# How to Delta Hedge with Futures?

The theory of delta hedging a short position in an option is based on trades in the stock and cash. I.e. I get the option premium and take positions in the stock and cash.

In the classical no-arbitrage theory I have the following if I am short an option, get the premium and hedge the structure with the stock: $$maketvalue(stock trading) + premium received = marketvalue(option)$$ which is equivalent to $$maketvalue(stock trading) = marketvalue(option) - premium received$$ where I assume zero interest rates.

Thus if I just Delta hedge an option that I did not short, thus the premium of which I did not receive, then the relation must be the same as the lhs does not change.

Thus if I want to "generate an option payoff" I just trade the underlying:

$$maketvalue(stock trading) = marketvalue(option) - premium received$$

If I do this with futures then all costs must be included in the pricing of futures and the margin payments.

Summing up: replicating the option without actually trading it, I will replicate it "minus" the premium ... I will get less than the pay-off.

Right?

What happens if I do the hedge with futures? Then I don't need cash (except for the margin account).

To ask the question differenetly: if I replicate a put or a call with futures can I make money (the pay-off) from nothing? Obviously not: I will miss the option premium. Furthermore I have a hedging error and additional risk (basis risk as Freddy mentions).

Is there anything more that I miss in this thought? Is the premium and the hedging error the only difference?

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 I just added the last paragraph. It should become clearer what I mean. – Richard Nov 26 '12 at 8:50

You are missing the futures basis and roll cost. Futures expire, and need to be rolled into the new expiry. The basis is not static and can vary considerably, depending on the specific underlying and contract. Quants may have a hard time to appreciate this but the basis is not at all fully quantifiable at all times: It can hugely vary entirely due to shifts in demand and supply. Those are the things you are definitely missing. I may in turn miss out on more but those two come to mind right away.

But more importantly, I sense from how you asked the question that you are not really fully on top of options pricing and hedging theory. To hedge part of an option exposure you need to fully understand which part you specifically do not want to take exposure to through implementing a hedge. You also need to understand that most all greeks are dynamic, which means that you need to consider re-hedging, specifically by how much and more importantly how often. I recommend you spend time reading through Taleb's "Dynamic Hedging" because it describes hedging from the practitioner's perspective.

Edit: Per request links to provide the replication approach for futures options:

http://ferrari.dmat.fct.unl.pt/personal/mle/DocCPG/Marek/LECS.pdf

and here one more interesting paper that shows how to deal with basis risk:

http://www.edhec-risk.com/edhec_publications/all_publications/RISKReview.2011-02-21.3709/attachments/EDHEC%20Working%20Paper%20Option%20Pricing%20and%20Hedging%20F.pdf

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 Thanks for your answer. For the roll costs: let's ignore them, if we just hedge for some days or a week - then there is no roll necessary - isn't it? I agree to the basis risk. I will certainly have a look at Taleb's book. But after all: ignoring margin and basis I can replicate an option to a certain degree without cash. If I adjust my position in high frequency can I produce an option pay-off with no (or little on the margin account, maybe less than the premium) money? What's the main argument why I can't?I am aware of changing implied volatility and such things...but what if I often adjust? – Richard Nov 22 '12 at 15:35 No you cannot, because otherwise you could, as you correctly pointed out, arbitrage the option against the replication. Futures trade against the underlying at a premium/discount (before settlement) for the precise reason of financing (borrow/loan) the underlying cash notional plus pos/neg. yields during the lifetime of the future. With positively yielding underlying assets the assumption is that forgoing positive cash flow plus its investment of such must result result in a lower futures price than underlying before expiration, the opposite applies for assets that cost to hold (storage,...). – Matt Wolf Nov 23 '12 at 8:51 ...sorry there is a lot more to this, because you sometimes have yields of opposing sign on the same asset. I do not believe this treatise should be part of the question as I assume most are familiar with the basics of futures/forward pricing. – Matt Wolf Nov 23 '12 at 8:54 thank you for your comments. Of course I am familiar with the cost-of-carry relation. But I am not sure if this explains everything. So I have cost-of-carry, margin payments and I get the option pay-off? In reality a Delta hedge is still very risky and imperfect, I know. But where is the striking argument? – Richard Nov 23 '12 at 16:20 I am confused what else you like to know: The fact that futures only require a margin outlay and not the full cash investment is already priced into the futures contract. Now you just price the option on it, that is how all futures options are priced. And that is how options on any asset are priced, whether bananas or futures, or stocks, or indexes. – Matt Wolf Nov 25 '12 at 10:01
I have implemented the set-up in the Black-Scholes world (setting $r=0$). The answer is yes.
E.g. if the option expires out of the money (pay off is zero) then the result of the hedging (left-hand side above) is given by the negative of the option premium. In "formulas" this is: $$maketvalue(stocktrading)=marketvalue(option)−premiumreceived$$ becomes
$$maketvalue(stocktrading)=−premiumreceived,$$ if the pay-off at expirs is zero. In the general case the pseudo formulas above apply.