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First, please make sure that when you resimulate sample paths, you are keeping your underlying random samples constant, as in this answer. For your delta, vega and rho there is some ambiguity in the definition of the greeks. Consider the simple case of delta in the presence of a skew $\sigma(K/S)$, and say that the underlying price right now is $S_0$. We ...

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In the Merton jump diffusion model, the stock price process consists of a continuous part and a discrete part (this one represents the jumps). While deriving the PDE for the riskless portfolio and imposing the riskless evolution, the discrete part can't be instantaneously hedged. In fact, you can assume that the effects of jumps can be nullified on average, ...

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Think of moving volatility in the other direction. As volatility approaches zero, any call strike strictly smaller than the ATM strike, $K<K_{ATM}$, will have zero probability of ending in the money, and the corresponding option value will be zero. An infinitesimally small change in stock price will not move $K$ past $K_{ATM}$, so the option value ...

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Victor123, let's start from $\Delta$. This is the expected change in the price of an option if the underlying asset moves by a currency unit, say 1 USD. For the case of a call option, the Delta varies between 0 and 1. Everything else been equal, the Delta of OTM calls will approach to 0 as the price moves out of the target barrier. Conversely for the case ...

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I think that the question in its current form is not really precise. The Greeks are quite easy to define for a given formula of a price: in that case they are simply partial derivatives, and of course linear (and in some case commute with integrals rather than just finite sums). However, they are crucially model-dependent - not only their expressions, but ...

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Well if we take a call option the gamma is non-zero. If we take the replicating portfolio for a call option, it consists of stock and bonds. Both of these have zero gamma. So in the form asked, I think the result is false for second derivatives.

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if the pay-off is continuous, the standard approach is to use the path-wise method also known as IPA. This essentially means that you differentiate along each path. It is the limit as the bump size goes to zero of finite differencing. The main downside of this method is that the differentiation can be fiddly and slow. The Smoking adjoints paper you mention ...

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