Infinity inc is a U.S. company whose shares are listed on and freely traded on the New York stock exchange.

You are the Global Head of Equity Options trading at Goldman Sachs. You are approached by a hedge fund that today wants to buy a security that has the following features:

As long as the Infinity share price never trades at or above $50 per share, the hedge fund receives nothing. However, if the Infinity share price ever gets to trade at or above $50 per share, the hedge fund will receive at that time, from Goldman Sachs, a payment in cash equal to $10,000,000, after which the security expires.

The contract is perpetual (it means it has no fixed end date and the security continues forever until it expires (i.e., until the price level of $50 per share is reached)).

Infinity inc can potentially go bankrupt in which case the shares in Infinity inc become worthless (and assume that the shares would remain worthless forever).

Assume that, in the absence of bankruptcy, the shares of Infinity inc will trade forever (no takeovers or delistings, for example).

The share price of Infinity inc today is $40 per share.

Assume that Infinity inc will never pay a dividend and will never do share buybacks. Assume, for simplicity, that the risk-free interest-rate is zero per cent and will be zero per cent forever.

Assume that Goldman Sachs can never go bankrupt and also assume that Goldman Sachs continues to exist for ever (so it can never renege on this contract).

a) The hedge fund wants to buy the above-mentioned security today. What price (in dollars) do you quote? How do you hedge this security so that Goldman Sachs has no risk? Carefully explain your strategy.

Assume the absence of arbitrage throughout. Assume that there are no transactions costs. Assume that you can freely trade Infinity inc shares and borrow or lend at the risk-free interest-rate (i.e., zero per cent) (forever, if needed).

b) How would your answer in part (a) change if, instead of assuming that the risk-free interest-rate is zero percent forever, I told you that the risk-free today is 3.85% per annum (continuously compounded) but will fluctuate through time as the Fed changes its monetary policy?