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Delta-Gamma Hedging and the Role of Option Position Sizing

Article Quant Q&A · Author: Kilkik
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Summary

The document asks whether machine learning needs to predict option gamma when a trader can rebalance delta hedges over time. It frames a portfolio containing a call, shares, and a second option, with target conditions for zero delta and zero gamma at a later time. The central conceptual issue is that a hedge set now can change as the underlying price and time change, so continually maintaining neutrality entails rebalancing; gamma describes how option delta responds to movements in the underlying. A question about predicting future hedge requirements is raised, but the source does not develop a forecasting method or demonstrate that machine learning is needed.

The answer explains that the second option’s position must be chosen to offset the original option’s gamma, which can require shorting it, and that practical trading generally uses whole contracts or scaled-up positions rather than fractional options. The required hedge ratio depends on the instruments’ gammas and available contract sizes. The brief response does not derive the full hedge equations, account for transaction costs, or resolve when predictions would improve on ordinary rebalancing.

Key ideas

  • Delta neutrality at one moment does not keep an option portfolio neutral as market conditions change.
  • Gamma measures how option delta changes with the underlying price.
  • A second option can offset gamma, with its position direction determined by the signs and sizes of the options’ gammas.
  • Practical option hedges are sized in contracts, often by scaling positions to approximate a desired ratio.
  • The document raises prediction questions but does not provide a forecasting framework or evidence for machine learning.

Tags

OptionsDerivatives pricingRisk managementPosition sizing
Full text
# What to predict in delta-gamma hedging?


# What to predict in delta-gamma hedging?












I am working in delta-gamma hedging with machine learning. I guess I have to predict gamma (since predicting gamma tells you how delta will behave) but I don't know why is it needed. I think that a mathematical framework can help me have good understanding of the context.

Suppose we currently have a call option with a current delta of $0.6$ at time $t$. At $t=t+1$, if the stock price goes up by $1$, then the option's value goes up by $0.6$ and we need to short $60$ shares and that's it I have a delta neutral portfolio at that time, and I keep doing this at each time step with the new deltas so that I always have a zero delta portfolio and there is nothing to predict and no need for gamma. By the way, when delta hedging the delta of the portfolio is constant equal to 0 and since gamma is the derivative of delta, isn't gamma forced to be 0 too as the derivative of a constant process ?

Of course I know I am wrong and my reasoning above has many issues so I tried to set up a mathematical framework for this. Let $S_t, D_t, G_t$ be respectively the stock price, Delta and Gamma of a call option at time $t$. These quantities are known at $t$.

Let $S'_t, D'_t, G'_t$ be similar quantities for another option on a same underlying, to be able to do neutral gamma hedging. What we want is at time $t+1$, $P_{d,t+1}=D_{t+1}+yD'_{t+1}-x=0$ and $P_{g,t+1}=G_{t+1}+yG'_{t+1}=0$ where $P_{d,t+1}$ is the portfolio's delta, $P_{d,t+1}$ its gamma, and $N_{t+1}$ the number of shares we short sell (that have a delta of 1). If the rebalancing is done before time $t+1$ I would understand why we are interested in predicting. Is it possible to buy a portion $y$ of an option ?

## Answer by KaiSqDist (score 0)

https://quant.stackexchange.com/a/79070

It does not make sense to buy the "prime" option, you would only obtain a gamma-neutral portfolio by shorting it, because you would need the option gammas to offset each other.

I don't think you can buy a portion of a option, it makes much more sense to size up the position. For example, if you need to hedge 1 $C$ with 0.4 $C'$, it is more feasible to long (short) 100 $C$ (40 $C'$). Then again, it really depends on what is offered on the platform or your broker.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.