I am trying to solve the following asset price model and wonder if the math is correct:
I consider a finite horizon economy with $T+1$ dates, $t=0,1,2, ...T$
There exists a risk-free asset with interest rate of 0 and a risky asset with supply $Q$ that has a claim to a single dividend $D_T$ with value
$D_T=D_0+\varepsilon\ _1+\ldots+\varepsilon\ _T, where \ \varepsilon_t\sim N\left(0,\sigma_\varepsilon^2\right), i.i.d over time$
and
$D_t=D_0+\varepsilon\ _1+\ldots+\varepsilon\ _t, \ where \ \varepsilon_t\sim N\left(0,\sigma_\varepsilon^2\right), i.i.d over time$
There are two types of utility maximizing traders with constant absolute risk aversion$\gamma$ defined over their next period’s wealth: fundamental traders making up $u^f$of the economy and extrapolators making up $u^e=1-u^f$ of the economy. Extrapolators’ time-t expectations about the risky asset’s future price changes is a weighted average of past price changes with more weight put on more recent changes:
$\ \mathbb{E}_t^e\left(P_{t+1}-P_t\right)=X_t\equiv\lambda_t\ast\left(1-\theta\right)\sum_{k=0}^{t-1}{\theta^k\left(P_{t-k}-P_{t-k-1}\right)}+\theta^tX_0$
Parameter $\theta$ is in the (0,1) interval. $X_0$ measures extrapolators’ expectations in $t\ =0$ and will be set to a neutral, steady value.
The demand function of fundamental traders is:
$N_t^f=\frac{D_t-\left(T-t-1\right){\gamma\sigma}_\epsilon^2Q-P_t}{{\gamma\sigma}_\epsilon^2}$
Demand function of extrapolators is:
$N_t^e=\frac{X_t}{{\gamma\sigma}_\epsilon^2}$
Market Clearing Condition:
$Q=u^e N_t^e+u_\ ^fN_t^f$
I would like to solve this for $P_t$. My attempt gave me this:
$P_t=\frac{D_t+\left(u^e/u^f\right)\left(\lambda_{t-1}\theta X_{t-1}-\lambda_{t-1}\left(1-\theta\right)P_{t-1}\right)-\left(\gamma\left(T-t-1\right)\sigma_\epsilon^2Q+\left(1/u^f\right)\gamma\sigma_\epsilon^2Q\right)}{1-\left(u^e/u^f\right)\lambda_t\left(1-\theta\right)}$
However, I would like to check somehow that I didn't make any mistake. My attempts to use Mathematica were to no avail, unfortunately.
The model is very similar to Da, Zhi and Huang, Xing and Jin, Lawrence J., Extrapolative Beliefs in the Cross-Section: What Can We Learn from the Crowds? (May 4, 2019). Available at SSRN: https://ssrn.com/abstract=3144849
However, they look at the cross-section and I only look at one stock. Additionally my $\lambda$ is time varying (If that makes any difference for the solution).