Question $2.($A Breeden$-$Litzenberger Exercise$)(35\text{'})$

Let's consider a hypothetical equity index. For simplicity, assume that the index price level,

denoted as $S_t,$ can only take positive integer values. That is$,$ it is only possible for the index

price to take values from $\{1,2,3,$dots,$n,$dots,$50\}.$ The index level cannot be zero or negative.

Historically, the index price has never exceeded $50$; therefore, investors do not believe it is

possible for the index to rise beyond $50.$ The current spot price $S_0=17.$

$(2\mathrm{.}1)$ What is the payoff diagram for a European call option on the equity index that

expires in exactly three months $(T=0\mathrm{.}25)$ with a strike price of $20?$ What about

European calls with the same $T$ and strike prices $21$ and $22?$

Please plot these three payoff diagrams in the same graph with MS Excel. Use a

scatter plot and do not interpolate.

$(2\mathrm{.}2)$ Now, you would like to examine the so$-$called Arrow securities that pay off in T$.$

An Arrow security for $S_T=n$ refers to a security with the following payoff:

Payoff $=1,if{S}_T=n$;

Payoff $=0,if{S}_{T}n.$

European calls with various strikes are traded on the market. Can you replicate an

Arrow security for $S_T=21$ using only European calls? What is the price of such a

security, expressed as a formula of involved call prices?

$($Hint: You only need options mentioned in $(2\mathrm{.}1).$ The price of a call with strike $K$ and

time$-$to$-$expiration $T$ can be denoted as $C(K,T).)$

$(2\mathrm{.}3)$ The current prices of European calls with T maturity are shown in $[$F$2023$

Midterm.xlsx$].$ Please calculate the prices of Arrow securities for $S_T=1,2,$dots,$50$

and present these fifty Arrow prices in a table. Also, plot these prices against $S_T$ in a

bar graph.

$($Hint: What is the call price with $K=0?$ What about $K=50?$ You may need these call

prices. It shouldn't take more than a few seconds to figure out these two call prices.$)$

$(2\mathrm{.}4)$ Using what you just learned from $(2\mathrm{.}3),$ you can now price any derivative with

payoffs contingent on $S_T!$ Let's give it a shot:

a$)$ What is the price of a security that pays off one dollar if $S_T=17,$ three dollars if

$S_T=40,$ and zero otherwise? $($Hint: Replicate it with Arrow securities$.)$

b$)$ What is the price of a put option with strike $30?$

c$)$ What is the price of a risk$-$free zero$-$coupon bond that pays one dollar at T$?$ What

is the continuously compounded risk$-$free interest rate in APR in this economy $?$ Question $2$ Call option price data

Strike prices Call prices

$116\mathrm{.}0148$

$215\mathrm{.}0300$

$314\mathrm{.}0470$

$413\mathrm{.}0693$

$512\mathrm{.}1028$

$611\mathrm{.}1549$

$710\mathrm{.}2339$

$89\mathrm{.}3476$

$98\mathrm{.}5026$

$107\mathrm{.}7042$

$116\mathrm{.}9561$

$126\mathrm{.}2601$

$135\mathrm{.}6169$

$145\mathrm{.}0259$

$154\mathrm{.}4857$

$163\mathrm{.}9941$

$173\mathrm{.}5485$

$183\mathrm{.}1458$

$192\mathrm{.}7832$

$202\mathrm{.}4574$

$212\mathrm{.}1654$

$221\mathrm{.}9043$

$231\mathrm{.}6711$

$241\mathrm{.}4634$

$251\mathrm{.}2785$

$261\mathrm{.}1142$

$270\mathrm{.}9685$

$280\mathrm{.}8394$

$290\mathrm{.}7252$

$300\mathrm{.}6243$

$310\mathrm{.}5354$

$320\mathrm{.}4571$

$330\mathrm{.}3883$

$340\mathrm{.}3280$

$350\mathrm{.}2752$

$360\mathrm{.}2293$

$370\mathrm{.}1893$

$380\mathrm{.}1548$

$390\mathrm{.}1250$

$400\mathrm{.}0995$

$410\mathrm{.}0779$

$420\mathrm{.}0596$

$430\mathrm{.}0444$

$440\mathrm{.}0320$

$450\mathrm{.}0219$

$460\mathrm{.}0140$

$470\mathrm{.}0081$

$480\mathrm{.}0039$

$490\mathrm{.}0012$