Consider a one period binomial model with two assets $S^{\text{A}\text{A}\text{P}\text{L}}($for Apple$)$ and

$S^{\frac{G}{O}OGL}($for Google$).$ There are two states in the future universe : ${}_1,{}_2.$ We have

$S_T^{\text{A}\text{A}\text{P}\text{L}}({}_1)=S_0^{\text{A}\text{A}\text{P}\text{L}}{u}_1$

$S_T^{\text{A}\text{A}\text{P}\text{L}}({}_2)=S_0^{\text{A}\text{A}\text{P}\text{L}}{d}_1$

$S_T^{\frac{G}{O}OGL}({}_1)=S_0^{\frac{G}{O}OGL}{u}_2$

$S_T^{\frac{G}{O}OGL}({}_2)=S_0^{\frac{G}{O}OGL}{d}_2$

Let $u_1=1\mathrm{.}02,u_2=1\mathrm{.}05,d_1=0\mathrm{.}95,d_2=0\mathrm{.}98,r=0,T=1.$ Note that $S_0^{\text{A}\text{A}\text{P}\text{L}}=S_0^{\frac{G}{O}OGL}=100S_0^{\text{A}\text{A}\text{P}\text{L}}=200,S_0^{\frac{G}{O}OGL}=100$tilde$(E)(e^{-rT}{S}_T)=S_{0}S{u}_{1}so$ Google has higher percentage return than Apple regardless $of$ outcomes,

which $isan$ arbitrage condition $aswe$ mentioned $in$ class.

$a.$ Create $an$ arbitrage opportunity when $S_0^{\text{A}\text{A}\text{P}\text{L}}=S_0^{\frac{G}{O}OGL}=100.$

$b.$ Create $an$ arbitrage opportunity when $S_0^{\text{A}\text{A}\text{P}\text{L}}=200,S_0^{\frac{G}{O}OGL}=100.$

$c.$ The risk neutral probability $is$ such that tilde$(E)(e^{-rT}{S}_T)=S_0$ where $S$ can $be$ either

Apple $or$ Google. Show that there $isno$ risk neutral probability $in$ the above model.