Equity derivative strategy 2012 Q1 update and Trading Volatility

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1 Equity derivative strategy 212 Q1 update and Trading Volatility Colin Bennett (+34)

2 Contents VOLATILITY TRADING FOR DIRECTIONAL INVESTORS Call overwriting Protection buying Choosing strike of option ESSENTIAL FACTS OF VOLATILITY TRADING Volatility is not as expensive as you think Hedging equity with volatility Stretching Black-Scholes Variable annuity and structured products impact on the market ADVANCED VOLATILITY TRADING Dividends and correlation Advanced volatility measures Term structure and skew 2

3 Call overwriting can yield enhance returns Call overwriting Reasons why volatility is usually overpriced Demand for protection Unwillingness to sell low premium (near dated) options High gamma of near dated options has gap risk premium Index implied lifted by structured products Call overwriting improves portfolio performance Selling expensive implieds can lift performance, but note that the delta of position is lower which reduces benefit of equity risk premium On balance call overwriting is a winning strategy in most market environments (except in very bullish markets) BXM index gives performance of S&P5 1m ATM call overwriting BUT is total return (need to compare it to SPXT not SPX) Return 15% 14% 13% 12% 11% 1% 9% 8% 7% 6% 5% 5% 6% 7% 8% 9% 1% 11% 12% 13% 14% 15% Price (rebased) Call overwriting Enhance performance selling OTM calls when volatility is high Equity Equity - call BXM is a total return index, so needs to be compared to S&P5 total return index for a fair comparison BXM (1m 1% Buy Write) S&P5 S&P5 total return Strike S&P5 1m ATM call overwriting performance 3

4 Overwriting with 1 month 14% strike is best Call overwriting Strike of optimal strategy depends on period of time examined Overwriting with near dated options outperform as can sell 12 one month options in a year, but only 4 three month options. BUT selling multiple short dated options can be seen as more risky (if markets rise one month, then fall) Equities must have a realistic positive return during back test period (negative return optimum strike is < ATM). In these periods a strike of 13-14% is best for 1 month SX5E options (17-18% for 3 month options). Strike should be higher for higher volatility stocks (rule of thumb is use c25% delta calls) Call overwriting 3.% 1% 11% 12% 13% 14% 15% 16% Exact peak strike for overwriting depends on period of backtest 18% 11% 2.5% 2.% 1.5% 1.%.5% Call overwriting return - index return Only upside risk is reduced (use Sortino ratio rather than standard dev) Index.% -8% -7% -6% -5% -4% -3% -2% -1% % Call overw riting volatility - index volatility 4

5 Performance depends on market environment Call overwriting Overwriting performs best on an index rather than single stock Overwriting with 1 month ATM call overwriting has outperformed, but there were periods where it underperformed As index implieds are more overpriced than stocks (implied correlation too high) best to overwrite using indices As call overwriting less attractive for single stocks, there is greater chance enhanced call overwriting can lift returns Relative performance (rebased) S&P5Call 1m overwriting ATM performance call overwriting depends on performance market environment since trough 29 trough Call overwriting outperforms Start of late 9's bull market Credit crunch Call ovewriting underperforms Asian crisis Outperform Significantly Breakeven Significantly Underperform Significantly Significantly Underperform Outperform Outperform Underperform TMT peak BXM / S&P5 total return 5

6 Markets can crash, correct or enter bear market Protection buying Examining previous declines is relevant to current crisis DAX declines since 1959 can be grouped into 3 categories: crash, correction and bear market Crash has a high annualised decline (c9%) for a period of 3 months or less Bear markets are multiple year declines of 23% or more Corrections are remaining declines of up to a year and up to 22% Type of correction protection is required against, can help determine which strategy to choose Average Duration Average decline Average annualised decline Duration range Decline range Crash 1 month 31% 96% < 3 months 19% - 39% Correction 3 months 14% 58% <= 1 year 1% - 22% Bear market 2.4 years 44% 26% 1-5 years 23% - 73% 6

7 Option structures incorporate delta and vol view Protection buying Choice of protection strategy depends on type of decline to be hedged Short dated puts are most appropriate for crashes, (rolling) put spreads are best for corrections (and bear markets) Protection has to be paid for through premium, loss of upside (collar) or potential losses on downside (1x2 put spd) Bullish strategies are the reverse of protection strategies. View on equity and volatility markets guides choice of optimum option structure Implied expensive Bearish Market View Volatility View Market View Bullish Implied cheap 7

