We discussed the professional arbitrage model and illustrated it using a numerical ex$-$ ample in the lectures. This question asks you how the professional arbitrage model works and what its implications are in an alternative numerical example. In this example, the fundamental value of the asset is set at V $=1.$ The resource under management by arbitrageurs at t $=1$ is F$1=0\mathrm{.}1.$ The resource under management by arbitrageurs at t $=2$ is F$2=$ F$1$ G$($x$),$ where x is the arbitrageurs$$ gross return between time $1$ and time $2$ and G$($x$)=$ ax $+1$ a with a $=1\mathrm{.}1.$ The pessimism of noise traders at t $=1$ is S$1=0\mathrm{.}2.$ The pessimism of noise traders at t $=2$ is S$2=0\mathrm{.}3$ when the pessimism deepens. It turns out that for these parameters, there is a q$=0\mathrm{.}4083$ such if the probability that the pessimism of the noise traders widens at t $=2,$ q$,$ is less than q$,$ the arbitrageurs would not hold cash at t $=1.$ If q is greater than q$,$ the arbitrageurs would hold some cash at t $=1.1.$ If q $<0\mathrm{.}4083,$ we have the following in the model: $$ Arbitrageurs are fully invested at t $=1,$ and D$1=$ F$1=0\mathrm{.}1,$ where D$1$ is the amount that arbitrageurs invest in the asset at t $=1$; the asset price at t $=1$ is p$1=0\mathrm{.}9$; $$ F$2=0\mathrm{.}0861$ and the asset price is p$2=0\mathrm{.}7861$ if noise trader sentiment deepens at t $=2$ ; $$ F$2=0\mathrm{.}1122$ and p$2=$ V $=1$ if noise trader sentiment recovers at t $=2.$ Please answer the following questions. $($a$)(3$ points$)$ What are the quantity of asset and the amount of cash held by ar$-$ bitrageurs at t $=1?$ What is the value of this portfolio at t $=2$ when the noise trader sentiment deepens? What are the gains or losses due to the price movements between time $1$ and time $2?($b$)(2$ points$)$ When the noise trader sentiment deepens at t $=2,$ is the value of the portfolio held by arbitrageurs from t $=1$ the same as the resource under management by arbitrageurs t $=2?$ Explain. $($c$)(4$ points$)$ What are the quantity of asset and the amount of cash held by ar$-$ bitrageurs from t $=2$ to t $=3$ if the noise trader sentiment deepens at t $=2?$ What are the adjustments of the portfolio made by arbitrageurs at t $=2$ in this $1$ case? What are the motivations for these adjustments? Are these motivations consistent with the market conditions? $($d$)(2$ points$)$ What is the value at t $=3$ of the portfolio held by arbitrageurs from t $=2$ when the noise trader sentiment deepened? What are the gains or losses due to the price movements between time $2$ and time $3?($e$)(1$ points$)$ Do arbitrageurs have gains or suffer losses overall between period $1$ to period $3$ if the noise trader sentiment deepens at t $=2?2.$ If q $=0\mathrm{.}55,$ we have the following in the model: $$ D$1=0\mathrm{.}0740$ and p$1=0\mathrm{.}8740$; $$ If noise trader pessimism deepens at t $=2,$ then F$2=0\mathrm{.}0924$ and p$2=0\mathrm{.}7924$; $$ If noise trader sentiment recovers at t $=2,$ then F$2=0\mathrm{.}1117$ and asset price p$2$ returns to V $=1.($a$)(3$ points$)$ Comparing this case $($q $=0\mathrm{.}55)$ with the previous case q $<0\mathrm{.}4083,$ is the asset price at t $=1$ more or less efficient? Explain the reason.