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Speaking on a high level, in the Black-Scholes model the $f\left(T,S_{T}\right)$ payoff's value dynamic is given by

$$df\left(t,S_{t}\right)=\left(\frac{\partial f}{\partial t}\left(t,S_{t}\right)+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}\left(t,S_{t}\right)\sigma^{2}S_{t}^{2}\right)dt+\frac{\partial f}{dx}\left(t,S_{t}\right)dS_{t},$$

where $S$ is the spot price process of an underlying, $\sigma$ is a constant volatility parameter. We try to catch this dynamic with a replicating portfolio of the form

$$dX_{t}=r\left(X_{t}-\Delta_{t}S_{t}\right)dt+\Delta_{t}dS_{t},$$

where $r$ is the constant risk free rate, $\Delta_{t}$ is the amount of underlying to hold in order to construct the replicating portfolio. To have $X=f$ $\forall t$, $dX_{t}=df_{t}$, we have $\frac{\partial f}{\partial x}\left(t,S_{t}\right)=\Delta_{t}$ and get a partial differential equation to equate the $dt$s term

$$\frac{\partial f}{\partial t}\left(t,x\right)+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}\left(t,x\right)\sigma^{2}x^{2}=r\left(f(t,x)-\frac{\partial f}{\partial x}\left(t,x\right)\right),$$

which PDE's solution give the value of the $f\left(T,S_{T}\right)$ payoff at $t$, when the underlying price is $S_{t}$.

So in the “derivation” above, it is reasonable to say that via holding $\Delta_{t}=\frac{\partial f}{\partial x}\left(t,S_{t}\right)$ at every $t$, the risk sourcing from the spot price change is hedged. But how are the $\frac{\partial f}{\partial t}\left(t,S_{t}\right)$ theta and $\frac{\partial^{2}f}{\partial x^{2}}\left(t,S_{t}\right)$ gamma risk are hedged? Is it fair to say that (theoretically) these “risks” are hedged via the act of asking for $X_{0}=f\left(0,S_{0}\right)$ price for the trade in order to construct a replicating portfolio?

How are these risks (gamma, theta, kappa etc) are hedged in practice/ in real life?

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    $\begingroup$ In the BS framework, profits(losses) from gamma are equal to the losses(profits) from theta. In reality, IV > RV most of the time, so you’d profit from just selling options. If you want a literally example; path dependent options can be used to hedge gamma, e.g. barrier options. Or you can setup a gamma-neutral spread such that the multiple options you have give a neutral gamma. I.e. portfolio of short and long an options such that the sum of their gamma’s equal 0. $\endgroup$ Commented Feb 17 at 12:38

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In the Black-Scholes model Gamma and theta do not need to be hedged because the BS PDE says that they balance each other (I'll take $r = 0$): $$ \frac{\partial f}{\partial t} + \frac12 \sigma^2 S^2\frac{\partial^2 f}{\partial S^2} = 0 $$

The need for Gamma hedging comes from model mis-specification. What that means is this: You price an option assuming that the spot price satisfies $$ dS = \sigma S dW $$ where $\sigma$ is some constant. However, the actual dynamics of the spot price may be $$ dS = \sigma_R S dW, \quad \sigma_R \neq \sigma $$ where for simplicity let's assume that $\sigma_R$ is also constant, and the subscript $R$ stands for realized.

Then the change in the value of a delta-hedged position is (by Ito-calculus) $$ df - \frac{\partial f}{\partial S} dS = \frac{\partial f}{\partial t} dt + \frac12 \frac{\partial^2 f}{\partial S^2} (dS)^2 $$ Now $(dS)^2$ is $\sigma^2_R S^2 dt$, since we mis-spcified the model, and by the BS PDE $\frac{\partial f}{\partial t} = - \frac12 \sigma^2 S^2 \frac{\partial^2 f}{\partial S^2}$ since we priced $f$ assuming that $dS = \sigma S dW$.

So the P/L from the delta-hedged position is $$ df - \frac{\partial f}{\partial S} dS = \frac12 S^2 \frac{\partial^2 f}{\partial S^2} \left( \sigma_R^2 - \sigma^2 \right) dt $$

So if you want zero p/l you'll need to hedge Gamma by trading another option in such a manner that delta hedging that option gives an offsetting Gamma P/L.

In reality $\sigma_R$ is not even constant and you get all kinds of other P/L terms, but the main point you should internalize is that models are almost surely mis-specified and that other options are needed to hedge away the mis-specification, should you wish to do so.

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  • $\begingroup$ Kinda make sense, but not fully. Using your reasoning and according to the PDE, they only balance each other perfectly when $r=0$, as your example suggests. Isn't theta the same as (-"dollarized gamma")+(infinitesimal change in the bank account) and in this case indeed the balance is perfect? $\endgroup$
    – Kapes Mate
    Commented Feb 17 at 20:35

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