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You would simply calculate the prices of various strike options using your parameters, then calculate the black scholes implied vol of each option. Did I miss the point of your question ?

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Indeed parameters are selected so that the quoted option prices are as close as possible to the model option prices. Alternatively, quoted and model implied volatilities can be used instead of prices.The first category are those that minimize the error between quoted and model. The second category,are those that minimize the error between quoted and model ...

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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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Hint By application of Ito's lemma, we have $$d(e^{kt}v_t)=\kappa e^{\kappa t}v_t\,dt+e^{\kappa t}dv_t+d(e^{\kappa t})dv_t$$ therefore $$v_t=v_0e^{-\kappa t}+\theta(1-e^{-\kappa t})+\sigma\int_{0}^{t}\sqrt{v_s}e^{-\kappa(t-s)}dB_{s}^{v}+\int_{0}^{t}e^{-\kappa(t-s)}J^v\,dN_{s}$$ $J_v$ is random jump size occurring at time $t_i$ and $N_t=N_t-N_0$ is the ...

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