# Tag Info

9

Yes it is a better way. Just take a look to figure 3, from Buss and Vilkov (2012, RFS):

9

Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see them and then deal with them one by one. Note, it is simpler for me to talk about variance instead of volatility. This has no material impact on the answer. ...

7

For How VIX works you can read this wonderful blog : http://onlyvix.blogspot.com/2011/09/intuitive-understanding-of-vix-formula.html It provide wonderful non mathematical explanation of the how vix is actually computed. Now comes to your last answer why vix is inversely related to market movement ? In simple words, if market is more volatile then ...

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 +\...

5

Note that total implied variance defined as $$V(T,K) = T\Sigma(T,K)^2$$ should be an increasing function of $T$. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and buy the cheap longer one). If you interpolate linearly your implied volatility is $$\Sigma(T,K) = w\Sigma(T_i,K) + (1-w)\Sigma(T_{i+1},K)$$ with weight w = \... 4 please go to {drvd} BVOL Equity Implied Volatilities Calculations paper. Disclamer: I was working for Bloomberg, that is as far we disclosed. 4 Note that \begin{align*} (K-S_T)^+ \ge K-S_T. \end{align*} Then \begin{align*} p &\equiv E\Big(e^{-rT} (K-S_T)^+ \Big)\\ &\ge E\Big(e^{-rT} (K-S_T) \Big)\\ &=K\, e^{-rT} - S_0\\ &= 670 \times e^{-0.05 \times 55/365} - 563.48\\ &=102.49. \end{align*} However, the option price is 101.375, which is smaller. This is the reason that you have ... 4 Constant Vega Requires Options Weighted Inversely Proportional to the Square of the Strike. E.g. if you have the following portfolio of options: $$\int_{S_i(t)}^{\infty}\frac{2\Big(1-\log[\frac{K}{S_i(t)}]\Big)}{K^2}C_i(t,\tau,K)dK+\int_{0}^{S_i(t)}\frac{2\Big(1-\log[\frac{K}{S_i(t)}]\Big)}{K^2}P_i(t,\tau,K)dK$$ You have a ... 4 It is always better to use some closed form approximation first to get initial guess. Corrado and Miller (1996) produced a solution that is quite accurate across a range of moneyness ( though it can be applied to BS model only and can’t be used for plain vanilla options or exotic options). The formula for implied volatility\sigma$is :$\sigma = \frac{1}{...

4

Do you really need to do this yourself? The absolute state of the art is Peter Jaeckel's work, where he makes an implied vol function as good as exp, cos, and log special functions. And he pulished source code and algorithm details with careful numerical analysis of errors and convergence. This is a wheel you don't have to reinvent, any more than you ...

4

Peter Jaeckel has written various papers on this. "by implication" and "Let's be rational" are the most recent ones. He also provides code on his website www.jaeckel.org. (Note: the question asked for literature.)

3

Directly from the paper: We assume that the representative agent in the economy is equipped with Epstein–Zin–Weil recursive preferences. Consequently, the logarithm of the intertemporal marginal rate of substitution, $m_{t+1} \equiv log(M_{t+1})$, may be expressed as $$m_{t+1} =\theta log\delta−\theta\psi^{-1}g_{t+1}+(\theta−1)r_{t+1}, (4)$$ […] ...

3

Generally speaking, in the real world, you'd always want to use the correct implied vol. But you should think of your question in terms of: (1) Vega mark-to-market (m2m) PnL vs. theta/gamma profile (2) Change in risk and PnL due to higher order risks (vanna, volga) Vega mark-to-market PnL vs. theta/gamma profile In a simple, pure Black Scholes world ...

