Changing Measure for a Nonlinear Asset Return Process

Summary

The document poses a discrete-time asset-pricing problem with log returns that depend on the previous return and a fresh Gaussian shock. It asks how to choose a time-indexed exponential tilt so that a normalized, tilted asset price is a martingale under the original probability measure, then how to construct a new measure under which the asset price itself is a martingale. The conditional moment-generating function is identified as the normalizing quantity.

No solution, derivation, or numerical evidence is provided; the text is a request for guidance. A response would need to clarify the filtration and indexing, derive the conditional tilt from the martingale condition, and verify that the proposed density defines a valid probability measure. The setup illustrates the link between exponential tilting and risk-neutral measures, but does not establish that the required tilt exists or that the price process is arbitrage-free.

Key ideas

• The return dynamics include dependence on the previous period’s return.
• The question uses a conditional moment-generating function to normalize an exponential tilt.
• It seeks a measure under which the asset price is a martingale.
• The document poses the problem without supplying a derivation or confirming that a valid measure exists.

Tags

# Question about Stochastic Calculus,(change of measure)?


# Question about Stochastic Calculus,(change of measure)?












Can any one give some hint for this question?

> Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $S_t$ follows a discrete time stochastic process under the probability measure $\mathbb{P}$: $$ \begin{split} y_t &\equiv \log S_t - \log S_{t-1} \\ y_t &= 0.5 y_{t-1} + \eta_t \sqrt{1 + y_{t-1}^2} \end{split} $$ where $y_0=0,S_0=1$ and $y_i \sim \mathcal{N}(0,1)$ are iid. Find one $\theta_t\in F_t$ such that $$\frac{e^{y_t \theta_t}S_t}{M_t(\theta_t)}$$ is a martingale with respect to $F_t$ under the probability measure P, where $M_t(z)$ be the conditional moment-generating function of $y_t$, given $F_{t-1}$ Give a new probability measure $Q$ such that $S_t$ is a martingale with respect to $F_t$ under $Q$.

Many thanks!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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