Investing in NonLinear Solutions
Derivativesbased Portfolio Solutions
DIRECTIONAL VOLATILITY TRADINGDirectional investors can use options to replace a long position in the underlying, to enhance the yield of a position through call overwriting, or to provide protection from declines. We help evaluate these strategies for our clients and explain how to choose an appropriate strike and expiry. We show the difference between delta and the probability that an option expires in the money and explain when an investor should convert an option before maturity. Option trading in practice. Using options to invest has many advantages over investing in cash equity. Options provide leverage and an ability to take a view on volatility as well as equity direction. However, investing in options is more complicated than investing in equity, as a strike and expiry need to be chosen. This can be seen as an advantage, as it enforces investor discipline in terms of anticipated return and ensures a position is not held longer than it should be. We examine how investors can choose the appropriate strategy, strike and expiry. We also explain the hidden risks, such as dividends, and the difference between delta and the probability an option ends up inthemoney.
 Maintenance of option positions. During the life of an option, many events can occur where it might be preferable to own the underlying shares (rather than the option) and exercise early. In addition to dividends, an investor might want the voting rights or, alternatively, might want to sell the option to purchase another option (rolling the option). We investigate these lifecycle events and explain when it is in an investor’s interest to exercise, or roll, an option before expiry.
 Call overwriting. For a directional investor who owns a stock (or index), call overwriting by selling an OTM call is one of the most popular methods of yield enhancement. Historically, call overwriting has been a profitable strategy due to implied volatility usually being overpriced. However, call overwriting does underperform in volatile, strongly rising equity markets. Overwriting with the shortest maturity is best, and the strike should be slightly OTM for optimum returns.
 Protection strategies using options. For both economic and regulatory reasons, one of the most popular uses of options is to provide protection against a long position in the underlying. The cost of buying protection through a put is lowest in calm, low volatility markets but, in more turbulent markets, the cost can be too high. In order to reduce the cost of buying protection in volatile markets (which is often when protection is in most demand), many investors sell an OTM put and/or an OTM call to lower the cost of the long put protection bought.
 Option structures trading. While a simple view on both volatility and equity market direction can be implemented via a long or short position in a call or put, a far wider set of payoffs is possible if two or three different options are used. We help our clients investigate strategies using option structures (or option combos) that can be used to meet different investor needs.
VOLATILITY AND CORRELATION TRADINGWe help our clients investigate the benefits and disadvantages of volatility trading via options, volatility swaps, variance swaps and gamma swaps. We also show them how these products, correlation swaps, basket options and covariance swaps can give correlation exposure. Recently, options on alternative underlyings have been created, such as options on variance and dividends; the distribution and skew for these underlyings is different from those for equities and should be modeled differently at the total portfolio level. Volatility trading using options. While directional investors typically use options for their equity exposure, volatility investors delta hedge their equity exposure. A deltahedged option (call or put) is not exposed to equity markets, but only to volatility markets. We demonstrate how volatility investors are exposed to dividend and borrow cost risk and how volatility traders can ‘pin’ a stock approaching expiry. We also show that while the profit from delta hedging is based on percentage move squared (i.e. variance or volatility^{2}), it is the absolute difference between realized and implied that determines carry.
 Variance is the standard, not volatility. Partly due to its use in BlackScholes, volatility has historically been used as the measure of deviation for financial assets. However, the correct measure of deviation is variance (or volatility squared). Volatility should be considered to be a derivative of variance. The realisation that variance should be used instead of volatilityled volatility indices, such as the VIX, to move away from ATM volatility (VXO index) towards a variancebased calculation.
 Volatility, variance and gamma swaps. In theory, the profit and loss from delta hedging an option is fixed and based solely on the difference between the implied volatility of the option when it was purchased and the realised volatility over the life of the option. In practice, with discrete delta hedging and unknown future volatility, this is not the case, which has led to the creation of volatility, variance and gamma swaps. These products also remove the need to continuously delta hedge, which can be labourintensive and expensive.
 Options on variance. As the liquidity of the variance swap market improved in the middle of the last decade, market participants started to trade options on variance. As volatility is more volatile at high levels, the skew is positive (the inverse of the negative skew seen in the equity market). In addition, volatility term structure is inverted, as volatility mean reverts and does not stay elevated for long periods of time.
 Correlation trading. The volatility of an index is capped at the weighted average volatility of its constituents. Due to diversification (or less than 1.00 correlation), the volatility of indices tends to trade significantly less than its constituents. The flow from both institutions and structured products tends to put upward pressure on implied correlation, making indeximplied volatility expensive. Hedge funds and proprietary trading desks try to profit from this anomaly either by selling correlation swaps or through dispersion trading (i.e. short index implied and long single stock implied strategies). Basket options and covariance swaps can also be used to trade correlation.
