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For a mathematical model you can have a look at this paper: The Valuation of Compound Options by Robert Geske where after equation (17) it is shown that $\partial \sigma_s/\partial S<0$.


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If $R$ and $r$ are the return on the portfolio after currency hedging and on the currency, if I write $V(\cdot)$ for variance, and a fraction of $t$ of the portfolio is exposed to currency risk, then the return of the unhedged portfolio is $R+tr$. Then: $$V(R+tr) = V(R) + 2t\mathrm{Cov}(R,r) + t^2 V(r)$$ so the marginal contribution (derivative with respect ...


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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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There are tons of market where vol smile doesn't exist - either because no one makes a market on the call/put options (private equity, physical real estate comes to mind) or only the ATM option gets traded infrequently. You can't have volatility smile without a vol market. On the other hand (and maybe more relevant to what you are trying to get at), if ...


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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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The motive is indeed to construct a constant gamma portfolio. A position in a VVIX portfolio replicates the volatility of VIX forward prices. VVIX portfolio prices have usually been at a premium relative to future realized volatility. The discount is a volatility risk premium. For nearby expirations, these prices have also tended to surge at the same time ...


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Historically, there has been little correlation between the VVIX and the VIX except at extreme values of the VIX. You are correct it will definitely lead to different greeks, but it does not matter at lot as your Portfolio objective would be fulfilled as methodology for both are same.


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If a USD based investor owns shares of Toyota Motor in Japan, the variance of USD based returns is approximately equal to the variance of Toyota in yen, plus the variance of USDJPY plus twice the covariance between Toyota and the exchange rate. The last term could be positive or negative; if I had to guess for a big exporter like Toyota it is probably ...


2

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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As you can tell, there are many ways to estimate volatility (standard deviation, range, etc.). What is better or worse depends on the use-case. What all volatility estimators have in common are that they try to measure variability. If this is for trading strategy development, you'll probably want to backtest a variety of methods to see what works best. Some ...


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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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Beta is volatility in relation to a benchmark whereas Standard Deviation is volatility in relation to actual returns vs expected returns


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Let $$q (S) := \frac{d\mathbb {Q}(S_T \leq S)}{dS} $$ denote the probability density function of the stock price at time $T>0$ under the risk-neutral measure. By definition, the price of a European call then writes \begin{align} C (K,T) &= P (0,T) E_0^{\mathbb {Q}}[(S_T-K)^+] \\ &= P (0,T) \int_K^\infty (S - K) q (S) dS \end{align} with $P ...



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