17

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 Policy Framework (March 2021) UBS. Suni Harford. Investing in an ESG world - A practitioner’s guide (2020) AQR. Clearing the Air: Responsible Investment (...


11

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 $$\Delta \frac{S}{V} \lessgtr 1$$ which is what Joshi is saying in words: compare $\frac{1}{S}$ (delta to price for the stock, the delta of the stock is 1) to $\frac{\Delta}{V}$ (delta to price ...


9

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 OG-Platform codebase referenced in the question and turns it into a library - no need for databases, servers or similar. Ease of use is a big focus and there ...


9

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 $$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$ then diversified could mean that these $\sigma_i$ are evenly spread over the assets in the portfolio. You find this approach and more in this paper ...


8

This problem can be expressed as the original Merton's portfolio problem. Consider wealth process defined by SDE $$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 } $$ where $\alpha_t$ is proportion of the investment in the risky ...


7

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 $b_i$, $\Sigma_{i=1}^n b_i =1$ be the risk budgets, $y_i$ the unscaled portfolio weights and $S$ the variance covariance matrix and $c$ arbitrary. $$ y^* = \text{...


7

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

I am a risk taker and I can say with confidence that you will never convince those individuals, you cited in your question, that they incur too much risk, because there will always be certain traders who prefer lottery tickets over longevity with the same firm (running high risk books unfortunately in the current environment runs equal to a free option; blow ...


6

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 risk parameters: estimation, validation, and stress testing. Springer Science & Business Media, 2006. Particularly, look at the chapter 4 and 5; the ...


6

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: $$ X = \begin{bmatrix}-1\\0\\1\\2 \end{bmatrix} \quad Y = \begin{bmatrix}0\\-1\\1\\2 \end{bmatrix} \quad X + Y = \begin{bmatrix}-1\\-1\\2\\4 \end{bmatrix}$$ What's the ...


6

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 variable X. Then blue part corresponds exactly to $\mathbb{E}[(X-x)_+]$, while the orange surface corresponds to $\mathbb{E}[(X-x)_-]$. In the picture $x=1$. Now if ...


6

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: momentum over the trailing 1 year? Trailing 6m? Looking over 3-5y periods where mean-reversion is more at play? Generally, in the absence of a clearer ...


6

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 $k$ is $RC_k=w_k \sum_j V[k,j]w_j$ in words this is "the weight of asset k times the inner product of the k-th row of $V$ and the weight vector". (Sometimes ...


6

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 $$ \lambda \approx \frac{\frac{V_1 - V_0}{V_0}}{\frac{S_1 - S_0}{S_0}} $$ which is as @noob2 said just the ratio of the relative returns for option and stock.


6

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 $\Lambda=\frac{\frac{\partial V}{V}}{\frac{\partial S}{S}}=\frac{\partial V}{\partial S}\frac{S}{V}$. An economist would call $\Lambda$ an elasticity. It tells you ...


5

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 once you are in TF, you can easily distribute your computations over a grid of computers (including GCP or AWS) if you are a machine learning enthousiastic: use ...


5

I would create categories, and work on risk parity among the categories. Otherwise, variance is not really a good measure of downside risk: Change your risk measure, use a rolling window historical VaR or Expected Shortfall at some horizon that matches your investment style. downside semi-variance could do the trick too if don't want to change your algo ...


5

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.


5

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 new ways of calculating risk, he looks at ways of communicating risk to a general public that doesn't have any knowledge of stats. I Linked an interesting video-...


5

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.


5

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 you observe 5 violations. Overestimation: you over-estimate the risk, i.e the risk is less important that you expect. You observe less VaR violations that you ...


5

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 course the normal distribution is among the elliptical distributions family. The latter can be helpful for analytical VaR modelling as an elliptical model is ...


5

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, see page 4 of these lecture notes. It's basically doing the same thing with a few additional complications. In CVaR optimization, there are more things to sum ...


5

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 distribution also with $\nu$ degrees of freedom. To see this, assuming mean returns have been subtracted, the multivariate t distribution decomposes as the distribution ...


5

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 gathers a number of statistics, among which asset exposures. The table below shows figures as of 2019Q4: So, at least for insurance companies that are ...


4

Another approach to construct a risk parity portfolio would be to use the formulation proposed by Spinu [1]: $$\begin{array}{ll} \underset{\mathbf{w}}{\textsf{minimize}} & \frac{1}{2}\mathbf{w}^{T}\Sigma\mathbf{w} - \sum_{i=1}^{N}b_i\log(w_i)\\ \textsf{subject to} & \mathbf{1}^T\mathbf{w}=1. \end{array}$$ where $\mathbf{w}$ is the vector of portfolio ...


4

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

I think you need to go even one step further than vonjd went in his reply. If liquid trading of the underlying is not possible, not only the arbitrage argument underlying risk neutral pricing breaks down. In that case there is simply no reason why the prices of those two assets (the option and its underlying) should be related in any way at all. So in my ...


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