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8

Dimensional analysis is the key: The change in option price is in dollars. The change in option price is of course the sum of its changes (partial derivatives) with respect to its underlying risk factors. However you cannot add terms with different dimensions, that would literally be trying to add apples and oranges. Let's look at delta, which (in finite ...

7

They are not the same, but they are related. Gamma is sensitivity to realized volatility. Vega is sensitivity to implied volatility. Vanilla options are always long gamma and long vega, so they are "long vol" and saying "I am a buyer of vol/gamma/vega" means that you are taking a position that benefits from a rise in volatility (either ...

4

In addition to the good answer given below: think of options with almost no maturity left (i.e. 1-day before expiry): for these options, the time value of the option is almost zero, so the option value as a function of the underlying has almost fully converged to the "step-function" of the pay-off at maturity (i.e. the classical "hockey-stick&...

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Your greeks are derivatives, not absolute price differences, when your underlying changes by 1%. Also, the put price and delta are not linear in the underlying (in fact in your case they are highly non linear) and you cannot expect for a large change (which the 1% change is) that the delta increases by the linear amount that you expected. You need to look at ...

3

The greeks are non-linear and only give you the instantaneous rate of change. The larger the change in underlying is, the less accurate the change based on the greeks will be. A doubling of the underlying will certainly not be predictable by the greeks alone. Delta and Gamma can be used to estimate the new price using a second-order taylor series ...

3

Agree with @Brian B. With BS, you cannot have the issue in (1). Tree, grid, Monte Carlo could all result in errors though. (2) is a likely reason. I just tried in Julia for ATM, 0 div and rates plus 0.2 vol and 1 year tenor. Shifts smaller than ~ 0.00008 result in an error for Gamma. Delta seems to be less sensitive for this, and it is fine for at least 1e-7 ...

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(Welcome to Quant SE. It looks like you haven't made up your mind on whether your shock is $dS$ or $S_0dS$. Also, $V_0$ doesn't belong in the denominator. You probably mean $S_0$. Please try to use Latex on this site next time you visit.) It's better to start with Taylor's theorem with remainder to convince yourself of the validity of these finite difference ...

3

Good question! The answer to this is no. Let us work through a simple example to see why. Assume that the Gamma is $10$ and that the break-even move is $1$. For simplicity, also assume that, these are unchanged by price moves in the underlying (this is reasonably accurate for small price moves), so: $\Gamma = 10$ $\delta S_{Break-Even} = 1$ Note that we ...

3

It's my understanding that indeed the cross partial derivative terms do have a contribution - so you're correct to say that what u really have to work with is a gamma matrix $$\gamma=[\frac{d^2V}{dr_idr_j}]_{i,j}.$$ In essence, the cross partial terms $\frac{d^2}{dr_idr_j}$ allude to the correlation between the various rates $(r_1,...,r_n)$. Assuming a high (...

3

The relationship between theta and gamma is the Black-Scholes PDE. Let's take normal B-S dynamics with $r=0$: $dS_t = \sigma S_t dW_t$ The pricing PDE for a derivative $g(S_T)$ is (with terminal condition $g$): $\frac{dp}{dt} = \frac{1}{2}\sigma^2S^2 \frac{d^2p}{dS^2}$ Or $\Theta = \frac{1}{2}\sigma^2S^2 \Gamma$ This PDE has a solution (Feynman-Kac Theorem): ...

2

Achieving gamma neutrality refers to your whole portfolio situation. If you have a portfolio $P$ made up of $n_S$ shares of stock $S$, and of $n_1$, $n_2$ option calls $C_1$, $C_2$ on $S$ (options 1 and 2 differ in strike price or expiration), pursuing a gamma hedging strategy would imply to achieve neutrality on the Delta $\Delta_P$ and Gamma $\Gamma_P$ ...

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Exact replicating portfolio for constant product AMM here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3550601 It's irritating that people use a new term 'impermanent loss' for something that happens with any option and has been well understood for decades!

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Also, I do not think Gamma = (Dplus-Dminus)/(Splus-Sminus) is correct, rather it is Gamma = (Dplus-Dminus)/(dS). You must use the increment dS throughout, for consistency. By making the denominator be a 2*dS step you are not correctly differentiating with respect to the variable "S".

1

You are generally correct in thinking that this strategy should make money most of the time, however I would warn you to be very careful with strategies that earn a small amount of money most of the time but lose many multiples of that when things go wrong. These are examples of “picking up pennies in front of a steamroller”. The options market is full of ...

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A calendar spread is nothing more than a short call and long call with TTM_1 and TTM_2. If you can find the quotes of those underlying options; you can retrieve the IV and hence the greeks.

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