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
13
votes
Accepted
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}{\...
- 16.5k
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-...
- 14.5k
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 ...
- 14.5k
10
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 ...
- 1,068
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
$$\Delta \frac{S}{V} \lessgtr 1$$
...
- 10.9k
9
votes
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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
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 ...
- 14.5k
7
votes
Accepted
European Call Option Delta Upper Bound
It is false. Here is an example. Let
$$
dS_t = rS_t dt + f(S_0) S_t dW_t,
$$
$$
dB_t = r dt.
$$
The price is then the Black-Scholes price with volatility $f(S_0).$ The delta is the BS delta plus
$$
f'...
- 6,853
7
votes
Accepted
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/...
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
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
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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 ...
- 1,356
7
votes
Accepted
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 $\...
- 15.2k
6
votes
Accepted
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 ...
- 21k
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{...
- 5,008
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
6
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....
- 10.9k
6
votes
Accepted
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 ...
- 8,143
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.
- 1,231
5
votes
Accepted
Find call and put volatilities using ATM, Risk reversal and Butterflies volatilities
The strangle vol defined in your formula
\begin{align*}
Strangle(∆) = 0.5[Call Vol(∆) + Put Vol(∆)] - ATM Vol
\end{align*}
is the smile butterfly volatility. Then you have the volatility quote. Your ...
- 21k
5
votes
Accepted
What is the difference between par delta and zero delta?
Delta is a linear approximation of the change in price due to a small move of the relevant interest rate. Typically a parallel move of the whole interest curve is assumed here. This applies to all ...
- 153
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
- 376
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 ...
- 16.5k
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 $$C = Se^{-qT}\Phi(d_1)-Ke^{-rT}\...
- 15.2k
5
votes
Accepted
FRTB Delta risk sensitivity definitions
The goal of dividing by the bump $\delta$ is to rescale the sensitivity (slope) $s$ to a 100% bump. If $i$ is just a linear cash position with notional $n$, $V(nx)=nx$, and $$s=\frac{V((1+\delta)nx)-V(...
- 11.9k
4
votes
European Call Option Delta Upper Bound
In any arbitrage free model, you can define the BS implied volatility $\sigma_{BS}(S;T,K)$ of the model by writing call prices as
$$
C_{Mdl}(S;T,K) = C_{BS}(S;T,K;\sigma_{BS}(S;T,K))
$$
So the ...
- 3,936
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 ...
- 16.5k
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