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.

  • $\begingroup$ 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 $\endgroup$ – Probilitator Mar 22 '14 at 14:01
  • $\begingroup$ 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 $\endgroup$ – adam Mar 22 '14 at 15:59
  • $\begingroup$ 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 ? $\endgroup$ – Probilitator Mar 22 '14 at 16:46
  • $\begingroup$ where $df(X_t)$ is not following Ito diffusion process $\endgroup$ – adam Mar 23 '14 at 7:49
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    $\begingroup$ 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. $\endgroup$ – pbr142 Mar 23 '14 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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  • $\begingroup$ Probilitator: Can you give a reference for the smoothness result for $g(t,x)$? I was just looking for something in that direction. $\endgroup$ – g g May 28 '14 at 12:09
  • $\begingroup$ @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 ? $\endgroup$ – Probilitator May 28 '14 at 13:12

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