Characteristic Functions in the COS Method and Lévy Processes

Summary

The document raises a question about why the COS method presents a characteristic-function relation specifically for Lévy processes. It notes that if a log price is represented as an initial log-moneyness plus a random log-price change, the characteristic function factors into the characteristic function of that change and a deterministic exponential shift. This algebraic factorization holds for any random variable representing the change, not only a Lévy process.

The question points toward a distinction between a general identity and the broader modeling assumptions used by the COS method. A Lévy process has stationary, independent increments, which can supply tractable time evolution and distributional structure; the excerpt does not include a response clarifying which part of the paper relies on those properties. It therefore identifies a useful modeling distinction but does not establish the full conditions under which COS pricing results apply.

Key ideas

• A deterministic shift in log-moneyness multiplies a characteristic function by a complex exponential factor.
• That shift identity does not by itself require the log-price change to come from a Lévy process.
• Lévy process assumptions can provide additional structure for modeling increments over time.
• The excerpt poses the distinction but does not give a complete account of the COS method's assumptions.

Tags

# Confusion re COS method and Levy Processes


# Confusion re COS method and Levy Processes












In the COS method (http://ta.twi.tudelft.nl/mf/users/oosterlee/oosterlee/COS.pdf), they say that

What do they mean that this holds for "Levy processes"? Surely this holds for all processes?

Here, $x = log(S_0/K)$, and $S_t = S_0 e^{L_t}$, and $\phi(\omega;x)$ is the characteristic function of $\log(S_t/K) = L_t + x$, and so

$$\phi(\omega; x) = E e^{i \omega (L_t + x)} = e^{i \omega x} E^{i \omega L_t} = \phi_{L}(\omega) e^{i\omega x}$$

.... this holds no matter what $L_t$ is, Levy or not. So why do they state it only holds for Levy processes?

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