Problem $1\mathrm{.}4.$ Consider a fixed time $T$ in future. Let $S$ be the price of a stock at time $T$ in

$$.$ For positive integer $n,$ let $c_n$ be the price of a binary call option that will pay $$1$ at time

$T$ if $Sn$ and $0$ if $p_n$$$1TSn{c}_7$$$1TS7p_5$$dots$$1nS$ using binary put

prices.

$(c)$ Equating the two expressions you calculated for the same state price, determine $a$

relation between binary call prices and binary put prices.$nS$ and $0$

otherwise. Hint: Buy $an\frac{d}{o}r$ sell different calls $to$ create a portfolio with the required

payoff. The payoff from selling $an$ option $is$ the negative $of$ the payoff from buying the

option.

$(b)$ Write $an$ expression for the state price for the state: $nS$ using binary put

prices.

$(c)$ Equating the two expressions you calculated for the same state price, determine $a$

relation between binary call prices and binary put prices.$nS$ using binary call

prices. Recall, this $is$ the price $of$ a portfolio that pays $$1ifnS$ and $0$

otherwise. Hint: Buy $an\frac{d}{o}r$ sell different calls $to$ create a portfolio with the required

payoff. The payoff from selling $an$ option $is$ the negative $of$ the payoff from buying the

option.

$(b)$ Write $an$ expression for the state price for the state: $nS$ using binary put

prices.

$(c)$ Equating the two expressions you calculated for the same state price, determine $a$

relation between binary call prices and binary put prices.$S$ and $0ifSn.$ For example, $c_{7}is$ the price $ofan$ option that will pay

$$1at$ time $TifS7$ and nothing otherwise. Similarly, $p_{5}is$ the price $ofan$ option that will

pay $$1at$ time $TifS<mn\; id="EditorElement\_97682">5</mn>$ and nothing otherwise. Suppose these prices are available for all

positive integral values $ofn.$ That $is,c_1,p_1,c_2,p_2,c_3,p_3,$dots are available.

$(a)$ Write $an$ expression for the state price for the state: $nS$ using binary call

prices. Recall, this $is$ the price $of$ a portfolio that pays $$1ifnS$ and $0$

otherwise. Hint: Buy $an\frac{d}{o}r$ sell different calls $to$ create a portfolio with the required

payoff. The payoff from selling $an$ option $is$ the negative $of$ the payoff from buying the

option.

$(b)$ Write $an$ expression for the state price for the state: $nS$ using binary put

prices.

$(c)$ Equating the two expressions you calculated for the same state price, determine $a$

relation between binary call prices and binary put prices.$S.$ Similarly, let $p_{n}be$ the price $of$ a binary put option that will pay

$$1at$ time $TifS$ and $0ifSn.$ For example, $c_{7}is$ the price $ofan$ option that will pay

$$1at$ time $TifS7$ and nothing otherwise. Similarly, $p_{5}is$ the price $ofan$ option that will

pay $$1at$ time $TifS<mn\; id="EditorElement\_99548">5</mn>$ and nothing otherwise. Suppose these prices are available for all

positive integral values $ofn.$ That $is,c_1,p_1,c_2,p_2,c_3,p_3,$dots are available.

$(a)$ Write $an$ expression for the state price for the state: $nS$ using binary call

prices. Recall, this $is$ the price $of$ a portfolio that pays $$1ifnS$ and $0$

otherwise. Hint: Buy $an\frac{d}{o}r$ sell different calls $to$ create a portfolio with the required

payoff. The payoff from selling $an$ option $is$ the negative $of$ the payoff from buying the

option.

$(b)$ Write $an$ expression for the state price for the state: $nS$ using binary put

prices.

$(c)$ Equating the two expressions you calculated for the same state price, determine $a$

relation between binary call prices and binary put prices.