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Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, and then discuss how SLV tries to improve things. Although there are many stochastic vol models, I limit the discussion here to the Heston model to keep things ...


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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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"match the prices of vanilla options" It means you need to reprice all vanillas with your LV model using monte carlo sims. If all prices are exactly equal to the market prices, your LV model is well calibrated I think there is a typo in your formula, it should be $$S_{t+1}=S_t\ exp((r-\frac{\sigma(S_t,t)^2}{2})dt+\sigma(S_t,t)\sqrt{dt}N(0,1))$$ &...


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The local volatility graph tomorrow doesn't change, unless the implied volatility surface tomorrow is not the same as today. LV takes the implied vol surf today as input, and outputs a instantaneous volatility function of spot and time, which can price vanilla options today exactly the same as the market prices. In local volatility world, you assume the ...


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