21
votes
How do different models impact option Greeks?
This is an interesting and not so easy question. Here's my 2 cents:
First, you should distinguish between mathematical models for the dynamics of an underlying asset (Black-Scholes, Merton, Heston ...
- 14.5k
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
17
votes
Accepted
How much can be said about the Greeks without picking a model?
Find the topic of model-independent properties of option prices very interesting as well. Here are some results that I am aware of and the respective references in the literature. Some are already ...
- 5,959
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
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
10
votes
Accepted
Relationship between Vega and Gamma in Black-Scholes model
Consider any option, vanilla or exotic. In between fixing dates it satisfies the Black & Scholes PDE (for simplicity zero interest rate and dividends)
$$
\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 ...
- 5,622
9
votes
Accepted
Mathematical underpinnings of the square root of time rule
For any process with independent increments, by the very fact of statistical independence the variance of $x_{t3}-x_{t1}$ is going to be the sum of the variances of $x_{t2}-x_{t1}$ and $x_{t3}-x_{t2}$ ...
- 9,332
8
votes
Accepted
How does Algorithmic Differentiation work and where can it be applied?
Automatic Differentiation (aka AD) is a family of methods that are used to evaluate the derivative of a coded function. These methods are far more accurate than finite differences, since they are ...
- 511
8
votes
Who has introduced the term 'vega' and why?
Joseph de la Vega wrote Confusion of Confusions in 1688, probably the World's first descriptive text on stock market processes and volatility.
I'm not sure that this is why Vega is thus named, but I ...
- 2,996
8
votes
Accepted
Derivation of BS PDE problem using Delta hedging
This question has been asked many times and some clarifications appear needed.
As pointed out in an answer to this question, the portfolio
\begin{align*}
\Delta_t^1 S_t + \Delta^2_t C,
\end{align*}
...
- 21k
8
votes
Accepted
Numeric example to understand the effect of option gamma
Using our good friend Taylor, we know that
\begin{align*}
C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2,
\end{align*}
where $\Delta_C$ and $\Gamma_C$ are the call's ...
- 15.2k
8
votes
Accepted
Negative theta for a short put
Theta on a European Put option on a non-dividend paying stock is:
$$\Theta=-\frac{S_t \sigma}{2\sqrt{\tau}}N'(d_1)+rKe^{-r\tau}N(-d_2) $$
For deep in-the-money Puts, $d_1$ and $d_2$ go to negative ...
- 5,998
8
votes
How to calculate theta/rho for interest rate derivatives?
Interest rate traders/quants do not really talk about rho, as in the sensitivity of the Black Scholes price to $r$. The reason, I guess, being that we use Black (not Black-Scholes) formula for options ...
- 930
8
votes
What are "greeks" in general for non-standard options (swaptions, capfloors, etc)
Practically, few things in real life have convenient closed-form calculations.
Instead, you price some exotic, then you bump the various inputs, one or several at a time, up and down, by various small ...
- 11.9k
7
votes
Accepted
why Delta increases as interest rate increases
[Mathematically]
Risk-neutral pricing means that
\begin{align}
C_0(K,T) &= \mathbb {E}_0\left[\frac{1}{B_T} (S_T - K)^+\right] \\
&= \mathbb {E}_0\left[\left(\frac {S_T}{B_T} - \frac {K}{B_T}...
- 14.5k
7
votes
Accepted
Do Perpetual American Options have closed form functions to compute the Greeks?
The Black-Scholes differential equation is a second-order PDE in two dimensions and reads as
\begin{align*}
\frac{\partial f}{\partial t} + rx\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2 x^2 \...
- 15.2k
7
votes
How to adjust delta hedging if stock price decreases?
You are long a vanilla option, so long gamma (positive gamma). If the stock price decreases, so does the delta of your option.
Since you short-sold the stock to hedge, you now have short-sold too ...
- 2,570
7
votes
Accepted
How to adjust delta hedging if stock price decreases?
You would be over hedged in your call position if it was delta neutral before the stock cratered. Since you are long delta on the call, you would have shorted stock to make the original position ...
- 5,662
7
votes
Good references on PNL explain?
I'm not aware of any great reference. However Peter Nash Effective product control: controlling for trading desks. Wiley (2018) chapter 10 Review of Mark-to-Market P&L is a good start. Andrew ...
- 11.9k
6
votes
Accepted
Derive vega for Black-Scholes call from this formula?
Note that,
\begin{align*}
\frac{\partial{C}}{\partial{\sigma}} &=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\frac{-1}{\sigma})(d_-)-\frac{Ke^{-rt}}{\sqrt{2\pi}}e^{\frac{-d_-^2}{2}}(\frac{-1}{\...
- 21k
6
votes
Who has introduced the term 'vega' and why?
I was one of the floor traders in bond options in the early 80's. Knowledge of options was growing fast at the time primarily lead by the O'Connor brothers who were grain traders from the CBOT. They ...
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
Accepted
What is the name (Greek) for sensitivity of an option's Theta to the Time to maturity?
No
Because the P&L it generates is in $O(dt^2)$. Ito's lemma tells you that you can ignore this P&L.
$$PnL = \frac{\partial^2 V}{\partial t^2}dt^2 = 0$$
- 401
6
votes
Accepted
Black Scholes theta as function of time to maturity
With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise ...
- 15.2k
6
votes
VIX OTM put options decrease value after sharp decrease of underlying
Yes, the VIX took a sharp downfall on 2020/03/02, from 40.11 to 33.42 (-6.69).
But that is not what the 2020/04/15 Put options are based on, they are based on the 2020/04/15 VIX Futures (VIJ20), ...
- 10.9k
6
votes
Accepted
Confusion about replicating a call option
In Black Scholes
$$\frac{dS}{S}=rdt+\sigma dW$$
$dC_{BS}(S,t)=\underbrace{\frac{\partial C_{BS}}{\partial t}dt}_{Theta PnL}+\underbrace{\frac{\partial C_{BS}}{\partial S}dS}_{DeltaPnL}+\underbrace{\...
- 401
6
votes
What are "greeks" in general for non-standard options (swaptions, capfloors, etc)
If the question is how one defines Greeks for interest rate options, then it is a relatively straightforward extension of the concept from the basic idea for say equity options. They are defined as ...
- 930
6
votes
Effect of Implied volatility on option delta
In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend ...
- 6,425
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