8 ITM options trade like a future Choosing strike of option ITM options have highest delta, hence highest return if investor is confident Typically investors trade ATM or OTM options as they are cheapest Highest return for a given market move occurs for ITM options, as their higher delta more than outweighs their higher cost (ITM options are similar to futures) ITM options do not have much convexity (compared to ATM), hence is a risky strategy as the high cost of ITM options could be lost Return 6% Profit of 1 year call if markets rise 1% ITM options have highest profit 5% 4% 3% OTM options have low profit due to low delta 2% 1% % 6% 64% 68% 72% 76% 8% 84% 88% 92% 96% 1% 14% 18% 112% 116% Strike 8

9 Contents VOLATILITY TRADING FOR DIRECTIONAL INVESTORS Call overwriting Protection buying Choosing strike of option ESSENTIAL FACTS OF VOLATILITY TRADING Volatility is not as expensive as you think Hedging equity with volatility Stretching Black-Scholes Variable annuity and structured products impact on the market ADVANCED VOLATILITY TRADING Dividends and correlation Advanced volatility measures Term structure and skew 9

10 Implied should be above realised Volatility is not as expensive as you think Assuming a positive equity risk premium, implied vol should be above realised Implied volatility is on average 1-2 pts above realised volatility Short volatility strategies are effectively long equity risk (assuming negative spot vol correlation) If long equity is expected to earn more than the risk free rate (i.e. positive equity risk premium) then short volatility should also be profitable (as exposed to the same risk) Fair value of implied volatility is therefore above realised volatility Structured products selling variance contain equity risk Shorting implied volatility is an opportunity, but returns are likely to be similar to going long equity There are many structured products based on selling variance swaps, their returns have suffered in the downturn as volatility spiked as equities fell 1

11 Vol is not as expensive as you think Hedging equity with volatility Implied volatility is negatively correlated to equity market, but is a poor hedge There is an R 2 of 56% between weekly SX5E and vstoxx returns. Hence implied volatility can be used as a low cost hedge. Strategy is not zero cost as implied volatility is on average expensive (trades above average realised volatility) as short vol is implicitly long equity risk (and equities have an equity risk premium) Hedging with volatility (or variance swaps) is less effective than with futures, as volatility has less than 1% correlation with equities and is expensive (on average) SX5E and vstoxx weekly returns vstoxx 1% 8% 6% R 2 =.56 4% 2% SX5E % -3% -2% -1% % -2% 1% 2% -4% Return 6% 5% 4% 3% 2% 1% % -1% -2% -3% -4% SX5E hedged with futures or variance swap Risk free rate (SX5E 1% hedged with futures) % 5% 1% 15% 2% 25% Volatility SX5E + 1 year variance swap Add increasing amount of variance swaps to 1% SX5E SX5E + futures 1% SX5E 11

12 Continuous delta hedging with known volatility Stretching Black-Scholes Black-Scholes has unrealistic assumptions Known future realised volatility Ability to hedge continuously Black-Scholes also assumes volatility is constant, but results are the same if this condition is relaxed P&L known under Black-Scholes Payout of delta hedged option is known under Black-Scholes Profit (or loss) is value of option using future (known) volatility less price paid for option (i.e. value of option using implied volatility) P&L of delta hedging an option under Black-Scholes vs Realised vol implied vol is a straight line Continuous delta hedging with known vol As underlying has constant volatility, the amount earned from gamma is exactly equal to the loss of time value (theta) Gamma 9% Theta 95% 1% 15% 11% Straddle T= Straddle T=1 Profit (& loss) from delta hedging Hedging with delta calculated using known volatility means profit (or loss) is constant P&L (%) Realised vol - implied vol (%)

13 Continuous delta hedging with unknown volatility Stretching Black-Scholes If future volatility is unknown delta is incorrect If future volatility is unknown delta has to be calculated using the implied volatility Delta calculated is only correct if future volatility = implied volatility (i.e. when option trading at fair price). Hence P&L line has to go through origin If there is a difference between future realised volatility and implied volatility then the payout is uncertain Always profit from delta hedging cheap option While the exact profit is uncertain, delta hedging a cheap option will always give a profit (P&L line has to go through origin) While bigger the difference between implied and realised means the delta is less accurate, this effect is dwarfed by the additional cheapness of the option Continuous delta hedging with unknown vol Profit (& loss) from delta hedging When realised volatility = implied volatility, delta from implied volatility is correct hence profit is always Delta hedging a cheap option with delta calculated from implied volatility (as volatility is unknown) is always profitable, but profits are spot dependent 1 Small Large profit profit 9% 95% 1% 15% 11% Straddle T= Straddle T=1 P&L (%) Realised vol - implied vol (%) Average profit Profit +/- 1σ 13

