# 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.

• As Mark Joshi pointed out your questions seem philosophical. Vanilla option prices are nothing more than a reflection of the market's take on implied volatility. How you arrive at such volatility estimate is entirely up to you. If you believe you have a superior model to arrive at implied volatilities (aka, if you think you are able to better predict future price variation of the underlying) then employ whatever you like and trade it against market prices. You should over time extract alpha if your model is indeed superior. – Matt Nov 7 '14 at 4:47
• Thanks for the comment. Isn't coming up with an opinion on BS IV is equivalent to having an opinion on absolute prices of the vanilla options, since all other inputs for the BS model are available? If my understanding is right, then your comment reads to me pretty much as: to determine prices of vanilla options (in dollar units) we need to determine prices of vanilla options (in volatility units), but I am not sure whether it's what you meant. – Ulysses Nov 7 '14 at 8:12
• Partially, instead of occupying your time to predict future option prices you can "simplify" by predicting the implied volatility. Not that it is any easier but you somewhat boil it down to the essential. Some option traders believe they are more successful at modeling volatility than modeling asset prices. Hence they hedge out other risk exposures and focus on volatility trading. Btw, other BS inputs are anything but static over time. Consider the underlying price, consider even something as "trivial" as dividend curves of single name equity options. – Matt Nov 7 '14 at 9:31
• @MattWolf: thanks, that's was my idea. So essentially, volatility in this sense is a useful parametrization of price that allows focusing mostly on pricing this parameters. According to this reasoning, trading vanilla options is direct - you can't really always say whether the market prices options wrong or right, unlike in the BS world (I hope I clarified this in my comment to Mark Joshi). – Ulysses Nov 7 '14 at 9:36
• @experquisite: that's exactly my point, and I don't question the usefulness of the BS parametrization. Rather, I wonder whether it has some value for pricing vanilla options beyond being a useful parametrization. Also, what's v/42 model? Or do you mean the vol divided by the ultimate answer? – Ulysses Nov 10 '14 at 8:02

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.

• Thanks for the answer and sorry, I should have indicated that I used CRR instead of BS since even if the stock prices would follow BS, we still won't be able to hedge them continuously, whereas in CRR case we can do perfect hedging. That's a minor point, though. I do agree with your second paragraph. To clarify, I wonder if there exists a model to price vanilla options with the following BS feature: if market price is far from the model price, the model would also tell me how to make an (almost) risk-free profit from that situation. – Ulysses Nov 7 '14 at 8:06
• well BS gives you a hedging strategy that will realize the replication price if you have the correct volatility. You have to hedge continuously which is impossible but if you hedge frequently you only get a very small error so you can realize the arbitrage. – Mark Joshi Nov 8 '14 at 19:48
• Leaving variance caused by imperfect hedging errors, could you clarify what do you mean by having the correct volatility? AFAIK, if the underlying price follows GMB with deterministic time-dependent diffusion coefficient $\sigma(t)$ (in particular, if $\sigma$ is constant) then the knowledge of $\sigma$ indeed tells us how to realize the replication price. However, I am not sure whether this even theoretically applies already to the case of stochastic $\sigma$ since the market is incomplete. – Ulysses Nov 10 '14 at 8:09