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22 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 ...
Quantuple's user avatar
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20 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 ...
Chris Taylor's user avatar
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17 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 ...
LocalVolatility's user avatar
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 \...
Gordon's user avatar
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12 votes
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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 ...
Jan Stuller'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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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 ...
Antoine Conze's user avatar
9 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 $x_{t3}-x_{t2}$ ...
Alex C's user avatar
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9 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 ...
Kevin's user avatar
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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 ...
StackG's user avatar
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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*} ...
Gordon's user avatar
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8 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 ...
Jan Stuller's user avatar
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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 ...
piterbarg's user avatar
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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 ...
Dimitri Vulis's user avatar
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 ...
Dimitri Vulis'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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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 \...
Kevin's user avatar
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7 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 $T$ -...
Chris Taylor'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
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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 ...
AlRacoon's user avatar
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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 ...
Michael Shutze's user avatar
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}{\...
Gordon's user avatar
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6 votes
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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. $$PnL = \frac{\partial^2 V}{\partial t^2}dt^2 = 0$$
ryc's user avatar
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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 ...
Kevin's user avatar
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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), ...
nbbo2's user avatar
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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{\...
ryc's user avatar
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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 ...
piterbarg's user avatar
  • 940
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 ...
Kermittfrog's user avatar
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6 votes
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Mixed greeks in Python - How plot the following

Something like this? ...
David Duarte's user avatar
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