Estimating Bonus Certificate Returns from Stock Return Distributions
Summary
The document proposes estimating the return distribution of a bonus certificate from the distribution of the underlying stock’s returns. The certificate is described as a combination of the underlying and a down-and-out put. The suggested bin-based procedure starts with stock-return probabilities, removes probability associated with outcomes between the bonus level and barrier, reflects selected bins around the barrier to represent barrier-triggered outcomes, and uses the remaining probability to set the bonus outcome’s bin height. It then converts log returns to simple returns and adjusts for the option premium.
The author asks whether this construction is valid and requests comparison with a probability-density calculation based on a partial differential equation. No answer, validation, or comparison appears in the document. The proposed bin manipulation is therefore a hypothesis rather than an established method; it does not show how path-dependent barrier crossing is captured, how binning error is handled, or what assumptions govern the certificate payoff and premium. The plot is mentioned but its data are not supplied.
Key ideas
- The proposed method derives certificate returns from the underlying stock’s return distribution using bins.
- It treats the certificate as an underlying position combined with a down-and-out put.
- The bin procedure adjusts probability around the bonus level and barrier, then converts log returns to simple returns.
- The method’s validity is not confirmed, and the document provides no PDE comparison or evidence.
- Barrier crossing and discretization assumptions are not explained in enough detail to reproduce or assess the estimate.
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# Probability Density of Returns of Bonus Certificates # Probability Density of Returns of Bonus Certificates Could anyone please help me with the following? 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: - divide the probability density of stock returns in a certian amount of bins (so the histogram gets a certain bin width), - sum up all the bins that are between the bonus-level and the barrier-level and remove them from the generated BC-histogram, - 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 - 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. - In order to transform from log- to ordinary returns simply calculate the exponential of log-returns and subtract one. - 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? Link to the plot: http://s7.directupload.net/images/121016/x37oi9ro.png Perhaps someone could post a plot he has generated using his own solution, like PDE-solution for instance. I would like to compare the results of the plots especially for a given bonus-level of 1.22 (i. e. 222% of current underlying price) and a given barrier-level of -0.01 (i. e. 99% of current underlying price). Any useful help will be greatly appreciated.
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