What information about the stochastic process is available from path-dependent options?

Question

Assume the stock follows a process, which is defined by the following stochastic differential equation $$\frac{dS}{S}=r(t)dt+\sigma(S,t)dW,$$ so that the stock price process has local volatility.

European options: based on the prices for European options for all strikes and maturities, I can compute a probability density distribution at all times conditional on the current spot price. The idea is that I can differentiate the differentiate the price of the option $C(S,K) = \int_{0}^{\infty} max(S-K,0)\phi(S)ds$ two times, to get a formula for the transition probability density function: $$\frac{\partial^2C}{\partial K^2}(K,T)=-\phi(S)$$

Asian options: In this setting, is there additional information about the process, which I can extract from the prices of path-dependent options?

Answer 1

There is no difference in information, though the fitting algorithm may increase in complexity.

First note that in practice you never have an entire curve or surface of prices $C(K,T)$ of any kind of option. You only have a finite number of observations and even those typically have a bid and an offer.

I would therefore argue that the correct picture of the problem is as follows: given an $n$-parameter specification $\sigma(S, t; \vec{\mu})$ based on parameters $\mu_1,\dots,\mu_n$, which surface best fits a set of $M$ market observations $V_1,\dots,V_M$?

The case you cite on European calls takes some unspecified intermediate formula of call prices at a given maturity $T$, then differentiates it to get a snapshot of the integrated variance to $T$. Doing this for several maturities $T_1,\dots,T_N$ sort-of specifies local vols $\sigma(S, t)$ but only by adding a few further assumptions. So, you see, even the European option case you cite is not so clear-cut, having missed both the initial price curve spec and the inter-maturity spec. If you consider those price curve parameters, you see that they are elements of your $\vec{\mu}$.

Moving back to our more general picture, once you have chosen your functional form $\sigma(\cdot, \cdot; \vec{\mu})$ (whatever it may be), you can use it along with a nonlinear optimizer (perhaps simulated annealing) to fit your local vols to any set of market data you like, including exotics prices.

Answer 2

The Price of an American option may contain information on the expected behaviour of it's holder. When might he/she exercise the option ? Contrary to European options that don't.

Thus when you are primarily interested in "reconstructing" the transition density - I would stick with the European-Option-Prices.

If however you were to price path dependant options it would be wise to calibrate you model to these.