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{\bf Question}
Two dealers in financial derivatives set out to make millions.
Both start with \$1000. The first is very careful with his
investments and manages to increase his wealth at a rate
proportional to the amount of money he has. The second dealer is
much more speculative in her approach and manages to increase her
wealth at a rate proportional to the square of the amount of money
she has. At the end of the first year of trading both dealers,
although they have used different strategies, have each managed to
be worth \$2000. If they both continue with their own strategies
how much will they each be worth at the end of the third year?
(please explain any unusual behaviour in your answers). (*)
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{\bf Answer}
Analysis for Dealer 1. Take $Q(t)=$ money measured in \$.\\
Equation for money is $\ds \frac{dQ}{dt}=kQ$ with $Q(0)=1000$ and
$Q(1)=2000$.\\ Solving the equation give $\ds Q=Ae^{kt}$\\
Applying conditions give $A=1000$ and $2000=1000 e^k$\\ Hence
$k=\ln 2$\\ $\ds q(t)=1000e^{t\ln2}=1000 (2)^t$\\ Hence at $t=3$
Q(3)=\$8000.\\
Analysis for Dealer 2. \\ Equation for money is $\ds
\frac{dQ}{dt}=KQ^2$ with $Q(0)=1000$ and $Q(1)=2000$.\\ The
equation is separable so that\\ $\ds \int \frac{dQ}{Q^2}= K \int
dt$\qquad\qquad $\ds -\frac{1}{Q}=Kt+c$ \qquad\qquad $\ds
Q=\frac{1}{-C-Kt}$\\ Using the conditions implies
$1000=\frac{1}{-C}$ hence $C=\frac{-1}{1000}$\\ Also $\ds 2000 =
\frac{1}{\frac{1}{1000}-K}$ implying $K=\frac{1}{2000}$\\ So that
$\ds Q(t)=\frac{2000}{2-t}$\\ Hence at $t=3$ we have
Q(3)=-\$2000\\ The negative amount of money occurs because,
although $Q(t)$ appears to be a monotonic increasing function $\ds
\left(\frac{dQ}{dt}=Q^2>0 \right)$ the function becomes unbounded
at $t=2$ and the model is not physically valid beyond that time.
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