23
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}\...
- 21.2k
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 ...
- 5,931
17
votes
Accepted
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 ...
- 2,591
9
votes
Accepted
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 $x_{t3}-x_{t2}$ ...
- 9,392
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}}\\
...
- 21.2k
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 &...
- 5,931
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.
...
- 17.5k
5
votes
Accepted
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 ...
- 6,064
5
votes
Accepted
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 ...
- 3,700
5
votes
how to calculate vega in stochastic vol?
This is an interesting question. Peter A is correct that SV is typically combined with LV these days to get the so called SLV (stochastic local vol model).
There is no obvious definition for Greeks ...
- 9,349
5
votes
Accepted
How do we hedge option vega practically?
If you are a market maker, your primary Vega hedge is to sell Vega to other clients. You do this by being the best offered side price in the market, so you will attract the next piece of business. ...
- 17.5k
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 ...
- 17.5k
4
votes
Accepted
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 ...
- 6,064
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: ...
- 6,652
4
votes
Accepted
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 ...
- 14.7k
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 ...
- 1,896
4
votes
Vega in the Heston model
Zhu makes sense to me.
The vega cash in Black-Scholes corresponds to a shift of the vol surface by 1%.
If you bump only $v_0$ in Heston, you bump only the short maturities, and if your structure is ...
- 1,434
4
votes
Accepted
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}(...
- 16.2k
4
votes
Accepted
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. ...
- 6,308
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 ...
- 9,349
4
votes
Purpose of Vega Hedging
Doing away with jargon for a bit, it is wise to hedge every risk factor present in your portfolio. Implied vol is one of them, just like the spot.
- 2,501
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....
- 6,993
3
votes
American Options relation between greeks
One way to think of American-exercise options is to break their value $V_A$ down into a value due to the european exercise, $V_E$, and a "premium" due to the possibility of early exercise, $V_P$
$$
...
- 14.9k
3
votes
Accepted
European option Vega with respect to expiry and implied volatility
To keep notations uncluttered, consider that $r=q=0$ in what follows, while focusing on the particular case of an ATM option i.e. $K=S$ (otherwise use the same reasoning with $K=F(0,T)=Se^{(r-q)T}$ i....
- 14.7k
3
votes
Accepted
Calculating Implied ATM Volatility with Vega
Assuming you mean inverting the Black Scholes Vega, it does seem possible:
Take the Vega formula:
$V = S \sqrt{\tau} n{\left (d_{1} \right)}= S \sqrt{\tau} \frac{1}{\sqrt{2 \pi}}e^{-0.5 d_1^2}$
...
- 6,222
3
votes
Accepted
What is vega, really?
Both equations for $S, v$ should remain the same as they govern the evolution of these quantities over time regardless of initial conditions. It is the initial condition (unstated here) that must ...
- 1,396
3
votes
how to calculate vega in stochastic vol?
The stochastic volatility model is calibrated to (a subset of) vanilla option prices. When the implied volatility is shifted to calculate vega, the model is calibrated again.
Although pure stochastic ...
- 494
3
votes
Accepted
Relationship between time decay and gamma
The relationship between theta and gamma is the Black-Scholes PDE.
Let's take normal B-S dynamics with $r=0$:
$dS_t = \sigma S_t dW_t$
The pricing PDE for a derivative $g(S_T)$ is (with terminal ...
- 674
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