$6$ Two$-$Period Binomial Tree $(20$ points$)$ There are three periods $t=0,1,2,$ and two assets with

known prices, a stock $(S)$ and a money market fund $(B).$

Period $0is$ today. The stock price $is{S}_0,$ and the money market fund starts $at{B}_0=1.$ The short$-$term

interest rate $is{r}_0,$ meaning that $$1$ invested $in$ the money market fund grows $to1+r_0$ dollars $att=1.$

From period $0to$ period $1,$ there are two possibilities. With probability $p_0,wegoto$ the $"up"$ state, $in$

which case the stock price rises $to{S}_1=u{S}_0.$ With the remaining probability $1-p_0,wegoto$ the "down"

state, $in$ which case the stock price goes $to{S}_1=d{S}_0.$ You may assume $u>1$ and either case,

the money market fund grows $to{B}_1=1+r_0.$

$So$ far everything $is$ the same $asin$ the two$-$period model $in$ class. Let's change the dynamics from

$t=1tot=2to$ make life more interesting. $In$ both cases, the stock will either continue $to$ rise $by$

another multiplicative factor $uor$ decrease $by$ a multiplicative factor $d,$ but the probabilities and short$-$

term interest rates are different.

$If,att=1S_1=u{S}_0{u}^2{S}_0{p}_u,$ and falls $todu{S}_0$ with probability $1-p_u.$ The short$-$term interest rate $is{r}_u.$ We'll call the state

where the stock price has risen twice the $"up-up"$ state, and the state where stock price falls the

$"up-$down" state.

$If,att=1,we$ are $in$ the "down" state $(S_1=d{S}_0)\{$:$ud{S}_0)$ with probability

$p_d,$ and falls $to{d}^2{S}_0$ with probability $1-p_d.$ The short$-$term interest rate $is{r}_d.$ We'll call the two

states described here $att=2$ the "down$-up"$ and "down$-$down" states.

These two short$-$term rates are again given with continuous compounding, $so$ $$1in$ the $up$ state invested

$in$ the money market fund grows $to1+r_u$ dollars $att=2.$ Likewise, $$1in$ the down state invested $in$ the

money market fund grows $to1+r_d$ dollars $att=2.$

For labeling, $do$ the same $asin$ class. Subscript $0$ indicates the initial period. $u$ and $d$ denote the $up$

and down states $att=1.uu$