# Probability Density of Returns of Bonus Certificates

I need to generate a histogram (resp. probability density) of returns of a bonus-certificate. A bonus-certificate can be replicated by an underlying and a down-and-out-put option. I tried to do that with Matlab but not by calculating the right PDE using conditional probability. Instead I've chosen a far easier way. I derive the wanted histogram (BC-histogram) from the histogram of stock returns. At first log-returns (that are normally distributed) are considered only. Then a transformation of log-returns to simple returns generates the BC-histogram of lognormally distributed returns. To achieve that following steps had to be made:

1. divide the probability density of stock returns in a certian amount of bins (so the histogram gets a certain bin width),
2. sum up all the bins that are between the bonus-level and the barrier-level and remove them from the generated BC-histogram,
3. mirror the bins at the point of the barrier to take into account that if the stock touches the barrier the investor gets the exact payment as someone would get by simply buying the underlying stock
4. and finally subtract the sum of the mirrored bins in point 3. from the sum of point 2. to get the bin height of the bonus bin.
5. In order to transform from log- to ordinary returns simply calculate the exponential of log-returns and subtract one.
6. Finally in addition to 5. subtract the premium payed for the down-and-out-put and divide by the Premium (in percent of the underlying stock) plus one to account for the decrease in return because of the option component of the bonus-certificate that had to be payed.

I've generated a plot that shows BC-Returns for various barrier- and bonus-levels. Could anyone please verify the validity of the plot and the steps I've mentioned above?