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15

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*} ...


12

It seems that you are thinking of the volatility as some sort of standard deviation of your stock price. It is not. In the BS model, $\sigma\sqrt{T}$ is the standard deviation of the log-return $\log(\frac{S_T}{S_0})$. There is no mathematical upper bound to its standard deviation. There is also no mathematical problem with returns being negative either. ...


9

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


8

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. ...


6

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

No. Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated. Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have ...


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

A very popular choice for mean reversion is the Ornstein–Uhlenbeck process (here in discretized form): $$L_{t+1}-L_t=\alpha(L^*-L_t)+\sigma\epsilon_t$$ Here you see that the level change is governed by some parameter $\alpha$, the mean reversion rate (or speed), and the distance between the long run mean $L^*$ and the actual level $L_t$ plus some noise. A ...


4

Simply put, no. Vega depends on a variety of factors (including the level/price of the underlying asset). However, vomma/volga/vega convexity (whatever you want to call dVega/dIV) is always positive. So as IV increases, the vega of an option increases - I think this might have been what you were getting at. It's important to understand that IV is an input ...


4

Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?


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 ...


3

First, as far as I can tell, you are not taking into account dividends. Second, If you simply take the forward price of the SPX @ $5.5\%$ which is what you are using, you get $1411 \cdot \text{exp}(0.055 \cdot 2.99) = 1663$. Given a strike of $1300$, the call should have an intrinsic value of $1663-1300= 363$. You have a price of $272$. The price is less ...


3

IV is one of the inputs for your option pricing model, vega measures the actual impact (e.g. in Dollars, Euros...) of any change in IV. Intuitively IV is the price of the option while vega is the sensitivity to IV. Bottom line: There is a clear distinction!


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

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

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

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 ...


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

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 ...


2

The most used equity volatility models in the industry are the Black-Scholes model (including its time dependent version) and the local volatility model. It always come along with stochastic rates, discrete dividends and quanto effects (a must-have when pricing even simple payoffs) so the calibration/pricing process is much more involved than what you might ...


2

The central limit theorem guarantees, under fairly general assumptions, that the sum of returns becomes more normally distributed as the number of returns grows (technically, defining a return as $\mathrm{log}(S_{t+\Delta t}/S_t)$, $\sum_i ^n \mathrm{log}(S_{t+\Delta t i}/S_{t+\Delta t (i-1)} \to \mathcal{N}(\cdot,\cdot)$ as $ n \to \infty $). Thus, as $T$ ...


2

Because there are several non-linearities involved this depends very much on where you are concerning the level of volatility and time to expiry. But I think what you really want is to get some feel for the sensitivities involved, right? With the following demonstration you can play with all kinds of combinations of all parameters to get some intuition for ...


2

The main thing to keep in mind with all these different option combination strategies is that you are really trading option greeks! I think the answer to why the calender spread is so popular lies in the special combination of gamma and vega risk: Calendar spreads are the one type of trade where gamma can be negative while vega is positive (and vice versa ...


2

The answer of AFK is very good and accurate in a BS setting. Thinking of jumps I would add the following: If we assume that stocks sometimes move in jumps (usually downwards) then it is clear that ATM or OTM options shortly before expiration with a price that accounts for the possibility of a jump - which is therfore quite high - only fit in the BS ...


2

The main difference is that one approach assumes that a certain dynamical structure properly describes the underlying instrument, while the other approach is really only a re-writing of the price in terms of an implied volatility. Implied volatility Implied volatility really only needs two things: the underlying stock price and the call option price (apart ...


2

First note that the price of binary call is related to the price of an ordinary call in any model by $$ BinC(T,K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[1_{S_T>K}] = - \frac{\partial}{\partial K}e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)_+] = - \frac{\partial}{\partial K}C(T,K) $$ Now the volatility smile is implicitly defined by $$ C(T,K) = ...


2

I think this extremely hard to do (to the point where I think that every hedge fund that trades vol should be avoided like the plague). The fundamental value of volatility would be a quantity that's related to the speed at which new news comes available to the market, the significance of news, the extent to which this news can be traded, general market ...


2

The most common use for implied volatility in valuation is for asseing options or option like postions. A volatile instrument is likley to activate or put an option postion in the money just on the basis of its volatility rather than any fundamental change in the intrinsic or fair market value of the underlying. This needs to be taken into account when ...



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