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I think what you are missing is simply the Vega-Gamma relation in the Black-Scholes model. Namely: $$Vega = \frac{\partial v}{\partial \sigma} = \sigma(T-t)S^2 \frac{\partial^2 v}{\partial S^2} = \sigma \tau S^2 \Gamma$$ Plugging this into your coverage error, you get its expression in terms of the Vega which is the most natural measurement of your ...

3

Since the bonds are independent we have one of three things that can happen (1) With probability 0.98*0.98 both bonds lose 1 Million, the total loss is 2 Million (2) With probability 2*0.98*0.02 one bond loses 1 million and the other 10, for a combined loss of 11 million (3) With probability 0.02*0.02 both bonds lose 10, overall loss 20 Now we need to ...

3

Note that \begin{align*} f(m) &= \int d_{\gamma}(m,x)\mu(dx)\\ &=\int \big[\gamma(m)-\gamma(x)-\gamma'(x)(m-x)\big]\mu(dx)\\ &=\gamma(m) - \int \big[\gamma(x)+\gamma'(x)(m-x)\big]\mu(dx). \end{align*} Then, \begin{align*} \frac{df}{dm} = \gamma'(m) - \int \gamma'(x)\mu(dx), \end{align*} and the critical point is given by \begin{align*} b = \big(\...

3

Well, if you assume $X$ has volatility $\sigma_X$ and $Y$ has volatility $\sigma_Y$, then $$\sigma_{X+Y} = \sqrt{ Var( X + Y) } = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho }$$ Then, you want to show $$\sigma_{X+Y} = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho } \leq \sigma_X + \sigma_Y$$ Squaring both sides: $$\sigma_X^2+\... 2 Insofar as a standard exists, it would be a Sharpe ratio from monthly returns using arithmetic rather than log returns. As a rule of thumb, arithmetic returns should always be used in any kind of reporting since log returns are an approximation (and one we tolerate for ease of use despite being slightly off, particularly for larger moves). Also return is ... 2 The question isn't simply answered and the short answer is it depends on a number of factors. The GIPS standard for investment managers is the only performance reporting standard AFAIK and it can be found here: https://www.cfainstitute.org/en/ethics-standards/codes/gips-standards/firms There are certain rules and requirements depending on the frequency you ... 2 E(X|X\geq b)=\frac{\int_b^{\infty}X dP}{P(X\geq b)}=\frac{\int_b^{a}X dP+\int_a^{\infty}X dP}{P(X\geq b)} \leq \frac{a\int_b^{a} dP+\int_a^{\infty}X dP}{P(X\geq b)}=\frac{a\int_b^{a} dP+\int_a^{\infty}X dP}{\int_b^{a} dP + P(X\geq a)} Now since a \leq \frac{\int_a^{\infty}X dP}{P(X\geq a)}, the right hand side of the equation above is smaller than or ... 2 This is a not a theoritcal/academic answer relating the two by an equation. But from a practicioners stand point. The relationship between vol and gamma depends on the strategy your putting on. For example. In a Short Straddle/Strangle/Butterfly/Iron Condor. Your short theta and the risks your taking are gamma risk, even though your delta neutral, and ... 2 2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ... 2 The entropic value at risk (EVaR) is a coherent risk measure, developed to tackle some computational inefficiencies of the CVaR. It is the tightest possible upper bound for traditional VaR and CVaR, obtained from the Chernoff inequality. EVaR can also be represented by using the concept of relative entropy, better known in statistics as the Kullback-Leibler (... 2 We define a convex risk measure as$$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2), $$for \lambda \in(0,1) . A coherent risk measure is subadditive and homogeneous thus for coherent \rho we get:$$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \rho( \lambda X_1) + \rho( (1-\lambda) X_2) $$by subadditivity and$$ \...

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As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works.

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I have solved it myself. The key was to realize that for $X \geq 0$ and $S_X(t) = \mathbb{P}(X>t)$ $$\int_0^\infty S(t) dt = \int_0^1 F_X^{-1}(u) du = \mathbb{E}\left[X \right].$$ This is elegantly explained in Characterization of $\mathbb{E}$. Now this relationship can be extended for the whole real line, thus $$\int_0^1 F_X^{-1}(u) du = \int_0^\... 1 Maybe prove that$$CVaR_\alpha (X) = \frac{1}{\alpha} \int_0^\alpha F^{-1}_X(u) du$$has the distortion function$$ g(u)= \begin{cases} \frac{u}{\alpha}, \quad \; u \leq \alpha \\ 1, \qquad u > \alpha\end{cases}$$would be easier? 1 Translation invariance of a risk measure \rho is defined as$$ \rho(X+k) = \rho(X)-k,  where $X$ is a random variable such that $\rho(X)$ exists and $k$ is a constant. The meaning is that if I add an amount $k$ to my risky positions then the risk is reduced by this amount. For VaR we consider the case that $X$ has a continuous distribution and that it ...

1

From Ziegel (2013) : The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called ...

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As I know, I think that spectral risk measure is a new kind of measure developed from the CVaR (weighted average value of VaR) and in the framework of coherent risk measures. If you can prove that a risk measure is coherent then you can add any types of weighted function $\phi$ to make it a spectral risk measure. The underlying idea is that the sum of any ...

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I believe that Prospect Theory (as defined by Kahneman, Amos, and Tversky) implicitly makes use spectral risk measures. Though I am not able to find any literature linking the two, I think there is clear link between the intuitions regarding loss aversion. The key difference is that spectral risk measures are normative; we assume that the utility function is ...

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Did the portfolio manager have the option of investing in emerging markets? If yes, use MSCI All-World. If the portfolio has holdings based in countries with "developed markets" yet has has emerging markets exposure to revenue/earnings, the convention is to use MSCI World.

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I have never seen such an adjustment. While monthly data are irregularly sampled in time (in every way...calendar days, trading days, seconds, etc), that irregularity is likely to be a smaller effect than your choice of data frequency (monthly, weekly, daily data). That said, your question is intriguing because in other fields they do have to deal with ...

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Normalize for trading days if possible.

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@Paul, I think you are correct. Your expression relates Gamma and Volatility Risk, as volatility risk is the risk of mis-estimating the future realised volatility. My only comment relates to your last bullet point: I have always viewed this formula outside of hedging strategies that target Gamma, as we only have the replication strategy. If you are ...

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