PPPN: participation rate, stocks and premium

I'm a student of financial engineering and am very new to all of this stuff. Now, I'm trying to make an "example of a beginners exercise", but alas, I don't have any clue on how to solve or even on how to begin this one. The exercise goes like this:

Suppose you have a PPPN where the invester recieves at maturity date $80 \%$ of his investment plus a premium, defined by: $$p \cdot N/S_0 \cdot (S_T - S_0)^{+},$$ where $(S_T - S_0)^{+} = \max(S_T - S_0,0)$ is the positive stock return over the period $[0,T]$ ($t=T$ is the time to the Maturity date), an investment $N$ and where $p$ is the participation rate. Now, set $p$ such that the product is attractive for investors and you have a certain margin.

In order for this exercise to get more real, I've chosen a stock at random, say Facebook, and assumed a maturity date of 13/12/2016. Here is the information of the stock found today, credits to yahoo finance:

I thought it would be wise to choose $N= S_0= 102.12$, so that the equation of the premium simplifies to:

$$p \cdot (S_T-102.12)^{+}.$$

Unfortunately, that's where my insights end. I don't have any clue on how to make further progress on this problem. Personally, I would just set $p =100 \%$, so you get the maximum possible return, but that can't be right. Any ideas/pointers/solutions?

In general, PPN is the short form for principal protected notes. Here, the principal, or notional, $N$ is generally return in full. I am a little confused why only 80 % is returned. It may be a contractual specification, and it is also called a PPN.
Regarding the variable interest, or premium in your term, is the return that the investor will achieve. In your specification, the variable interest is defined by \begin{align*} p N/S_0 (S_T-S_0)^+ &= pN \left(\frac{S_T}{S_0} -1 \right)^+. \end{align*} Note that $\left(\frac{S_T}{S_0} -1 \right)^+$ is the positive part of the return. Here, $p$ is usually called the participation rate, and $N$ is the notional.
In summary, this PPN will return 80 % of the notional $N$ to the investor, and will pay the investor a ratio of $p$ of the final equity return if it is positive. The value of this PPN is the sum of the value of a zero-coupon bond plus a ratio of a European option. The participation rate $p$ can be determined so that the PPN value is higher than the value for a direct deposit into a banking account.
• Ok, so far so good, thanks. But how does one determine the value $p$ and moreover, since $S_T$ is unknown, how do you even begin to tackle this problem? – Riley Dec 13 '15 at 18:09
• Using the market volatility quote, you can value the option with payoff $(S_T-S_0)^+$ using Black-Scholes's formula. For $p$, it is based on business judgment, depending on what you want to do with your client, but that is out of my knowledge. – Gordon Dec 13 '15 at 18:25
• In my product, I want to make us of the Put-Call Parity formula. So I need the interest rate. I was wondering, because the maturity date is in more then over a year, which interest rate $r$ do I have to choose? – Riley Dec 16 '15 at 17:40