18 votes

Quantifying climate change risk

Here are some resources that I found useful when learning about this subject, in which I'm very interested. (Some may be more general ESG than just just climate.) Citigroup. Environmental and Social ...
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 $$\Delta \frac{S}{V} \lessgtr 1$$ ...
  • 9,647
10 votes
Accepted

Kelly criterion for normally distributed returns

This problem can be expressed as the original Merton's portfolio problem. Consider wealth process defined by SDE $$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac ...
9 votes

What is the best alternative of Quantlib library

The Strata project is the new pure Java market risk quant library from OpenGamma. For more information, see the documentation and GitHub. It is Apache v2 licensed. Strata takes the experience of the ...
9 votes
Accepted

What to use as portfolio diversification measure?

If you measure risk by the standard deviation of the portfolio return $$ \sigma = \sqrt{w^T \Sigma w}, $$ then it is usual to define risk contributions for each asset by $$ \sigma_i = w_i (\Sigma w)_i/...
  • 13.3k
9 votes
Accepted

Intuitive explanation for expectiles

No reply has been given so I wanted to at least give a visualisation of the expectiles. Suppose the curvy dashed line in my picture represents a cumulative distribution function of some random ...
  • 1,487
7 votes

How to construct a Risk-Parity portfolio?

I am very happy with the following equivalent formulation for the risk budgeting problem (as presented in Bruder, Roncalli, 2012, Managing Risk Exposures using the Risk Budgeting Apporach): Let $b_i$,...
  • 2,894
7 votes
Accepted

Why is the variance of a portfolio a quadratic form?

if you take the variance of a single asset it scales as a quadratic, $$ var(\lambda X) = \lambda^2 var(X) $$ so it's not surprising that the general case gives a quadratic form.
  • 6,763
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 = \...
  • 6,394
7 votes
Accepted

How to check if a portfolio has momentum bias

It kind of depends what your objective is. First, momentum 'bias' isn't well-defined. Are you looking to eliminate momentum exposure for some reason? Momentum itself isn't even well-defined really: ...
  • 1,608
7 votes
Accepted

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 $\...
  • 14k
6 votes

What is the best alternative of Quantlib library

I did not tested it by now, but Google released a library similar to quantlib written in TensorFlow (tf-quant-finance). It may be worthwhile to test it (and to post here your views on it), because ...
  • 10.7k
6 votes

References for PD / LGD estimates of low-default portfolios

I cannot suggest some reference particularly, since the field is going to develop day by day, but, generally, you could take a look to: Engelmann, Bernd, and Robert Rauhmeier, eds. The Basel II ...
  • 2,446
6 votes
Accepted

Variance attribution calculation from a covariance matrix

Suppose the covariance matrix is $V$ (which is n by n) and the weights are $w$ (of length n). Then the Portfolio Variance is $V_p = w^T V w$ and the Risk Contribution (in terms of variance) of asset ...
  • 9,647
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{...
  • 5,028
5 votes

Works of Nassim Taleb

On the N. N. Taleb's website, you can find all his papers collected in the bibliography he updates on his own site. Hope this will help.
  • 2,446
5 votes
Accepted

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 ...
  • 330
5 votes

Stressed Value at Risk vs Value at Risk

The most important difference is that the calculations are based on a "stressed" historical period in the markets as opposed to the most recent X number of years.
  • 51
5 votes
Accepted

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 ...
  • 2,534
5 votes
Accepted

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 ...
  • 1,209
5 votes
Accepted

Question on Rockafellar's Paper for optimisation of CVaR

On 1, I suspect that is a typo and that the second formula should sum to r. On 2, that is applying well-known techniques in how to handle piece-wise linear functions in an optimizer. For instance, ...
  • 5,311
5 votes
Accepted

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 ...
  • 3,470
5 votes
Accepted

Book recommendation

First, a word of caution One of the problem with mathematical finance, as well as the related field of financial economics is that there is more than enough to learn to fill your schedule several ...
  • 2,336
5 votes

Book recommendation

I am going to deviate from the "Shreve camp" :) If, like me, you like books that are less than 300 pages long, then for an introduction to quantitative finance concepts, I recommend the following two ...
5 votes
Accepted

Why do (life) insurance companies face equity risk?

The publication is made by the UK institute of actuaries so I'm answering from the perspective of the insurance industry in the European union. Is it allowed? For European Insurance companies EIOPA ...
  • 7,731
4 votes

What is the best alternative of Quantlib library

QSTK is nice and open source , it is the QuantSciTookKit and it has some good functionality if you are interested in python programming. Here is the GitHub repo.
4 votes

How to construct a Risk-Parity portfolio?

Another approach to construct a risk parity portfolio would be to use the formulation proposed by Spinu [1]: $$\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}...
4 votes

Is there anyone still using Markowitz modern portfolio theory?

Lots of wealth management firms still use MPT; in my experience regulators like it because they understand it. If asset returns are normally distributed, the standard deviation of the portfolio is ...
  • 1,329
4 votes

What is the difference between asset management and wealth management?

The definitions will vary from organization to organization but generally: Wealth Management is the management (either direct or through distribution to other managers) of an individual investor’s ...
  • 53
4 votes
Accepted

How to extrapolate VaR?

It depends on the method by which you calculate VaR. Some models (t-distributuion, normal) lead to a form of VaR such that it is just scaled volatility: $$ VaR = c \sigma $$ with some proper $c$ (e.g. ...
  • 13.3k

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