2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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*}

$\endgroup$
  • $\begingroup$ thanks for your answer. I made the same reasoning. However, if you apply Ito's formula you get $dX_{t} = \dfrac{\alpha_{t}}{S_{t}} dS_{t} + S_{t} d\dfrac{\alpha_{t}}{S_{t}} + d \langle S_{t} , \dfrac{\alpha_{t}}{S_{t}} \rangle$, and then you need to have $ S_{t} d\dfrac{\alpha_{t}}{S_{t}} + d \langle S_{t} , \dfrac{\alpha_{t}}{S_{t}} \rangle = 0$ (to have the dynamics of the wealth process) what seems very convenient and do not have a financial sense. $\endgroup$ – Ivan Aug 30 '18 at 0:53
  • $\begingroup$ Generally, you need another asset, such as the deposit account, to make the investment self-financing. But here we can only heuristically argue like that given that you have only the risky asset. $\endgroup$ – Gordon Aug 30 '18 at 12:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.