Implied Dividend from American Options (in practice)

Question

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.

Answer 1

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.