14 Discrete delta hedging with known volatility Continuous trading is unrealistic assumption Presence of weekends and less than 24 hour trading makes continuous trading an unrealisic assumption Trading costs make it unlikely delta hedging is done continuously throughout the day Ideal time to delta hedge is before an uncertain annoucement or potential change in direction of market Discrete hedging introduces noise Noise of discrete delta hedging is independent of how cheap (or rich) the option is It is possible to lose money when buying a cheap option when discrete delta hedging Noise from discrete hedging is halved if frequency of delta hedging is 4x as frequent σ P&L = σ x Vega x (π/[4n]) Stretching Black-Scholes Discrete delta hedging 9% 95% 1% 15% 11% Initial delta hedged straddle Changing hedging frequency has changed the profit made Straddle rehedged after +5% move Profit (& loss) from delta hedging Hedging a 4x frequency halves the noise from discrete delta hedging P&L (%) Realised vol - implied vol (%) Average profit Profit +/- 1σ Profit +/- 1σ with 4x frequency 14

15 Discrete delta hedging with unknown volatility Stretching Black-Scholes Real life has unknown vol and discrete hedging Errors in real life are combination of unknown volatility and discrete hedging 15 1 Discrete delta hedging Implied vol at inception = 22% Realised vol over life = 42% 4 35 Possible to lose money when delta hedging a discrete option 5 3 If you had bought an ATM option in April 28 with 22% implied, you would have lost money despite realised being almost twice as large (42%) Loss due to fact delta was up to 24% different (when using future realised volatility instead of implied) As market declined before vol spiked, delta using implied was near 1% but should be less. Hence trader bought too many futures to delta hedge and suffered loss. Should calculate delta using expected vol If volatility is seen as 5pts too cheap, should bump vol surface 5pts to calculate deltas Error is most significant for long vol, as markets tend to decline when vol rises 25 Apr-8 May-8 Jun-8 Jul-8 Aug-8 Sep-8 Oct-8 Nov-8 Dec P&L using delta from future vol P&L using delta from implied vol SX5E (RHS) Profit (& loss) from delta hedging Errors from discretely hedging with unknown volatility is the sum of error due to discrete hedging and error due to unknown volatility P&L (%) Realised vol - implied vol (%) Average profit Profit +/- 1σ

16 Variable annuity often give investors a put option Variable annuity hedging Variable annuities often sold with protection to make them more attractive With fixed annuities, the insurance company invests proceeds and guarantees a fixed return Variable annuities allow the purchaser to pick the investments, but leaves investor exposed to the downside To make variable annuities more attractive they were often sold with forms of downside protection Hedging of variable annuities lifts index term structure and skew While products can be up to 2+ years long, position are dynamically hedged with 3-5 years puts for liquidity reasons When modelling dynamic strategies, future implied volatility is modelled with a confidence interval, e.g. 95% to ensure only 1 in 2 chance of a loss. As volatility rose to levels greater than seen in great depression, cost of hedging has weighted on margins The constant bid from variable annuity hedging lifts term structure and skew, particularly for the S&P5 (but also for other major indices due to relative value traders) Volker rule prohibits proprietary trading, which has reduced the number of counterparties for long dated protection causing skew to rise (particularly at the far end of volatility surfaces) 16

17 Structured products can cause volatility overshoot Structured products Hedging of structured products can exaggerate implied volatility moves Sale of structured products causes investment banks to be short skew (vanna) and short vega convexity (volga) When markets decline, the skew skew position causes sellers to become short vol. To hedge this position traders buy volatility, lifting implieds. As implieds rise, the short vol position increases in size due to vega convexity. Traders then have to buy more vol, causing a structured product vicious circle and an implied volatility overshoot 1. Market declines 2. Traders become short vol as are short skew 3. Traders buy vol Price Implied Vol Vicious Circle Time 8% 9% 1% 11% Strike 4. Vega convexity means traders become shorter vol as volatility rises 17

18 Contents VOLATILITY TRADING FOR DIRECTIONAL INVESTORS Call overwriting Protection buying Choosing strike of option ESSENTIAL FACTS OF VOLATILITY TRADING Volatility is not as expensive as you think Hedging equity with volatility Stretching Black-Scholes Variable annuity and structured products impact on the market ADVANCED VOLATILITY TRADING Dividends and correlation Advanced volatility measures Term structure and skew 18

19 Dividends have lower vol, but higher skew Dividends and correlation Realised dividends are less volatile than equities Constant dividend yield implies dividends have same vol surface as equities Realised dividend volatility is 5-7% of equity volatility as: A) companies suppress dividend volatility due to less than 1% payout and; B) equity volatility is too high compared to fundamentals ATM dividend volatility should be lower than equity ATM implied Dividends have higher skew than equities While realised dividends are less volatile than equities, implied dividends can be more volatile due to imbalances caused by structured product sellers (get longer dividends as market falls) Underlying of options on dividends is implied dividends (realised dividends cannot be traded) Dividends have higher skew (3rd moment) than equities as dividends are cut to zero before equity prices reach zero Implied vol 21 dividends % 28% 26% 24% 22% 2% 18% 16% 14% 12% 1% SX5E 21 Dividends vs Spot Long SX5E 21 dividends traded similar to long SX5E and short SX5E 3 strike put (Jan-8 to Sep-8) 21 (Oct-8 onwards) Dividend volatility surface Low strike implieds are greater than high strike implieds SX5E Negative skew Strike ( ) 19

