13
votes
Accepted
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 ...
- 6,924
11
votes
Accepted
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 ...
- 5,041
10
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
$$\Delta \frac{S}{V} \lessgtr 1$$
...
- 10.9k
10
votes
Accepted
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-...
- 4,186
9
votes
Accepted
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 ...
- 6,924
8
votes
Accepted
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 ...
- 2,218
8
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:
$$ X = \...
- 6,924
7
votes
Accepted
Mathematical Derivation of Residual Risk
Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is
\begin{align*}
r_P = \alpha+\beta r_B + \varepsilon,
\end{align*}
where $\varepsilon$ is the residual. The ...
- 21k
7
votes
Accepted
EUR/CHF fx rate drop on the 15th of January 2015
What happened was totally unexpected end of peg against the euro @ 1.2CHF regime that Swiss central bank aborted. See some articles about it. As far as I know nobody in the markets knew, there was no ...
- 732
7
votes
Accepted
Which is riskier: a call option or the underlying?
As @ir7 did, I only briefly want to add to @noob2's spot-on answer. He's of course right and $\Lambda=\Delta\frac{S}{V}$ decides how risky the option is compared to the stock.
Firstly, note that $\...
- 15.5k
6
votes
Accepted
Calculate VaR for a liabilty taking a exponential distribution?
The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$.
Thus we need a $q$ such that
$$
P[L>q] = 1-\...
- 13.6k
6
votes
Accepted
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 = \...
- 1,886
6
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, ...
- 27.3k
6
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 ...
- 1,219
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
$$ \lambda \approx \frac{...
- 5,008
6
votes
Accepted
Risk-Neutrality: Discount factors of the $P$ world according to risk preferences?
You're right. Euler's equation states $$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) $M$....
- 15.5k
6
votes
Conceptual problem with risk neutrality-What is a 'risk-neutral world', exactly?
A little bit of history. This goes back to the early days when the Black Scholes formula for options was proposed but was still new and somewhat mysterious.
It was (and is) widely accepted in Finance ...
- 10.9k
6
votes
Conceptual problem with risk neutrality-What is a 'risk-neutral world', exactly?
I have masters degree in mathematics so the math isn't the problem; but, trying to get my head around financial math, I keep having problems with the concept of 'risk-neutrality'.
I suspect you might ...
- 161
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).
- 1,507
5
votes
Accepted
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 ...
- 363
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. ...
- 16.6k
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 ...
- 27.3k
5
votes
Accepted
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 ...
- 13.6k
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 ...
- 2,552
5
votes
Accepted
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, ...
- 27.3k
5
votes
Accepted
What is an accepted method to calculate percent PnL from a short position?
A short position is a liability on your books, as the borrowed asset has to be returned to the owner. The return is then the percentage return of that liability.
Assume that the shorted asset at ...
- 1,034
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 ...
- 3,595
5
votes
Accepted
Calculating beta to market
In a word, yes. That's a correct and valid view to take but, as you'll always find in finance, it really depends on context and the question that you're trying to answer. This is the case in markets ...
- 131
5
votes
Accepted
What is the industry standard way of calculating and annualizing performance metrics?
To give you an idea of industry standards for funds (although not hedge-fund specific), Morningstar and Trustnet both use monthly returns and annualize their data. See, for an example plucked at ...
- 1,396
5
votes
Accepted
What does 5 year OIS actually mean?
When people say OIS swap they mean an exchange of some sort of fixed cash flow and in return the receipt of daily OIS based on the "Fed Effective Rate" (FEDL01 Index on Bloomberg).
The floating ...
- 2,588
Only top scored, non community-wiki answers of a minimum length are eligible