I want to invest in a certain stock portfolio using long down-and-out barrier options. But I am not sure in what ratio to buy the barrier options to retain the same exposure to the volatility of the underlying stocks. Suppose that an XYZ stock portfolio contains 20% company X, 30% company Y and 50% company Z. The aim is to retain the diversification effect of the portfolio, yet leverage the investment. In what ratio should I buy the barrier options to preserve the diversification effect of the underlying stock portfolio? The barrier options do not expire unless the stop-loss barrier is hit.
My reasoning is as follows, but I am not sure if this is correct. Let:
p = Option price S = Stock price F = Financing level. So that: p = S - F
As a result, when buying the option you own the following fraction of the underlying stock:
own = p / S
So my sense is that the investment would be as follows:
investment = (weight / own) * p
This means I would end up with the following numbers. I not sure whether this is correct though. Any suggestions?