 Dividend volatility trading. If a constant dividend yield is assumed, then the volatility surface for options on realised dividends should be identical to the volatility surface for equities. However, as companies typically pay out less than 100% of earnings, they have the ability to reduce the volatility of dividend payments. In addition to lowering the volatility of dividends to between ½ and ⅔ of the volatility of equities, companies are reluctant to cut dividends. This means that skew is more negative than for equities, as any dividend cut is sizeable.
OPPORTUNITIES, IMBALANCES AND MYTHSThe impact of hedging both structured products and variable annuity products can cause imbalances in the volatility market. These distortions can create opportunities for investors willing to take the other side. We examine the opportunities from imbalances and dispel the myths of overpriced volatility and using volatility as an equity hedge. Overpricing of volatility is partly an illusion. Selling implied volatility is one of the most popular trading strategies in equity derivatives. Empirical analysis shows that implied volatility or variance is, on average, overpriced. However, as volatility is negatively correlated to equity returns, a short volatility (or variance) position is implicitly long equity risk. As equity returns are expected to return an equity risk premium over the riskfree rate (which is used for derivative pricing), this implies short volatility should also be abnormally profitable. Therefore, part of the profits from short volatility strategies can be attributed to the fact equities are expected to deliver returns above the riskfree rate.
 On the efficiency of long volatility positions as an equity portofolio hedge. An ideal hedging instrument for a security is an instrument with 1.00 correlation to that security and zero cost. As the return on variance swaps can have up to a 0.70 correlation with equity markets, adding long volatility positions (either through variance swaps or futures on volatility indices such as VIX or vStoxx) to an equity position could be thought of as a useful hedge. However, as volatility is on average overpriced, the cost of this strategy far outweighs any diversification benefit.
 Corollaries of variable annuity hedging. Since the 1980s, a significant amount of variable annuity products have been sold, particularly in the US. The size of this market is now over USD 1 Trillion. From the mid1990s, these products started to become more complicated and offered guarantees to the purchaser (similar to being long a put). The hedging of these products increases the demand for longdated downside strikes, which lifts longdated implied volatility and skew.
 Corollaries of structured products hedging. The sale of structured products leaves investment banks with a short skew position (i.e. gap risks equivalent to holding short OTM put strips in order to provide longterm capitalprotected products). Whenever there is a large decline in equities, this short skew position causes investment banks to be short volatility (i.e. as the short OTM put becomes more ATM, the vega increases). The covering of this short vega position lifts implied volatility further than would be expected. As investment banks are also short vega convexity, this increase in volatility causes the short vega position to increase in size. This can lead to a ‘structured products vicious circle’ as the covering of short vega causes the size of the short position to increase. Similarly, if equity markets rise and implied volatilities fall, investment banks become long implied volatility and have to sell. Structured products can therefore cause implied volatility to undershoot in a recovery, as well as overshoot in a crisis.
FORWARD STARTING PRODUCTS AND VOLATILITY INDICESForward starting options are a popular method of trading forward volatility and term structure as there is no exposure to nearterm volatility and, hence, zero theta (until the start of the forward starting option). Recently, trading forward volatility via volatility futures such as VIX and vStoxx futures has become increasingly popular. However, as is the case with forward starting options, there are modelling issues. Forward starting products. As the exposure is to forward volatility rather than volatility, more sophisticated models need to be used to price forward starting products than ordinary options. Forward starting options will usually have wider bidoffer spreads than vanilla options, as their pricing and hedging is more complex.
 Volatility indices. While volatility indices were historically based on ATM implied, most providers have swapped to a variance swap based calculation. The price of a volatility index will, however, typically be 0.20.7pts below the price of a variance swap of the same maturity, as the calculation of the volatility index typically chops the tails to remove illiquid prices. Each volatility index provider has to use a different method of chopping the tails in order to avoid infringing the copyrights of other providers.
 Futures on volatility indices. While futures on volatility indices were first launched on the VIX in March 2004, it has only been since the more recent launch of structured products and options on volatility futures that liquidity has improved enough to be a viable method of trading volatility. As a volatility future payout is based on the square root of variance, the payout is linear in volatility not variance. The fair price of a future on a volatility index is between the forward volatility swap, and the square root of the forward variance swap. Volatility futures are, therefore, short vol of vol, just like volatility swaps. It is therefore possible to get the implied vol of vol from the listed price of volatility futures.
