Trading Long Gamma

Exhibit 9.2 shows a EUR/USD spot ladder from a long EUR20m 1wk ATM position, with current spot highlighted in the middle of the ladder. Values in this spot ladder are generated assuming market data (apart from spot) remains unchanged and no additional trades are executed as spot moves from its current level. This is a long vanilla option position and is therefore long gamma. Peak gamma is at current spot because it is an ATM contract. The contract is delta neutral by construction so initial delta is zero.

As spot moves, P&L change occurs faster the further spot moves in either direction. This is characteristic of being long the second derivative (i.e., long gamma). In trading positions, gamma is quoted per 1% move in spot, so, for example, on a 1% spot move higher (from 1.3000 to 1.3130), delta increases by approximately +EUR5.6m (current gamma).

Assume EUR/USD spot rises from 1.3000 to 1.3065, as shown in Exhibit 9.3:

• Spot higher and long gamma leads to an increasing long delta position as spot rises.
• Spot higher and long delta produces positive P&L change (i.e., making money, a good thing) as seen in the spot ladder.

The trader now has a decision to make:

1. Sell spot in the market to reduce the long EUR2.8m delta exposure, known as “taking profit.” Selling EUR2.8m EUR/USD spot will hedge (offset) the delta position back to approximately flat and the delta hedging process can start again. After selling EUR2.8m EUR/USD spot at 1.3065, the position is shown in Exhibit 9.4. Note that the P&L change is now approximately equal for a same-sized up or down move in spot; the position is “balanced.”
2. Do not hedge the delta (i.e., let the position run). If spot continues higher, the position will make even more money, but if spot retraces back lower, the positive P&L will be lost. If delta is not hedged, the position is not balanced. By allowing a delta position to accumulate the trader has an increased exposure to the future direction of spot.

Now, from the same starting point, assume EUR/USD spot falls from 1.3000 to 1.2935, as shown in Exhibit 9.5:

• Spot lower and long gamma leads to an increasing short delta position as spot falls.
• Spot lower and short delta produces positive P&L change again, as seen in the spot ladder.

Once again, spot has moved, a delta position results from the long gamma, and the trader has two choices:

1. Buy spot in the market to reduce the delta exposure. Again, this is taking profit but the trader does not have to completely hedge the delta exposure. For example, after buying back just EUR2m EUR/USD spot at 1.2935, the position is shown in Exhibit 9.6. By not buying back the full delta amount, the position is not completely balanced, but it is more balanced than it was.
2. Do not hedge the delta (i.e., let the position run). If spot continues lower, the position will make even more money, but if spot retraces back higher, the positive P&L will be lost.

If the delta is initially balanced, long gamma causes a positive P&L change if spot moves either up or down, just so long as it moves somewhere. Hedging the delta resulting from a long gamma exposure naturally leads to buying spot when it goes lower and selling spot when it goes higher. Buying low and selling high locks in P&L over the course of the day in a process known as trading the gamma. When leaving orders in the market to trade a long gamma position, looped take-profit orders are often used (e.g., sell EUR5m at 1.3130, if done, buy EUR5m at 1.3000, if done, sell EUR5m at 1.3130).

Positions with larger gamma exposures are delta hedged more frequently. In this single option example, if spot stays close to the strike as days pass, the gamma from the vanilla option increases, which will lead to more frequent delta hedging. Then, on the expiry date of the long ATM contract, Exhibit 9.7 shows the trading position with the delta balanced around the strike.

Trading risk is viewed as of a specific horizon date (usually today) and the risk on options that expire on the horizon is viewed at expiry time. Therefore, the option is viewed as a strike; an instantaneous delta jump equal to the option notional rather than gamma exposure. With spot at 1.3001 (above the strike), delta exposure is +EUR10m. With spot at 1.2999 (below the strike), delta exposure is –EUR10m: a EUR20m delta change, the notional of the option. Trading this position is similar to trading gamma, except that all the delta change occurs at the strike.

