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25

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} \...


13

Statistically you would apply Bessel's correction to address the bias you point out. However, that misses the point that the variance-covariance matrix is non-stationary, suffers from the curse of dimensionality, and that the noisy mean return estimates have significantly more impact than a biased covariance matrix on portfolio weights. The best ways to ...


10

I just ran across an interesting presentation comparing the effectiveness of Normal, Cauchy, and Student's-t distributions in modeling the S&P. It concludes that the normal distribution underestimates extreme movements, the Cauchy overestimates them (although a comment on the presentation points out that Mandelbrot used different parameters than the ...


10

Derman et al has a long note on this from 1999. Variance swaps are actually the more natural choice. It has nothing to do with leverage. From the linked article: Although options market participants talk of volatility, it is variance, or volatility squared, that has more fundamental theoretical significance. This is so because the correct way to ...


9

I am implementing a method in Java to calculate the variance, covariance, and value at risk for a portfolio, which should be flexible for use with any number of assets in a portfolio. I am struggling with how to calculate the covariance of the assets as I can only find formulae to do so for two or three sets of values. Are you sure you are ...


9

Volatility = Variance^1/2 = Standard Deviation


9

Var and vol swaps are very similar products, with the leverage (convexity) being the biggest theoretical difference, yes. In the actual market however they are very different. After the 2008 debacle var swaps in the single stock space are not too common, whereas single stock vol swaps are regularly quoted. One interesting perspective is trading one versus ...


9

Let’s take a simple example to answer a broad but interesting question: Imagine that we have a daily return serie denoted $r_{t}$ ( which is assumed to be stationary) and let's take a little time to define main concepts : Mean Process (First moment process) The unconditional mean of $r_{t}$ denoted $u$ is just its expectation $E(r_{t})$. It is not time ...


9

The best answer to your question is probably given by the Nobel prize committee itself in "The Prize in Economic Sciences 2003 - Advanced Information" document. You should read it in full. Below is an excerpt. According to the committee: Financial economists have long since known that volatility in returns tends to cluster and that the marginal ...


8

As you know both var swap & vol swap are traded on vol. The difference comes in convexity. Although variance swap payoffs are linear with variance they are convex with volatility. Because of the convexity, a variance swap will always outperform a contract linear in volatility of the same strike. This convexity is the reason that variance swaps strikes ...


8

As I've mentioned in a comment, it would be wrong to think that entering a variance swap specifically amounts to being "long skew". What you can say however is that, in the absence of jumps (i.e. in a pure diffusion framework, see here and here for further info), the fair variance strike $K_{var}$ at which a variance swap with notional $N$ and payoff $$ N ...


8

If you take Quantuple's stuff a little further, you can really see whether you're long skew. You can pretty easily see the dependence on convexity too (though it should be obvious that you're long convexity). So first off, we need some smile parametrisation that lets us easily control convexity and skew. I just went with a made up one; $$\mathrm{convexity} ...


7

The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap. However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense: Consider a notional of 1 and a ...


7

$$ Variance \, strike = E_t \left[ \int_t^T \sigma_u^2 du \right ] $$ $$ Volswap\, strike = E_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] $$ $$ VIX = \sqrt{E_t \left[ \int_t^T \sigma_u^2 du \right ]} $$ $$ VIX \, future = E_t \left [\sqrt{E_T \left[ \int_T^{T'} \sigma_u^2 du \right ]} \right ] $$ $$ Forward\, variance\, strike = E_t \left[ \int_T^{T'}...


7

Here are two approaches that you could take to compute the variance of $X_t$. I am not making the conditioning explicit as it just complicates the notation but doesn't really add any additional insights. Compute $\mathbb{E} \left[ X_t \right]$ and $\mathbb{E} \left[ X_t^2 \right]$. You can then you use that \begin{equation} \text{Var} \left( X_t \right)...


7

Usually the formula for the sample variance of a stock is given by: \begin{equation} Var(R_{i}) = E (R_t - E(R_t))^2 \end{equation} If you are using daily data to compute the variance then the second term: $E(R_t) \approx 0$, therefore you can drop it from the computation. Which yields: \begin{equation} Var(R_{i}) \approx E (R_t)^2 \end{equation} ...


