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For example, when we price options on assets under the Heston model, we often compute the volatility of the volatility of the price of those assets implied by the market at time $t=0$ using the market price of options at time $t=0$. I've read that this is done for "hedging reasons", such as hedging exotic options using vanillas(?).

When we adjust our hedge at time $t=t'$, the vol of vol implied by market prices may be different from what we computed at time $t=0$. So, when we price exotics at time $t=0$, aren't we more concerned with our (the hedger's) expectations of the implied volatility at time $t=t'$, which may be different from the time $t=0$ implied volatility from the market?

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Let $\Theta$ be the vector of parameters on which a specific pricing model $\mathfrak{M}$ depends upon. For example, for the Black-Scholes model, $\Theta=\sigma$ where $\sigma$ is the implied volatility, whereas for the Heston model $\Theta=(\varkappa,\alpha,\nu)$ where $\varkappa$ is the vol's mean-reversion, $\alpha$ its long-run average and $\nu$ the vol of vol (assuming zero correlation between the underlying and its volatility).

Let $\theta^M$ be the value taken by the vector $\Theta$ if we imply values from the market, whereas $\theta^\prime$ the value taken using another method, for example the hedger's expectation of the implied parameters at a future time. Let $f$ be the pricing formula for the model $\mathfrak{M}$ for a certain product. Then the market price $P^M$ of the product verifies: $$P^M=f(\theta^M)$$ That is, the price is equal to the model's pricing function evaluated at the implied values $\theta^M$. Hence for any other method to determine model parameters, we have: $$f(\theta^\prime)\neq P^M$$ That is, the model will give a price which is different from the market price. This will generate an arbitrage. For example, if $\theta^\prime$ is such that: $$P^\prime:=f(\theta^\prime)>f(\theta^M)=P^M$$ Then if the hedger quotes the price $P^\prime$, another market participant might generate a riskless profit by selling the product to the hedger, then taking a long position in the market at the price $P^M$: he will cash in the amount $P^\prime-P^M>0$ and, because the products are the same, the terminal payoffs will also be the same, so there is no risk in the position. The hedger is therefore forced to use the market's implied parameters to avoid quoting arbitrageable prices.

Some additional thoughts:

  • Aside from quoting arbitrageable prices, if the hedger uses his own assessment of future implied parameters and trades at that price, then it should normally mean he is quoting a price which is more favourable that the market price. Therefore, he will have to record a negative PnL at time zero, equal to $P^M-P^\prime$.
  • The hedger might leverage his expectation of the future value of implied parameters when choosing his hedging strategy: once he has sold the product at a price $P^M$, if he expects the values of the implied parameters will evolve in his favour, he might tweak his hedging strategy accordingly, for example by not hedging at all.
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  • $\begingroup$ What about stability of the parameters? If the pricing model like Heston assumes fixed values of the parameters during the life of the trade, but in real life the parameters change day to day, how does it affect hedgers position? $\endgroup$
    – emot
    Jul 30, 2021 at 20:46
  • $\begingroup$ @emot in general you need to update the parameters to reflect their implied values. Your trade should be accounted at fair (i.e. market) value in the accounts, which is achieved by using the implied params. $\endgroup$ Aug 2, 2021 at 9:20
  • $\begingroup$ I assumed that we should update the parameters. When we calculate sensitivies to some risk factors (for example delta and implied vol) we hold the parameters of Heston model constant, right? But the next day we fit the model to the market and find that the parameters changed, doesn't it mean that our hedging is ineffective because in our hedging we assumed that parameters are constant? Or does the hedging strategy assumes also that the parameters change and we hedge that as well? $\endgroup$
    – emot
    Aug 2, 2021 at 9:29
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    $\begingroup$ @emot you might also include sensitivity of implied parameters to underlying risk factor, if your model allows. As for next day, you will need to update the parameters to the new implied values. You might also be interested in sticky-delta / sticky-strike rules, see the Goldman Sachs QS Research note on "Regimes of Volatility" (Derman, 1999). $\endgroup$ Aug 3, 2021 at 16:46
  • $\begingroup$ thank you very much! $\endgroup$
    – emot
    Aug 4, 2021 at 17:35

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