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First of all I would not recommend the $M\cdot t$; it is fragile to the choice of $t_0$, isn't it? Nevertheless your specification seems to be close to the one of a linear regression (if $\epsilon$ is Gaussian and you metric is the Mahalanobis' one): just organize your dataset as a nice matrix and perform a linear regression.

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Have you solved it yet? For example in the drift parameter, the dt needs to be vector of time from 0 to 1 by dt. My code is: GBM<-apply(BM,2,function(x) 100*exp((cumsum((r-0.5*sigma*sigma)*time)+sigma*x))) where I'm using GBM on already cumsummed Brownian Motion (x).

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Have you look at copula package! Maybe you could get ideias from it https://www.jstatsoft.org/article/view/v021i04/v21i04.pdf http://finzi.psych.upenn.edu/R/library/copula/html/copula-package.html

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You can have a look at Andrew Patton's "Copula toolbox for Matlab". It contains his code for the "Time-varying Symmetrised Joe-Clayton copula".

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In order to compute $$P_0 = \mathbb {E}[C (\hat{V})]$$ where $$\hat{V} = \frac {1}{T} \int_0^T \sigma^2_s ds$$ and $$d\sigma_t = \sigma_t (\alpha dt + \gamma dW_t)$$ using Monte Carlo, you should: Generate stochastic volatility paths over $[0,T]$ by discretising the above SDE (which here defines a GBM, not a Hull & White diffusion) Calculate the ...

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