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13 votes
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Skew and shadow delta

Basically, the author is saying that the delta of an option, $dC/dS = \frac{\partial C}{\partial S} + \frac{\partial C}{\partial v}\frac{\partial v}{\partial S}$, where the $\frac{\partial C}{\...
dm63's user avatar
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11 votes
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Mathematical equation relating $\frac{dV}{dS}$ to $\frac{dV}{dK}$

If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is space homogeneous, you have that the price of a European vanilla option is itself a space-...
Quantuple's user avatar
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11 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
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10 votes
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Different versions of sticky strike, moneyness and delta

I feel like your notations are not accurate enough to write what you would like to write. Let $\Sigma(S;K,T)$ denote the implied volatility of a European vanilla of strike $K$ and maturity $T$ now ...
Quantuple's user avatar
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10 votes

Which is riskier: a call option or the underlying?

A better, clearer, answer is to compute Lambda (leverage) of the option (link) and see if it is bigger or smaller than 1. Lambda is $\Delta \frac{S}{V}$ so we test $$\Delta \frac{S}{V} \lessgtr 1$$ ...
nbbo2's user avatar
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8 votes
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Stochastic Volatility and Sticky Delta

Intuitively, in a (log)-space homogenous diffusion model $$ S_t \propto S_0, \forall t \geq 0 $$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot ...
Quantuple's user avatar
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7 votes
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Calculate strike from Black Scholes delta

This is a little more complicated than the answer provided above since this is FX and the convention for determining the strike matters. https://www.researchgate.net/publication/...
FinanceGuyThatCantCode's user avatar
7 votes
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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}...
Quantuple's user avatar
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7 votes
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Strike / delta relationship for FX options

In FX world, the ATM strike is the delta-neutral strike, that is, the absolute delta values of a call and the corresponding put are the same. Moreover, the delta can be premium adjusted or not ...
Gordon's user avatar
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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 ...
siou0107's user avatar
  • 2,680
7 votes
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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 ...
AlRacoon's user avatar
  • 6,632
7 votes
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Possibility of delta greater than 1

Only constrained to be <1 in the simplified Black-Scholes setting with zero cost of carry on the underlying. In the more realistic and common setting where the cost of carry of the underlying is ...
Ivan's user avatar
  • 1,396
7 votes
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Which is riskier: a call option or the underlying?

As @ir7 did, I only briefly want to add to @noob2's spot-on answer. He's of course right and $\Lambda=\Delta\frac{S}{V}$ decides how risky the option is compared to the stock. Firstly, note that $\...
Kevin's user avatar
  • 16k
7 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 ...
Kermittfrog's user avatar
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7 votes

How is delta defined as a unit?

If you want to split hairs, as I like to do, there are 2 ways to express Delta. "Pure Delta" is a fraction, i.e. a number between 0 and 1 (for a Call). In terms of units, it is a pure number....
nbbo2's user avatar
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6 votes

Derivation of Call Delta from Black Scholes Model

Here's a mathematical derivation of the Black-Scholes delta. The call option price under the BS model is $$ C = S_0 N(d_1) - e^{-rT} K N(d_2) \quad\text{with}\quad d_{1,2} = \frac{\log(S_0\,e^{rT}/K)}...
Najee's user avatar
  • 101
6 votes

Which is riskier: a call option or the underlying?

Just a small addendum to @noob2's answer. The discrete shape of $\lambda$ is: $$\lambda \approx \frac{V_1 - V_0}{S_1 - S_0} \times \frac{S_0}{V_0} $$ which can be rewritten as $$ \lambda \approx \frac{...
ir7's user avatar
  • 5,043
6 votes
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Bartlett's delta gives wrong signs for calls and puts

Bartlett's delta as computed in your code is a simple finite difference (FD), also called bump and reprice, of the Black values. I do not think there is anything wrong here, besides the fact that you ...
AKdemy's user avatar
  • 9,024
6 votes

Delta of Black formula vs numerical

Your Delta_fd is forward delta (you're bumping the fwd). Delta is spot delta. Hence the discount factor.
user35980's user avatar
  • 1,426
5 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
milkmotel's user avatar
  • 376
5 votes

Calculate strike from Black Scholes delta

Just to skip to solution from the aforementioned paper: For a volatility surface of Delta $\Delta$ vs volatility $\sigma$, we can calculate the strike $K$ with underlying $f$,$\phi$ is 1 for call, -1 ...
rbonallo's user avatar
  • 171
5 votes

Why isn't the delta of a slightly in the money American option 1?

The delta is only 1 if the option is certain to be exercised. This is not the case if it is ‘slightly in the money’. If it is deep in the money, such that immediate exercise is optimal , then the ...
dm63's user avatar
  • 17.2k
5 votes

What is delta of an option signaling?

The comments already give links to many top answers and articles outlining the answer. Here's the summary: The Black-Scholes formula for European-style call options is $$C = Se^{-qT}\Phi(d_1)-Ke^{-rT}\...
Kevin's user avatar
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5 votes
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FRTB Delta risk sensitivity definitions

The goal of dividing by the bump $\delta$ is to rescale the sensitivity (slope) $s$ to a 100% bump. If $i$ is just a linear cash position with notional $n$, $V(nx)=nx$, and $$s=\frac{V((1+\delta)nx)-V(...
Dimitri Vulis's user avatar
4 votes

why Delta increases as interest rate increases

Just to strengthen the intuition in the perfect answer above: With r going very high (and hence F), all prices on cash instruments are expected to gain fast with time (to compensate for the carry) and ...
Mats Lind's user avatar
  • 1,412
4 votes

Interest Rate Risk - The Greeks

Receiving fixed on an IRS is both long delta and long gamma. The delta is obvious. The gamma is because the long position in delta increases as rates go down, and decreases as rates go up. Swaps ...
dm63's user avatar
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4 votes
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Option delta - Conditional probability definition?

IMHO the 'definition' you mention is not a mathematical definition per se, but rather an approximation used by some practitioners. Mathematically, it is $N(d_2)$ in the BS formula which figures the ...
Quantuple's user avatar
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4 votes
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Why " Even if the underlying asset price remains unchanged, the option delta for an in-the-money option increases as expiration nears"

@Kiwiakos gave you the intuition. Here is the corresponding analysis that you asked for. The European plain vanilla call delta is given by \begin{equation} \frac{\partial C_0}{\partial S_0} = \...
LocalVolatility's user avatar
4 votes
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What is the relationship between Time-To-Expiry and Delta?

Here is a page from The Options Guide with an understandable picture. They explain, As the time remaining to expiration grows shorter, the time value of the option evaporates and correspondingly, ...
rajah9's user avatar
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