Why Option Pricing Uses Market-Implied Parameters

# Why are implied parameters preferred over expectations of future implied parameters?


# Why are implied parameters preferred over expectations of future implied parameters?












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?

## Answer by Daneel Olivaw (score 2)

https://quant.stackexchange.com/a/66243

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.