# List of financial derivatives Ito's Lemma does not apply

According to Ito's Lemma there is no restriction on the continuity of the stochastic process. The restrictions are on the continuity of the pay-off so that second derivatives with respect to underlying exists.

What are the list of financial instruments where their evolution (derivative) cannot be explained by Ito's Lemma? I have thought about barrier but the PV of those options are also continuous.

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I you talking about vanillas ? In the OTC case you can construct whaterver you want and thus you are free in your choice of pay-off function $f(X_T)$- meaning that $f$ can be easily designed to not fit the conditions of Itô's Lemma –  Probilitator Mar 22 at 14:01
Please give an example real trade that we cannot use Ito. Here I believe one considers the value of the derivative from initiation to final time point. Thus discontinuity at maturity does not make application of Ito inappropriate –  adam Mar 22 at 15:59
okey perhaps I am just misundersting you and would like to understand you queston correctly :) - could you thus please explain what it is exactly that you mean by "Applying Ito's Lemma to a financial instrument" ? - are you talking about something like Feyman-Kac where the Lemma is used in the proof ? –  Probilitator Mar 22 at 16:46
where $df(X_t)$ is not following Ito diffusion process –  adam Mar 23 at 7:49
The question doesn't make much sense. If the underlying dynamics of $X_t$ contain a discontinuous part, then $f(X_t,t)$ will also be discontinuous in general; even when $f$ is a smooth function. –  pbr142 Mar 23 at 9:36

adam I still think that your question is a bit vague but perhaps the following will be of some help to you.

First of all Itô's theorem is a tool. It will never give you the price by itself. While working out the concrete formula one might end up using it in one context or another.

In case of a european option, a borel measurable function $h$ and $X_t$ being an Itô Process one has $$g(t,x)=\mathbb{E}[h(X_T)|X_t=x]$$ It can be shown that $g(t,x)$ is smooth and thus we can apply Itô.

In the case of american options we can exercise whenver we want. Let $\Phi(s,X_s)$ be the value if the option is exercised at time $s$. The generic price-formula for an american type option is given by

$$v(t,x)=\sup_{t\leq \tau \leq T}\mathbb{E}[\Phi(\tau,X_\tau)|X_t=x]$$

Due to the supremum one can no longer simpli apply Itô directly to $v(t,X_t)$. There are some cases where the $\sup$ of a function will also be smooth but that must not necessarily be the case.

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Probilitator: Can you give a reference for the smoothness result for $g(t,x)$? I was just looking for something in that direction. –  g g May 28 at 12:09
@gg I will have to look through my text-books. The last time I looked for it, it did take me a while to find the proof. The argument is however trivial if you assume $h$ to be bounded - would this suffice to meet your needs ? –  Probilitator May 28 at 13:12