Consider the $3-$period binomial model on $$a stock with current

price S$0=$$$100.$There are three time steps, and at $$each time step, the stock price either

moves up$$by$20\%$or $$moves down by$20\%.$The risk$-$free interest rate is$$r$=10\%.$

Find the current price and compute the number of $$shares of $$stock which should be

held by $$the replicating portfolio at $$time $0$only $(0)$for:

$($a$)$Lookback call option with a strike price of $$$$100$that pays off $($at $$time three$)$

V$3=($max$0$n$3$Sn$-100)+.$

i got V$\_0$to be $38\mathrm{.}585$but i don't know

i could not find delta$\_0$

i ran a program and got this but it confuses me

Stock Prices at t$=3$: $[51\mathrm{.}20000000000001,76\mathrm{.}80000000000001,115\mathrm{.}2,172\mathrm{.}79999999999998]$

Payoffs at t$=3$: $[0,0,15\mathrm{.}200000000000003,72\mathrm{.}79999999999998]$

Current Price of Lookback Call Option $($V$0)$: $27\mathrm{.}89256198347109$

Delta at t$=0($Delta$\_0)$: $-0\mathrm{.}7871900826446282$

$($b$)$Lookback put option with a strike price of $$$$100$that pays off $($at $$time three$)$

V$3=(100-$min$0$n$3$Sn$)+$