Bullish Strategy, With A Twist (Buy a Put)

If you are bullish on a given underlying, you probably think about buying a naked call. However, sometimes, the only option available to buy is a put. Should you buy it? Short answer: YES.

However, to replicate the call payoff, you actually need to delta hedge the put and then buy even more shares, to an amount equivalent to 100 delta... Which means that if you buy 1000 puts, then you buy 1000 shares.

To illustrate: If the call you wanted to buy initially has a 40 delta, then put at the same strike will have a 60 delta (put/call parity...) To equate the 40 delta of the call, then buy 40 delta more worth of shares... As simple as that.

Then, once the position is in place, if the the combination of put + shares will replicate a position on the naked call:
- when the underlying moves higher
     - the put delta value will decrease, as well as the delta of the option (at a decreasing rate)
     - the long shares position will gain in value
- when the underlying moves lower
     - the put value will increase, as well as the delta 
     - the long shares position will lose money

The greeks profile in terms of gamma, theta and volatility will be the same.
Differences will arise if the option is american and there are dividends as the put is never exercised, while a deeply in the money call will be exercised before the dividend ex-date.

Below, using a Black Sholes pricer, comparison between the values of a 3month 50 naked call with a 50 spot price, volatility = 25% and an equivalent put, over-hedge (according to the description above) with respect to spot price changes.

And voila, under your eyes, you can see that we have a replicating portfolio.