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20 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 ...
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16 votes
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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 ...
16 votes
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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 ...
  • 5,638
12 votes
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Why gamma for ATM option decreases as volatility increases

Think of moving volatility in the other direction. As volatility approaches zero, any call strike strictly smaller than the ATM strike, $K<K_{ATM}$, will have zero probability of ending in the ...
  • 14.5k
12 votes
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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 \...
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11 votes
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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 ...
9 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 ...
  • 1,018
8 votes

The greeks: where do they come from?

First, my notation. $K$ is the strike price, $S$ is the stock price, $r$ is the continuously compounded risk-free rate, $T$ is time at expiration, $t$ is time at issue, $\sigma$ is volatility, $\delta$...
8 votes
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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 ...
8 votes
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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 $x_{t3}-x_{t2}$ ...
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8 votes
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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*} ...
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8 votes
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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 ...
  • 14k
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 ...
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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 ...
7 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 ...
  • 2,856
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}...
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7 votes
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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 \...
  • 14k
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 ...
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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 ...
  • 5,452
7 votes
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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 ...
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6 votes
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Can I add the greeks of individual postions to obtain greeks for the portfolio

most models in financial maths are linear so prices and Greeks just add. This is in particular true of Black--Scholes so Yes. However, once one starts taking into account value adjustments non-...
  • 6,763
6 votes
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Does higher vega imply higher IV and vice versa

Simply put, no. Vega depends on a variety of factors (including the level/price of the underlying asset). However, vomma/volga/vega convexity (whatever you want to call dVega/dIV) is always positive. ...
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6 votes
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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}{\...
  • 20.5k
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 $T$ -...
  • 5,638
6 votes
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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 ...
  • 14k
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), ...
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6 votes
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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{\...
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6 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 ...
6 votes

Relationship between Vega and Gamma in Black-Scholes model

I'll give a heuristic "proof" for general European claims which will cause mathematicians to feel sick, but which physicists / practitioners would probably be quite happy work with: Write ...
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 ...
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