Sampling from distribution

[afoster28]

Check whether sampling from distribution implemented in the code is mathematically and conceptually sound:

• 5% mean return and 20% standard deviation from the task are assumed to follow a normal distribution
• These are converted muLog and sigmaLog for a lognormal distribution
• Random sampling of returns from this lognormal distribution - these should be lognormally distributed
• The time to maturity is not considered; all options will have the same time to maturity so we need one distribution of prices; we're assuming 5000 spot price will continue into the future and standard deviation around this will produce bell curve same as today - is this sensible or should we evolve spot price into the future using 5% mean return (scaled down to 60 days as per option time to maturity)? Edit: proposed solution is to rescale mu and sigma inputs to 60 days because they are currently annual values, so:

• Conversion to log returns by taking the natural log of each element - these should be normally distributed log(return) sufficient or log(1 + return)?
• These have a mean of 3.2% and a standard deviation of 18.7% - is this expected?
• Log returns are applied to the spot price of 5000 independently (i.e. without sequencing) to generate a point-in-time distribution of possible prices - makes sense? spot price (1 + log(return)) sufficient or spot price exp(log(return))?
• Resulting price distribution has a mean of 5170 and standard deviation of 933

[afoster28]

Fixed price sampling by applying the following:

muLog = log(spot) + (mu - 0.5 sigma ^ 2) t sigmaLog = sqrt(sigma ^ 2 * t) sampled log prices from the normal distribution exponentiated to get lognormal distribution of prices at maturity