Just follow the requirements of the question and do it$.$ Prove the pricing formula for European call option. THEOREM $($BLACK$-$SCHOLES FOR EUROPEAN CALL OP$-$

TIONS$)$: Suppose Lucas CCAPM holds. Consider a European Call option

written on an asset $k$ at period $t$ with a strike price $S$ and expiration date

$t+T.$ Random variables

$x_k=ln(pro{d}_l=1^T{R}_{kt+l}),$

$y_i=ln(b_i^T\frac{u_i^{\text{'}}(c_{it+T})}{u_i^{\text{'}}(c_{it})})$

are bivariate normally distributed with expectations

$(E[x_k],E[y_i])=(T{}_k,T{}_{c_i})$

and the variance$-$covariance matrix

$V=([T{}_k^2,T{}_k{}_{c_i}],[T{}_k{}_{c_i},T{}_{c_i}^2]),$

where $$ is the correlation coefficient between random variables $x_k$ and $y_i.$

Moreover, the underlying asset $k$ pays dividends such that the dividend$-$price

ratio is constant over time, i$.$e$.,$

$\frac{d_{kt+T}}{p_{kt+T}}\frac{?}{b}=ar(D{)}_k$AAT$=0,$dots,$.$

Show that

Call$(p_{kt},S,{}_k,r_f,T\frac{,}{b}ar(D{)}_k)=\frac{1}{(1\frac{+}{b}ar(D{)}_k{)}^T}{p}_{kt}N(Z_{ks}+\sqrt[\phantom{2}]{T}{}_k)-\frac{S}{(1+rf{)}^T}N(Z_{ks}),$

where

$Z_{ks}=\frac{ln(\frac{p_{kt}}{S})+ln(\frac{1}{(1+\frac{1}{D_k}{)}^T})+ln(1+r_f{)}^T}{\sqrt[\phantom{2}]{T}{}_k}-\frac{1}{2}\sqrt[\phantom{2}]{T}{}_k.$

HINT: Use the fact that

$\frac{p_{kt+T}}{p_{kt}}=\frac{1}{(1\frac{+}{b}ar(D{)}_k{)}^T}pro{d}_l=1^T{R}_{kt+l}.$