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Normalizing Delta-Hedged Option P&L by Underlying Notional

Article Quant Q&A · Author: Fadmad
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Summary

The document examines how to interpret a study’s reported returns for a short gamma strategy in S&P 500 options. The strategy’s daily dollar profit and loss is divided by the prior day’s index level, rather than by the initial option premium or trade cost. The explanation is that one option contract’s notional value scales with the index, so this normalization adjusts for changes in contract value over the historical sample.

This ratio is proportional to a return if the capital required for the trade is assumed to remain a constant fraction of the option’s notional. The document describes this as a plausible but imperfect measure, rather than a conventional return on invested capital. It also notes that a lack of historical margin requirement data may make capital-based comparisons difficult. The explanation offers a way to read the paper’s calculation, but does not establish that the chosen normalization is universally appropriate.

Key ideas

  • Dividing option strategy P&L by the prior index level normalizes for the changing notional of an index option.
  • The resulting measure is not necessarily a return on the capital invested in the trade.
  • It is proportional to a capital return only if required capital is a stable fraction of option notional.
  • Missing historical margin data limits the ability to calculate a direct return on capital.

Tags

OptionsFuturesRisk managementBacktesting
Full text
# Computing Delta-Hedged Option Returns


# Computing Delta-Hedged Option Returns












I was reading some papers on delta-hedged option returns and came across an intriguing paper that I found quite interesting.

However, I was a bit confused on the authors' methodology of computing option returns for their strategies. For some context, the paper is looking at the returns of a gamma selling option strategy with S&P500 as the underlying. The formula is on page 5 of the paper

Roni Israelov, Harsha Tummala: Which Index Options Should You Sell? SSRN, 28 Jun 2017

They use the following formula:

I found this confusing because I would expect the return of a delta-hedged short option position to be the daily change in price divided by the initial cost of entering the position (just like all other papers I have read on this subject). In this case, they simply divide the daily P&L by the underlying.

Is there something I'm missing?

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/70562

It sems a plausible though not perfect measure.

The numerator is the dollar p&l of one option hedged with futures.

The notional value of one option is proportional to the S&P level (specifically 250 times S&P futures for CME options, 100 times SPX for CBOE options). The study goes from 1996 to 2015, during which the notional value has changed (increased) considerably.

The division by SPX(t-1) is a normalization to take into account the changing value of 1 option. It is not perhaps a "return" but it is proportional to the return if you assume the capital needed to do this trade is a constant fraction of the notional value of one option.

(In the absence of historical data on margin requirements, which I believe is hard to find, it is perhaps the best you can do. Though it won't impress academics).

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.