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I am going to supply an answer that is quite similar to SRKX's (which is very very good) because I want to discuss in more detail a few important things. First, you cannot use a stochastic volatility model for the SDE that you've provided as that's GBM with constant diffusion. However, based on what you've said it's obvious you wish to model a discretized ...

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I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as $$\frac{dS_t}{S_t} = \sqrt{v_t}dW_t$$ $$dv_t = -\kappa(v_t-\eta)dt + \theta \sqrt{v_t}dZ_t$$ with $d\langle W,Z\rangle_t = \rho dt$ and $v_0 = \sigma_0^2$. Using Itô's lemma we can derive  \log\left(\frac{S_t}{S_0}\right) = \int_0^t ...

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For pricing, there are a few products whose prices are sensitive to the forward smile and when you compute that with just local vol, it is not realistic. So if you are a seller, you go to the next church and find something that looks kindof reasonable, and that kind of can reconstruct a reasonnable forward smile structure. The game in pricing is to not ...

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I am not an expert in this field, but it would be best to consider a full multivariate GARCH model. This paper by Engle and Sheppard should be a good start. I think the constant correlation matrix approach is covered to a certain extent too. I hope this helps.

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Consider the following analogy: you can hedge a derivative in a deterministic-volatility model using either futures, or spot underlying. The hedge ratio will change, but all the mathematics to effectively eliminate stochastic portfolio PL is the same, and must work out to be equivalent. A similar situation applies here: any triangle of (nontrivial) ...

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Dealing with model error under stochastic volatility (in a more formal way) you could use the UVM (Uncertain Volatility Framework). Here are what i think are the most seminal references: Avellenada et al (1995) Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3736 ...

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