Deriving Risk-Neutral Log-Price Dynamics with Multiplicative Noise
Summary
The document sets up an equity log-price as a function of firm asset value and time, plus a mean-reverting market-noise component. It gives real-world dynamics for the asset value and noise, then applies Itô’s lemma to obtain the drift and two diffusion terms for the log price. The question asks how to derive the risk-neutral dynamics and presents a thesis claim: if the discounted stock is a martingale and volatility is unchanged across measures, the log-price drift must equal the risk-free rate less half the total instantaneous variance.
The excerpt supplies the claimed result but does not provide the derivation or an answer to the question. The key mathematical route is to convert log-price dynamics to stock-price dynamics with Itô’s lemma, then impose the discounted-martingale condition on the stock drift. This conclusion presumes the specified volatility structure and a valid risk-neutral measure; the treatment does not explain how the underlying asset and noise Brownian motions are jointly changed or whether the model parameters admit such a measure.
Key ideas
- Itô’s lemma gives the real-world log-price drift and diffusion from the asset-value and noise processes.
- The log price contains separate diffusion contributions from firm value and market noise.
- A discounted stock martingale condition fixes the stock drift under the risk-neutral measure.
- The log-price drift then includes a correction equal to half the total instantaneous variance.
- The excerpt states the target dynamics but leaves the measure-change derivation unresolved.
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Full text
# Derive Q-dynamics of $\ln S_t$ having multiplicative error structure
# Derive Q-dynamics of $\ln S_t$ having multiplicative error structure
From Kwon, T. Y. (2012). Three essays on credit risk models and their bayesian estimation (Doctoral dissertation):
Assume the following log equity price model: $$\ln S_t = g_S(V_t,t,\Theta_V) + Z_T$$
where $g_S$ is a function of $V_t$ (value of firm's assets), $t$ (time) and $\Theta_V$ (parameters governing the asset dynamics); and $Z_t$ - market noise.
We also model $V_t$ and $Z_t$ as follows:
$$ \begin{align} dV_t &= \mu V_t dt + \sigma_V d W_t^{\mathbb{P}V}\\ dZ_t &= -\theta_Z Z_t dt + \sigma_Z dW_t^{\mathbb{P}Z} \end{align} $$
By applying Itô's lemma, we can derive the $\mathbb P$ dynamics of $\ln S_t$:
$$ \begin{align} d \ln S_t &= d g_S + d Z_t \\ &= \frac{\partial g_S}{\partial t} dt + \frac{\partial g_S}{\partial V} dV + \frac{1}{2}\frac{\partial^2 g_S}{\partial V^2} (dV)^2 -\theta_Z Z_t dt + \sigma_Z d W_t^{\mathbb{P}Z}\\ &= \left(\frac{\partial g_S}{\partial t} + \frac{\partial g_S}{\partial V} \mu V_t + \frac{1}{2}\frac{\partial^2 g_S}{\partial V^2} \sigma_V^2 V_t^2 -\theta_Z \left(\ln S_t- g_S\right) \right) dt + \\ &\qquad+ \frac{\partial g_S}{\partial V}\sigma_V V_t d W_t^{\mathbb{P}V} + \sigma_Z d W_t^{\mathbb{P}Z} \end{align} $$
> Task: Derive the $\mathbb{Q}$ dynamics of $\ln S_t$.
The text of the thesis then says:
> Since the discounted $S_t$ is a martingale under $\mathbb Q$ and its volatility term remains unchanged under both measures $\mathbb P$ and $\mathbb Q$, we now conclude that under $\mathbb Q$, $S_t$ follows: $$ \begin{align} d \ln S_t &= \left(r - \frac{1}{2}\left(\frac{\partial g_S}{\partial V} \sigma_V V_t\right)^2 - \frac{1}{2}\sigma_Z^2 \right) dt +\\ &\qquad+ \frac{\partial g_S}{\partial V}\sigma_V V_t d W_t^{\mathbb{Q}V} + \sigma_Z d W_t^{\mathbb{Q}Z} \end{align} $$
It is not entirely clear for me why this result follows from that. I would like to see the maths.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.