Stock $Y$ is currently priced at $25$ per share and it pays dividendsA stock index pays dividends continuously proportional to its price.

The rate is $3\%$ during the next $4$ months. Afterwards, the

rate is $1\%.$

Howard invests in the index at time $0$ and reinvest all dicidends

by purchasing extra units of the index. After $8$ months, he ouns

$120$ units of the index. If the curvent index value is $25,4,$ what

is the initial cost for his position? Itime $0$

Hint:

$cost=S_0{n}_0$

$=25,4n_0$

continuously at a rate propertional to its price at a constant rate of $3,5\%.$

You purclase $100$ shares of stock $Y$ and invest all dividends by purclasing

extra shares of stock Y$.$ After $4$ months, you close out all positions

when the stock price is $S_{\frac{1}{3}}.$

If the $4-$month profit is $0$ when the continuously compourded

risk$-$free interest rate is $2\mathrm{.}5\%,$ find $S_{\frac{1}{3}}.$

Hint: prefit $=$ value at time $\frac{1}{3}-$ FV of cost at time $0.$

$0=S_{\frac{1}{3}}$ubrace$(100e^{0\mathrm{.}035\frac{1}{3}}$ubrace${)}_{\frac{?}{?}\text{\#}\text{s}\text{h}\text{a}\text{r}\text{e}\text{s}}-25100e^{0\mathrm{.}025\frac{1}{3}}$