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28

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*} \...

23

\begin{align*} \text{Variance strike} &= \mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ] \\ \text{Volswap strike} &= \mathrm{E}_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] \\ \text{VIX} &= \sqrt{\mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ]} \\ \text{VIX future} &= \mathrm{E}_t \left [\sqrt{\mathrm{E}_T \left[ \int_T^{T'} \...

14

Volatility = Variance^1/2 = Standard Deviation

13

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 ...

11

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: $$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\... 9 The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap. However, if you are to trade a VIX future (i.e. a delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense: Consider a notional of 1 and a ... 9 The best answer to your question is probably given by the Nobel prize committee itself in "The Prize in Economic Sciences 2003 - Advanced Information" document. You should read it in full. Below is an excerpt. According to the committee: Financial economists have long since known that volatility in returns tends to cluster and that the marginal ... 8 If you take Quantuple's stuff a little further, you can really see whether you're long skew. You can pretty easily see the dependence on convexity too (though it should be obvious that you're long convexity). So first off, we need some smile parametrisation that lets us easily control convexity and skew. I just went with a made up one;$$\mathrm{convexity} ...

8

Usually the formula for the sample variance of a stock is given by: $$Var(R_{i}) = E (R_t - E(R_t))^2$$ 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: $$Var(R_{i}) \approx E (R_t)^2$$ ...

8

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 ...

7

In any finite sample, it is always possible for the Zhou estimator to return a negative number, even though we know the unobservable parameter being estimated is non-negative. This is a well known issue in the academic literature. There are several approaches to dealing with this problem: 1) Ignore it. (I don't like this one). It is particularly nefarious ...

7

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 ...

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 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 \text{Var} \left( X_t \right)... 6 If you have access to intraday data, they are better ways to estimate the bid-ask spread. If you have Open, High, Low and Close price on each 5min bin b (or any other interval): the Close of the previous bin and the Open of this one are consecutive. Hence dP(b)=C(b-1)-O(b) allows to define an estimate \psi(b) of the bid-ask spread\psi(b):=\min_{b:\, ...

6

Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic function of the realised interest rate $R_t=\int_0^t r_s\text{d}s$ is \begin{align*} \varphi_{R_t}(u)=\mathbb{E}\left[e^{iuR_t}\right] = A_t(u)e^{B_t(u)r_0}, \end{...

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

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).

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.

5

From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price. Put differently, you are you confusing $$v_t \approx \text{var}(\ln(S_{t+\delta t}/S_t))/\delta t$$ with $$\text{var}(\ln(S_t))/t$$ presumably because in the Black-Scholes framework these ...

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

I'm not a Python programmer, however, reading the reference manual of np.var, you're using the "biased" version of the variance estimator. Instead use the unbiased variance estimator: import numpy as np from numpy.random import randn X = randn(1000,3) Sigma = np.cov(X.T) w = np.array([0.2,0.3,0.5]) print(np.var(X@w, ddof=1)) ...

5

The probablility of a jump of $J = \phi$. ( in either direction so I'll assume $\frac{\phi}{2} =$ probability of J and $\frac{\phi}{2} =$ probability of -J ). The probability of a jump of $0 = (1-\phi)$. So, the expectation of the of jump amount, MM, $= E(MM) = \frac{\phi}{2} \times J + \frac{\phi}{2} \times -J + (1-\phi) \times 0 = 0$ The variance, \$\...

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 ...

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