Final answer:
The question asks about calculating the probability of a lognormally distributed stock price exceeding a certain value and finding a confidence interval for the stock price after 3 months. These concepts relate to probability and statistics, and involve the use of the lognormal distribution and confidence interval computations.
Step-by-step explanation:
Lognormal Distribution and Confidence Intervals for Stock Prices
The first part of the question involves calculating the probability that a lognormally distributed stock price St exceeds K = $90 in 3 months. Given parameters α = 0.3, σ = 0.5, S0 = 80, T = 0.25, δ = 0, one would typically use the lognormal distribution formula and standardize it using the natural logarithm to find this probability. However, since specific steps are not given to calculate this, we cannot provide an exact answer without further statistical methods or computational tools.
The second part of the question is asking for a confidence interval where the stock price lies with 50% confidence after 3 months, given S0 = 100, α = 0.1, σ = 0.1. A 50% confidence interval corresponds to finding the median, or the 25th and 75th percentiles, as 50% of the data lie between these values. Using the lognormal distribution properties, one can calculate these percentiles.
The information provided for both lognormal distribution and the computation of a confidence interval belongs to the field of probability and statistics, which are advanced mathematical topics typically studied in a college setting.