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$\backslash$times $$e$$dt$\backslash$times $$N$($d$1)$E$\backslash$times $$e$$Rt$\backslash$times $$N$($d$2)$

d$1=[$ln$($S$/$E$)+($R$$d$+\backslash$sigma $2/2)\backslash$times $$t$](\backslash$sigma $$t$)$

d$2=$d$1\backslash$sigma $\backslash$times $$t$$

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$\backslash$times e$$dt$+$P$=$E$\backslash$times e$$Rt$+$C

where d is the continuously compounded dividend yield.

A stock is currently priced at $$82$ per share, the standard deviation of its return is $56$ percent per year, and the risk$-$free rate is $3$ percent per year, compounded continuously. What is the price of a put option with a strike price of $$78$ and a maturity of six months if the stock has a dividend yield of $3$ percent per year? $($Do not round intermediate calculations and round your answer to $2$ decimal places, e$.$g$.,32\mathrm{.}16.)$