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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 ...


11

Volatility = Variance^1/2 = Standard Deviation


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

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


7

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 ...


7

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

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} ...


7

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\...


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

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 ...


6

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)...


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

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 +\...


5

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


5

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 ...


5

An example of non-overlapping one month returns: the return in January, the return in February, the return in March, etc. An example of overlapping 30 day returns: the return from January 1 to January 30, the return from January 2 to January 31, the return from January 3 to February 1, the return from January 4 to February 2, and so on. There are far fewer ...


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.


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

An Axioma research paper from August 2011, Using Multiple Risk Models for Superior Portfolio Management… A Practice Not Just For Quants, answers exactly your question, I believe. Note the graphs at the top of page 8. They compare their medium-horizon fundamental and statistical factor models from January 2008 to January 2009. At the start of the period, ...


4

There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More Mathematical Finance among other places. It all started with the work of Anthony Neuberger on the Log Contract in 1994, which seemed to be a theoretical result ...


4

The solution provided can be derived using the CAPM. For asset $A$ you have: $$R_A-R_f = \alpha_A +\beta_A(R_M-R_f)+\epsilon_A$$ Similarly for asset B: $$R_B-R_f = \alpha_B +\beta_B(R_M-R_f)+\epsilon_B$$ Calculate the covariance: $$\text{Cov}(R_A, R_B) = \text{Cov}(\beta_AR_M, \beta_BR_M)$$ Here I have dispensed with all the constant terms, and also ...


4

The options will form a static replication - and yes - they should expire on the same day as the variance swap. You should be sure to do all of your analytics in business time. Also, typically a year is considered 252 business days by default - even in FX where there are no holidays (though there are no WMR fixings on New Years, Christmas, Good Friday). ...


4

$V_L$ is the long-run variance (or the unconditional variance) if and only if $\gamma=1-\sum_{i=1}^n \alpha_i$, because the long-run variance compatible with the model $$ \sigma_n^2 = \gamma V_L + \sum_{i=1}^n \alpha_i u_{n-i}^2 $$ is $$ \sigma^2=\frac{\gamma V_L}{1-\sum_{i=1}^n \alpha_i}. $$ The presence of the intercept $\gamma V_L$ restricts $\sigma_n^2$...


4

I believe you want to know why the VIX is a weighted portfolio of calls and puts with weights proportional to $\frac{1}{K^2}$ (NB: obviously the T is there to adjust for time to maturity, hence is not very enlightening). Let's start with the basics. As shown by Breeden and Litzenberger (1978) the second derivative of an option price with respect to the ...


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