You are sitting on a gold mine: after a good game of chess against the one and only Lionel Messi you had won the rights to produce diamonds with his bright smile engraved in them for the next two years. Well, the deal came with a caveat: you can only sell $10,000$ of them. Lionel does want to keep some residual rights after all.

After some preliminary research you estimate that you can sell each Messi$-$engraved$-$diamond for PB $=$ $$2500.$ This price is fixed, no matter what happens to world diamond prices the willingness to pay for a Messi$-$engraved diamond is $$2500.$ The variable cost per diamond is basically the actual diamond itself, which are running these days for around Pd $=$ $$1200.$

The costs of setting up production and making the $10,000$ Messi$-$engraved diamonds is around $12$m$.$ Further assume that once you decide to produce the Messi$-$engraved diamonds $($at any

point in the following two years$)$ things will go quickly enough so you can ignore discounting of operating costs and revenues $($it is not that hard to engrave diamonds after all$).$

The expected price appreciation of the raw diamonds you will use in your production process over the next two years is $15\%($in annual terms$),$ and its volatility is expected to run around

$50\%($also in annual terms$).$ You also know the risk$-$free rate is $5\%.$

You have the following information on put option prices traded at the CBOE. All the options have a two$-$year maturity and the underlying asset is the type of diamond you plan to use in your operations.

Strike European Put American Put

$1100251\mathrm{.}4260\mathrm{.}6$

$1200293\mathrm{.}1305\mathrm{.}1$

$1300337\mathrm{.}2351\mathrm{.}5$

$1400377\mathrm{.}2396\mathrm{.}2$

Assume you live in a world in which there are no taxes.

What would you say is a fair value for the project?