 Volatility future ETN/ETF. Structured products based on constant maturity volatility futures have become increasingly popular, and in the US have at times had a greater size than the underlying volatility futures market. As a constant maturity volatility product needs to sell neardated expiries and buy fardated expiries, this flow supports term structure for volatility futures and the underlying options on the index itself. The success of VIXbased products has led to their size being approximately twothirds of the vega of the relevant VIX futures market (which is a similar size to the net listed S&P 500 market) and, hence, appears to be artificially lifting neardated term structure. The size of vStoxx products is not yet sufficient to significantly impact the market, hence they are a more viable method of trading volatility in our view. We recommend shorting VIXbased structured products to profit from this imbalance, potentially against long vStoxxbased products as a hedge. Investors who wish to be long VIX futures should consider the frontmonth and fourthmonth maturities, as their values are likely to be depressed from structured flow.
 Options on volatility futures. The arrival of options on volatility futures has encouraged trading on the underlying futures. It is important to note that an option on a volatility future is an option on future implied volatility, whereas an option on a variance swap is an option on realized volatility. Both have significantly downward sloping term structure and positive skew. Implied volatilities for options on volatility futures should not be compared to the realized of volatility indices.
LIGHT EXOTICSAdvanced investors can make use of more exotic equity derivatives. Some of the most popular are light exotics, such as barriers, worstof/bestof options, outperformance options, lookback options, contingent premium options, composite options and quanto options. Barrier options. Barrier options are the most popular type of light exotic product as they are used within structured products or to provide cheap protection. The payout of a barrier option knocks in or out depending on whether a barrier is hit. There are eight types of barrier option, but only four are commonly traded, as the remaining four have a similar price to vanilla options. Barrier puts are more popular than calls (due to structured product and protection flow), and investors like to sell visually expensive knockin options and buy visually cheap knockout options.
 Worstof/bestof options. Worstof (or bestof) options give payouts based on the worst (or best) performing asset. They are the second most popular light exotic due to structured product flow. Correlation is a key factor in pricing these options, and investor flow typically buys correlation (making uncorrelated assets with low correlation the most popular underlyings). The underlyings can be chosen from different asset classes (due to low correlation), and the number of underlyings is typically between three and 20.
 Spread options. Spread options are an option on the difference between returns on two different underlyings. They are a popular method of implementing relative value trades, as their cost is usually cheaper than an option on either underlying. The key unknown parameter for pricing spread options is implied correlation, as spread options are short correlation.
 Lookback options. There are two types of lookback options, strike lookback and payout lookback, and both are usually multiyear options. Strike reset (or lookback) options have their strike set to the highest, or lowest, value within an initial lookback period (of up to three months). These options are normally structured so the strike moves against the investor in order to cheapen the cost. Conversely, payout lookback options tend to be more attractive and expensive than vanilla options, as the value for the underlying used is the best historical value.
 Contingent premium options. Contingent premium options are initially zeropremium and only require a premium to be paid if the option becomes ATM on the close. The contingent premium to be paid is, however, larger than the initial premium would be, compensating for the fact that it might never have to be paid. Puts are the most popular, giving protection with zero initial premium.
 Composite and quanto options. There are two types of options involving different currencies. The simplest is a composite option, where the strike (or payoff) currency is in a different currency than the underlying. A slightly more complicated option is a quanto option, which is similar to a composite option, but the exchange rate of the conversion is fixed.
ADVANCED VOLATILITY TRADINGAdvanced investors often use equity derivatives to gain different exposures; for example, relative value or the jumps on earnings dates. We demonstrate our clients how this can be done and also reveal how profits from equity derivatives are both path dependent and dependent on the frequency of delta hedging. Relative value trading. Relative value is the name given to a variety of trades that attempt to profit from the mean reversion of two related assets that have diverged. The relationship between the two securities chosen can be fundamental (different share types of same company or significant crossholding) or statistical (two stocks in same sector). Relative value can be carried out via cash (or delta1), options or outperformance options.
 Relative value volatility trading. Volatility investors can trade volatility pairs in the same way as trading equity pairs. For indices, this can be done via options, variance swaps or futures on a volatility index (such as the VIX or vStoxx). For indices that are popular volatility trading pairs, if they have significantly different skews this can impact the volatility market. Singlestock relative value volatility trading is possible, but less attractive due to the wider bidoffer spreads.
 Trading earnings announcements/jumps. From the implied volatilities of neardated options it is possible to calculate the implied jump on key dates. Trading these options in order to take a view on the likelihood of unanticipated (low or high) volatility on reporting dates is a very common strategy. We examine the different methods of calculating the implied jump and show how the jump calculation should normalise for index term structure.