3

In general, $v = \frac{\partial C}{\partial \sigma} > 0$ and $\theta = \frac{\partial C}{\partial t} < 0$. If maturity $T$ increases than $C$ increases. Suppose volatility is non-constant. Then if $T$ increases, the option value is more volatile, since the stock price is more volatile. Since $v > 0$ the option price must increase. He claims that \... 3 You are asking about the term structure of lognormal implied volatilities for European swaptions, which is a two dimensional function (expiration and tenor). First expiration: typically (but not always), implied volatilities are increasing in the 0 to 6 month sector, because the immediate future is often more predictable than the medium term. At some ... 3 A possible reason may be your computation of maturity period. Exchange compute the maturity in minute till expiry and then divide it by total trading minute in a year to arrive at maturity. An another possible reason may be your choice of risk free interest rate. There are various proxy for risk free interest rate like Treasury rate and LIBOR of different ... 3 They expire 30 days before the expiration of the S&P monthly options. The latter usually expire on the third Friday of the month (however, in rare cases the S&P opts. expire on Thursday because the Friday is a holiday; the last time it happened was April 17, 2014 since April 18 2014 was a NYSE holiday). Neglecting the holiday thing, the expiration ... 3 In the BS model there is the upper bound of the stock price, which can be proven by the fact the stock price bounds the call option pay-off. Here we are seeing a similar effect: the discounted rate corresponds to the stock price. 3 Let \begin{align*} L(t; T, T + \Delta) = \frac{1}{\Delta} \left[ \frac{P(t,T)}{P(t, T+\Delta)} - 1 \right] \end{align*} be the forward Libor rate at timet$for the period$[T, T+\Delta]$. Consider a caplet with payoff at$T+\Deltaof the form \begin{align} \Delta\max\big(L(T; T, T + \Delta) -K, \, 0 \big) &= \Delta\max\big((L(T; T, T + \Delta)-\alpha)... 3 You can show that "the implied variance of an ATM short maturity option is equal to the expectation under the risk neutral measure of the integrated variance over the life of the option." As you move away from the assumptions: ie not ATM, longer maturity, risk neutral measure far from true, then the forecasting power diminishes. (Google 'stochastic ... 3 I would say that\log K/F$points towards a log-normal type model. If I were you I would experiment with the moneyness defined as$K-F$instead. This would make it consistent with normal dynamics. An alternative would be to define an 'interest rate floor', say$L=-200bp$and take relative changes relative to that rather than zero, ie define moneyness as$\...

2

Basically there are three steps to accomplish this. 1 - collect time series of options for several expirations and strikes. 2 - calculate implied volatility surface for every time period, and use model-based or model-free interpolation to create continuum of strikes / expirations. 3 - from the continuous surfaces you can calculate series of any specific ...

2

I don't believe you will necessarily find a cite-able source as, I believe, this comes from a practical rather than theoretical motivation. As you know option prices are a function of: future prices, discount rates and implied volatility, volatility surface skew and other supple/demand factors. So when you are trading these instruments, you need to ...

2

It is all a matter of frequency. For instance if you want to get annual realized volatility you multiply your last expression by $\sqrt{(N*251)}$ or the second to last expression by $\sqrt{(251)}$. In other words, your last expression is the 5-min realized volatility whereas the second to last expression is the daily realized volatility.

2

This is a somewhat ill-posed question. The "components" in your question are not components, they are just different options and all have different implied volatilities - all for the same underlying. If you are looking to get single number volatility a-la VIX without the whole VIX calculation, you should use ATM (at-the-money) implied volatility, which is ...

2

I'll address your questions in order: 1a) For TSRV constructed using high frequency returns from NYSE market open to market close on a single day, the output should be numbers on the order of magnitude of 1e-4 to 1e-5. In other words, your numbers look about right. I got these number from calculating TSRV for IBM data myself using Kevin Sheppard's MatLab ...

2

It implies negative forward variance. I have the book, and went through the section following your quote. In math terms, he is making a proof by contradiction. He first assumes that you can interpolate Iinearly, and comes to the conclusion that it is not a good assumption. The argument does involve some calculus. I don't think I have a better explanation, so ...

2

You can guesstimate by vega weighted implied vol. This is why: Say that you have a portfolio of options with prices $P_j$. Each one of them has a different pricing function $f_j$ (as function of vol) and a different implied vol $\sigma_j$. For each option $f_j(\sigma_j)=P_j$. Now you put them together in a single product. If the implied vol of the product ...

2

Using that data the best way to compute implied volatility is tho use the methodology to approximate the variance swap rate closely following the model-free estimate proposed by Demeter et al. (1999) and Carr and Madan (1998) who show that if one owns a portfolio of options across all strikes inversely weighted by the squared strike then one gets a variance ...

2

It is difficult to gain intuition by just looking at the price surface, and it is also easier to calibrate models on the volatility surface rather than on the price surface because with the later you are dealing with numbers of very different sizes (depending on the moneyness and maturity) which is not good for minimization algorithms. However low and high ...

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