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1. weighted Milstein Scheme We assume $\{X_t\}_{t\geq0}$ described by the following stochastic differential equation $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ Under the Ito version of this scheme Equation $(1)$ becomes $$dX_{t+\Delta t}=X_t+[\alpha\,\mu(t,X_t)+(1-\alpha)\mu(t+\Delta t,X_{t+\Delta t})]\Delta t+\sigma\sqrt{\Delta t ... 0 Girsanov'Theorem let \theta_t be an adapted procee such that the solution of SDE$$dL_t=-L_t\, \theta_t \,dW_t , \, L_0=1$$is a Martingale.We set Q{{|}_{\mathcal{F}_t}}=L_t\,P{{|}_{\mathcal{F}_t}},then$$W_{t}^{Q}=W_{t}^{P}+\int_{0}^{t} \theta_s\,ds is a standard wiener process under Q measure. Result Now we assume $\{S_t\}_{t\geq0}$ be a ...

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$r-\frac{\sigma^2}{2}$ for the drift only applies to the log-returns. The Euler discretisation simply discretises the SDE directly. You'd use the risk-free rate for you drift under the risk-neutral measure for your question. For your reference: Please read the wikipedia for more details.

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I have approximate the integrals by Monte Carlo Method but you can use several method such as Newton-Cotes formulas and Gaussian quadrature. Function Example Solutions Call = 34.0976 Put = 4.8941 Parameters were extracted from Jianwei Zhu(2008),Page 10,Table 4

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Well, log-normality does not allow you to have negative earnings and companies do have negative earnings. I suggest you to download the earnings data and perform a Jarque-Bera test for normality.

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