let's take delta; What can the delta of a Heston model be used for? I know it can used for hedging strategies, but can we say something about the market and the model by looking at the delta. Can we conclude anything by comparing the deltas of two models? Or other Greeks

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    $\begingroup$ It is important to understand what factors contribute to the movement in the price of an option, and what effect they have.Options traders often refer to the delta, gamma, vega and theta of their option positions.These terms may seem confusing and intimidating to new option traders, but broken down, the Greeks refer to simple concepts that can help you better understand the risk and potential reward of an option position. $\endgroup$ – user16651 Aug 9 '16 at 6:36

Delta measures the sensitivity of an option's theoretical value to a change in the price of the underlying asset. It is normally represented as a number between minus one and one, and it indicates how much the value of an option should change when the price of the underlying stock rises by one dollar.Indeed $$\color{red}{\Delta \approx \frac{U(S+dS,v,t,T,K)-U(S,v,t,T,K)}{dS}}$$ where $U$ denotes the option price and $v_t$ is the stochastic volatility.

Mathematically speaking

In the Heston model, we have $$\frac{\partial U}{\partial t}+\,r{{S}_{t}}\frac{\partial U}{\partial S}+[\kappa (\theta -{{v}_{t}})-\lambda {{v}_{t}}]\,\frac{\partial U}{\partial v}-rU+\\\frac{1}{2}{{v}_{t}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}+\rho \sigma \,{{v}_{t}}{{S}_{t}}\frac{{{\partial }^{2}}U}{\partial S \partial v}+\frac{1}{2}{{\sigma }^{2}}{{v}_{t}}\frac{{{\partial }^{2}}U}{\partial {{v}^{2}}}=0\tag 1$$ As you know $$\Delta =\frac{\partial U}{\partial S}\,,\,\Gamma =\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}\,,\,\rho =\frac{\partial U}{\partial r}\,,\,\Theta =\frac{\partial U}{\partial t},\\ \vartheta =\frac{\partial U}{\partial v},\operatorname{Vanna}=\frac{{{\partial }^{2}}U}{\partial S\partial v},\operatorname{Volga}=\frac{{{\partial }^{2}}U}{\partial {{v}^{2}}}\tag 2$$ $(1)$ and $(2)$ $$rU=\color{red}{\Theta} +\,(r-q)S\color{red}{\Delta} +[\kappa (\theta -v)-\lambda v]\color{red}{\vartheta} +\frac{1}{2}v{{S}^{2}}\color{red}{\Gamma} +\rho \sigma \,vS\,\,\color{red}{\operatorname{Vanna}}+\frac{1}{2}{{\sigma }^{2}}v\,\color{red}{\operatorname{Volga}}$$

Therefore the Greeks are so important.


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