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How about letting the FX rates remain fixed, and recalculate the portfolio volatility. That seems very obvious - am i missing something?

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Strictly speaking, indices such as the VIX are built to approximate the expected variance (of log-returns) that would effectively realise under a pure diffusion setting (i.e. no jumps) $$\frac{dX_t}{X_t} = \mu(t) dt + \sigma(t,.) dW_t^{\mathbb{Q}}$$ Writing out the equations (*) yields the famous static replication formula in terms of strike-weighted OTMF ...

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You are on the right track IMO, except that there are no exceptions to "the greater the expected value, the higher the option value" rule, since an option premium's is precisely the discounted expectation of its payout by absence of arbitrage - at least under the risk-neutral measure. As far as the influence of volatility on the option value is concerned: ...

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it certainly works best at the money. Why? I think it comes from the fact that Black's formula is approximately linear at the money. The approximation $$\frac{1}{\sqrt{2\pi}} \operatorname{SR} \sigma \sqrt{T} A,$$ with $A$ the annuity is remarkably good. One way of deducing these formulas is to do an asymptotic/Taylor expansion about $\sigma=0.$

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There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More Mathematical Finance among other places. It all started with the work of Anthony Neuberger on the Log Contract in 1994, which seemed to be a theoretical result ...

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Exploiting an arbitrage is straightforward. Constructing and noticing one is the hard part. In your case if you know that Swptn(K,T1,T2)+Swptn(K,T2,T3) >= Swptn(K,T1,T3), Simply sell Swptn(K,T1,T2)+Swptn(K,T2,T3) and buy Swptn(K,T1,T3). Sell the most expensive and buy the cheapest. L.

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