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I don't think there is a closed formula for the non-perpetual American option greeks. You'll have to compute them numerically or approximate them just like the option price itself. http://www.wilmott.com/messageview.cfm?catid=34&threadid=65417


Note that the vega you derived is called the local vega. However, the vega people usually called is the term vega which is the sensitivity with respect to the changes in the Black's implied volatilities, for which you will need to have the relationship between the local volatilities and the implied volatilities.

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