For a long strike position:

• If spot goes above the strike, the trader can sell spot in the market to hedge the long delta exposure.
• If spot goes below the strike, the trader can buy spot in the market to hedge the short delta exposure.

Once the delta position is fully hedged back to zero, it is not possible to hedge again until spot goes back through the strike.

In general, P&L volatility from a single vanilla option increases as time to expiry gets shorter if spot remains near the strike. Plus, for a given ATM vanilla option, P&L volatility from trading the strike at expiry tends to be larger than P&L volatility from trading the gamma prior to expiry.

Trading a long gamma position looks easy—buy vanilla options to get long gamma, and if spot moves higher or lower, you make money. Long gamma naturally means spot can be bought low and sold high, hence generating positive P&L. Of course, there is a cost for this and that cost is theta , also known as time decay or just decay. For a long gamma position, theta is the cost of holding the trading position from one trading day to the next.

If realized volatility is larger than implied volatility there will be more opportunities to delta hedge, which usually results in larger P&L from trading a long gamma exposure than is paid in theta, hence generating positive P&L overall.

If realized volatility is smaller than implied volatility there will be fewer opportunities to delta hedge, which usually results in smaller P&L from trading a long gamma exposure than is paid in theta, hence generating negative P&L overall.

Delta Hedging in Practice

If no delta hedging is performed throughout the trading day (and assuming no P&L from any other source), the daily P&L will be a function of the difference between yesterday's end-of-day spot and today's end-of-day spot (end-of-day market data is the official reference used for P&L generation, limit checking, etc.). Alternatively, if delta hedging were performed continuously (not possible in practice), P&L would be a function of implied volatility and realized volatility (recall from Chapter 5 that this is the essence of the Black-Scholes formula derivation). An FX derivatives trader's main focus, skill, and exposures are to FX volatility. Therefore, option positions are delta hedged frequently throughout the day.

If spot followed a mathematical random walk, the primary factor in determining the optimal delta hedge frequency of a long gamma position would be the spot bid–offer spread size. The Euan Sinclair book “Volatility Trading” has a good section on this (see Further Reading). In practice, though, decisions on when and how to trade the delta exposure are based on different factors:

• Spot bid–offer spreads. The wider the spreads, the less often traders like to delta hedge because they must cross a spread each time they do so. If spot bid–offer spreads are wide, it often makes more sense to leave orders in the spot market rather than transacting directly. Plus, outside the main book-runner's time zone, leaving orders ensures that opportunities to delta hedge are not missed.
• Expected spot volatility and jumps. During quieter times, traders may hedge smaller spot moves. But if spot jumps when important economic data is released, traders often let spot run further to take advantage of their positive second derivative.
• Time of day. At the start of the trading day, traders tend to let deltas run further in an attempt to generate larger P&Ls, but toward the option expiry they “scalp” (trade the delta) more aggressively.
• Key levels in the spot market (i.e., rounded spot levels where barriers will be positioned or recent low and high spot levels). If spot breaks through key levels, this often causes spot to run further and traders may let larger deltas accumulate before hedging.
• How spot is moving. If spot is whipping up and down, traders will delta hedge more aggressively, whereas if spot is trending strongly, they may let the delta run further before hedging.

Many derivatives traders love trading spot to manage their delta exposures throughout the day and they believe that they add value by doing this. A friend at another bank reported their boss proclaiming (with a straight face) “trading spot is like playing Mozart” and selling EUR/CHF spot shortly before the Swiss central bank unexpectedly intervened and sent spot over 8% higher. In stable market conditions it is debatable how much value a trader can add, but when unexpected events occur, the ability of a human trader to rapidly process new information in context can be an advantage.

Trading Short Gamma

Exhibit 9.8 shows a EUR/USD spot ladder from a short EUR20m 1wk ATM position.

P&L change, delta, and gamma are all equal and opposite (negative) to the long ATM position:

• Spot higher and short gamma leads to an increasing short delta position as spot rises.
• Spot higher and short delta produces negative P&L change.

while:

• Spot lower and short gamma leads to an increasing long delta position as spot falls.
• Spot lower and long delta again produces negative P&L change.

