When Itô’s Formula Applies to European and American Option Values

Summary

The document clarifies that Itô’s formula is a mathematical tool for analyzing a process, rather than a pricing method on its own. For a European option, it describes the value as a conditional expectation of the terminal payoff given the current state. Under the stated assumptions, this value function is smooth enough to apply Itô’s formula, even when the payoff itself is only measurable.

For an American option, the value is instead expressed as the best conditional expected payoff over possible exercise times. Taking a supremum can produce a value function that is not smooth, so direct application of Itô’s formula to that function may fail. The answer emphasizes that smoothness can hold in some cases, but it is not guaranteed. It does not provide a list of instruments where Itô’s formula is inapplicable, nor does it discuss barrier options specifically; its main lesson is to check regularity of the value function and account for the early-exercise feature.

Key ideas

• Itô’s formula analyzes stochastic processes but does not determine an instrument’s price by itself.
• A European option value can be represented as a conditional expectation of its terminal payoff.
• Under the stated conditions, the European value function is smooth enough for direct application of Itô’s formula.
• The supremum over exercise times for an American option may produce a nonsmooth value function.
• Whether Itô’s formula applies to an American option value depends on the smoothness of that value function.

Tags

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


# 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.

## Answer by Probilitator (score 1)

https://quant.stackexchange.com/a/10730

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.