^{tx}\lambda e^{-\lambda x}dx = \lambda/(\lambda-t) \] \[ g^n(0) = n! / \lambda^n \] \[ g'(0) = 1 / \lambda \] \[ g''(0) = 2 / \lambda^2 \] \[ \mu = 1 / \lambda \] \[ \sigma^2 = 2 / \lambda^2 - (1 / \lambda)^2 = 1 / \lambda^2\]