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

Option Greeks

European Vanilla Option Greeks

Delta

Interpretations

  1. Option price sensitivity to spot (Spot Delta)

gradient of option price tangent line

  1. Proxy for Probability of option finishing ITM.

Delta value is between 0 and 1. But this is just a proxy for exercising probability for interpretation purpose.

In fact, the risk neutral Probability Density Function (PDF) is the 2nd derivative of call price with respect to Strike, i.e.

    \[\frac{\partial ^{2}C}{\partial K^{2}} = e^{-r(T-t)}\pi (K)\]

  1. Hedge ratio

Delta used on Vol Smile

Conventionally in FX Option space, the x-axis of Vol Smile plot is denoted as Delta (10d, 25d) instead of strikes.

Adapted Delta

Adapted Delta is the “real delta”, i.e. the actual hedge ratio, taking into account the shape of the vol smile.

Black-Scholes assumes constant vol.

Spot Delta vs Forward Delta

We normally refers Delta as Spot Delta, i.e. spot sensitivity \frac{\partial C}{\partial S}.

Forward Delta is the sensitivity to Forward price, i.e. \frac{\partial C}{\partial F} which captures interest rate risk implicit in forward points.

Forward Delta is typically used for NDF currencies and long-dated options.

Impact of Spot

Impact of Time to expiry

As we see from the below plot, as time passes, the option price curve moves closer to the At-Expiry payoff. Therefore, ITM Delta moves closer to 1 and OTM Delta moves closer to 0. ATM Delta has greater uncertainty (high Gamma) near expiry.

Impact of Vol

For ITM options, higher vol means less certainty that it will finish ITM, i.e. smaller Delta.

For OTM options, higher vol means higher probability that it will finish ITM, i.e. higher Delta.

A doubling of vol has roughly the same effect on an option’s Delta (and its price) as a quadrupling of time. For example,

    \[Call\, Option\, with\, S = 102, K = 100, T = 1m, \sigma = 5\%, Delta = 0.92\]

    \[ \sigma = 5\% \rightarrow 10\% \Rightarrow Delta = 0.92 \rightarrow 0.76\]

    \[ T = 1m \rightarrow 4m \Rightarrow Delta = 0.92 \rightarrow 0.76\]

Gamma

Impact of Time to expiry

Impact of Vol

A doubling of vol has roughly the same effect on an option’s Gamma as a quadrupling of time. For example,

    \[Call\, Option\, with\, S = 100, K = 100, T = 1w, \sigma = 5\%, Delta = 0.42\]

    \[ \sigma = 5\% \rightarrow 10\% \Rightarrow Delta = 0.42 \rightarrow 0.27\]

    \[ T = 1w \rightarrow 1m \Rightarrow Delta = 0.42 \rightarrow 0.27\]

Gamma Trading

If we long an Option with Delta hedged, we will have positive P/L from long Gamma.

However, this Gamma P/L comes at cost of Theta decay as we are long option.

Theta

Theta measures the Option value decay as time passes. We say Theta is positive, meaning, as time passes (time to expiry decreases), Option price also decreases.

But can European Put Option Theta be negative?

Impact of Time to expiry

Impact of Vol

Vega

Impact of Time to expiry

Impact of Vol

Continuous Barrier Option Greeks

Reversed Knock-Out (RKO): Up and Out Call with Barrier > Strike OR Down and Out Put with Barrier < Strike

Reversed Knock-In (RKI): Up and In Call with Barrier > Strike OR Down and In Put with Barrier < Strike

Reversed means the Barrier level is In-The-Money (ITM).

RKO Call / Put + RKI Call / Put = Vanilla Call / Put

This is true for option price and all greeks.

Delta

Impact of Spot

Delta is the gradient of curve. We notice there’s a Delta gap on barrier trigger, for both RKI and RKO.

RKI Call Option

RKO Call Option