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Why using the swap curve as riskfree rate and no longer gov bonds?

I guess it depends on what they're referring to... The traditional swap curve (LIBOR-based) is certainly not risk free, as evidenced by the experience of the financial crisis and the resulting ...
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13 votes
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cvxpy portfolio optimization with risk budgeting

The underlying problem: your ACTR constraints aren't convex The $i$th constraint on your risk contribution can be written: $$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$ And this isn't a convex ...
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11 votes

Which is riskier: a call option or the underlying?

A better, clearer, answer is to compute Lambda (leverage) of the option (link) and see if it is bigger or smaller than 1. Lambda is $\Delta \frac{S}{V}$ so we test $$\Delta \frac{S}{V} \lessgtr 1$$ ...
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11 votes
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Knightian uncertainty versus Black Swan event

I'm sure this falls short of proper philosophical precision, but here goes. Hark back to a slightly modified rehash of Donald Rumsfeld's infamous: Reports that say that something hasn't happened are ...
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9 votes

Where can I find a list of VaR and CVaR formulas for continuous distributions?

Values of VaR are just the inverses of the cumulative distributions. CVaR is not a very commonly used term, its more frequently ...
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9 votes
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Realized Variance (realized volatility)

The TLDR; to your question: How can one use realized volatility as a volatility model to do out-of-sample prediction? You extend known models to incorporate additional information procured from high-...
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8 votes
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Portfolio Risk Decomposition - different methodologies

Different portfolio risk decompositions answer different questions. Before discussing what method to use, first ask why you want a decomposition and what definition of risk are you using. Is the ...
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7 votes
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Risk neutral drift vs real world

The risk neutral drift is the risk free rate for an asset with no dividends, no cost of carry, no repo cost, etc. Otherwise the drift has to be adjusted to take these into account, and the easiest way ...
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7 votes
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Dollar-Neutral Strategy

Long-short strategy is generally used by hedge funds. In simple words, an equity long-short strategy means buying an undervalued stock and selling(shorting) an overvalued stock. In normal ...
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7 votes
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Mathematical Derivation of Residual Risk

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The ...
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7 votes
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EUR/CHF fx rate drop on the 15th of January 2015

What happened was totally unexpected end of peg against the euro @ 1.2CHF regime that Swiss central bank aborted. See some articles about it. As far as I know nobody in the markets knew, there was no ...
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  • 720
7 votes

non-subadditivity of VaR

Simple example where sub-additivity fails Let there be four possible outcomes $i=1,2,3,4$ that occur with equal probability $\frac{1}{4}$. Payoffs for $X$, $Y$, and $X + Y$ are given by: $$ X = \...
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7 votes
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Which is riskier: a call option or the underlying?

As @ir7 did, I only briefly want to add to @noob2's spot-on answer. He's of course right and $\Lambda=\Delta\frac{S}{V}$ decides how risky the option is compared to the stock. Firstly, note that $\...
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6 votes
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Why would there be a positive risk-free rate?

Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 ...
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6 votes

Which is riskier: a call option or the underlying?

Just a small addendum to @noob2's answer. The discrete shape of $\lambda$ is: $$\lambda \approx \frac{V_1 - V_0}{S_1 - S_0} \times \frac{S_0}{V_0} $$ which can be rewritten as $$ \lambda \approx \frac{...
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6 votes
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Risk-Neutrality: Discount factors of the $P$ world according to risk preferences?

You're right. Euler's equation states $$p_t=\mathbb E^\mathbb P_t[M_{t+1}X_{t+1}],$$ that is pricing under $\mathbb P$ requires you to know the stochastic discount factor (SDF, aka pricing kernel) $M$....
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5 votes

Lévy alpha-stable distribution and modelling of stock prices.

I asked this question 6 years ago, and in the meantime I came across this little volume: Lévy Processes in Finance: Pricing Financial Derivatives by Wim Schoutens (2003).
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5 votes

questions on VAR manipulation

First, I am quite sure that this is a typo and it should be $$ 0 < VaR_1 < VaR_0 $$ then $$ -VaR_0 < -VaR_1 $$ and the plot is correct. Second, the put strategy does not change only the ...
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5 votes
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New ways of communicating risk

Try to give David Spiegelhalter a read/listen to David Spiegelhalter's work and research. He is a statistician and a Professor of the Public Understanding of Risk at Cambridge England. Rather than ...
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5 votes
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Calculate VaR for a liabilty taking a exponential distribution?

The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$. Thus we need a $q$ such that $$ P[L>q] = 1-\...
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5 votes
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Why are investors risk-averse?

Below you find some observations... In CAPM, we assume people are risk-averse and people get compensated for the systematic risk they suffer. The assumption that most people are risk-averse ...
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5 votes

What are the some good measures of risk for options?

I don't have a reference for you but I have some experience. Risk management departments at hedge funds and banks would primarily look at the Var in order to capture the risk of an options portfolio. ...
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5 votes

Are smart beta and risk-parity the same?

This is a very good question. It can be argued that risk parity is one example of a smart beta strategy. Yet it is important to understand that both are coming from two different directions: risk ...
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5 votes
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how can we know the residual return will be uncorrelated with the market return

Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS ...
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5 votes
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Overestimating or underestimating risk?

Yes, it is correct. Underestimation: you under-estimate the risk, so you have more VaR violations than what your model predicts. Ex: With 100 observations, and a 99% VaR, you expect 1 violation but ...
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5 votes
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Risk, required return and expected volatility - what is the relationship?

I think you may be interested in this QJE forthcoming article by Ian Martin. The key idea of the article (page 5) is that the expected return on the market can be decomposed as $E_t[R_{t+1}]-R_f = \...
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  • 1,856
5 votes
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Calculate risk measures (book recommendation)?

A good starting point is the following paper: Risk Measures in Quantitative Finance by Sovan Mitra (2009) From the abstract: "[...] Despite risk measurement’s central importance to risk management, ...
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5 votes

Which risk-free interest rate to use in Black-Scholes equation

In theory, $r$ is a short-term safe interest rate, and it is constant through time though the theory does goes through with $\bar{r}$ (average $r$ from $t$ to $T$) in place or $r$. In practice, ...
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5 votes
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non-subadditivity of VaR

VaR is not sub-additive in general. Relying on Mark Joshi comment, there are particular cases where it can be. Such cases occur for portfolios containing elliptically distributed risk factors. Of ...
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  • 1,219
5 votes
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Can portfolio Value-at-Risk be calculated analytically for multivariate t-distributed returns?

Let the $n-$dimensional vector of returns $\mathbf{r}$ have a multivariate t distribution with $\nu$ degrees of freedom. The marginal distribution of any component $r_i$ has a univariate t ...
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