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consider adjoint algorithmic differentiation to get an exact derivative here. Works especially well for monte carlo. Here is an example paper: http://luca-capriotti.net/pdfs/Finance/jcf_capriotti_press_web.pdf

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the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from the tiny number of paths within $\Delta S$ of the strike, and these paths have huge value. This is a well-known problem. As the bump size goes to zero, the ...

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It's a combination of too few sample paths and/or too small an increment. Your estimation error on the price is magnified by the $dS^2$. Try using a larger sample or a larger increment. Alternatively, you can use a multiplier instead of a fixed increment; in my experience, it usually yields better results.

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Consider an instrument value $f(S_0^1, \ldots, S_0^n)$ that depends on $n$ spot levels. Let $$\overrightarrow{S}_0=[S_0^1, \ldots, S_0^n]^T$$ be an $n$-dimensional vector representing the spot levels. We can approximate the cross gamma \begin{align*} \frac{\partial^2 f\big(\overrightarrow{S}_0\big)}{\partial S_0^i \partial S_0^j} \end{align*} by a finite ...

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Your portfolio composition is not clear. To simplify, we assume that it consists of units of a stock and options on this stock. What you can do is to sell 4000 units of options that will bring it to gamma neutral, and then to balance the delta, you can buy 2,400-450=1,950 units of the stock.

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