$1$ Exotic Options: The Log Contract

A $log$ contract is an important building block in volatility derivatives and in the

calculation of the VIX. As you will see later during this course, the log contract allows

you to achieve pure exposure to fluctuations in volatility.

The payoff from a log contract at maturity $T$ is simply the natural logarithm of the

underlying asset divided by the strike price, $ln(\frac{S_T}{K}).$ The payoff is thus nonlinear

and has many similarities with options. The value of this contract at time $t=0$ for

a non$-$dividend paying asset is

$L_0=e^{-rT}[ln(\frac{S_0}{K})+(r-\frac{1}{2}{}^2)T].$

In this exercise, you will find the price of a $log$ contract using a binomial model.

We will consider a $log$ contract that pays $L_T=1000ln(\frac{S_T}{K})($which is equivalent

with $1000$ individual log contracts$).$ Report answers with $4$ decimal digits.

Consider the following inputs for a non$-$dividend paying stock

The initial stock price $S_0$

The payoff function $L_T=1000ln(\frac{S_T}{K})$

The interest rate $r$

The length of the period $h$

The up and down factors $u$ and $d$

The number of periods $n$

a$.$ Compute the initial value of a $log$ contract with $n=4,r=0\mathrm{.}05,h=1,$

$u=e^{rh+\sqrt[\phantom{2}]{h}},d=e^{rh-\sqrt[\phantom{2}]{h}},S_0=100,=0\mathrm{.}3,$ and strike $K=90.$

Show in a table the values of the log contract on all nodes of the tree. What is

the replicating portfolio at time $0?$

What is the difference between the closed$-$form price $($use Eq$.(1))$ and the initial

value of the $log$ contract that you found above?

b$.$ Compute the initial value of a $log$ contract with $n={48}_{,}r=0\mathrm{.}05,h=\frac{1}{12},$

$u=e^{rh+\sqrt[\phantom{2}]{h}},d=e^{rh-\sqrt[\phantom{2}]{h}},S_0=100,=0\mathrm{.}3,$ and strike $K=90.($I have

underlined the parameters that are different from the previous point.$)$

What is the replicating portfolio at time $0?$

What is the difference between the closed$-$form price $($use Eq$.(1))$ and the initial

value of the log contract that you found above?