We're talking about how we price every financial instrument: by discounting the payoff, that is, we take future cash flows and we discount them by a proper rate which takes into account the risk of not receiving those flows. I was thinking about these two points:
- to keep it simple, when we're dealing with unknown future cash flows (e.g. floating rate notes) we use forward prices to be consistent with an arbitrage-free setting;
- Put-Call parity yields implied dividends paid by the underlying in the future.
Let you have an array of implied dividends coming from Put-Call parity: these are the future cash flows paid by the underlying according to the arbitrage-free rules for which Put-Call parity holds true.
So if the present value, $P(0)$, of any security is the discounted sum of its future cash flows on date $t$, $c(t)$, here we could write
Aside from typos, last cash flow is the terminal value and $d(t)$ terms are the implied dividends paid on date $t$. If we knew $k(t)$ term structure, we would know something like a yield term structure for the underlying risk: unfortunately, we don't have that. However, we have $P(0)$.
- does it make sense to solve for $\bar k$ like we do for vanilla bonds yield to maturity? This method is currently used by practitioners for everything which it shouldn't be used for: variable rate notes, callable bonds and so forth. Then I feel authorized to do it.
- What would this $\bar k$ represent? I would say it has the same properties of the yield to maturity: it is the return of the security only under very irrealistic assumptions... nonetheless, it's useful for comparisons and rankings purposes.
- Why isn't this model consistent with arbitrage-free valuation? Today I buy shares at $P(0)$ and I hedge my position by building (short) parity positions: I sell Call options and buy Put options struck at the same price. I get dividends from the underlying shares, same dividends that I "lose" when my short Calls expire and I have to deliver.
Any thoughts about the reason for which I'm wasting my time by building a screener for such "implied dividend yields"?