Question $8$

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$$dt $\backslash$times N $()$ E $\backslash$times $\backslash$times N $()$ d$1$ e$$Rt 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 exceptfor 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 $$81$ per share, the standard deviation of its return is $44$ percent per year, and the risk$-$free rate is $4$ percent per year, compounded continuously.What is the price of a put option with a strike price of $$77$ and a maturity of six months if the stock has a dividend yield of $2$ percent per year?