Degrees of Freedom in Delta, Gamma, and Vega Hedging
Summary
The document asks whether a portfolio can be hedged simultaneously for value, delta, gamma, and vega, after considering hedges that match pairs of sensitivities. The answer frames the problem in terms of degrees of freedom: a hedge needs enough independent instruments or positions to offset each target exposure. It uses vector combinations to illustrate that independently available hedges for the component sensitivities can be combined into a joint hedge.
This is a conceptual argument, not a worked options portfolio or proof that a particular market’s instruments provide the required independent exposures. The notation labels the third component as theta while the question asks about vega, and the answer’s conclusion therefore relies on an informal degree-of-freedom interpretation. In practice, feasibility depends on the instruments’ sensitivity vectors being sufficiently independent, as well as constraints such as liquidity and position limits.
Key ideas
- A simultaneous sensitivity hedge requires enough independent degrees of freedom in the available instruments.
- Separate hedges can be combined when their exposure vectors span the target sensitivities.
- The argument does not establish that a given set of traded options can produce the required exposures.
- The answer uses theta in its vector notation while the question concerns vega, leaving an ambiguity.
Tags
Full text
# Is it possible to construct a hedge that matches value Delta Gamma and Vega? # Is it possible to construct a hedge that matches value Delta Gamma and Vega? Given a strike price, current price, risk free rate, dividend yield and volatility, I have been asked to calculate: - a hedge which matches the value Delta and Gamma - a hedge which matches the value Delta and Vega I have managed to calculate both of these, however I have also been asked whether it is possible to calculate a hedge which matches the value Delta, Gamma and Vega? My gut instinct is no, but I am not entirely sure why, and I cannot find anything online to help me. If someone could help me understand why it is yes or no, it would be greatly appreciated! ## Answer by Attack68 (score 1, accepted) https://quant.stackexchange.com/a/51757 If you consider delta, gamma and vega as three variables, and you are able to construct a portfolio with any values, i.e. with three degrees of freedom: $$ [\delta, \gamma, \theta]$$ And you have a space of products which allow you to construct a hedge for any such delta and vega then you must have at least these two degrees of freedom (in some basis): $$ [\delta, \gamma, \theta] + x [1, 0, 0] + y [0, 0, 1] = [0, \gamma, 0] $$ Similarly if you can do it for delta and gamma then: $$ [\delta, \gamma, \theta] + x [1, 0, 0] + z [0, 1, 0] = [0, 0, \theta] $$ So then by definition you must have the necessary degrees of freedom, i.e the combination of products to hedge the full suite: $$ [\delta, \gamma, \theta] + a [1, 0, 0] + b [0, 1, 0] + c [0, 0, 1] = [0, 0, 0] $$ Failing that explanation you can always hedge your portfolio by doing exactly the same but opposite trades!!
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.