In the spirit of Aristotelian logic and the doctrine of categorical syllogisms, rather than starting the job explanation with talk of stochastic processes, Ito's lemma and the derivation of the Black-Scholes equation, its best to start by removing those inherent complexities and breakdown the description to its most simple form (one everybody ultimately understands and relates to): buying low and selling high.

Unfortunately when most people learn about derivatives in their work or studies, they often get caught up in the memorization of valuation formulas rather than what those formulas actually mean. For instance, it's incredible how many people can actually price a forward, but when asked how it would be hedged remain speechless. People get familiar with terms like delta hedging, gamma, vega, and convexity, but don't really know what they mean in practice. In the summer of 2000, I was having dinner with my partner and a very seasoned Swedish stock and derivatives broker with whom we executed most of our Scandinavian trading. Earlier that year we had amassed a large long volatility position of Nokia options. The stock was wildly trading in a range and our delta hedging activity in that period had us repeatedly buying the stock as it made its way down and selling stock as it came back to the high end of its range. Over our smorgasbord, the broker started to praise us about our prophetic Nokia stock trading. We were shocked at his uninformed statement – even though we had purchased our Nokia options position via the same broker, it became obvious that our Swedish derivative salesman had skipped over the section about dynamic delta hedging during his training program. To our amusement we did not shed light on the subject and modestly accepted his acclaim. Its time to confess – what follows is a brief explanation on how value is extracted when trading volatility positions.

An option's delta is the sensitivity of the option value to changes in the underlying asset. For example, Nokia is trading at €12 and you buy a 3 month call struck at €12 on 100,000 shares of Nokia. The delta is 50% and so a €0.12 move down in the stock would decrease the value of your call option position by €0.06 multiplied by 100,000 shares, or €6,000. In order to hedge your call option's exposure to the price movement of Nokia you would sell 50,000 shares so that a €0.12 move down in the stock would earn €6,000. The gain in the stock position offsets the loss in the option position. Thus, at that point in time the option position is "delta hedged". The reason that delta hedging (the verb) is dynamic is because when Nokia moves from €12, or any other parameters relating to the valuation of the option changes, you are no longer delta hedged and need to "rebalance". If Nokia moved suddenly to €13 and the delta of the option changes to 75%, you would sell 25,000 more shares at €13. If the stock then moved back to €12 and the delta is again 50% then you only need to have a short 50,000 share position to be delta hedged in which case you would buy back the 25,000 shares that you had sold at €13. Therefore, as the stock moves down your rebalancing activity has you buying stock, and as the stock moves up you are selling stock – continuously buying low, selling high, which makes your stockbroker think you are the Oracle of Nokia. The critical part they miss is that the buying low, selling high profile of a delta hedged long call position comes at a cost – the value of your call option decays with time. A stock is expected to move at a certain rate (implied volatility) every day and if that rate is met then the profits from your rebalancing activity that day should cover the loss in the option's time value. If it doesn't move then your realized loss is capped that day to your time decay. If, however, the stock moves significantly then you can make multiples of your time decay – another alluring profile to a long volatility position.

A key concept that is often overlooked is that the buying low, selling high profile is the product of being long volatility rather than being long calls or long puts – because you are delta hedging whether your long volatility position comes from puts, calls, or a combinationof the two becomes irrelevant. If you change the Nokia example to a purchase of put options instead of call options, then the initial delta hedge would have been to buy Nokia stock. When the stock moves up you would need to sell some stock since the option's delta would decrease, and if the stock then moved down you would need to buy it back since the delta would increase.

Effectively money is made trading volatility positions in two ways:

- Buying implied vol low and selling implied vol high,
- In cases of buying options: Buying implied vol low and realizing higher volatility via delta hedging with the underlying asset over the life of the option
- In cases of selling options: Selling implied vol high and realizing lower volatility via delta hedging with the underlying asset over the life of the option.

