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:

  1. 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;
  2. 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 $P(0)$.

My questions:

  1. 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.
  2. 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.
  3. 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"?


2 Answers 2


Since you have not defined a probability measure for $\mathbb{E}\left[d(t)\right]$, I don't think your model can qualify as risk neutral.

In order for a discounting model to qualify as risk neutral, it must define a distribution or measure. In quant finance parlance, a risk free measure exists only if one can:

a) define a probability measure such that the price is equal to the expected net present value (i.e., equivalent martingale measure) which is absolutely continuous with respect to the original measure (e.g., via Girsanov's theorem); and,

b) demonstrate that the chosen probability measure cannot be incorrect (via the dynamic hedging argument and/or the fundamental theorem of asset pricing).

It seems that what you are actually trying to value qualifies as an annuity, which falls under the umbrella of actuarial science rather than quantitative finance. Generally, actuarial sciences deal with real world probability measures since it is difficult to infer the concrete relationship which generally hold for other financial derivatives, such as:

  1. Contingent payoff conditions
  2. Probability measure for the underlying of dividends (profits?)
  3. Arbitrage relationship
  4. Terminal conditions such as time
  5. Boundary conditions such as strike

As such, valuation models for annuities, equity (e.g., DCF and DDM), and other underlying assets are usually taken with regard to real world measures. For a better explanation of real world versus risk neutral measure, please see this Wiki.

If you are willing to forego a strict interpretation of the annuity model, Samuelson and McKean (1965) provide a closed form approach for pricing American-style warrants which are analogous to pricing under the DDM. Also, the Merton Model (1974) proceeds to value corporate liabilities in the Black-Scholes world. In fact, Moody’s flagship model, the Kealhoffer-Merton-Vasicek (KMV) model, is basically a snazzier version of Merton.

However, if you want to get into the wonky actuarial math, there have been a number of papers on stochastic cash flows and stochastic annuities. Many of the foundational works, in my opinion, are from Daniel Dufresne and Mark Yor. Also notable is: (Moshe Arye Milevsky. The present value of a stochastic perpetuity and the Gamma distribution Feb 1997).

My key insight is that quant finance values contingent claims only by assuming an integrated measure of underlying asset value is the right one whereas actuarial science may attempt to derive the integrated asset value from the cash flows themselves.

I have hunch that the chasm between these two worlds can be bridged through path integral approaches which were developed for quantum mechanics. The main difference is that the integration occurs pathwise in path integration whereas the foundations of modern quant finance (Ito calculus) proceeds as the limit of Riemann sums with finite quadratic variation. Indeed, promising inroads have been made (Devreese, Lemmens, and Tempere 2009).


In regards to 1 & 2, I think k(t) in this case represents more of the spot rate than the YTM. If you are assuming that the term structure is static, then k(t) can be interpolated for n periods of discounting dividends, but this is not realistic. I think this is ultimately why your model is not consistent with arbitrage free valuation.


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