In addition to the five factors discussed in the chapter, dividends also affect the price of

an option. The Black$-$Scholes option pricing model with dividends is:

$C=S{e}^{-dt}N(d_1)-E{e}^{-Rt}N(d_2)$

$d_1=[ln(\frac{S}{E})+(R-d+\frac{{}^2}{2})t](-\sqrt[\phantom{2}]{t})$

$d_2=d_1-\sqrt[\phantom{2}]{t}$

All of the variables are the same as the Black$-$Scholes model without dividends except

for the variable $d,$ which is the continuously compounded dividend yield on the stock.

The put$-$call parity condition is altered when dividends are paid. The dividend$-$adjusted

put$-$call parity formula is:

$S{e}^{-dt}+P=E{e}^{-Rt}+C$

where $d$ is the continuously compounded dividend yield.

A stock is currently priced at $$85$ per share, the standard deviation of its return is $42$

percent per year, and the risk$-$free rate is $4$ percent per year, compounded continuously.

What is the price of a put option with a strike price of $$81$ and a maturity of six months if

the stock has a dividend yield of $2$ percent per year? $($Do not round intermediate

calculations and round your answer to $2$ decimal places, e$.$g$.,32\mathrm{.}16.)$

Put price