22 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}\...
Gordon's user avatar
  • 21.1k
20 votes
Accepted

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 ...
Chris Taylor's user avatar
  • 5,911
12 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 \...
Gordon's user avatar
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12 votes
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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 ...
Jan Stuller's user avatar
  • 6,098
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 ...
Chris Taylor's user avatar
  • 5,911
10 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 ...
NBF's user avatar
  • 1,068
9 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 ...
OGC's user avatar
  • 271
9 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 ...
AlRacoon's user avatar
  • 6,532
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}}\\ ...
Gordon's user avatar
  • 21.1k
9 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 ...
Kevin's user avatar
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9 votes
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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 = \...
ir7's user avatar
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8 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 &...
Chris Taylor's user avatar
  • 5,911
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 ...
LocalVolatility's user avatar
7 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 ...
Bram's user avatar
  • 812
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 ...
Daneel Olivaw's user avatar
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. ...
dm63's user avatar
  • 17.1k
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 $T$ -...
Chris Taylor's user avatar
  • 5,911
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, ...). ...
Kevin's user avatar
  • 15.9k
5 votes

What is the intuitive reason why the Gamma and the Theta tend to have the opposite sign?

I think I've found the answer to my question (I'm waiting for confirmation from you in the comments) The intuitive difference in this negative sign correlation depends on the position taken on ...
Carlo's user avatar
  • 143
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: $...
Daneel Olivaw's user avatar
5 votes
Accepted

How to compute gamma for at-the-money regular calls and puts when they approach expiration to avoid explosion of portfolio's gamma?

Many traders build a spreadsheet of how their delta changes across a range of market moves (-10%, -8%, ,....+8%, +10%) for example. That is a lot more useful than a single gamma number. It also ...
dm63's user avatar
  • 17.1k
5 votes

Gamma PnL Formula and Break-Even volatility

Good question! The answer to this is no. Let us work through a simple example to see why. Assume that the Gamma is $10$ and that the break-even move is $1$. For simplicity, also assume that, these are ...
Misha Wolynski's user avatar
5 votes

Optimal delta-hedging frequency when gamma scalping

The model I quite like as a base-case/rule of thumb is the Hoggard, Whalley, and Wilmott (1994) model. Assuming GBM - the number of shares, $N$, per interval is: $$N = Δ(S+dS,t+dt)- Δ(S,t)≈ Γ*dS$$ ...
Newquant's user avatar
  • 769
4 votes

Swaptions Gamma Interview Questions

Using Taylor polynomials of 2nd order:$$V(r+h)\approx V(r) + \frac{\partial{V}}{\partial{r}}h +\frac{1}{2}\frac{\partial^2{V}}{\partial{r}^2}h^2$$ $$V(r-h)\approx V(r) - \frac{\partial{V}}{\partial{r}}...
MaPy's user avatar
  • 263
4 votes

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

There are serious issues with how this graph is drawn, which impede understanding. The y axis is unlabeled and should be labeled "profits" or $\pi$ on a hedged position. The other axis should be ...
Alex C's user avatar
  • 9,372
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 ...
fni's user avatar
  • 1,886
4 votes

What is the intuitive reason why the Gamma and the Theta tend to have the opposite sign?

I think this is very well explained (with almost no maths) in the first chapter of Lorenzo Bergomi's book "Stochastic Volatility Modeling" (sample available here for download). Note that he explains ...
Quantuple's user avatar
  • 14.6k

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