Estimating Implied Volatility for Multi-Leg Options Positions

# Implied volatility of a complex options position


# Implied volatility of a complex options position












Assume I have a "complex" options position like a straddle, strangle, or iron condor. In other words, several options traded together as a single position against one underlying asset (not a basket option).

I know the implied volatility of each of the options within the position. Is there an accepted method of generating an implied volatility for the overall position?

## Answer by Kiwiakos (score 12)

https://quant.stackexchange.com/a/22775

You can guesstimate by vega weighted implied vol. This is why:

Say that you have a portfolio of options with prices $P_j$. Each one of them has a different pricing function $f_j$ (as function of vol) and a different implied vol $\sigma_j$. For each option $f_j(\sigma_j)=P_j$.

Now you put them together in a single product. If the implied vol of the product is $\sigma$ then $\sum f_j(\sigma)=\sum P_j$. Now, approximately each pricing function will satisfy $f_j(\sigma)\approx P_j+V_j (\sigma-\sigma_j)$ as a linear expansion around its price, with $V_j$ the Vega.

If you substitute and solve you end up with the vega weigted vol $$\sigma \approx \frac{\sum V_j\sigma_j}{\sum V_j}$$

## Answer by Phun (score 4)

https://quant.stackexchange.com/a/21239

First note that implied volatility only makes sense with respect to an etablished pricing model, like the Black-Scholes or Bachelier model, and it is the quantity which has to be put into a closed form pricing formula obtained in one of those models to get the market dollar price.

For a straddle with strike $ K $ holds $$ \text{Price_Straddle}(K) = \text{Price_Call}(K) + \text{Price_Put}(K) \\ = 2\text{Price_Call}(K) - \text{UnderlyingPrice} + K,$$ due to the Put-Call parity. So the implied vola of a straddle equals the implied vola of a call.

To my knowledge, for strangles such a formula does not exists the same as for iron condors.

## Answer by JBerstein (score 4)

https://quant.stackexchange.com/a/58871

Yes, and in fact this is a quoting convention in FX derivatives; for flys straddles and reversals. It is applied to Legs by often by symmetric delta, and strike-by-delta formulas is used to convert out. Keep in mind that put call parity as it relates to option type is relevant here.

Reference a see: "Foreign Exchange Option Pricing: A Practitioner's Guide by Iain J. Clark". The method is simple but long for a thread here.

The weighted approach previously answered, is a clever approach, and will give better results in relation to sensitivity as things change which is ultimately what I think you are interested in.

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/21757

In general, the implied volatility is based on vanilla European or American options. In your case, since the positions depend on only a single underlier, if you can have an analytical formula, or approximation, for each individual position, then, in principle, you can compute an implied volatility based on the market price of your portfolio. However, note that, in general, there is no liquid market price quote for such "complex" options.

## Answer by L. Francis Cong (score 0)

https://quant.stackexchange.com/a/81915

The high-rated answer is brilliant! Let me discuss it a little bit more rigorously.

Let $F(f,\sigma)$ be the option pricing formula where $f(x)$ is the payoff function of the option at maturity where the single argument of this function is the underlying asset price at maturity (e.g. $f(x)=max\{x-K,0\}$ for a European call option with strike $K$) and $\sigma$ is the volatility.

Then, from the risk-neutral pricing formula, we know that $F(f,\sigma)$ is linear in $f$, i.e. $F(a_1f_1+a_2f_2,\sigma)=a_1F(f_1,\sigma)+a_2F(f_2,\sigma)$ where $f_1$ and $f_2$ are any two different payoff functions.

Now consider a portfolio of options with price $P=\sum_in_iP_i$ where $P_i$ is the $i$th option in the portfolio. Then, the payoff function of this option portfolio is $\sum_in_if_i$. Our goal is to calculate the option portfolio's implied volatility $\sigma$, such that \begin{equation} F\left(\sum_in_if_i,\sigma\right)=P=\sum_in_iP_i=\sum_in_iF(f_i,\sigma_i) \end{equation}

As proposed in the high-rated answer, we assume $\sigma_i\approx\sigma$ so that we have the following first-order approximation: \begin{equation} F(f_i,\sigma)\approx F(f_i,\sigma _i)+\frac{\partial F(f_i,\sigma_i)}{\partial \sigma_i}(\sigma-\sigma_i) \end{equation}

Multiplying by $n_i$ and taking the sum, we have \begin{equation} \sum_in_iF(f_i,\sigma)\approx \sum_in_iF(f_i,\sigma _i)+\sum_in_i\frac{\partial F(f_i,\sigma_i)}{\partial \sigma_i}(\sigma-\sigma_i) \end{equation}

Let $\mathcal{V_i}=\frac{\partial F(f_i,\sigma_i)}{\partial \sigma_i}$ be the vega of option $i$ and use the linearity of $F(f,\sigma)$ with respect to $f$. We finally have \begin{equation} P=F\left(\sum_in_if_i,\sigma\right)=\sum_in_iF(f_i,\sigma) \end{equation} \begin{equation} \approx \sum_in_iF(f_i,\sigma _i)+\sum_in_i\mathcal{V}_i(\sigma-\sigma_i)=P+\sum_in_i\mathcal{V}_i(\sigma-\sigma_i) \end{equation} As a result, we have $\sum_in_i\mathcal{V}_i(\sigma-\sigma_i)\approx0$ which gives \begin{equation} \sigma=\frac{\sum_in_i\mathcal{V}_i\sigma_i}{\sum_in_i\mathcal{V}_i} \end{equation}

## Answer by Newquant (score 0)

https://quant.stackexchange.com/a/81918

The implied volatility of any non-linear payoff instrument is given by: $$\sqrt{\frac{2\Theta}{\Gamma S^2}} $$

So for a group of options your overall implied volatility can be described as: $$\sqrt{\frac{2\sum|\Theta_i|}{S^2 \sum|\Gamma_i|}}$$

Then scale this by the sign of gamma to know if you are net long or short realised volatility.

This can be useful for things like risk reversal where the difference in IVs on the skew can allow one to create a strategy which is locally long gamma AND long theta.

Also consider a calendar spread trade where there is a spread between different IVs, you can figure out your breakeven move in the underlying using this method.