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 ...
- 5,971
12
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 ...
- 6,490
11
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 ...
- 1,113
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)
$$
\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 ...
- 5,702
9
votes
Accepted
Numeric example to understand the effect of option gamma
Using our good friend Taylor, we know that
\begin{align*}
C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2,
\end{align*}
where $\Delta_C$ and $\Gamma_C$ are the call's ...
- 16.3k
8
votes
Accepted
Derivation of BS PDE problem using Delta hedging
This question has been asked many times and some clarifications appear needed.
As pointed out in an answer to this question, the portfolio
\begin{align*}
\Delta_t^1 S_t + \Delta^2_t C,
\end{align*}
...
- 21.3k
8
votes
Who has introduced the term 'vega' and why?
Joseph de la Vega wrote Confusion of Confusions in 1688, probably the World's first descriptive text on stock market processes and volatility.
I'm not sure that this is why Vega is thus named, but I ...
- 3,066
8
votes
How to prove Gamma is the same for a European call and European put with the same inputs?
Put-call parity says that a call and put (worth $C$ and $P$ respectively) with the same strike $K$ have the following relationship with the spot rate $S$, risk-free rate $r$, and time to maturity $T$ -...
- 5,971
8
votes
Accepted
Negative theta for a short put
Theta on a European Put option on a non-dividend paying stock is:
$$\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 ...
- 6,490
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 ...
- 940
8
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 ...
- 13.2k
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 ...
- 13.2k
8
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 ...
- 7,140
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 \...
- 16.3k
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 ...
- 2,700
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 ...
- 6,672
7
votes
Accepted
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
What is the relationship between Time-To-Expiry and Delta?
You are looking for the Greek commonly referred to as Charm. This is a quick visualization with a good chart I found on Google:
https://www.optiontradingtips.com/greeks/charm.html
- 386
6
votes
Accepted
Attempt of an analytical proof that a call price decreases as its strike increases
Something went wrong in the third equality of the equation where you compute $\partial C_0 / \partial K$. Starting from the second equality, you can use that
\begin{equation}
S_0 \mathcal{N}' \left( ...
- 6,094
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$$
- 401
6
votes
Accepted
Black Scholes theta as function of time to maturity
With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise ...
- 16.3k
6
votes
VIX OTM put options decrease value after sharp decrease of underlying
Yes, the VIX took a sharp downfall on 2020/03/02, from 40.11 to 33.42 (-6.69).
But that is not what the 2020/04/15 Put options are based on, they are based on the 2020/04/15 VIX Futures (VIJ20), ...
- 11.8k
6
votes
how to calculate vega in stochastic vol?
This is an interesting question. Peter A is correct that SV is typically combined with LV these days to get the so called SLV (stochastic local vol model).
There is no obvious definition for Greeks ...
- 9,884
6
votes
Accepted
Confusion about replicating a call option
In Black Scholes
$$\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 ...
- 940
6
votes
Accepted
Numeraire explanation on currency greeks
The best way is to start with definitions (instantaneous and their finite difference versions) of Greeks.
For a currency pair $(FOR,DOM)$ with FX rate $S$, the number of $[DOM]$ (domestic, numeraire, ...
- 5,173
6
votes
Numeraire explanation on currency greeks
Everything (warning: I have not checked 3rd order greeks) that is not delta is in terms of ccy2 in the standard Garman Kohlhagen model. Gamma is not in CCY1 by default either (some vendors like ...
- 9,884
6
votes
Mixed greeks in Python - How plot the following
Here is another solution using Plotly.
First of all let me correct a typo in your code
...
- 12.6k
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