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I'd say your question is answered here: How should I convert FX Volatility Surface from one base currency to another? As they say, you need to make correlation assumptions. For an intuitive explanation of why this is the case, there's a nice framework for understanding the connection between volatilities and correlations in cross rates which you can find ...

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EDIT: Apologies, one more edit, but an important one: Note, as kindly pointed out to me by an interested reader a short time ago: there is a potential issue with the simple model I proposed. Namely, as it stands the model implies that the illiquid asset$Y_t$ is not a martingale. But all is not lost; the model could potentially still be used if the illiquid ...

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What you are given is a linear combination in instruments and corresponding (benchmark) prices, what you need are invert the linear combinations to arrive at the benchmark prices back out implied vols from benchmark prices apply the vols to your new products. For the first step, I'd go as  \begin{align} Ax&=b\\ \begin{pmatrix}1&0&0&0&...

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This is a problem commonly faced by investment banks and buy-side firms (such as hedge funds) that deal in lots of derivatives. There isn't much more one can do than employ a few rules of thumb, and those rules have not changed much over the decades. In this case, those tricks look something like the following: First, let's assume you have your stock $S$ ...

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