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2

Your vector $a=(1,\ldots,1)^T$ does not satisfy $$\| a \|^2=a_1^2 + \ldots + a_m^2=1,$$ as assumed by authors in Result 2. (Q1) For $m=2$, we see it is needed when computing the eigenvalues of $I_2 - aa^T$, that is the roots $\lambda$ of equation $$0=\det \begin{pmatrix} 1-\lambda- a_1^2 & -a_1 a_2 \\ -a_1 a_2 & 1-\lambda- a_2^2 \end{pmatrix} = (... 1 If i understand this correctly, you want to be able to infer a future volatility surface, given the current simulation parameters you have. What you're essentially trying to do it include the modelling of forward vol/skew in your MC. Getting the forward vol surface vaguely correct is quite important to price some types of derivative - i.e. anything that has ... 2 Use the risk free rate for pricing You use the risk free rate (using the risk neutral measure \mathbb{Q}) so that you can use the formula$$ V(t) = \underbrace{\exp(-r(T-t))}_{\text{because we used $\mathbb{Q}$}} \mathbb{E}^{\mathbb{Q}}(P(S_T)),  where because we used $\mathbb{Q}$ we were able to discount the expectation after doing all the MC ...

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Based on Quantuple comments (thank you), I fixed many mistakes and I came up with the following code: import numpy as np import scipy.stats nb_simuls = 5_000_000 # parameters from https://file.scirp.org/pdf/JMF_2014050615380663.pdf Table 1 / cell 1 S1, S2, S3 = 50, 60, 150 v1, v2, v3 = 0.3, 0.3, 0.3 rho_12, rho_23, rho_13 = 0.40, 0.20, 0.80 K = 30.0 T = ...

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Monte Carlo is more natural to perform a forward induction (think TARNs), whereas trees are more natural to do a backward induction/dynamic programming (think Bermudans). Forward induction may be the way to go in case you have a trade that is path dependent (i.e. The price at a time depends on the past history of the process (in the sense that at a ...

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