A stock has current price S$0=50.$ A butterfly spread on the stock is the payoff

fT $=$

$$

$$

$$

$0,$ if ST K$1,$

ST $$ K$1,$ if K$1$ ST K$2,$

K$3$ ST $,$ if K$2$ ST $$ K$3,$

$0,$ if ST $>=$ K$3,$

at maturity time T$,$ where K$1$ K$3,$ K$2=($K$1+$ K$3)/2,$ and ST is the stock price at maturity time.

Suppose a trader wants to create a butterfly spread with K$1=30,$ K$3=60$ and maturity in $0\mathrm{.}5$ years,

but the options needed to do this are not available on the market. Instead they can create a replicating

portfolio for this derivative. The volatility of the stock is $2$ log$(1\mathrm{.}25)$ p$.$a$.,$ and the risk$-$free rate is $5\%$ p$.$a$.$

The stock price process follows a $2-$step binomial tree with u $=1/$d$,$ and to put this in the discrete time

market model, the times are t $=0,1,2,$ where t $=1$ corresponds to $0\mathrm{.}25$ years, and t $=2$ corresponds to

$0\mathrm{.}5$ years.

$($a$)$ Show that u $=1\mathrm{.}25,$ d $=0\mathrm{.}8,$ and the risk$-$neutral probability is pb $=0\mathrm{.}4724.$ Then draw the binomial

tree. $[2$ marks$]$

$($b$)$ Construct a replicating portfolio for the derivative. Note that this means you should provide the

units of stocks AS$($t$)$ and units of bonds AB$($t$)$ to hold at time t $=1,2,$ where $($AS$(1),$ AB$(1))$ are

determined at time $0$ at node $0,$ and $($AS$(2),$ AB$(2))$ are determined at time $1$ at node u and node d$.$

You may alternatively give the dollar value of the bonds as opposed to the units of bonds. $[5$ marks$]$

$($c$)$ Numerically verify that the replicating portfolio is self$-$financing. $[2$ marks$]$

$($d$)$ Numerically verify that the initial investment needed to set up the replicating portfolio is the no

arbitrage price of the derivative. Note that $($AS$(0),$ AB$(0))=($AS$(1),$ AB$(1)).$