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9
Here couple pointers that may make it clearer:
Drift can be replaced by the risk-free rate through a mathematical construct called risk-neutral probability pricing.
Why can we get away with that without introducing errors? The reason lies in the ability to setup a hedge portfolio, thus the market will not compensate us for the drift above and beyond the ...
2
Being on the sell side and selling options you can intuitively think of it like this:
An option is like any other product that is being produced out of ingredients and because of the competitive situation of the producer this is done by the cheapest possible production process.
The ingredients are in a simple (Black Scholes) setting a stock and and a risk ...
2
$$dS / S = \mu dt + \sigma dW \\
\\ dS / S -r dt= \mu dt - rdt + \sigma dW \\
\\ dS / S -r dt= [\frac{(\mu - r)}{\sigma}dt + dW]\sigma \\$$
Then, Girsanov tells us that, as long as the risk premium is bounded from below, we can write
$[\frac{(\mu - r)}{\sigma}dt + dW]\sigma$ as $\sigma d\tilde{W}$ where $\tilde{W}$ is simply another brownian motion with ...
2
There are essentially two approaches you can take:
Approximate changes in IV by establishing a relationship between IV and option prices through a function of IV solely dependent on option price. While it is computationally very convenient it introduces huge estimation errors in certain cases. As pointed out in my comment above one such case is a slide in ...
2
If you can get anywhere close to the same open-interest and volume using European options as the corresponding American ones, you'll have a much easier time just using them.
American options with high probability of early exercise don't contain information about that back end of the vol surface, and it's kind of hard to decide just what to do with their ...
2
Here is a paper by the infamous Mark Rubinstein that should get you started.
http://www.haas.berkeley.edu/groups/finance/WP/rpf232.pdf
And here the trinomial tree version:
http://www.ederman.com/new/docs/gs-implied_trinomial_trees.pdf by no lesser than Derman and Kani.
This may also help with the actual computations:
...
1
You don't need an algorithm to solve that - just program a simple BS option calculator using standard BS with dividend in Excel and fix all the inputs except the volatility. Then use goal seek/solver to change the volatility to get the given price and as a result you will have the implied volatility of the price.
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