Mini Chapter Five
Let's speak Greek(s)
Option Greeks
Risks embedded in an option contract that appear with movements in its variables and determine its pricing are categorised as option greeks. Let’s tackle each of them below:
- Delta – is the measure of the change in the price of the option due to change in the price of the underlying. Few other ways to think about it:
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- Graphically it can be understood as the slope of the option price plotted against the asset price, in the curve shown below.
- One can also understand delta to mean the probability of the option being in the money.
- It’s the fraction of the option contract’s notional that one buys/sells of the underlying to neutralise market price risk of the option.
- A long call option delta ranges between 0 to +1 while a long put option delta ranges between -1 to 0; note the sign on delta denotes the direction of exposure on the underlying asset. As the option’s moneyness increases, so does its absolute delta; for deep in the money this eventually tends to +1 for a call option and -1 for a put option.Â
- In addition to being sensitive to underlying’s price, delta is also sensitive to volatility of the underlying and time to expiration of the option. An at/out of the money option on a more volatile asset would have a wider probability distribution and hence a higher probability of ending in the money at expiry → higher delta. Time to expiry brings in the time value effect (extrinsic value) on the option price. As the option approaches expiration its delta tends to either 0 or 1 depending on its moneyness. For an option approaching expiry at the money, delta remains at ~0.5.
- Delta Bleed – in common trading parlance is the erosion or addition to the existing delta of a portfolio as it ages by a day (refer to the graph on Delta vs time to maturity) with no change in underlying market. This is one of the parameters used by traders to decide the amount of delta to hedge keeping in mind the bleed on it in case all else is constant.Â
- Below are graphical representations of the greek profiles for a long call option with:
- Strike @ 110
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- Notional – each option contract notional is made of 1 stock
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- Option expiries – 1wk, 1m, 3m, 6m, 1y
- Implied volatilities – 5%, 15%, 25%, 30%
Graph 8 – Delta profile across Option expiries
Source: Pandemonium.
- Delta ranges between 0 & +1 for a call option, ATMF delta is around +0.5.
- As option maturities are longer delta curves are flatter.Â
- Delta increases with maturity for out of the money options as there is more time available for the option to end up in the money and vice versa.
Graph 9 – Delta profile across Implied Vols
Source: Pandemonium.
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- Delta profiles flatten out at higher Implied Vols.
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- For Out of the money Options Delta would increase as Implied Vols go up – Intuition is that the chances of option ending in the money increase as Vol goes up.
- Similarly for In-the-Money options Delta would go down as Vols increase as the probability of the option going out of money increases.
- Gamma – is the change in an option’s delta due to the change in the price of the underlying. Refer to the delta chart above – slope of the delta curve increases as the underlying approaches the strike price. Hence gamma is maximum for an option when the underlying is at the strike price. In other words, for a market price further away from strike gamma is lower (tends to 0 just before expiry) and a market price close to strike gamma is higher (tends to infinity at strike just before expiry).Â
- Graphically then, all else being the same, Y axis now changes to dollar value of gamma (or change in Delta) for every $1 move in the underlying’s priceÂ
Graph 10 – Gamma profile across option expiries
Source: Pandemonium.
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- Peak Gamma is observed around the ATMF strike
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- Amount of gamma is higher around the ATMF when the option is nearer maturity
- For options closer to maturity, gamma falls off faster as the underlying moves away from ATMF as compared to a longer tenor option
Graph 11 – Gamma profile across Implied Vols
Source: Pandemonium.
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- Gamma profile narrows as IV decreases – intuitive as ATMF option will have more sensitivity to underlying changes if volatility is lower
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- Amount of gamma is higher for strikes at our around ATMF
- For options where IV is higher gamma is more spread out
- Vega – is the change in the option’s price due to change in the underlying asset’s implied volatility; dollar value change for every 1% move in implied vol. A more volatile asset price would increase the probability of its option’s payoff to increase, thereby increasing the value of a long position. In other words a long option position has a positive Vega that’s maximised for at the money (ATMF) strikes.