6

I know you're really looking for some empirical work on this topic, but I think the following theoretical paper puts your question into proper perspective.* Risk-Based Asset Allocation: A New Answer to an Old Question by Wai Lee, JPM 2011. Overall, he finds that supposedly risk-based approaches to portfolio construction are really making implicit ...


6

PCA gives you a decomposition of the covariance matrix of the form $$ \Sigma = V \Lambda V^T $$ where $\Lambda$ is diagonal with the eigenvalues in the diagonal. Your portfolio variance is $$ w^T \Sigma w = (V^T w )^T \Lambda (V^T w) $$ On the other hand if you take your return matrix $R$ and define $$ F = V^T R $$ then the covariance matrix of these so ...


6

We first list the assumptions. \begin{align*} g_{t+1} &= \mu_g + \sigma_{g, t} z_{g, t+1}, \tag{1}\\ \sigma_{g, t+1}^2 &= a_{\sigma} + \rho_{\sigma} \sigma_{g, t}^2 + \sqrt{q_t} z_{\sigma, t+1}, \tag{2} \\ q_{t+1} &= a_{q} + \rho_q q_t + \varphi_q \sqrt{q_t} z_{q, t+1}. \tag{3} \end{align*} Moreover, \begin{align*} r_{t+1} &= -\ln \delta +\...


6

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 $E_t[R_{t+1}]-R_f = \frac{1}{R_f}Var^Q(R_{t+1}) + \text{extra terms}$ As you correctly pointed out the expected return should be related with the risk neutral variance. The issue ...


6

If you really believed the CAPM's prediction that $\alpha=0$, then imposing $\alpha=0$ in your estimation would indeed lead to your 2nd formula. The problems? The CAPM doesn't work so imposing a false restriction during estimation is problematic. More generally, taking factor models extremely seriously and imposing $\alpha=0$ in estimation to gain ...


5

The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement. In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric ...


5

To answer your questions: Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl. how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma? He is short gamma but long theta. He is initially making money ...


5

The vega of an option is very dependent on the spot price. The vega of a variance or volatility swap is not.


5

Here's another take on the question: \begin{align} \int_0^t W_s^2 ds &= \int_0^t \int_0^s d(W_u^2) ds \\ &= 2 \int_0^t \int_0^s W_u dW_u ds + \int^t_0 \int^s_0 du ds \tag{Itô's lemma}\\ &= 2 \int_0^t \int_u^t W_u ds dW_u + \frac{t^2}{2}\tag{Stochastic Fubini}\\ &= 2 \int_0^t W_s (t-s) dW_s + \frac{t^2}{2} \end{align} Now you can use Itô's ...


5

We have weights $w_A$, $w_B$ and $w_C = 1 - w_A - w_B$ that sum to $1$. With de-meaned returns $r_A$, $r_B$, and $r_C$, the portfolio variance is $$E\{[w_A r_A + w_B r_B + (1 - w_A - w_B)r_C]^2 \} = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2 w_A w_B\rho_{AB}\sigma_A \sigma_B,$$ assuming the cash volatility $\sigma_C$ is zero.


5

Below are my 2 cents only, but this was too long for a comment. As he shows in the next lines (see also Variance Swaps chapter of Bergomi's book) $$ \sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \tilde{\sigma}^2(z,T) \phi(z) dz \tag{0} $$ where $\sigma_{VS}(T)$ denotes the volatility of a fresh-start variance swap of maturity $T$; $\phi(\cdot)$ the standard ...


5

The piece you are missing is an approximation via the Taylor formula of the logarithm: $$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$ Apply this to the first term in the final formula of the technical paper: $$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\...


4

In terms of ARCH conditional variance is the variance conditional on past information (i.e. the history of the process). This is useful for modeling a process that exhibits volatility clustering. Perhaps he means that starting with the standard deviation (unconditional volatility) of stock returns one can then use that as an input to estimate the conditional ...


4

the only difference between volatility and variance is the square. everything else is bs, as concept that apply to one applies to the other (historical vs implied, blabla)


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