 Stretching BlackScholes assumptions. The BlackScholes model assumes an investor knows the future volatility of a stock, in addition to being able to continuously delta hedge. In order to discover what the likely profit (or loss) will be in reality, we stretch these assumptions. If the future volatility is unknown, the amount of profit (or loss) will vary depending on the path, but buying cheap volatility will always show some profit. However, if the position is deltahedged discretely, the purchase of cheap volatility may reveal a loss. The variance of discretely deltahedged profits can be halved by hedging four times as frequently. We also show why traders should hedge with a delta calculated from expected – not implied – volatility, especially when long volatility.
SKEW AND TERM STRUCTURE TRADINGWe examine how skew and term structure are linked and the effect on volatility surfaces of the square root of time rule. The correct way to measure skew and smile is examined, and we show how skew trades only breakeven when there is a static local volatility surface. Skew and term structure are linked. When there is an equity market decline, there is normally a larger increase in ATM implied volatility at the near end of volatility surfaces than the far end. Assuming sticky strike, this causes neardated skew to be larger than fardated skew. The greater the term structure change for a given change in spot, the higher skew is. Skew is also positively correlated to term structure (this relationship can break down in panicked markets). For an index, skew (and potentially term structure) is also lifted by the implied correlation surface. Diverse indices tend to have higher skew for this reason, as the ATM correlation is lower (and low strike correlation tends to 1.00 for all indices).
 Square root of time rule can compare different term structures and skews. When implied volatility changes, typically the change in ATM volatility multiplied by the square root of time is constant. This means that different (T2T1) term structures can be compared when multiplied by √(T 2T1)/(√T2√T1), as this normalises against 1Y3M term structure. Skew weighted by the square root of time should also be constant. Looking at the different term structures and skews, when normalised by the appropriate weighting, can allow us to identify calendar and skew trades in addition to highlighting which strike and expiry is the most attractive to buy (or sell).
 How to measure skew and smile. The implied volatilities for options of the same maturity, but of different strike, are different from each other for two reasons. Firstly, there is skew, which causes lowstrike implieds to be greater than highstrike implieds due to the increased leverage and risk of bankruptcy. Secondly, there is smile (or convexity/kurtosis), when OTM options have a higher implied than ATM options. Together, skew and smile create the ‘smirk’ of volatility surfaces. We look at how skew and smile change by maturity in order to explain the shape of volatility surfaces both intuitively and mathematically. We also examine which measures of skew are best and why.
 Skew trading. The profitability of skew trades is determined by the dynamics of a volatility surface. We examine sticky delta (or ‘moneyness’), sticky strike, sticky local volatility and jumpy volatility regimes. Long skew suffers a loss in both a sticky delta and sticky strike regimes due to the carry cost of skew. Long skew is only profitable with jumpy volatility. We also show how the best strikes for skew trading can be chosen.
APPENDIXTechnical details and other areas related to volatility trading. Local volatility: While BlackScholes is the most popular method for pricing vanilla equity derivatives, exotic equity derivatives (and ITM American options) usually require a more sophisticated model. The most popular model after BlackScholes is a local volatility model as it is the only completely consistent volatility model. A local volatility model describes the instantaneous volatility of a stock, whereas BlackScholes is the average of the instantaneous volatilities between spot and strike.
 Measuring historical volatility: We examine different methods of historical volatility calculation, including closetoclose volatility and exponentially weighted volatility, in addition to advanced volatility measures such as Parkinson, GarmanKlass (including YangZhang extension), Rogers and Satchell and YangZhang.
 Modeling volatility surfaces: There are a variety of constraints on the edges of a volatility surface, and this section details some of the most important constraints from both a practical and theoretical point of view. We examine the considerations for very shortdated options (a few days or weeks), options at the wings of a volatility surface and very longdated options.
 BlackScholes formula: The most popular method of valuing options is the BlackScholesMerton (BSM) model. We show the key formulas involved in this calculation. The assumptions behind the model are also discussed.
 Greeks and their meaning: Greeks is the name given to the (usually) Greek letters used to measure local risks. We give the BlackScholes formula for the key Greeks and describe which risk they measure.
 Shadow Greeks: How a volatility surface changes over time can impact the profitability of a position. Two of the most important are the impact of the passage of time on skew (volatility slide theta) and the impact of a movement in spot on OTM options (anchor delta).
 Sortino ratio: If an underlying is distributed normally, standard deviation is the perfect measure of risk. For returns with a skewed distribution, such as with option trading or call overwriting, the Sortino ratio is more appropriate.
 Capital structure arbitrage: The high levels of volatility and credit spreads during the bursting of the TMT bubble demonstrated the link between credit spreads, equity, and implied volatility. We examine four models that demonstrate this link (Merton model, jump diffusion, put vs CDS, and implied nodefault volatility).
