Mean Correction and Equivalent Martingale Measures for Time-Changed Lévy Prices

Summary

The document asks whether normalizing a time-changed pure-jump Lévy price process by its conditional expected exponential makes it risk-neutral in the stronger sense of defining an equivalent martingale measure. The proposed price scales the exponential process by the risk-free growth net of dividends and divides by the conditional expectation of that exponential, with the initial state of the time-change process specified.

The question distinguishes a discounted price process having the martingale property from proving that its probability measure is equivalent to a physical measure. It points out that the cited treatment does not define the physical measure, leaving the equivalence claim unclear. No derivation or resolution is supplied, so the document frames a measure-theoretic caveat rather than establishing that mean correction alone guarantees an equivalent martingale measure.

Key ideas

• Mean correction normalizes the exponential of a time-changed Lévy process by its conditional expectation.
• The pricing setup includes the risk-free rate and dividend yield in the price scaling.
• A discounted price being a martingale does not by itself establish equivalence to a physical measure.
• The document leaves the existence and equivalence of the relevant measures unresolved.

Tags

# Mean correcting martingale argument


# Mean correcting martingale argument












Let $X=(X_t,t\in\mathbb{N})$ be a pure jump Lévy process (i.e $X_t$ does not have a diffusion component). Let further $Y=(Y_t,t\in\mathbb{N})$ be a process that is used to time-change the process $X$ (there are several examples in Schoutens (2003) of these type of processes, such as the integrated CIR model). E.g, the time-changed Lévy process is given by $X_Y=(X_{Y_t},t\in \mathbb{N})$.

In Schoutens (2003), it is argued that for a price process $S=(S_t,t\in\mathbb{N})$ given by

$S_t=S_0 \frac{\exp((r-q)t)}{\mathbb{E}(\exp{X_{Y_t}}\vert y_0)}\exp{X_{Y_t}}$

where $q$ is the dividend yield, $r$ the interest rate, and $y_0$ the initial value of the underlying process for which $Y$ is based on, puts us into the risk-neutral world by mean-correcting arguments (because of the factor $\frac{\exp({r-q}t)}{\mathbb{E}(\exp{X_{Y_t}}\vert y_0)}$).

Later on Schouten suggests to use the process $S$ to price options directly in the risk neutral world. It seems that this is all based on arbitrage-free principles.

So now to my question: In what way does the mean correcting argument in this case yield an Equivalent Martingale measure? They don't even define the Physical measure? Even though the discounted price process is a martingale(?) how can it be sure that it is also equivalent to some physical measure?

Schoutens (2003) Lévy processes in finance, Wiley.

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