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

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