Volatility Surface

Vol Surface: Term Structure

Term Structure usually refers to ATM implied vols by Time to maturity. And of course, these vols are all annualised vols for consistency.

Term Structure is usually upward sloping. But front-end vols are more sensitive to changes of realised vols and anticipated events (e.g. French election etc.). So market turmoil could lead to Term Structure inversion.

The Term Structure shape tend to be mean-reverting in nature. Trading strategies exploiting this mean-reverting feature involves buying and selling two options in Vega-neutral amounts, so that we have exposure to only the vols curve shape, not the level shift. This is similar to Duration-neutral Yield curve trades in Fixed Income.

Front-end vols are primarily Gamma plays, so views on Gamma is essential to formulating Term Structure. ?

Back-end vols are usually considered as sum of Front-end vols and vols curve. ?

Vol Surface: Vol Skew (Risk Reversal)

By FX market convention, Risk Reversal is quoted as

    \[\sigma _{Call, 10d} - \sigma _{Put, 10d}\]

    \[\sigma _{Call, 25d} - \sigma _{Put, 25d}\]

Risk Reversal represents directional variation of implied vol with Strike. This corresponds to the Third Standardised Central Moment of underlying spot distribution.

For reference, the n-th moment of Probability Density Function f(x) about value c is defined as:

    \[\mu _{n} = \int _{-\infty}^{\infty} (x-c)^{n}f(x) dx\]

Risk Reversal is well correlated to the correlation between spot and vol moves. We can think of Risk Reversal as the implied skew while spot-vol correlation as realised skew. And this means:

Positive skew: Option market expects spot rallies to be more volatile than sell-offs. E.g. USD/EM pairs. The option-implied spot distribution is tilted to the right.

Negative skew: Option market expects spot sell-offs to be more volatile than rallies. E.g. JPY cross pairs. The option-implied spot distribution is tilted to the left.

Skews are also often to be valued by comparing to ATM vols, i.e. \frac{RR}{ATM vol} ratio.

Vol Surface: Vol Fly (Butterfly)

By FX market convention, Butterfly is quoted as

    \[\frac{1}{2}(\sigma _{Call, 10d} + \sigma _{Put, 10d}) - \sigma _{ATM}\]

    \[\frac{1}{2}(\sigma _{Call, 25d} + \sigma _{Put, 25d}) - \sigma _{ATM}\]

Butterfly represents undirectional variation of implied vol with Strike or convexity of vol curve / smile. This corresponds to the Fourth Standardised Central Moment of underlying spot distribution.

We can think of Butterfly as the dimension of vol curve / smile that richens “wings” or low-delta options compared to ATM options. So non-Zero Butterfly means underlying spot distribution deviates from log normality assumed by Black-Scholes, and wing / low-delta vols are priced at premium to ATM vol.

Butterfly is well correlated with volatility of ATM vol. Thus, it is often considered as the parameter capturing vol-of-vol.

Vol Surface Arbitrage


Vol Surface Interpolation

9 thoughts on “Volatility Surface”

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