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Why Option Delta Bands Differ for Long and Short Positions

Article Quant Q&A · Author: peterram
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Summary

This note addresses why optimal delta hedging bands may differ when holding long versus short options, even when the cited approximation appears to depend on the absolute value of gamma. It reports a distinction from Zakamouline’s treatment: the effective variance is adjusted upward for a long call and downward for a short call, with the adjustment depending on the position’s sign. This provides a reason the resulting bands need not match despite the apparent symmetry in the gamma term.

The note attributes this rule to a paper discussing Leland’s option hedging strategy with transaction costs. It gives no derivation, parameter definitions, numerical example, or evidence comparing the resulting bands, so the claim cannot be independently checked from this text alone. The explanation is limited to calls and the referenced setup; applying it to other options or hedging assumptions would require consulting the underlying source.

Key ideas

  • The note says the effective variance adjustment differs for long and short call positions.
  • The stated distinction depends on whether the option position is long or short.
  • The source cited is a paper on option hedging with transaction costs.
  • The document provides no derivation or numerical validation of the adjustment.

Tags

OptionsVolatilityDerivatives pricingRisk management
Full text
# Optimal Hedging of Options - asymmetry between long and short vol positions


# Optimal Hedging of Options - asymmetry between long and short vol positions












Going over Zakamouline's Approximation method for optimal delta hedging of options, it is claimed that the result remains valid for both buying options (long vol positions) or selling options (short vol positions). Furthermore, when the final delta bands are plotted for these two different cases, they are clearly different charts (see image):

However, mathematically looking at the result, I can't see why the bands will be different. My reasoning is that the main difference between both positions are the sign of the Gamma, since all the other inputs are the same. However, the results of Zakamouline, seem to only depend on the absolute value of the Gamma:

All images were used from Euan Sinclair's book. The Zamakouline papers cited in the book are:

Optimal Hedging of Options with Transaction Costs

Efficient Analytic Approximation of the Optimal Hedging Strategy for a European Call Option with Transaction Costs

European Option Pricing and Hedging with both Fixed and Proportional Transaction Costs

My question is: why are the delta bands different for long/short option positions, when looking at the mathematical equations above? Thanks in advance!

## Answer by asdffdsa (score 1)

https://quant.stackexchange.com/a/71054

For a long call use:

$\sigma_m^2 = \sigma^2(1+K)$

For a short call use:

$\sigma_m^2 = \sigma^2(1-K)$

Source: V. Zakamulin: Yet Another Note on the Leland's Option Hedging Strategy with Transaction Costs, 2005

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.