If spot moves in either direction, the delta change from a short gamma position produces a negative P&L change. The decision for the trader therefore becomes whether to stop-loss the delta (i.e., buy high/sell low) or let it run and potentially lose even more at an increasing rate due to the short second derivative. In practice, trading short gamma is often made easier by accepting that at least half the theta earned will be lost through stopping out. Trading with close stop-loss orders helps avoid large negative P&Ls.

A short gamma position earns theta but loses money throughout the trading day as spot moves. Again, if the position is delta hedged frequently, the overall P&L generated is primarily a function of implied volatility and realized volatility, but this time the position will generate higher positive P&L if realized volatility is below implied volatility.

P&L Balance

P&L balance involves assessing whether the position generates similar P&L changes for similar-sized up and down spot moves. This, of course, assumes the trader has no strong opinion on future spot moves. P&L balance can be checked in the spot ladder.

The P&L and delta positions should tie up:

• Long delta and spot higher → P&L increases
• Long delta and spot lower → P&L decreases
• Short delta and spot lower → P&L increases
• Short delta and spot higher → P&L decreases

Any discrepancies should be investigated; unexpected P&L jumps are most likely caused by exotic risk in the position.

Assuming the trader has no opinion on spot direction, it can be checked that the P&L change on equal-sized up and down spot moves is roughly the same. Traders usually look at a one-day spot move of approximately one-and-a-half standard deviations up and down. In currency pairs with implied volatility somewhere close to 10%, spot moves around 0.75% to 1% are often considered within P&L balance. In practice, though, traders look over a wider spot range, and if there are extreme negative P&Ls at spot levels which might conceivably be reached, it is more important to concentrate on controlling the P&L in those areas rather than on pure spot up/spot down P&L balance.

In the example AUD/USD position, the P&L is roughly balanced, but a bit of additional spot could be bought to balance it better as highlighted in Exhibit 9.13.

To increase the P&L at 0.9233 by USD8k and hence reduce the P&L by USD8k at 0.9068 (the equivalent spot move lower), 8,000/(0.9233 − 0.9150) = AUD960k spot should be bought. Buying spot changes the delta exposure and hence adjusts the P&L profile over spot as shown in Exhibit 9.14.

In practice, the delta and hence the P&L profile over different spot levels can be adjusted by buying or selling spot but full P&L balance is a multi-dimensional problem. The gamma profile must be considered as well as impacts from ATM volatility and the volatility smile.

Theta

In trading positions, theta is quoted as the cumulative change in value from one trading day to the next for all deals in the position.

If a trading position is mainly long strikes and long gamma, the long option values reduce over time and theta will be negative. The negative theta is roughly the maximum that can be lost from the short-date position if the delta is initially balanced since delta hedging the long gamma position will make money back.

If a position is mainly short strikes and short gamma, the short option values reduce over time and theta will be positive. The positive theta is roughly the maximum that can be made from the short-date position if the delta is initially balanced since delta hedging the short gamma position will cost money. A trader friend at another bank was once told, “You can have any gamma position you like, so long as you don't pay any theta!”

In Black-Scholes world, gamma and theta are proportional: Higher theta implies higher gamma exposure and hence higher P&L volatility. Put another way, with more positive or negative gamma the trader is more exposed to how spot moves. If a trader decides that theta and hence P&L volatility is too large, short-dated vanilla options should be transacted to reduce the existing gamma position.

It is worth noting here that factors other than gamma can also generate theta, for example, smile decay, roll down the ATM curve, funding cash balances, and so forth. These factors are explored in Chapter 14.

Gamma

The spot ladder shows how the gamma exposure changes at different spot levels. Recall that:

• Long gamma → want spot to move more than implied volatility suggests it will, the more the better.
• Short gamma → want spot to move less than implied volatility suggests it will, the less the better.

