Management Of Negative Gamma - Part I
Dec 21, 2004
This article deals with the management of negative gamma in a portfolio. Before we delve into the management of negative gamma, it's imperative that we understand the intricacies of gamma and how it affects the portfolio.
Definition of Gamma
The gamma is the rate of change of delta of an option, with respect to the spot. The gamma is also referred to, as the curvature of an option, is the rate at which the delta of the option changes, as the value of the underlying changes. The gamma is usually expressed as deltas gained or lost per one point change in the underlying, with the delta increasing the amount of gamma when the underlying increases, and falling by the same amount of gamma when the underlying decreases. Mathematically it is the second derivative with respect to the asset price.
Nature of gamma
Gamma versus spot
Gamma versus time
Trading the Gamma
Let's say a portfolio is long an over-night option, in 10 million USD. The underlying is usdyen, which is at 105.00. The strike is long 105.00 USD Call JPY Put.
Case 1: Spot = 105.00
The Option delta = 50 percent
Theta = the price of the options, as the options is overnight, so all the value of the options will be realized the next day, when the option get exercised or expired.
Vega = Negligible, overnight option Rho = Zero.
To be delta neutral, the trader has to sell 5 million USD.
Case 2: Spot goes to 105.75
Delta = 82, as the options goes in the money.
To be delta neutral, the trader has to sell 3.2 million USD more.
Case 3: Spot goes back to 105.00
Delta = 50, as the option is again at the money.
To be delta neutral the trader can buy back the 3.2 million USD he had sold at 105.75
Net the trader makes 0.75 on 3.2 million USD = 7582 USD.
Case 4: Spot goes to 104.45
Delta = 20, as the options get out of the money.
To be delta neutral the trader can buy 3 million USD at 104.45.
Case 5: Spot moves back to 105.00
Delta = 50, as the option again becomes at the money.
The trader can now sell the 3 million USD at 105.00, thus making a profit of 15,700 USD.
So, when a portfolio is long gamma, there will be take profit orders on both sides of movement of the underlying, making profit every time the underlying moves and comes back. However the risk is that the portfolio looses the theta. It's only profitable if by trading the gamma the portfolio makes more that what it looses on the theta. This is also called gamma trading or gamma scalping. Long an option, delta-hedged, essentially means long realised vol (long gamma), short implied vol (short theta). If vol re-marks lower (i.e. implied vol is lower), then you gain on your theta (as you're short), and depending on how realised vol moves your loss on gamma may/may not offset the gain on the theta.
Vega changing to Gamma
The vega of an option slowly changes to gamma with time. This is dependent on the movement of spot with respect to the strike of the options. However if we consider that spot remaining same, the vega decreases, while the corresponding gamma of the option increases. Hence when managing gamma dynamically, vega buckets should be kept in mind. Long 1 month vega, will provide more gamma in two weeks time, as long as spot stays near the strikes.
Out of the money options will have different volatility from the atm options as the risk reversals and the butterflies affect the price of the out of money options. This skew can vary between different maturities, and is know as the volatility smile. When evaluating a portfolio of options, the gamma emanating from the smile can have effect on the portfolio.
Managing Negative Gamma
Managing negative gamma is a bit tricky than managing negative vega. Negative gamma can work very well over holidays, where chances of spot movement are minimal, however, there is always a gapping risk if something untoward happens during the holidays, and the underlying asset might gap higher or lower at the open of the next market session.
Few things to consider when managing negative gamma
1/ The change of gamma with respect to spot, which means, with higher value of the underlying asset your gamma can be more negative or it can turn to flat or positive gamma. In this case the rate of delta change will be more when the gamma turns more negative, than when it is flat. This can happen when you have a short dated risk reversal position on the portfolio. Hence, the p/l variation will be more as gamma turns more negative, and it is for you to stop the delta as the underlying goes in that direction
Gamma and delta behavior in a short dated risk reversal
2/ Keeping the stops at reasonable levels: If one is earning 20,000 USD as theta due to your negative gamma position, then its essential one should keep the stops such a way as to minimize the loss to at most the level of theta decay. In general, the stops are placed to minimize loss in case of violent spot movement, however, while placing the stops, a balance should be made between the amount of theta decay and the level of volatility of the spot.
3/ The gamma can change over time, which means that negative gamma can be more negative, or can turn to positive gamma with movement of time. This is what is better known as bleed. The gamma bleed results in corresponding delta bleed too, and a portfolio might get shorter, or longer delta depending on which side the underlying close with respect to the short strike (the origin of negative gamma). Now if the negative gamma is slowly increasing in absolute value, then one can consider closing the strike risk (if it does not look very benign), or one can buy shorter dated options whose gamma increase over time offsets the portfolio.
4/ The negative gamma is the highest for overnight at the money option. If there is no event risk, it makes sense to run this risk. However, the risk/reward profile might be pretty high. Here the denominator is limited to the price of the option, which is collected over the next day if the underlying does not move. However the risk can be theoretically unlimited, as the underlying can move either up or down to any extent. Hence referring to the second point discussed, it's advisable to keep stops balancing between the level of theta and the movement of spot.
5/ Levels of volatility of short dated options: Running negative gamma at higher implied volatility, will accumulate more theta, however the inherent risk remains that the underlying will move more, as the higher implied volatility suggests. Correspondingly, under low levels of implied volatility, even though the portfolio collects lesser decay, the chances of getting stopped out on either side is lower too. Hence it is a fine balance, to operate negative gamma under different levels of volatility, and when to take advantage or higher levels of implied volatility and vice versa. Sometimes implied volatility will fall before the holiday season begins and will rise as the holiday season nears the end. In such cases negative gamma might prove profitable, when the probability of any event risk remains low.
6/ Negative gamma resulting from the smile can be hazardous: A portfolio might be flat gamma and still loose time decay. This happens when the portfolio is long gamma and short smile gamma. The positive gamma coming from short dated at the money options, while it is offset by the negative gamma emanating from longer dated out of the money options in bigger size. In such situations, the portfolio will loose decay, however unable to take advantage of movement in spot. It makes sense to buy the option (which is emanating negative gamma) at reasonable levels of volatility, than run the position.
However the crux of the solution is to get the option back at a reasonable level of volatility, as the seller of the same option will face a similar situation in his portfolio.
7/ Negative gamma near a barrier - This can be interesting situation, where the gamma emanates from the barrier of an exotic option. The gamma generated from a trigger cannot be hedged accurately through vanilla options, however the only way to hedge is perfectly is offsetting it with an equivalent digital risk through another exotic option. This requires extensive deliberation, and due diligence will be done on request.
Apart from these seven points, there are a couple of other things to look at. We just dealt the seven points at the basic level. To have an in-depth grasp on the subject, it is advised to go into more in-depth analysis, based on real market conditions and an updated volatility smile.