20 Vega weighted dispersion is best Dividends and correlation Dispersion traders need to decide how to weight short index & long single stock legs Theta (or correlation) weighted: Vega x volatility is equal for both legs. This weighting assumes implieds move by same percentage amount (eg. if index vol is 2% and increases to 3%, single stock vol of 25% rises to 37.5%). This is the purest dispersion trade as payout = difference between realised correlation and implied correlation MULTIPLIED by weighted average variance of stocks. Due to the payout being multiplied by weighted average variance dispersion is short vol of vol (as correlation is correlated to volatility). Vega weighted: Vega is equal for both legs. This weighting assumes implieds move by the same absolute amount (eg. if index vol is 2% and increases to 3%, single stock vol of 25% rises to 35%). As correlation is correlated to volatility, the payout of a theta weighted dispersion trade is (negatively) correlated to volatility. To remove this sensitivity it is better to go long more single stock volatility, as vega weighted dispersion does. Arguably vega weighted dispersion has a single stock leg 2-5% too large, but over hedging could be seen as an advantage (as funding etc could dry up in a crisis and the position exited prematurely). Gamma weighted: Rarely used, as difficult to justify using more single stock vega than index vega when stocks have higher volatility Greek Theta-weighted Vega-weighted Gamma-weighted Type of correction protection is required against, can help determine which strategy to choose Theta pay pay a lot Vega Short Long Gamma Very short Short Single-stock vega Less than index Equal to index More than index 2

21 Yang-Zhang is best measure for small samples Advanced volatility measures Using intraday prices can improve historical volatility measurement When comparing volatility between regions, weekly volatility is better than daily to reduce effect of different time zones. This is only appropriate for large data samples, if this is not available / practical an advanced volatility measure is better. Close to close volatility needs c2 or more days of data to be accurate, for smaller periods e.g. 5 days close to close volatility is very noisy. An advanced measuring Open (O), High (H), Low (L) and Close (C) is better for small samples. Estimate Prices Taken Handle Drift? Handle Overnight Jumps? Efficiency (max) Close to close C No No 1 Parkinson HL No No 5.2 Garman-Klass OHLC No No 7.4 Rogers-Satchell OHLC Yes No 8 G-K Yang-Zhang ext OHLC No Yes 8 Yang-Zhang OHLC Yes Yes 14 21

22 Surfaces move by square root of time Term structure and skew Volatility move weighted by square root of time is roughly constant Near dated implieds move more than far dated implieds. Can adjust whole surface by adjusting implieds for maturity T by one year implied vol move / T p. P is the power of the move. Typically volatility move weighted by square root of time is approximately constant (power.5). Surfaces also sometimes move in parallel (power ). On average surfaces move power.44, hence usually square root of time but sometimes parallel. Volatility moving by square root of time 23% 22% 1 year implied moves half amount of 3 month implied Implied vol 21% 2% 19% 18% +2% +1% -1% -.5% 4 year implied moves half amount of 1 year implied 17% 3 months 6 months 1 year 2 years 3 years 4 years Rise in implied Flat term structure Fall in implied 22

23 Can compare different term structure & skew Term structures can be normalised If assume term structure is a fixed vol for infinite maturity and a square root of time bump, then different term structures can be compared Multiplying standard V 2 V 1 term structure by (T 2 T 1 )/ ( T 2- T 1 ) allows different term structures to be compared Normalised term structure puts term structure in same units as 1 year 3 month term structure Skew multiplied by square root of time constant Skew is greater for near dated implieds than far dated Can compare different skews when multiply by square root of time Term structure x (T2T1) ( T2- T1) Skew (normalised T) Dec-6 Apr-7 Term structure and skew Term structure (normalised) Aug-7 Dec-7 Apr-8 Aug-8 Dec-8 Apr-9 Skew (normalised) In 21 Q2 skew spiked, particularly at the far end, due to changes in US regulation 2 Dec-9 Mar-1 Jun-1 Sep-1 Dec-1 Mar-11 Jun-11 Sep-11 Dec-11 Aug-9 Dec-9 Apr-1 Aug-1 Dec-1 Apr-11 3 month skew (9-1%) 6 month skew (9-1%) Aug-11 6 mths - 3 mths (normalised) 1 year - 6 mths (normalised) Dec-11 23

24 24

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