Managing Risk-Option Greeks

Option GreeksOptions are subjected to risks from, option underlying changes, option volatility and passage time. These three forces act together but not in unison to impact on the price changes in everyday trading. Individual forces can be viewed per se to analyse the dynamics. To do so, we have 4 sensitivities, namely Delta, Gamma, Vega and theta respectively.Delta: Change in Option price for an infinitesimal change (1 bp or 1/100 bp) in the underlying (futures price or swap rate or Libor rate)Gamma: Change in the Delta of the option for a shift in the underlying.Vega: Change in the option price for an infinitesimal change in the volatility of the underlying.Theta: Change in the option price due to passage of time.Vega skew: Change in the Volatility due to change in the underlying.Main sources of change in the option price can be attributed toa) Change in underlying (Yield Curve)b) Change in volatility (Implied Volatility)c) Change in option Moneyness (expiring in the money or out of the money)d) Passage of time (carrying value)
Analysis of DeltaDelta of an option connotes,· Probability that option will expire in the money or out of the money· Hedge ratio (underlying, to be bought or sold to make position Delta neutral)Delta Range: -1 to +1 for call and puts respectivelyDelta ITM ATM OTMLong Call +0.5 to +1 +0.5 0 to +0.5Long Put -0.5 to -1 -0.5 0 to -0.5Governing forces for the Deltaa) Delta dynamics with passage of timeb) Delta dynamics with yield curve changesc) Delta dynamics with volatility changesDelta dynamics with passage of time: Delta determines probability that option might settle ITM or OTM. As we approach expiration, there will be less chance for the underlying to move around. This will cause an OTM option to expire OTM by changing its current delta (probability) less than 0.5 to zero. Similarly, an ITM option will attain delta (probability) of 1. Tricky element will manifest through an ATM option. This option will have delta of 0.5 until before expiry and on the day of expiry this option delta will move from 0.5 to 1 and expire in the money or move to 0 and settle unexercised. This kind of radical movement causes option gamma to manifest wide swings causing huge PL movements.Option type Moneyness Before Expiry on ExpiryLong Call ITM between 0.5 to 1 1Long call ATM 0.5 1 or 0Long Call OTM between 0 and 0.5 0Delta Dynamics with Yield curve changes: Yield curve (underlying) changes will impact on the option by changing the moneyness. Rising yield curve may cause the moneyness of a payer to increase or decrease for a receiver swaptions.Option type Moneyness Rates Rise ChangeLong Call ITM Delta moves towards 1 from above 0.5 Delta increasesLong Call ATM Delta moves towards 1 from 0.5 Delta increasesLong Call OTM Delta moves towards 0 from below 0.5 Delta decreasesOption type Moneyness Rates Fall ChangeLong Put ITM Delta moves towards -1 from above -0.5 Delta increasesLong Put ATM Delta moves towards -1 from 0.5 Delta increasesLong Put OTM Delta moves towards 0 from below -0.5 Delta decreasesDelta Dynamics with Volatility Changes: Volatility impacts, delta by increasing the uncertainty with in the deltas. So higher volatility should suggest, an ITM option can move to ATM or OTM and OTM can move into ITM and so on. Therefore, Volatility’s impact is viewed as causing the ITM delta to go down; TM delta to increase and ATM delta remains same.Option type Moneyness Volatility Rises ChangeLong Call ITM Delta moves towards 0.5 from near 1 Delta decreasesLong Call ATM Delta remains at 0.5 Delta increasesLong Call OTM Delta moves towards 0 from below -0.5 Delta decreasesPL Impact: PL impact from the delta can be estimated by applying market change to the delta.PL = Delta* Market ChangeConventions for swaptions· Purchase payer equal to long call in terms of rates· Purchase of receiver is equal to long put in terms of rates· Sell a payer is equal to short call in terms of Rates· Sell a receiver is equal to short put in terms of ratesDelta can be measured in terms of yield rather than in terms of price. In a sell off positive delta position (short market) will gain and in a rally negative Delta position (Long Market) will make loss.Delta Direction for SwaptionsSwaption Type Delta Market positionPurchase Payer Positive Short the MarketPurchase Receiver Negative Long the marketSell Payer Negative Long the MarketSell Receiver Positive Short the MarketAnalysis of GammaGamma risk predicts how delta changes with