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

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?

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What is the density are you talking about exactly - could you elaborate ? Do you mean the transition density with the parameters you calibrated from the vol-surface ? –  Probilitator Feb 18 at 9:12
@Probilitator I would like to compute the transition density probabilities form option prices directly. I have added more information. –  user1157 Feb 18 at 11:00
okey but to do that you will need to do some curve fitting to the price-surface and then differentiate this function. Am I correct ? –  Probilitator Feb 18 at 11:56
Curve fitting could be one way to go. Currently, I am not so much concerned in the numerical procedure, but in the difference in information which is contained in European and Asian options. –  user1157 Feb 18 at 12:39
Asian options are path dependant - so one would assume you would get some information on how $S_t$ will move as opposed to European Options where PayOf only denpends on $S_T$. Still I would argue that by going through several maturities for Europeans you will also incorporate most of this infomration. Why don't you apply you approach to both Americn Option prices and European option prices separatly and compare the resulting densities. –  Probilitator Feb 18 at 12:47

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.

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what are the $V_1, \dots V_M$ - volas or prices ? –  Probilitator Feb 25 at 8:12
If exotics, they would likely be prices, If vanilla options they might be in volatility terms but for these purposes you can think of them as then being transformed to prices before engaging in the fit. –  Brian B Feb 25 at 17:15

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.

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