Digital option delta

The delta in option pricing, also called the hedge ratiois expressed as the sensitivity of the option price to digital underlying price change. Digital analytical solution for the most common option pricing option, such as the Black-Scholes, Corrado and Su, and other frameworks can be found delta the internet or option books. However, I am dealing with a more complex model delta which the analytical solution is not that obviousand hence therefore want to obtain the delta and later also the other greeks by means of a numerical method.

More specific, when simply calculating the gradient of the Delta price with respect to the underlying Spot price, I get different values than from the analytical solution -- In case of the Black-Scholes model. Can someone explain digital the gradient does not equal the delta and what the numerical alternatives are for this issue?

As you option one way is to calculate delta by an analytic formula, i. The second way is to do it numerically, i. Which method is delta best very much depends on the dynamics of the delta surface and many digital have to be taken into consideration here. This is also related to the so called Sticky Delta vs.

For a good overview and introduction see here: Laughter in the Dark: An Digital to digital Volatility Smile by Emanual Derman esp. It may be the case with certain exotics option greeks are derived analytically through approximations.

In that case digital certain boundaries you may get different results from such approximation delta the numerical delta. Why do you not approach the numerical case similarly than most banks and hedge funds when they "shock" their options books: Simply shift your underlying, re-calculate the option price, derive a convexity adjustment factor, and from both approximate your delta.

More specific, when simply calculating the gradient of the Call price option respect to the underlying Spot price, I get different values than from the analytical solution -- In case of the Black-Scholes model Can someone explain why the gradient does not equal the delta and what the numerical alternatives are for this issue?

In this case numerical derivative might give result very different from analytical digital. Could it be your case? I never used gradient delta to option derivatives manually. When I try it it comes close option the blsdelta.

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