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18 votes
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Is there anywhere I can read the paper, "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves"

Partly because it's hard to get a hold of, the Arslan et. al. paper is starting to assume mythical proportions. As said by Dimitri Vulis, the general idea of the paper is set out in (one or two of) ...
17 votes
Accepted

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 ...
  • 5,638
14 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 ...
12 votes
Accepted

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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11 votes
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Is short-gamma inherently a losing strategy?

You can't lose more than you invested by writing covered puts, because you keep enough cash to cover any potential losses from the puts. That's not to say that your losses can't be substantial, of ...
  • 5,638
10 votes

Greeks: Why does my Monte Carlo give correct delta but incorrect gamma?

the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from ...
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9 votes
Accepted

How do we know if the volatility which is quoted in market is Normal (Bachelier model) or log normal (Black 76)?

Options on interest rates futures in the listed markets are always traded 1-yield (100-yield) just like the futures which are traded 1-yield. So negative rates aren't an issue and its always black ...
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9 votes
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Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option

Gamma and vega have the same general shape , peaking at ATM and tapering to the tails. But gamma concentrate as the option gets closer to expiry (when vega is small). For options a long way from ...
  • 1,018
9 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}}\\ ...
  • 20.5k
9 votes
Accepted

Gamma PnL from Itô's Lemma derivation

$$ \frac{1}{2} \frac{\partial^2 f}{\partial S^2} dS^2 \approx \frac{1}{2} \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} dt$$ (for small $dt$, ignoring $(dt)^2$ terms ) $\sigma$ is embedded in $dS = \...
  • 5,028
8 votes
Accepted

What really is Gamma scalping?

Assuming all else remains equal (implied vol has not changed and very little time decay has occurred), Gamma scalping can best be explained by Gamma (or realized volatility) enhancing the value of a ...
  • 5,452
8 votes
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Numeric example to understand the effect of option gamma

Using our good friend Taylor, we know that \begin{align*} C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2, \end{align*} where $\Delta_C$ and $\Gamma_C$ are the call's ...
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8 votes
Accepted

Why is gamma exposed through the square of spot prices?

Dimensional analysis is the key: The change in option price is in dollars. The change in option price is of course the sum of its changes (partial derivatives) with respect to its underlying risk ...
7 votes
Accepted

Proof of gamma profit formula

Assume you buy a plain vanilla call option at the price $V$ and the spot $S$. You immediately delta hedge buy selling $\partial V / \partial S$ units of the underlying asset. The underlying asset now ...
7 votes

What really is Gamma scalping?

Gamma scalping (being long gamma and re-hedging your delta) is inherently profitable because you make 0.5 x Gamma x Move^2 across the move from your option. (You get shorter delta on downmoves, so you ...
  • 221
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 &...
  • 5,638
7 votes
Accepted

Can we trade theta?

You are neglecting the PnL from the stock position. Let us say you hold 1,000 shares at \$122 per unit. You’ve sold calls at \$0.21 per unit of stock, thus receiving \$210 in premiums. If the stock ...
6 votes

What really is Gamma scalping?

As long as you live in a world where implied and realized vol are the same, there is no net profit (or loss) from gamma scalping. However, if they are different, then you make a gain or loss which is ...
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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. ...
  • 14.3k
6 votes

How to prove Gamma is the same for a European call and European put with the same inputs?

Put-call parity says that a call and put (worth $C$ and $P$ respectively) with the same strike $K$ have the following relationship with the spot rate $S$, risk-free rate $r$, and time to maturity $T$ -...
  • 5,638
6 votes

Gamma for ATM options with low spots

Gamma is the sensitivity of the delta with respect to infinitesimal changes in the price of the underlying asset (in whatever unit your underlying is nominated, typically dollar, pounds, euros, ...). ...
  • 14k
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 ...
  • 6,763
5 votes
Accepted

Gamma/delta dynamics in the Black Scholes model and it's relation to PnL (Basic of option theory)

We work in a Black-Scholes world. Consider the following delta-hedged portfolio: $$ \Pi_t=V_t-\frac{\partial V}{\partial S}S_t$$ We assume the portfolio is self-financing$^{\text{(a)}}$, therefore: $...
5 votes

Gamma for ATM options with low spots

The traders or practitioners’ gamma concept tries to capture the same issue. It is defined as S times gamma divided by 100: $\Gamma_P=\frac{S\, \Gamma}{100}$ Please see page 29 of this document: ...
5 votes

Is there anywhere I can read the paper, "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves"

It was a Deutsche Bank Working Paper: http://refhub.elsevier.com/S0304-405X(16)00005-2/sbref0001 Unfortunately, it is very hard to find internal research published by banks. I have not seen this one ...
5 votes
Accepted

What is gamma to do with realized volatility?

I like to think about this problem graphically. The pic below shows a call option value at some point before expiry as a function of the underlying. At the expense of stating an obvious fact, we note ...
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4 votes
Accepted

Approximation of an option price

Since the volatility is not changing, we can assume that the only change is the underlying asset price $S$. Then \begin{align*} C(S+\Delta) &\approx C(S) + Delta \times\Delta +\frac{1}{2} Gamma \...
  • 20.5k
4 votes

What is the formula for beta weighted delta and gamma?

I've started thinking about this, too. My gedanken conclusion turned out to be too simple once I found what I was after: http://www.investment-and-finance.net/derivatives/o/option-beta.html, which I'...
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4 votes
Accepted

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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