respect to larger underlying changes. Due to large movements in the underlying, option delta changes and this will cause significant PL impact. For instance, an ATM option before expiry if settles ITM or OTM (crossing the strike) on the day of expiration then delta changes from 0.5 to 0 or 1. This sharp change in delta causes gamma to increase significantly.How Gamma works: For a long ATM payer swaption (long Call) position is long gamma. Whenever market rallies (yields fall), this position will loose delta and becomes less short. In other words, option will move from ATM to OTM. In this transition, Long gamma acts as cushion by adding PL. Similarly, when market yields rise due to a sell off, the position will become, ITM and gamma makes the position shorter.a) Gamma Dynamics for Yield curve changesb) Gamma dynamics for the volatility changesc) Gamma dynamics for the passage of time.Gamma Dynamics for the yield curve changes: Options most common and direct and largest source of risk comes from the underlying changes. Underlying movement can make an OTM option into an ITM option and vice versa or leave both ITM and OTM options on the edge at ATM. This kind of change has major impact from gamma to the option.Option Type Gamma Rates Rise changeLong Call ITM moves towards 1 from above 0.5 (Gamma decreases)Long Call ATM moves towards 1 from 0.5 (Gamma decreases)Long Call OTM moves towards 0 from below 0.5 (Gamma decreases)Gamma dynamics with Volatility Changes: Volatility being indication of uncertainty, it drives Gamma for ATM option down because, rising volatility suggests that market is going to move from current point to somewhere. This will make ATM delta to move from 0.5 to higher or lower. So, Gamma decreases. Whereas, for OTM and ITM options rise in volatility means no more status quo and rates will move to somewhere and they will be getting new money ness, which would be obviously ATM. This would cause, their Gamma to rise.Gamma Dynamics for the passage of Time: Gamma grows, as time to expiry gets closer. This will create even unthinkable gammas. Passage of time is worst enemy of ATM option writer. As we get close to maturity, gamma rises so high that it becomes impossible to hedge the option. Whereas, OTM and ITM options together enjoy the luxury of gamma decreasing to zero.PL ImpactGamma = ((Delta (new) – Delta (old)) /2)*( Market Change)^2New Delta = old Delta + Gamma*(Market Change)Swaption Type Gamma position Market positionPurchase Payer Long Gamma Short MarketPurchase Receiver Long Gamma Long MarketSell Payer Short Gamma Long MarketSell Receiver Short Gamma Short MarketLong Gamma position will contribute towards more delta in rising markets and less delta in falling markets. This kind of situation can be paraphrased as Gamma Trading.Analysis of VegaVega risk predicts the change in the option price due to the volatility change in the market.A) Vega dynamics due to yield curve changesB) Vega dynamics due to passage of timeC) Vega dynamics due to change in VolatilityVega Dynamics due to passage of time: Vega of an ATM option is highest and ITM and OTM options have lower Vega. Vega of and option decreases with expiry date getting closer. In other words, we will see the certainty in the option getting expired ITM or OTM. This shows, Vega decreases for options of all moneyness to zero.Vega dynamics due to change in Volatility: Vega rises with increase in implied volatility. ATM Vega will change slightly but OTM and ITM Vega rises. Significantly.Vega Dynamics due to Yield Curve changes: Yield curve changes alter the moneyness of the option. This will cause and option to move from being ITM to OTM and so on. Accordingly, Vega will change due to these yield curve movements. This change is referred as Vega skewAnalysis of ThetaTheta risk predicts everyday cost of carrying the option. There is no imaginable hedge for this kind of risk. As we are all certain time passes by. But, still it warrants, to get grips with the dynamics of the theta.Theta Dynamics due to yield curve changesTheta dynamics due to Volatility changesTheta dynamics due to yield curve changes: Yield curve changes alter the moneyness of the options. This will alter the theta profiles. For an ATM option carry will be highest and lower for ITM and OTM options. Theta for options that are close to expiry and ATM will be highest and for deep ITM and OTM options close to zero.Theta dynamics due to volatility changes: Theta for options increase with rise in volatility.