21
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}\...
- 21k
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 ...
- 5,891
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 \...
- 21k
11
votes
Accepted
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 ...
- 5,891
11
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 ...
- 5,998
10
votes
Accepted
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 ...
- 1,068
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 ...
- 5,672
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 ...
- 792
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}}\\
...
- 21k
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 = \...
- 5,008
8
votes
Accepted
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 ...
- 15.3k
8
votes
Accepted
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 &...
- 5,891
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 ...
- 5,959
7
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.
...
- 16.6k
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 ...
- 241
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 ...
- 7,999
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 ...
- 802
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 $T$ -...
- 5,891
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, ...). ...
- 15.3k
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 ...
- 133
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:
$...
- 7,999
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 ...
- 16.6k
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 ...
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:
$$N = Δ(S+dS,t+dt)- Δ(S,t)≈ Γ*dS$$
...
- 749
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 ...
- 9,332
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 \...
- 21k
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 ...
- 1,886
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