Why Averaging Option Strikes and Volatility Can Misestimate Portfolio Delta
Summary
The document asks whether the average strike and, where relevant, average implied volatility of a set of calls can be entered into one Black-Scholes calculation to approximate the portfolio’s average delta. The scenarios hold the underlying, expiry, rates, dividends, and contract counts constant, while varying strikes and then volatility across strikes.
The accepted response frames the issue as whether option value is linear in strike and volatility inputs. Black-Scholes depends on strike through both a direct term and the normalized distance from the forward, so substituting averages generally does not reproduce the average of individually calculated option values or deltas. The response notes that limited near-linearity may occur in some parameter regimes, but offers no quantitative error bound. It also says the shortcut is not common in professional practice because calculating each option is inexpensive; accuracy should therefore be checked by summing individual deltas.
Key ideas
- Averaging option inputs before pricing is generally not equivalent to averaging outputs calculated option by option.
- Black-Scholes call value depends on strike both directly and through normalized distance from the forward.
- Volatility variation across strikes adds another source of nonlinearity to the proposed proxy.
- The response offers no error bound and recommends individual calculations as practical in ordinary workflows.
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# Calculating Delta of option portfolio using average of inputs
# Calculating Delta of option portfolio using average of inputs
Trying to think through two options portfolio scenarios, which are highly similar. I'm wondering if you can take a portfolio of options, all written against the same underlying product, and use average other inputs into Black Scholes to get the delta of the entire portfolio.
Scenario 1 - you have a portfolio of Calls ONLY, in which all the CALL options in the portfolio have the same time to expiry, interest rate, dividend, and underlying, and continuous/constant volatility across strikes, but different strike prices. You only have 1 contract at each of the various strike price (i.e. constant volume).
Can you take an average of all of the strikes in the portfolio, and use the average strike and average volatility value in Black Scholes to derive a good estimate of the portfolio delta? Or will this value differ substantially from the the portfolio delta that would be derived by calc'ing the delta of each option individually, and then summing, and dividing by # of contracts?
Scenario 2 - same as scenario 1 but there is no longer continuous volatility, so there are different implied volatility values at each strike. If you take the average of all of the implied volatility values and the average of all strike prices, and input into a single Black Scholes formula, is this a good proxy for portfolio delta?
Realize this question is subjective - how accurate does the portfolio delta need to be. I'm wondering if this proxy technique is used in professional practice, and also just subjective thoughts on whether this is a reasonable approach for retail trading.
Thank you for your help!
## Answer by lehalle (score 3, accepted)
https://quant.stackexchange.com/a/50458
Your question is in fact one on the linearity of the replication cost of an option. Let formulate it a general way: once you can express the replication cost $C$ of a payoff as a function of several factors $X$, the strike $S$ and the volatility $\sigma$ that you assume to be a function of the strike, you are asking if $$\frac{1}{N}\sum_\ell C\big(X, S_\ell, \sigma(S_\ell)\big)= \textstyle C\big(X, \frac{1}{N}\sum_\ell S_\ell,\frac{1}{N}\sum_\ell \sigma(S_\ell)\big).$$ This is clearly a question about the linearity of $s\mapsto C(F,s,\sigma(s))$.
Now have a look at the Black-Scholes formula of a call (taken from wikipedia): $$C(F, \tau) = D \cdot\left[ N(d_+) F - N(d_-) K \right].$$
The strike is involved (linearly) in the forward price $F$ and in the renormalized distances: $$d_\pm = \frac{1}{\sigma\sqrt{\tau}}\left[\ln\left(\frac{F}{K}\right) \pm \frac{1}{2}\sigma^2\tau\right]$$
It is not easy to infer all the configurations for which this formula is linear in $F$, but at least when the implied volatility is so large that $N(d_\pm)$ are more or less contants then $C$ become linear in $F$.
To fully answer to your question: I have never seen this kind of considerations used in practice because once you know numerically how to price one option, it is not very costly to price more of the same kind.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.