# Model-Free Option Pricing

From Breeden and Litzenberger (1978) and subsequent work, we may find the risk-neutral density $$q_{S_T}$$ of $$S_T$$ from European option prices - assuming there are enough traded options (e.g. SPX) via

\begin{align*} q_{S_T}(x) = \frac{\partial^2 C(K)}{\partial K^2}\Bigg|_{K=x}. \end{align*}

Having found this implied density, we could compute the price of every (European) derivative by numerically integrating the product of the payoff function with $$q_{S_T}$$. This would avoid choosing a specific stochastic model (Black-Scholes, Heston, SVJ, VG, NIG etc.) and hence it eliminates the model error.

So, instead of choosing some model and calibrating its parameter, we estimate the entire density curve and use it for pricing. This recovers the idea of relative valuaion, i.e. assuming all other traded derivatives are fairly priced, how can we price a new one without creating arbitrage by replicating the payoff with known instruments.

Since the density changes for every maturity, it is consistent within this framework to recalibrate''. On the other hand, suppose you pick a certain parameteric model, say Heston, which incorportas a long-term mean. Recalibration often leads to changed parameter values which is somewhat inconsistent as those values ought to remain constant.

I guess the weaknesses of the method include its sensitivity towards changes in the estimation technique of $$q_{S_T}$$, the lack of closed form solutions (despite the solutions for most models requires some numerical evaluation as well) and the requirement of decent available option data.

What is the opinion of the community? Is this a sensible idea or obvious nonsense for a reason I do not see? Has perhaps anyone already tried such an approach or have seen a paper/book which suggests a similar algorithm?

• I am not sure what precisely you're asking. For European path-independent options you only need to calculate the density once and afterwards in theory you could statically replicate the payoff using Carr-Madan. Of course if the next day you need to price another path-independent you'd indeed need to recalibrate the density again. For path-dependent you will need a model of some sort. Even if there is a very liquid market for forward start options I am not sure you can do without a model for path dependent options. – ilovevolatility Jul 16 at 8:48
• I think it is very reasonable to price path-independent European options as model-free as possible, which is exactly what Breeden-Litzenberger and Carr-Madan offers. But as you mentioned yourself, there isn't a continuum of strikes available so there will always be a bit of error. And then the hedging is also a different question since again trading a continuum of strikes to replicate a target claim is not feasible in pratice. – ilovevolatility Jul 16 at 8:57
• There are good methods available to smooth the IV surface and hence the density can be found as well, so Breeden-Litzenberger works for pricing. What I am saying is that hedging is different, so the cost of hedging error should be added to the theoretical price. I have to admit I read Kristensen and Mele long time ago and unfortunately didn't pay too much more attention to it. Upon re-reading I am more interested in it. I think there is a good chance of approximating prices of path-dependent claims especially if correlation between vol and spot is zero (in SV world), but that's just my hunch. – ilovevolatility Jul 16 at 10:07
• No problem. Yes IV surf needs to be extrapolated in strike and time to maturity. Take a look at Fengler or the SVI method. For the path-dependent, sorry I went off on a tangent, wasn't thinking about K&M but actually about put call symmetry and barrier option replication. In SV world with zero correlation put call symmetry still holds so barrier option replication is 'easier', and then maybe the path dependence can be tackled more readily. But ignore this vague comment of mine for now. – ilovevolatility Jul 16 at 10:40
• @KeSchn I once did something like that as part of an academic paper. I needed to extract the risk-neutral density from option prices, though the purpose was not pricing. I obtained an empirical CDF then fitted a range of parametric distributions such as Student , Lognormal, etc. to it. To assess the method, I tried to reprice the options and found extremely good results; only very, very deep OTM options exhibited errors between market and modelled price greater than 1%. – Daneel Olivaw Oct 24 at 13:01