Deriving Wealth Dynamics from a Dollar Investment in a Risky Asset
Summary
The document explains why a stochastic-volatility model writes wealth changes in terms of the dollar amount invested in a risky asset rather than the asset price itself. If an investor commits an amount to the asset, the number of units held is that amount divided by the current price. Applying the asset’s return over a small time interval to the invested amount gives the stated wealth dynamics, with drift and volatility inherited from the asset.
The setup includes an auxiliary stochastic process that affects the asset’s drift and volatility, as well as correlated Brownian shocks. The accepted explanation focuses on the risky holding and does not include a cash account or interest rate, consistent with the displayed wealth equation’s simplified setup. It sketches the differential using a short time increment but does not address portfolio rebalancing details, admissibility beyond the stated integrability condition, or the martingale expression used elsewhere in the question.
Key ideas
- The portfolio variable represents the dollar amount invested in the risky asset.
- Dividing invested dollars by the asset price gives the number of units held.
- The change in wealth over a short interval follows the return earned on the invested amount.
- The asset’s drift and volatility determine the corresponding drift and diffusion terms in wealth.
- The displayed formulation omits a separate riskless asset and interest rate.
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# How to deduce the formula of the wealth process of a stochastic volatility model?
# How to deduce the formula of the wealth process of a stochastic volatility model?
I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu.
The authors consider the utility function $U: \mathbb{R} \to \mathbb{R}$, with
\begin{align} U(w) = -\exp{\left( - \gamma w \right)} \end{align}
and the dynamics of the risky asset and the dynamics of the auxiliary process as follows.
\begin{align} dS_{t} = S_{t}[\mu(Y_{t}) dt + \sigma(Y_{t})dW_{t}^{1} \\ dY_{t} = g(Y_{t})dt + \beta(Y_{t})[\rho W_{t}^{1} + \sqrt{1 - \rho^{2}} dW_{t}^{2}] \end{align} where $\rho$ is the correlation of the two noises.
According to the article, they want to compute a utility-based price option. For that purpose they make use of the dynamics of the wealth process
$dX_{t} = \alpha_{t}(\mu_{t}(Y_{t})dt + \sigma(Y_{t})dW_{t}^{1}), X_{0}=x$.
Where $\alpha_{t}$ is a $\mathcal{F}_{t}$-adapted process representing the amount of money invested in the risky asset at time $t $ such that
$E \int_{0}^{T} \alpha_{t}^{2} dt < \infty$
Question: Does anyone knows how the authors deduce the formula for the wealth process? I mean how can they deduce the formula without mentioning the riskless asset, and the interest rate? Why do they use $\alpha_{t}$ in the wealth process instead of $S_{t}$ that is the risky asset?
By the way, they make use of the formula
\begin{align} M_{t} = \exp{\left\lbrace \int_{0}^{t} \left[ -\gamma \alpha_{u} \sigma(Y_{u}) dW_{u}^{1} - \dfrac{1}{2} \gamma^{2} \alpha_{u}^{2} \sigma^{2}(Y_{u})du\right] \right\rbrace} \end{align} that is a martingale.
I would really appreciate any hint or reference about how to deduce this formula. Thanks in advance.
## Answer by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/41496
By investing the amount of $\alpha_t$ at time $t$ in the risky asset, the wealth is given by \begin{align*} X_t = \frac{\alpha_t}{S_t} S_t, \end{align*} where $\frac{\alpha_t}{S_t}$ is the units of the risky asset. For $\Delta$ sufficiently small, the wealth at time $t+\Delta$ becomes \begin{align*} X_{t+\Delta} = \frac{\alpha_t}{S_t} S_{t+\Delta}. \end{align*} Then, \begin{align*} X_{t+\Delta}-X_t &= \frac{\alpha_t}{S_t}\left(S_{t+\Delta} - S_t\right)\\ &\approx \frac{\alpha_t}{S_t} S_t \Big(\mu(Y_t) \Delta + \sigma(Y_t)\big(W_{t+\Delta}^1 - W_t^1\big) \Big). \end{align*} That is, \begin{align*} dX_t = \alpha_t\Big(\mu(Y_t) dt+ \sigma(Y_t)dW_t^1 \Big). \end{align*}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.