# FLEX Volatility Interpolation

## Introduction

This paper presents the method used for calculating implied volatilities for FLEX options by interpolating between known SpiderRock surfaces. The technique involves using neighboring expiries to estimate the volatility at a given FLEX expiry, with detailed formulas for calculating weighted volatility averages, variance space, and adjustments for earnings events prior to expiration.

Volatility is a critical factor in option pricing, as it represents the uncertainty in the underlying asset’s price. For certain FLEX options, which offer customizable terms, calculating the volatility could become more complex due to the possible absence of relevant listed expiries. This paper outlines a method for interpolating volatility between neighboring standard options expiries using SpiderRock live volatility surfaces.

## FLEX Volatility Interpolation Approach

The goal is to compute the volatility for FLEX options using neighboring expiries as reference points. Since FLEX options may or may not involve listed expiries, we identify the two nearest standard option expiries (the neighboring listed expiries). Once identified, we calculate the weighted averages based on the time to expiry of the FLEX option in relation to its neighboring expiries and use these weights throughout the interpolations.

### Interpolation Weights

The weights for the neighboring expiries are calculated using the time remaining until the FLEX option’s expiry,

$ww_{Hi} = \frac{FLEXYearsToExpiry - yearsLo}{yearsHi - yearsLo}$ $ww_{Lo} = 1.0 - ww_{Hi}$

where *yearsLo* and *yearsHi* are both taken directly from the corresponding neighboring volatility surfaces.

These weights determine the contribution of the surface volatility multiples from the lower and upper neighboring expiries (Lo and Hi) when calculating the FLEX option’s volatility.

## Volatility Calculation in the Variance Space

Once the weights are established, the next step is to calculate the At-The-Money (ATM) censored volatility in the variance space. Variance, which is volatility squared, measures the overall risk of returns and scales linearly with time. We use the following formula to calculate the ATM-censored volatility without considering earnings:

$atmCen = \sqrt{\frac{wwLo \times yearsLo \times atmCenLo^2 + wwHi \times yearsHi \times atmCenHi^2}{FLEXYearsToExpiry}}$

## Earnings Adjustment and ATM Volatility Calculation

To adjust for the number of earnings events that might occur before the FLEX option's expiration, we apply the following model:

$σ^2T = σ_E^2n + σ_C^2T$

Where:

*σ*= volatility of the option*σ*= volatility of earnings events occurring prior to expiration_{E}*σ*= volatility of the option with all earnings events removed (censored volatility)_{C}*T*= time-to-expiration for the option*n*= number of earnings events occurring prior to expiration

The adjusted ATM volatility for the FLEX expiry can then be computed via:

$σ = \sqrt{σ_C^2 + \frac{n \times σ_E^2}{T}}$

Using the values from the curve records directly, this calculation becomes:

$atmVol = \sqrt{atmCen^2 + \frac{earnCnt \times eMove^2}{FLEXYearsToExpiry}}$

This formula ensures that both time and earnings-related volatility are accurately incorporated into the final volatility estimate for the FLEX option.

## Final Surface Volatility Calculation

Once the ATM volatility has been calculated, the final volatility (*sVol*) at the given strike price, including any FLEX strikes, can be computed by blending the volatilities from the neighboring expiries. This is done by finding the volatility multiples at the corresponding strikes for both neighboring expiries, then applying the weighted average formula:

$sVol = atmVol \times (ww^{Lo} \times \frac{sVolLo}{atmVolLo} + ww_{Hi} \times \frac{sVolHi}{atmVolHi})$

This step ensures that the volatility at the specific strike price and expiry reflects both the surrounding surfaces and the characteristics of the FLEX option.

## Edge Cases

The method described above works well in most cases where neighboring expiries are available. However, there are edge cases where adjustments need to be made:

- If there is only one neighboring expiry (either higher or lower), then the volatility is calculated using just that expiry and the formulas listed above by applying a weight of 1 to the existing expiry.
- If the strike is outside of the
*minXAxis*and*maxXAxis*(the valid moneyness range of the volatility surfaces), then the volatility will be interpolated from the wings.

## Conclusion

The approach to calculating volatility for FLEX options provides a robust way to interpolate between neighboring expiries, accounting for time, earnings events, and ATM volatilities. This method ensures a smooth transition between standard options and more customizable FLEX options.