Assume that we are working in an $N-$period binomial model with parameters $S_0>0ud=1F>0Fmi{n}_{i}n{S}_i{S}_n{G}_n=\frac{F{S}_n}{mi{n}_i}n{S}_i,$nin$\{0,1,$dots,$N\}.G_n=F{Y}_n{Y}_n=\frac{S_n}{mi{n}_i}n{S}_i{V}_n{V}_n=f_n(Y_n),$nin$\{0,1,$dots,$N\}.f_n(y)V_{0}N=3,u=2,d=\frac{1}{2},r=\frac{1}{2},S_0=4F=81(1+r)$ and $S_0>0.In$ addition, assume that $ud=1.$

A buy$-at-$min American option with face value $F>0$ allows the holder $to$ purchase $F$

dollars worth $of$ stock, but purchased $at$ the minimum price $mi{n}_{i}n{S}_i$ instead $of$ the

current price $S_n.$ That $is,$ the option has intrinsic value process given $by$

$G_n=\frac{F{S}_n}{mi{n}_i}n{S}_i,$nin$\{0,1,$dots,$N\}.$

$We$ can write this $as{G}_n=F{Y}_n$ where

$Y_n=\frac{S_n}{mi{n}_i}n{S}_i$

$1$

$(a)$ Show that the pre$-$exercise value $V_{n}of$ the stock admits a state representation $of$ the

form

$V_n=f_n(Y_n),$nin$\{0,1,$dots,$N\}.$

Derive the backward recurrence relation for the functions $f_n(y),$ including the termi$-$

nal condition.

$(b)$ Compute the arbitrage$-$free price $V_0$ when $N=3,u=2,d=\frac{1}{2},r=\frac{1}{2},S_0=4,$ and

$F=81\mathrm{.}0$

$(1+r)$ and $S_0>0.In$ addition, assume that $ud=1.$

A buy$-at-$min American option with face value $F>0$ allows the holder $to$ purchase $F$

dollars worth $of$ stock, but purchased $at$ the minimum price $mi{n}_{i}n{S}_i$ instead $of$ the

current price $S_n.$ That $is,$ the option has intrinsic value process given $by$

$G_n=\frac{F{S}_n}{mi{n}_i}n{S}_i,$nin$\{0,1,$dots,$N\}.$

$We$ can write this $as{G}_n=F{Y}_n$ where

$Y_n=\frac{S_n}{mi{n}_i}n{S}_i$

$1$

$(a)$ Show that the pre$-$exercise value $V_{n}of$ the stock admits a state representation $of$ the

form

$V_n=f_n(Y_n),$nin$\{0,1,$dots,$N\}.$

Derive the backward recurrence relation for the functions $f_n(y),$ including the termi$-$

nal condition.

$(b)$ Compute the arbitrage$-$free price $V_0$ when $N=3,u=2,d=\frac{1}{2},r=\frac{1}{2},S_0=4,$ and

$F=81.$