Delta Hedging with Futures: Premium, Basis, and Rebalancing
Article Quant Q&A · Author: Richi Wa
Summary
The document asks whether dynamically hedging an option exposure with futures can reproduce its payoff without paying an option premium. In an idealized Black–Scholes setting with zero interest rates, continuous rebalancing, and no transaction costs or basis risk, the cited analysis says a delta hedge’s terminal value equals the option payoff less the premium received. For someone who did not sell the option and collect that premium, the hedge therefore does not generate the full payoff from nothing.
The discussion then identifies practical complications in using futures: their basis can change, contracts expire and must be rolled, and margin and clearing arrangements matter. Delta and other option sensitivities also change over time, so the hedge must be rebalanced at a chosen frequency. The idealized result is presented as a theoretical benchmark, not a realistic profit strategy; transaction costs, basis risk, roll costs, and imperfect hedging can worsen outcomes.
Key ideas
- In the idealized setting described, a delta hedge replicates the option payoff less the premium received.
- A trader who did not receive the option premium cannot create the full option payoff by delta hedging alone.
- Futures hedges introduce changing basis and contract roll costs.
- Option sensitivities change over time, requiring decisions about hedge size and rebalancing frequency.
- Transaction costs and imperfect hedging can make real outcomes worse than the theoretical benchmark.
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Full text
# How to Delta Hedge with Futures? # 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: $$ \rm Market Value (stock \, trading) + Premium Received = MarketValue(option) $$ 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? ## Answer by Richi Wa (score 2, accepted) https://quant.stackexchange.com/a/4639 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. ## Answer by Matt Wolf (score 12) https://quant.stackexchange.com/a/4597 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 ## Answer by QFLearner (score 1) https://quant.stackexchange.com/a/80376 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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