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I just tried to price the implied dividend for a few active, liquid options markets using current prices and I am not convinced my results are accurate.

I am using American options, and using the put-call parity relationship that exists for European options. I've seen that at-the-money (or near-the-money) options will give a pretty accurate description of implied dividends. If I cannot use put-call parity, what methods are use by practitioners to get an implied dividend?

I used an interpolated treasury yield curve for accurate interest rate values, and priced IDIV with $$IDIV = \text{Stock Price } - \text{Strike } \times e^{-rT} - Call(K,T) - Put(K,T)$$

For AAPL: expiry 2016-11-11 -0.040236 2016-11-18 -0.053026 2016-11-25 -0.061683 2016-12-02 -0.065252 2016-12-09 -0.076144 2016-12-16 -0.029923 2016-12-23 -0.100593 2017-01-20 2.660728 2017-02-17 0.092540 2017-03-17 0.131359 2017-04-21 0.263763 2017-06-16 0.538302 2017-07-21 0.613789 2017-11-17 1.193600 2018-01-19 1.352709 2019-01-18 2.295825 For SPY: expiry 2016-11-09 0.006997 2016-11-11 0.008535 2016-11-16 -0.000494 2016-11-18 0.006222 2016-11-23 -0.004294 2016-11-25 0.002909 2016-11-30 -0.006724 2016-12-02 -0.008246 2016-12-07 -0.016802 2016-12-09 -0.013155 2016-12-16 0.799113 2016-12-23 0.741128 2016-12-30 0.519134 2017-01-20 0.872681 2017-02-17 0.850424 2017-03-17 1.253229 2017-03-31 1.446670 2017-06-16 2.063210 2017-06-30 2.285904 2017-09-15 2.853458 2017-09-29 2.841766 2017-12-15 3.393382 2018-01-19 3.920152 2018-03-16 4.540356 2018-06-15 5.096783 2018-09-21 5.609085 2018-12-21 6.897434

These seem far enough off that it's not due to computational errors. What else do I need to account for when using American options to price the implied dividend.

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  • $\begingroup$ Just out of curiosity, what is implied volatility curve of AAPL near ATM options as expiration date increases... vs the same curve for SPY (which obviously has no dividends). Second thought: the last time I checked, weekly options traded poorly, especially for non-indexes, so that might be a factor. $\endgroup$ Commented Nov 7, 2016 at 19:09
  • $\begingroup$ SPY does have dividends. IV curve for both flattens as time goes on. I agree that there is a liquidity issue, but how do practitioners estimate implied dividend on less liquid stocks? $\endgroup$ Commented Nov 7, 2016 at 19:20

1 Answer 1

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There are 2 ways to do it. The good-enough way, and the complete and complex way.

The Good-Enough Way

Here you will convert to a situation where you can apply put-call parity.

Begin by finding the strike $K$ where put and call prices are closest to each other. This might not end up being the closest-to-the-money strike, but it will do.

Now run the following algorithm until it converges on your dividend rate $q$ to sufficient accuracy:

  • Begin by setting "equivalent" European prices the same as the American prices
  • Use a pricing algorithm for European options and put-call parity to estimate $q$
  • Use $q$ to find the implied vols $\sigma_{P,C}$ for the put and call in the American algorithm
  • Generate new "equivalent" European prices using $q$ and $\sigma_{P,C}$
  • Go to step 2

This won't be quite correct, since the effective tenor of American options is naturally somewhat less than European, but it will work amazingly well.

Complete and Complex

For a more complete solution, you need to have a volatility model, and a term structure of available option prices that goes beyond your tenor of interest. For example, your model might be that Black-Scholes European volatility looks like

$$ \sigma_{BS}(K, T) = \sigma_0 + \frac{\mu_1}{T}\log\left(\frac{K}{S_0}\right) + \frac{\mu_2}{T^2}\log\left(\frac{K}{S_0}\right)^2 $$

From this you must work out the math for local volatility, and write an American option pricer capable of using those local volatilities.

You then run a nonlinear optimizer to fit this model and your term structure of dividends to the entire option market via the pricing algorithm you wrote.

Final Caveat

Using put-call parity provides us with some rate $q$ such that

$$ F = e^{(r-q)T}(C-P) $$

This does not necessarily mean that $q$ is the dividend rate.

In fact it is comprised of three pieces

$$ q = \epsilon_r + b + \delta $$

which are

  • $\epsilon_r$: The difference between the interest rates you are using and market interest rates
  • $b$: borrow cost of the underlying
  • $\delta$: dividend rate

The borrow cost in particular is often very significant.

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  • $\begingroup$ Thanks Brian. Both of these methods require transforming into the volatility space and back. Is there any research along these lines that approach this problem "model-free"? $\endgroup$ Commented Nov 7, 2016 at 20:17
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    $\begingroup$ For having worked on the same topic for a while, I completely agree. Very nice answer @Brian B. $\endgroup$ Commented Nov 7, 2016 at 21:25
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    $\begingroup$ No model-free approach will exist. The only reason it exists for European options is that they are a very special case. If you had different European exercise options, say some ratchet options, you still would not have a model-free approach. American exercise and the random stopping time for option tenor just make the problem even further from model-free. $\endgroup$ Commented Nov 8, 2016 at 3:32
  • $\begingroup$ Using the first method, I can get accurate expected dividends for near-term liquid expirations. Once I approach the 1- or 2-year mark (even on SPY which is very liquid) I get results like those I posted. If I am doing a separate calculation that requires the expected dividend (to incorporate in $e^{(r - \delta )T}$) what is the best way to proceed? $\endgroup$ Commented Nov 10, 2016 at 17:31
  • $\begingroup$ Perhaps that is borrow cost. I have updated the answer. $\endgroup$ Commented Nov 11, 2016 at 13:34

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