Fundamentally, the trader has a decision to make: If the position is long gamma, looking at the spot ladder and the theta paid, will spot move enough to make back the theta from delta hedging? If the position is short gamma, looking at the spot ladder and the theta earned, will spot cause a loss from delta hedging larger than the theta?

Strikes

For options that settle against a cut rather than a fix, at the expiry time of each option in the position, the owner of each option contacts the writer to tell them if they want to exercise the option. The most common cut time in G10 pairs is NY cut and the 20-minute period before NY cut is simply called expiries on most trading desks in London and New York.

In practice, from the start of the trading day, bank trading desks contact each other to exercise strikes that are very far from the current spot level. Over the course of the day, the strikes being dealt with get closer and closer to current spot and at some point responsibility passes from middle office to traders.

If spot is very close to a particular strike, communication between the option owner and the option writer will be established shortly beforehand, with the option owner asking to “hold” on the strike. Then as the cut time arrives, the decision to exercise or expire the contract is made based on the prevailing spot level.

This process is usually smooth but it can get dangerous when a trading position has multiple offsetting strikes at the same level. For example, a trader might be short a contract to one bank and long the same contract to a second bank. In order to keep an unchanged delta position the trader must wait for the bank that is long the contract to exercise or expire before they can pass the same decision onto the other bank. Often it is obvious what action should be performed, but if spot is exactly on the strike, this can be a difficult situation to manage. There are stories about traders hiding under the desk and refusing to come out until expiries are finished.

It is also possible for an option to be partially exercised. For example, if an option notional is AUD50m, the option owner could partially exercise it in only AUD30m. When this occurs, the new delta exposure within the trading position must be established as quickly as possible. Remaining calm when a partial exercise is requested on a large strike is a rare skill.

After expiries, if there were close strikes, the delta position may need to be rebalanced. For example, if a trading position is long USD100m USD/JPY 105.00 strike, delta may be positioned long USD50m above and short USD50m below the strike. After expiries, when the option has “rolled off,” the delta exposure will remain with no protection from the strike and hence the delta must be hedged back to flat by trading spot.

For options that settle against a fix rather than a cut, if the option is in-the-money at expiry, rather than generating a spot FX transaction, the cash-settled option generates a single cash payment. In practice, this means that the delta position from the option disappears and must be replaced with an FX trade which is usually transacted off the same fix as the option settled, hence minimizing risk.

Gamma/Strike Profile

A strike topography can be used to judge how the gamma profile will change over time. Strikes disappear from the position as they expire and options “behind” start to produce more gamma as they move closer to the horizon. Based on this, a trader can judge how their gamma position will evolve. Also, if there are significant option expiries on (what traders call) “good” or “bad” dates in the future, a trader might want to unwind or offset these positions.

So-called “good dates” have events (data releases) on them and spot is therefore expected to move more than usual, for example, days on which employment or GDP data is released. Such events can cause spot to jump as the market adjusts to new information. Traders must therefore know the big events coming up in their currency pairs for the next month at least.

So-called “bad dates” are expected to be quiet and therefore spot is expected to move less than usual, for example, days over Easter or between Christmas and New Year's.

It is easier to risk manage a short-date position that is long good days and short bad days because the position will be long gamma when spot is moving more and short gamma when spot is moving less. However, this must be considered with reference to the price paid to achieve the position. Anyone can buy a Non-Farm Payroll day at 25% volatility or sell a holiday Monday between Christmas and New Year's for 3% volatility, but does that represent good value? Analysis that answers questions like these is introduced in Chapter 17.

Also, some traders are happy to run lots of strike risk (i.e., have lots of open strikes in the book), while others prefer a much cleaner position:

• Many strikes in position → position harder to manage since delta changes more → higher P&L volatility but less spread cross paid away cleaning the position.
• Few strikes in position → position easier to manage since delta changes less → lower P&L volatility but more spread cross paid away to obtain the clean position.

Traders must know the big vanilla options they have in their position, so if there is an opportunity to close out a contract with clients or in the interbank broker market with minimal spread cross (or even spread earned), they can take it. Traders may also show improved prices to clients (called “axes”) to close out existing positions.