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Are we back to Forward Starts then?!

Yes! The option price is linear in St, the stock 3 months from now

@tibbar,
If you haven't seen it, Risk and Return of Equity Index Collar Strategies looks useful. Just scanning it briefly, the conclusions sounded right to me.

Are we back to Forward Starts then?!

No this is an asset management risk management strategy where you want to protect against portfolio losses by buying a 3 month put option that protects against losses above (say) 10%. When the option expires a new 3 month put is purchased to replace the expired one.

The problem is to calculate the expected cost of the option that you will purchase at the rollover date. But you will not pay a premium up front.

Why do i care about the expected cost? I want to assess the expected cost versus expected return of the strategy (ER assessed under real world measure). The key question is in assessing the expected cost of the option that will be purchased in 3 months time what implied vol should be used. I'm fairly sure it should be a forward vol since this will give the markets best estimate of implied vol in 3 months time.

This is probably closely related to pricing of forward start options in the choice of implied vol?

@tibbar,
If you haven't seen it, Risk and Return of Equity Index Collar Strategies looks useful. Just scanning it briefly, the conclusions sounded right to me.

Thanks will take a look

Suppose in 3 months time the stock is 100, and so you buy the 90 put and suppose it costs 3.

Now suppose the stock price three months from now is 200, and so you will but the 180 put. It very easy to prove that the price will be 6.(in the Black and Scholes world).

In general the price of the put you're going to buy will be 3% of the stock price 3 months from now.

Now, what is the expected value of 3% of the stock 3 months from now?

It's 3% * S(0) * e^(r*3/12)!

This also shows you that if you set 3% of your stock aside you will always be able to finance your put with that.

Sure but we don't know what it will cost in 3 months time. The question is what is:

E_P( E_Q( max(S(t+0.25) - K,0) | F_{t+0.25}) | F_t)

where t is current time, F are filtrations, Q is risk neutral measure, P is physical measure.

My shortcut for a back of the envelope approximation to this was to assume S at time t+0.25 is simply S(t) x exp(0.25g) where g is the expected return and then plug this into Black Scholes using sigma equal to the forward vol (see https://en.m.wikipedia.org/wiki/Forward_volatility)

I think the question of what implied vol to use is key.

You'll use the 3-to-6 months vol, that's the price you can buy it for today.

In the end, there's no magic associated with any passive option hedging strategy. Over time, the expected return vs. some (downside) risk measure (before transaction costs) is going to be roughly comparable to some other passive way of achieving the same risk reduction, such as a (rebalanced) stock/bond allocation.    

Given the extra costs, one issue is why bother? 

To make a more detailed comparison, the ideal solution would be to find the historical performance of a very low cost manager that has pursued the strategy in question over a *long* time period -- starting before the '87 crash, as that is relevant. For the collar strategy, I don't think you're going to find such a record.

The next best solutions are passive indices (such as those maintained by the CBOE), theoretical calculations based upon models, and other simulations based upon history. I would guess the minimal theoretical model even having a hope of matching history might be the Bates model, calibrating both a P-version and a Q-version. If you could more or less match the performance of the CBOE indices, then you might be able to use the model to answer some questions going forward for scenarios not seen in the history. But, as I understand the problem, tibbar wants to explore questions such as quarterly vs one-year rebalancing/rehedging. To do that with a model will mean the model will have to be good enough to match conditional market smiles up to one year in maturity. This may require a *much* better model.

But, to belabor the point, in the end one is going to get a reduced return roughly in line with the reduced risk. 
 

Sure but we don't know what it will cost in 3 months time. The question is what is:

E_P( E_Q( max(S(t+0.25) - K,0) | F_{t+0.25}) | F_t)

where t is current time, F are filtrations, Q is risk neutral measure, P is physical measure.

My shortcut for a back of the envelope approximation to this was to assume S at time t+0.25 is simply S(t) x exp(0.25g) where g is the expected return and then plug this into Black Scholes using sigma equal to the forward vol (see https://en.m.wikipedia.org/wiki/Forward_volatility)

I think the question of what implied vol to use is key.

Forward vol would be a good estimate to use. But since you are always going to roll at a fixed moneyness, you should calculate forward vols at moneyness level(or delta). You don't have to worry about strike anymore. Also, do check if these IV estimates are conservative enough to avoid future surprises.

Your payoff (hedging pnl) generated by rolling options will always be lesser than the case if you had bought a 1 yr put(given its not at ATM option). So naturally the cost of the options would be less(which is zero in your case since it is a collar). I believe that this hedging will be highly ineffective as you will not be able to hedge for (100 %- 90 %) of Spot value after every roll date.(because every roll date, you are buying a new 90% put).