Estimating Implied Dividends from American Option Prices
Summary
The document explains why applying European put-call parity directly to American option prices can produce implausible implied dividend estimates. It presents a practical approximation: choose a strike where call and put prices are close, treat American prices initially as equivalent European prices, estimate a dividend yield with European pricing and parity, then iterate by recalculating implied volatilities and equivalent European prices until the estimate stabilizes. The method is described as useful in practice, though not exact because American options have early-exercise features and therefore an effective tenor that differs from the European case.
For a more complete estimate, the answer proposes fitting a volatility model and dividend term structure to a broader set of option prices using an American pricer and nonlinear optimization. It also emphasizes that the yield inferred from parity may combine interest-rate error, stock-borrow cost, and dividends. Borrow costs can materially affect the result, so the parity-implied quantity should not automatically be interpreted as the actual dividend rate. The exchange gives methods and caveats but no validation against observed dividend payments.
Key ideas
- European put-call parity does not apply directly to American option prices.
- An iterative European-equivalent price procedure can provide a practical dividend-yield estimate.
- A fuller approach fits a volatility surface and dividend term structure to American option prices.
- The parity-implied yield can include rate differences and underlying borrow costs as well as dividends.
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Full text
# Implied Dividend from American Options (in practice)
# Implied Dividend from American Options (in practice)
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 by Brian B (score 10, accepted)
https://quant.stackexchange.com/a/30904
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.