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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: $$ \rm Market Value (stock \, trading) + Premium Received = MarketValue(option) $$ which is equivalent to $$ \rm MarketValue(stock \, trading) = MarketValue(option) - Premium \, Received $$ where I assume zero interest rates.

Thus if I delta hedge an option that I did not short, the premium of which I did not receive, then the relationship must be the same as the left hand side of the equation does not change.

If I want to "generate an option payoff", I trade the underlying:

$$ \rm MarketValue(stock \, trading) = MarketValue(option) - Premium \, Received $$

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

To summarize: Replicating the option without actually trading it, I will replicate it "minus" the premium ... I will get less than the pay-off. Is this correct?

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

To ask the question differently: 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).

Is there anything that I missed? Is the premium and the hedging error the only difference?

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  • $\begingroup$ I just added the last paragraph. It should become clearer what I mean. $\endgroup$
    – Richi Wa
    Commented Nov 26, 2012 at 8:50

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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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  • $\begingroup$ 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? $\endgroup$
    – Richi Wa
    Commented Nov 22, 2012 at 15:35
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    $\begingroup$ 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,...). $\endgroup$
    – Matt Wolf
    Commented Nov 23, 2012 at 8:51
  • $\begingroup$ ...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. $\endgroup$
    – Matt Wolf
    Commented Nov 23, 2012 at 8:54
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    $\begingroup$ 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. $\endgroup$
    – Matt Wolf
    Commented Nov 25, 2012 at 10:01
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    $\begingroup$ I am getting more and more confused what you try to achieve here. I cannot comment anymore for lack of understanding your point. If you deal with a futures option then the right way to delta hedge is with the future. You can still delta hedge any asset with a futures contract but it will possibly not be the most optimal hedge. $\endgroup$
    – Matt Wolf
    Commented Nov 26, 2012 at 9:23
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I have implemented the set-up in the Black-Scholes world (setting $r=0$). The answer is yes.

If I just take positions in the stock according to delta then the result (pathwise, i.e. in each realisation, not only on average) is the option pay-off minus the option premium. That is to say, if the option expires out of the money (pay-off is zero), then the result of the hedging is given by the negative of the option premium. In "formulas" this is: $$ \rm MarketValue(stock \, trading) = MarketValue(option) - Premium \, Received $$ so that

$$ \rm MarketValue(stock \, trading) = - Premium \, Received $$

In an ideal setting, the trade loses money. This is the theoretical answer in an unrealistic setting without transaction cost, with continuous hedging, no basis risk, no contango nor clearinghouse concerns. In a realistic setting, the trade loses a lot more money. But at least the setting in the "perfect world" is now clear to me.

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    $\begingroup$ A factor you have ignored is contango. A risk you have ignored is clearinghouse margin adjustments. There are others, but these two are sufficient to undermine the arbitrage. $\endgroup$
    – Matthew
    Commented Dec 5, 2012 at 0:31
  • $\begingroup$ Yes, thanks for the comment. In the question nevertheless I first wanted to settle the theoretical question. Practical issues arise of course, and their precise impact depends on various factors. $\endgroup$
    – Richi Wa
    Commented Dec 7, 2012 at 12:13
  • $\begingroup$ Contango and clearinghouse requirements are more than mere issues. As @Matthew said, if they undermine the arbitrage, they render the theoretical question pointless as currently stated. You need to determine whether the costs of contango and clearinghouse margin changes are quantifiable (seems unlikely...) and are less than the arbitrage value of the trade strategy. $\endgroup$
    – Ellie K
    Commented Dec 1, 2013 at 4:46
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I am facing the same question and I feel what Wa tries to do is to replicate an option using futures. On a theoretical basis, entering futures costs nothing. Therefore, if BS model says, Option value/prenium = Stock + Bond (missing all the position weights here), it must mean: 0 = Stock + Bond - Option Premium. I replace stock with stock future, bond with treasury future and dynamically rebalance. Under each simulation path and if following the delta hedge, at time T, we must have Stock_future_T + Bond future_T - Option Premium * exp(rT). (Stock_future_T + Bond future_T) should replicate the option payoff.

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