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17 votes
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Gamma Pnl vs Vega Pnl

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\...
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16 votes
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American Options relation between greeks

No, you should not expect such a relationship to hold in general. The reason is that American options have an "exercise barrier" which European options don't, and this results in different prices and ...
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14 votes
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Barrier option (autocallable) Vega profile

You have a multidimensional problem - there isn't an answer of "this is what the greeks look like" for all cases, because it depends on the various levels of the different parameters. For example, if ...
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13 votes

Expectation of Gamma times S$^2$ in Black-Scholes model

What you have to do is to show that the dollar gamma satisfies the Black-Scholes PDE. Using Feynman-Kac it then follows that the dollar gamma is an expectation of a "payoff", just like the ...
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11 votes
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Link between Vega and Gamma

Under the Black-Scholes model, \begin{align*} Gamma &= \frac{N'(d_1)}{S \sigma \sqrt{T-t}}\\ Vega &= SN'(d_1) \sqrt{T-t}. \end{align*} Then, it is easy to see that \begin{align*} Vega = S^2 \...
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8 votes
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Mathematical underpinnings of the square root of time rule

For any process with independent increments, by the very fact of statistical independence the variance of $x_{t3}-x_{t1}$ is going to be the sum of the variances of $x_{t2}-x_{t1}$ and $x_{t3}-x_{t2}$ ...
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8 votes

Expectation of Gamma times S$^2$ in Black-Scholes model

The conjecture is true when the interest rate is zero. Note that, from this question, under the Black-Scholes model, \begin{align*} \Gamma(t,S_t) &= \frac{N'(d_1(t))}{S_t \sigma \sqrt{T-t}}\\ ...
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7 votes
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Vol, Gamma, Vega -- essentially all the same?

They are not the same, but they are related. Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are &...
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6 votes
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Does higher vega imply higher IV and vice versa

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. ...
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6 votes

Gamma Pnl vs Vega Pnl

Not sure this is a valid question! Gamma p/l is by definition the p/l due to realized volatility being different from implied. Vega p/l is by definition the p/l due to moves in implied volatility. ...
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5 votes

is there an analytical proof that vega-neutral also provides (gamma & theta) neutral?

if you have a portfolio of calls and puts with the same maturity then your portfolio is gamma neutral if and only if it is vega neutral. The reasons is that the BS gamma divided by the BS vega is a ...
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5 votes
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Question about the vega of a stock

Vega is the partial derivative of the option price (as a function of parameters -- current stock price $S_t$, strike price $K$, implied volatility $\sigma$, etc.) with respect to $\sigma$ -- holding ...
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4 votes
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Volatility products and constant vega

Here is a document that will answer some of your concerns. There are many other good reads out there but this one is a nice one to get started with. In case the link is broken at the time one reads ...
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4 votes
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Vega and Gamma signs

Usually vega and gamma go in the same direction, but you can have opposite exposure in a calendar spread. For an ATM option, vega decreases closer to maturity while gamma increases. If you implement ...
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  • 1,856
4 votes

Long/Short Vega and Option Positions

No, you are incorrect. A deep in the money option is long vega. It's not just about the probability of being in the money, it's about how far in the money it is. Your reasoning is correct if we ...
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4 votes
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Long/Short Vega and Option Positions

The risk exposures/sensitivities of long and short positions always have different signs. This has to hold since derivatives are zero sum games. Vega is always positive for a long position in a ...
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4 votes

Is Complete Vega Elimination Possible?

Well , complete elimination of even Delta is not possible, forget about Vega. When I say this , I'm talking about the trouble you'd face if you keep dynamically hedging your position from time to time....
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  • 308
4 votes

Is Complete Vega Elimination Possible?

Constant Vega Requires Options Weighted Inversely Proportional to the Square of the Strike. E.g. if you have the following portfolio of options: \begin{equation} \int_{S_i(t)}^{\infty}\frac{2\Big(1-\...
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4 votes

Does higher vega imply higher IV and vice versa

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 ...
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4 votes

Why do traders think about options in terms of volatility?

Volatility is in effect what you are trading in options and the one unknown quantity in options valuation. The other inputs in option valuation: (1) Spot Price: observed in the markets, (2) Strike: ...
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4 votes
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Why do we need to calibrate vega?

It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. Thus, this hedge is not against ...
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4 votes
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Calculating vega in Heston?

Let $V$ denote the variance and $v$ the volatility, i.e. $V=v^2$. The natural argument for the option price under a stochastic volatility model is typically the variance, i.e. $C_\text{SV}=C_\text{SV}(...
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4 votes
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Relationship between VIX and Vega

Vega is the option's price sensitivity to the volatility (i.e. IV). In the graph below, vega is shown to be a strictly positive function in volatility, which means that at any point in the graph (i.e. ...
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4 votes

Relationship between VIX and Vega

VIX almost always only spikes when SPX goes down as @Jan Stuller also mentions in a comment. Insofar the question is a bit counterfactual. I frequently use twin axis in the charts that follow. The ...
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3 votes
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Why vega increases further out in time

Intuitive, no math explanation: Imagine two call options, option A expiring tomorrow and option B expiring in two months. Both of the options are way out of the money and have the same strike price. ...
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  • 1,335
3 votes

Vega hedging with implied volatility smile

Instead of just considering a parallel shift of the whole volatility surface, you can decompose the surface into maturities/strikes domains, so called buckets and consider Vega buckets which are ...
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3 votes

Is Complete Vega Elimination Possible?

In general only non-linear instruments, like options, posses vega. Vega is always positive, no matter the directional component. So when you are long either a call or a put option you are long vega ...
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3 votes

Simulation of the Vega in Heston model (for Asian Option)

Chan, Jiun Hong and Joshi, Mark S. and Zhu, Dan, First and Second Order Greeks in the Heston Model (December 26, 2010). Available at SSRN: https://ssrn.com/abstract=1718102 or http://dx.doi.org/10....
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