Implied Volatility as a Pricing Convention and Model Input

# Implied volatility and pricing of vanilla options


# Implied volatility and pricing of vanilla options












As far as I understood, implied volatility (IV) is a lucky parametrization of the vanilla option's price. That is, instead of deciding how much the call worth now, you can decide on its IV and put this in the Black-Scholes (BS) formula since all other inputs (underlying price, time to maturity etc.) are readily available. In that case, we use IV $\sigma$ as a free variable which we adjust to fit the market prices.

This parametrization and the choice of the free variable is by no means unique: for example, we can say that instead of BS price for call $V(S,\sigma,\dots)$ we invent $W(S,\sigma,\alpha,\dots) = \alpha\cdot V(S,\sigma,\dots)$. In that case, we may estimate $\sigma$ as a 30-days end-of-day volatility of underlying returns (so that it becomes a measurable from market data, fixed quantity) and let $\alpha$ be a new free variable. It will have a similar effect: raise $\alpha$ to raise call price, and we can talk in term of implied $\alpha$ surface rather that IV surface.

The popularity of the IV parametrization seems to be in the fact that it's simpler, we just use the BS framework and don't have to come up with new variables. Am I right, or am I missing some points here?

The IV is hence an inconsistent model: it's like we pick up a random formula (say BS formula) with one free variable, and just try to fit the output to the market prices by changing the value of this variable. Because of that, we can't do much against the market - in contrast, would the CRR binomial model predict statistics of underlying prices correctly, if we get a market price significantly different from the CRR price, we can trade it and make a risk-free profit by hedging. The IV approach does not even seem to have a potential here: you are relying on the market prices, and cannot say whether they are right or wrong. Am I right here as well?

For the reasons above, I have the following question. Gatheral writes that more consistent stochastic volatility models are used to derive values for exotic options, parameters being fitted over the vanilla options prices. Does it mean that we can't do better with vanilla option prices just by using the IV approach?

Please tell me if the question is not clear, I'd be happy to fix that.

## Answer by Mark Joshi (score 6)

https://quant.stackexchange.com/a/15349

CRR is just a numerical approximation to Black--Scholes. Its main use is in getting American option price. There is no real difference other than slight inaccuracy when using it for Europeans. So no it wouldn't do what you ask.

Your questions are philosophical. What is the purpose of the model? if you estimate the volatility from a time series then you can use it to assess the prices of vanilla options. If you fit it to vanilla options then you can't but you can then use it to price exotics.

You might find looking Rebonato's Volatility and Correlation helpful.