Consider a one period trinomial model with two assets $S^1,S^2.$ Again there are

three future states of the universe : ${}_1,{}_2,{}_3.$ Suppose that

$S_T^i({}_1)=S_0^i{u}_i$

$S_T^i({}_2)=S_0^i{m}_i$

$S_T^i({}_3)=S_0^i{d}_i$

for $i=1,2.$ Let $u_1=1\mathrm{.}05,m_1=1,d_1=0\mathrm{.}95,u_2=1\mathrm{.}15,m_2=0\mathrm{.}95,d_2=0\mathrm{.}90,r=$

$0,T=1,S_0^1=S_0^2=100.$ The risk neutral probability condition is

tilde$(E)(e^{-rT}{S}_T^i)=S_0^i,i=1,2$

where we designate $p_1=$tilde$(P)({}_1),p_2=$tilde$(P)({}_2),1-p_1-p_2=$tilde$(P)({}_3).$

a$.$ Show that there is a unique solution $p_1,p_2($that form a probability distribution$)$

in this model.

b$.$ Consider an option $V_T=(S_T^1-S_T^2{)}^{+}.$ Use the risk neutral pricing formula

$V_0=$tilde$(E)(e^{-rT}{V}_T)$ to find the no arbitrage price of this option.

c$.$ The replicating portfolio in this model consists of $x$ shares of $S^1,y$ shares of

$S^2$ and $z$ dollars in cash. Find the replicating portfolio for the option in part b and

use it to find the price of the option at time $0.$ Verify that the answer you get is the

same in part b$.$ The replicating portfolio satisfies:

$x{S}_T^1+y{S}_T^2+z=(S_T^1-S_T^2{)}^{+}$

for all $3$ outcomes.Consider a one period trinomial model with two assets $S^1,S^2.$ Again there are

three future states of the universe : ${}_1,{}_2,{}_3.$ Suppose that

$S_T^i({}_1)=S_0^i{u}_i$

$S_T^i({}_2)=S_0^i{m}_i$

$S_T^i({}_3)=S_0^i{d}_i$

for $i=1,2.$ Let $u_1=1\mathrm{.}05,m_1=1,d_1=0\mathrm{.}95,u_2=1\mathrm{.}15,m_2=0\mathrm{.}95,d_2=0\mathrm{.}90,r=$

$0,T=1,S_0^1=S_0^2=100.$ The risk neutral probability condition is

tilde$(E)(e^{-rT}{S}_T^i)=S_0^i,i=1,2$

where we designate $p_1=$tilde$(P)({}_1),p_2=$tilde$(P)({}_2),1-p_1-p_2=$tilde$(P)({}_3).$

a$.$ Show that there is a unique solution $p_1,p_2($that form a probability distribution$)$

in this model.

b$.$ Consider an option $V_T=(S_T^1-S_T^2{)}^{+}.$ Use the risk neutral pricing formula

$V_0=$tilde$(E)(e^{-rT}{V}_T)$ to find the no arbitrage price of this option.

c$.$ The replicating portfolio in this model consists of $x$ shares of $S^1,y$ shares of

$S^2$ and $z$ dollars in cash. Find the replicating portfolio for the option in part b and

use it to find the price of the option at time $0.$ Verify that the answer you get is the

same in part b$.$ The replicating portfolio satisfies:

$x{S}_T^1+y{S}_T^2+z=(S_T^1-S_T^2{)}^{+}$

for all $3$ outcomes.