Finding Implied Volatility with an Options Pricing Model
Summary
The document discusses how to infer volatility from an observed option price when the pricing model can handle American exercise and dividends. Its central numerical method is bisection: choose lower and upper volatility bounds, price the option at their midpoint, then replace the bound on the same side of the market price. Repeating this narrows the volatility interval until it meets a chosen tolerance. This approach can use an existing option pricer, including a binomial tree, without requiring a closed-form solution for implied volatility.
The responses also point to research on implied-volatility methods for binomial and trinomial trees, and to QuantLib as an implementation reference. A spreadsheet goal-seek approach is suggested for a Black–Scholes model with dividends, though that does not itself address American early exercise. The discussion gives no benchmark or implementation details, and bisection depends on suitable bounds and a price that the model can attain. It is a practical overview of solution approaches rather than a full treatment of model assumptions or numerical edge cases.
Key ideas
- Implied volatility can be found by repeatedly repricing an option while adjusting volatility.
- Bisection narrows a volatility bracket according to whether the model price is above or below the observed price.
- The pricing routine can be a binomial or trinomial tree that accommodates the contract features.
- A spreadsheet solver or a library implementation can provide an alternative route, subject to model fit.
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Full text
# How to calculate the implied volatility using the binomial options pricing model # How to calculate the implied volatility using the binomial options pricing model I want to calculate IV for american options with dividends. So far I have found algorithms to calculate the option price given a volatility. Please can you point me to paper or implementation (R, python or any other language) of an algorithm that can calculate the IV given option prices, risk free rate, dividends, etc. ## Answer by Matt Wolf (score 5, accepted) https://quant.stackexchange.com/a/8237 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: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2008-044.pdf ## Answer by Olorun (score 1) https://quant.stackexchange.com/a/8235 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. ## Answer by Vince (score 1) https://quant.stackexchange.com/a/8236 I tried to answer this in the comments but it got too long. simplest approach would be to guess a low and high volatility that is guaranteed to envelope the one to solve for. then compute the corresponding options prices at each of these guesses using your pricer. then while the difference between your guesses (the low/high volatility) is greater than some specified espilon, compute the price of an option at the average of your two guesses. Now adjust either your low volatility guess or high volatility guess depending on whether the price of the option at the average volatility is greater than or less than the price you are given from the market. This will then allow you to push up your low volatilty guess to where the average of the guesses was or push down your high guess to the average. Thus you bisect and iterate, and ultimately your two guesses are equal and given you the price of the option given from the market. This is the simple algo in so many words. ## Answer by unclepaul84 (score 0) https://quant.stackexchange.com/a/8323 Checkout QuantLib. It has an implied volatility calc. https://github.com/lballabio/quantlib/blob/master/QuantLib/ql/instruments/impliedvolatility.cpp?source=cc
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