Also from the graph below even when implied vols are different for the same underlying, the vega amount is similar for ATMF strikes corresponding to all implied vols. Intuitively this would hold true because even a marginal change in implied vol (regardless of its absolute value) would trigger as much of a change in the ATMF option’s price as suggested by its probability of being in or out of the money which is the same for a 0.50 delta strike. Alternatively one could think about no change to the vega amount at the ATMF strike with an underlying movement in spot (vanna – defined later – is zero) irrespective of the absolute level of implied vol. This is the same as saying that the rate of change of price of options due to change in Implied Vols remains the same when the options have 0.50 delta.
- But as Spot moves away from ATMF strike i.e. is in or out of the money, vega amount reduces.
- Vega profile for a long option is steeper to the left and stretches out to the right of ATMF – this is primarily on account of underlying ‘spot’ values that brings about linear changes and not lognormal values that correspond with normal/symmetrical underlying returns as discussed in Black Scholes. Equal moves on left hand strikes would therefore amount to a higher percentage move (given a lower base) versus those for the right hand strikes. Hence the drop in vega left of ATMF is much faster/profile is steeper while right of ATMF declines a lot slower i.e. stretches out.
- Volga and Vanna – these are second order greeks applied on vega. Volga is defined as the change in Vega due to change in the Implied Vol, analogous to gamma that refers to change in delta due to change in Spot.Â
- Thus Volga
- Vanna is the sensitivity of vega to change in spot, expressed as:
, which when expanded can also be written as
- And finally – rearranging the notation again,
or the change in Option’s Delta due to change in its Implied Vol
- Graphs below plot a long call option’s Vega profile across different expiries and implied volatilities.
Option: Call
Strike: 110
Implied Vols: 5%, 10%, 15%
Expiries: 3m, 6m, 1y
Notional: 100 Option contracts with each contract notional of 1 stock
Graph 12 – Vega profile across Option expiries
Source: Pandemonium.
Graph 13 – Vega profile across Implied Vols
Source: Pandemonium.
- Peak Vega is attained at or around the ATMF Strike
- Amount of Vega at ATMF strike is independent of the level of Implied Vols
- At higher Implied Vols the vega profile flattens out – ie amount of vega for strikes away from the ATMF increases as IV increases. This is intuitive as the probability of in-the-moneyness is higher at higher vols for the same strike.
- Theta – or an option’s decay is the rate at which the option’s value declines with passage of time; theta values are negative. It implies the value of an option that comes from the time remaining to expiry and independent of its moneyness.
For a certain underlying, assuming the same time to expiry, strike and flat implied vol (to say that volatility doesn’t change across different observations of the underlying), theta would be determined by the distance between ATMF and strike. In that case, closer the underlying is to the strike higher is the theta of the option. By way of magnitude an at the money option has the highest theta which increases as we approach expiry.
- Theta’s relationship with a stock’s volatility is a corollary from the explanation above – for the same option tenor a more volatile underlying (option with a higher time value) would lose more as time passes relative to a less volatile asset. Hence options on higher vol assets have more theta.
- Connecting the dots
Since theta signifies the option value decay/earned associated with the passage of time for long/short option positions one can think of it as a representation of carry for that position. So, if there is no change to the market as the option ages by a day the change in the option’s (or a portfolio’s) value is measured by its theta or carry loss/gain.
Rho – Defined as sensitivity of the option price to changes in the risk free interest rate, it is positive for calls and negative for puts. Conceptually speaking, as interest rates go higher so do the earnings on the short sale proceeds of a stock/asset motivating arbitrageurs to monetise the gap between deep in the money calls selling in discount (less than parity) and the ATMF vs call strike differential; arbitrage involves buying the discounted call and selling short the stock. In the same manner to close in the arbitrage between deep in the money puts selling in discount and the put strike vs ATMF differential one would need to buy the discounted put and buy the stock to hold it till the put can be exercised. Higher interest would increase the holding cost of the stock and hence dis-incentivise arbitrageurs, bringing down the price of puts. Simply then,  for put-call parity to hold a higher price of call as risk free rate increases would have to be met with a lower price of a put.
It’s intuitive now that Rho is largest for deeply in the money options and smallest for deeply out of the money. Longer expiry options also have a larger rho exposure (as discounting applies for a longer period), while its nearly zero for very short term options.