Like any asset, the price of implied volatility increases or decreases with changes in supply and demand for volatility. Consider having purchased an option which was valued based on an implied vol of 20 and selling the same option immediately at an implied vol of 22 – the difference in implied vol is the only determinant of p&l. If the volatility position is held for any period of time prior to unwinding the position then you need to factor in the realized volatility, or p&l from your delta hedging activity and time decay. Similar to the concept of duration for bonds, the longer the option then the higher the sensitivity to changes in implied vol. The shorter the option, the higher is the sensitivity to changes in time relative to the value of the option – for example a 1 day option, all else equal, loses its entire time value with the passage of 1 day, whereas a 10 year option, all else equal, loses almost no value with the passage of 1 day. This sensitivity dynamic of long dated versus short dated options highlights a critical point – trading long dated volatility and short dated volatility positions are two very different disciplines.

Success in trading long dated volatility positions emerge from having a significant insight into the supply and demand forces of longer dated implied vol. Supply and demand imbalances caused by structured products and illiquidity can exist for extended periods of time and thus mark-to-market fluctuations can be large.

Countless long dated volatility positions may at first seem absurdly cheap or expensive but unfortunately your investors are not going to stick around for 10 years to see it through. As a result, very few institutions are setup to trade these tenors effectively. Only the naïve gambler sits down at a blackjack table with $100 to lose and makes $25 bets. His bet size is too large relative to his capital, he has left no room for error, and it's extremely unlikely that he will survive for long. Even if his odds of winning each hand are 7525 in his favour his bet size is still too large and he will run out of capital quickly. If on the other hand he made much smaller bets then he can ensure he will reap the benefits of his favourable odds in the long run. The same goes with trading long dated volatility – size your positions according to vol moving to even more absurd levels against you and forget about theoretical vol arbitrage. Traders and quants often neglect how long supply and demand imbalances can persist in any asset when they cannot be arbitraged. Take the art world for instance, in 1961 Piero Manzoni, one of Italy's most controversial artists, canned his own faeces as an ironic statement on the art market. Each tin contained 30 grams of his faeces and he sold it for the same price per gram as gold. In 2002, London's Tate Gallery paid over £22,000 for one tin, more than 100 times the price of gold. Never underestimate bubbles and long dated imbalances.

On the other hand, imbalances in short dated implied vol (positions less than 1 year) can remain indefinitely, but the mark to market effect can be outweighed by the realized volatility that you can capture delta hedging. If held to maturity, it is ultimately the realized volatility that will gauge the success of your trade. Effectively shorter dated volatility trades are in fact a relative value trade of implied volatility versus realized volatility. Sounds simple and straightforward, but there is one added hitch of course – when dynamically hedging options your p&l is not only a function of what realized volatility was, but also how it moved and when it moved.

Going back to the Nokia three month call option example, assume that for the first two months of the life of the option that the stock price slowly and steadily increased. You broke-even on delta hedging p&l versus time decay and the option is now deep in the money with a delta of 100%. Positive earnings come out and the stock gaps up 25% on the news. At expiry of the option, the realized volatility over the prior three month period will seem very high given the huge move after earnings, but unfortunately the option was already deep in the money when that move occurred so there was no "optionality" left in the position to trade. Even though you had purchased 3 month implied vol and realized vol subsequently exploded during that period, you only broke-even on the position. Therefore strike selection becomes critical in effective volatility trading when positioning by dynamically hedging calls or puts. Some volatility products are path independent – in variance swaps, for example, the volatility exposure is constant for each day of the swap regardless of where the stock has moved.

The effect of path dependence is an area often neglected in derivatives research. Another area tied to path dependence that further complicates delta hedging is that Black-Scholes valuations assume that hedging is done in continuous time. When hedging in discrete time, decreasing the frequency of rebalances increases the distribution of hedging profits/costs significantly. Which then leads to a question that is rarely asked – what is the optimal hedging frequency for an option? The annoying answer is… that it depends.

