Why Discrete Option Portfolios Cannot Stay Price-Neutral Across All Moves
Article Quant Q&A · Author: bokabokaboka
Summary
The document examines whether a portfolio of options can capture a relative implied-volatility opportunity while remaining immune to changes in the underlying price throughout its holding period. The author constructs a four-option position with initial delta and gamma neutrality, then reports that both exposures reappear the following day as option prices and Greeks change. This illustrates that a hedge set at one moment does not remain fixed as market conditions evolve.
The replies distinguish local Greek neutrality from immunity across all price movements. One proposes combining synthetic long and short positions, while another argues that eliminating price sensitivity at every order with a finite set of options is generally unattainable; a continuum of options might resemble a volatility derivative, though spot-sensitive skew can undermine the intended exposure. The discussion also notes a possible vol-of-vol component. It offers conceptual guidance and a numerical example, not a general proof or a tested trading strategy.
Key ideas
- Delta and gamma neutrality at initiation do not guarantee those exposures remain neutral later.
- Option Greeks change as prices and market conditions evolve.
- Eliminating sensitivity to spot at every order is a stronger requirement than local delta and gamma hedging.
- A finite collection of options is unlikely to achieve complete price immunity while retaining skew exposure.
- Volatility-of-volatility may offer a distinct exposure, but the discussion does not develop a trading method.
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Full text
# Is it possible to establish a purely options-based portfolio that is completely neutral to changes in asset prices? # Is it possible to establish a purely options-based portfolio that is completely neutral to changes in asset prices? I found that the implied volatility of 10008162 is relatively high, while the implied volatility of 10007955 is relatively low. Therefore, I want to go long on 10007955 and short on 10008162, aiming to profit from the decreasing difference in implied volatility between the two options. However, I do not want to take on the risk of price fluctuations in the underlying asset. Therefore, I have constructed a delta-neutral and gamma-neutral pure options portfolio. This options portfolio also includes two long-dated call option contracts.My goal is to achieve immunity to price fluctuations in the underlying asset throughout the entire holding period of the portfolio, without needing to adjust the option positions in the portfolio! | code | price | IV | Delta | Gamma | Vega | Theta | Strike Price | | 10007955.SH | 0.082 | 0.2693 | 0.512 | 1.9204 | 0.0031 | -0.0022 | 2.75 | | 10008162.SH | 0.0109 | 0.4892 | 0.0691 | 0.3526 | 0.001 | -0.0013 | 3.4 | | 10008264.SH | 0.235 | 0.2411 | 0.5623 | 0.7097 | 0.0089 | -0.0006 | 2.75 | | 10008265.SH | 0.2151 | 0.2449 | 0.5273 | 0.7068 | 0.009 | -0.0007 | 2.8 | | code | position | | 10007955.SH | 12140610000 | | 10008162.SH | -65729200000 | | 10008264.SH | -47831560000 | | 10008265.SH | 47831560000 | The above data and positions are as of October 31, 2024. You will find that the delta and gamma of this options portfolio are indeed both zero！ The following are the delta and gamma data for this portfolio as of November 1. | code | price | position | delta | Gamma | Portfolio delta | Portfolio Gamma | | 10007955.SH | 0.0912 | 12140610000 | 0.5485 | 1.9142 | 6659124585 | 23239555662 | | 10008162.SH | 0.0102 | 65729200000 | 0.0671 | 0.3565 | -4410429320 | -23432459800 | | 10008264.SH | 0.2494 | 47831560000 | 0.5754 | 0.6883 | -27522279624 | -32922462748 | | 10008265.SH | 0.2301 | 47831560000 | 0.542 | 0.6858 | 25924705520 | 32802883848 | | | | | | sum | 651121161 | -312483038 | It is clear that this portfolio already has exposure to delta and gamma risks. Therefore, it is no longer possible to be immune to changes in the underlying asset's price. As a result, the purpose of constructing this portfolio cannot be achieved! However, I still want to know if it is possible to achieve immunity to price movements in the underlying asset by constructing a pure options portfolio. This portfolio must be long on 10007955 while being short on 10008162! Now I am concerned whether such a portfolio exists theoretically？ For example, can adding Greeks like DgammaDspot and DvegaDvol increase the number of equations in a linear system, and can adding more long-dated option contracts increase the number of columns in the linear system (effectively increasing the dimension of the null space of an underdetermined system)? Just considering delta and gamma neutrality is equivalent to a second-order Taylor expansion, which may not fit the "option price - underlying asset price" curve accurately enough. Would incorporating higher-order Greeks significantly reduce the sensitivity of the options portfolio to changes in the underlying asset price, ultimately achieving immunity of the options portfolio to fluctuations in the underlying asset price? ## Answer by Newquant (score 1) https://quant.stackexchange.com/a/81040 Sure you can. Enter a synthetic (long call, short put) long at one strike and enter a synthetic short (short call, long put) at a higher strike. You are now totally immunised against market movement. ## Answer by Frido (score 1) https://quant.stackexchange.com/a/81043 The chatGPT output unfortunately does not do justice to your question. I hope chatGPT doesn't become a regular guest here. Hence my vote to close. Having said this, if you want to have no sensitivity to stock price throughout the option holding period, I think that basically means you do not want sensitivity to the stock price at all orders. No delta, no gamma, no dgamma/dspot etc. That will be quite difficult to achieve with only a discrete number of options. In theory you'd need an infinite number of options and I suspect you'll end up then with some sort of pure volatility derivative. However that might defeat the purpose of your trade, as the skew is sensitive to the spot price. Conclusion: I don't think you can achieve skew exposure without some exposure to price movement. Nuance: part of the 'skew' is actually due to vol of vol. You can capture vol of vol without exposure to spot price, but this might be taking your question a step too far.
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