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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!

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  • $\begingroup$ A quick intuitive answer: your statement is approximately true for reasonable inputs. However I doubt anyone would do this in practice - it’s not like the Black Scholes formula is computationally very expensive. $\endgroup$ – dm63 Dec 27 '19 at 14:02
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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.

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