A number of factors need to be analyzed to assess at what frequency to rebalance. A large majority of option traders hedge close-to-close but I have yet to meet one that has provided a sensible explanation as to why they do that. A more appropriate approach is to first get acquainted with the distribution of your results by running a simulation that decreases the frequency of rebalances. The alpha you are trying to add trading vol can get muted if rebalances are too infrequent and the distribution of hedging costs/profits are too wide. That analysis should also factor in transaction costs, whether the asset is in a high or low vol environment or is trending, the liquidity of the underlying instruments and should give you a range of acceptable rebalance frequencies. Now all that remains is actually selecting which assets to trade volatility in.

A lognormal distribution assumption may describe some asset price behaviours well, but not all. Various techniques such as skew adjustments exist to compensate for fat tails and jumps in asset prices, but adjustments tend to be made generically as opposed to stock specifically. The same way actuarial models breakdown for small sample sizes, so do valuation models for some assets whose actual distribution patterns don't fit an option model's wide assumptions. There are reminders throughout history to expect the unexpected. Here are a few:

At the beginning of the 15th century, the Chinese assembled a powerful fleet – three thousand ships, including two hundred fifty massive galleons roughly five times the size of Portuguese vessels that sailed later in the century. The Chinese ships were armed with cannons more sophisticated than contemporary arms in Europe, carried crews of over 500 men, and embarkedon several expeditions around the globe including many along the east coast of Africa. Over 50 years before Portuguese ships first traversed the southern tip of Africa, the Chinese had begun to explore the globe. The distance from Portugal to West Africa is small; the distance from China to east Africa is almost halfway around the world. My point being that in the middle of the 15th century, there was every reason to believe that China would rule the world. That unexpectedly changed very quickly – an Emperor came to the throne, he had no interest in the outside world, and suddenly ended the expeditions permanently. In 1500 it was made a capital offence to go to sea in a ship with more than two masts. The largest fleet in the world was left to collect mothballs and was eventually destroyed in 1525. China fell into centuries of isolation. Nobody expected this.

In 1242 Europe was in serious trouble. In the six years prior, the Mongol's, led by Batu Khan, had accomplished the unimaginable having conquered Russia, Poland, Hungary and penetrated Germany; their conquest of Europe seemed inevitable. But unexpectedly, Europe's apocalypse didn't materialized when the Mongols suddenly stopped on the European fronts and retreated due to having learned of the death of their leader, Batu's father, Ogedai (the successor of Genghis), which by tradition required Batu to return home so that succession can be decided before commanders can be given orders. Nobody expected this.

In 1784 volcanic eruption of Laki, in southern Iceland, caused a quasi-atomic winter in much of the Northern Hemisphere. The eruption was also responsible for cold temperatures for many years following, which depleted crop yields in northern Europe. In 1789 the French Revolution began and was popularly attributed to outcries against a tyrannical government or the incompetence of King Louis XVI. However, it could be argued that it was actually the severe shortage of food that was the catalyst. Nobody expected the Laki eruption to trigger one of the most pivotal events in European history.

The greater the influence that a single individual or entity may have on the outcome of an asset price, then the added human element of illogical decision making, madness, or outright stupidity must be considered and should not be underestimated (which it often is in distribution assumptions.) Natural disasters are characteristically unpredictable. There are intelligent ways to capitalize on rich or cheap "fear premiums" but prudent risk management should accept that fat tails will occur. Extracting alpha from volatility positions requires that you both trade long and/or short volatility. Diversification is the key to trading short volatility – avoid any action with an unacceptable outcome and size your short positions accordingly.

Well, there you have it, the confessions of derivative maestros, long overdue to my Swedish broker. Alpha is plentiful in volatility trading and aforementioned nuances to delta hedging and dealing with path dependency are measures taken to maximize that alpha. As for fat tails – be aware they exist – the good news is that the same human element of illogical and unforeseen decision-making is what keeps markets inefficient and keeps hedge funds in business.

**Mr Castrounis** is a fund manager with Maple Leaf Capital. The firm's Macro Volatility strategy, started in January 1, 2004, has posted YTD returns of 20.4% with a Sharpe ratio of 2.3, and marked their expansion into volatility trading in multiple asset classes and without a dedicated directional bias.

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## Commentary

## Issue 132

## The Convexity of Trend Following

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