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American options Implied Dividend
1. Project objective:
The purpose of this project is to extract implied dividend from American options and
quantify the impact of de-Americanization.
Implied dividend for European options can be calculated analytically using Put-Call
parity and the below formula.
However, most US single stock options are generally American. So you need to
convert American prices to European (de-Americanization) before you can use putcall parity to imply dividends.
3.1 Starting with XOM and T (both have very liquid option markets), I’d like you
to back out market expected (option implied) dividend yield for 2021.
3.2 You can use options with approximate 1-year expiry for this exercise.
3.3 You can ignore short borrow cost (both these names are easy to borrow),
and compare to current dividend yields.
3.4 You can assume flat interest rate curve (constant r).
3.5 Using option data, apply de-Americanization technique (see the binomial
tree models in the reference paper) to get European equivalent option prices
3.6 Assuming interest rate r is flat, apply put-call parity to back out implied
3.7 Now assuming interest rate and volatility unknown. From step 3.5, you
should have de-Americanized option prices for different strikes. Perform
calibration (optimization) to find out the implied volatility (σ), implied
dividend (q) and interest rate (r).
Attached data includes option (with approximate 1-year expiry) prices, spot prices,
strike prices for all stocks in S&P 500.
6. Notes for Calibration:
In this framework, instead of solving directly for σ* you solve for a, b (linear fit – it is
a, b, c in a quadratic fit) in σ*(a,b) as well as r and q simultaneously across a bunch of
ATM options (if they are ATM then a linear skew fit will be better as skew exhibits
OTM convexity). If you do this then you will have a sensible shape for the skew,
which enables a fit across multiple option strikes, as well as the desired values for r
and (at the end of the rainbow) q. In the Liu paper they use a neural network to do
this which sounds like it is an efficient way of solving the problem at industrial scale
but you could just use a solver and an off the shelf American option pricer.
In other words, you solve simultaneously with a bunch of options for the fair implied
skew (as a function of strike) and the interest rate and the dividend. You need a
smooth low dimensional parameterization of skew (linear or quadratic) plus the
other two variables, meaning you have to run a non-linear solver to minimise fit
error on your fitted option prices as a function of those 4-5 (linear-quadratic)
variables. Means you ideally need say 10 option prices to have a robust fit.
Rather than a single parameter for implied volatility, you use a function of the
unknown skew parameters.