Lecture 10: Dispersion Trading
Marco Avellaneda G63.2936.001
Spring Semester 2009 What is dispersion trading?
• Dispersion trading refers to trades in which one
-- sells index options and buys options on the index components, or
-- buys index options and sells options on the index components
• All trades are delta-neutral (hedged with stock)
• The package is maintained delta-neutral over the horizon of the trade
Dispersion trading:
-- selling index volatility and buying volatility of the index components -- buying index volatility and selling volatility on the index components Why Dispersion Trading?
Motivation: to profit from price differences in volatility markets using index options and options on individual stocks
Opportunities: Market segmentation, temporary shifts in correlations between assets, idiosyncratic news on individual stocks Index Arbitrage versus Dispersion Trading
Stock N Index Arbitrage: * Reconstruct * an index or ETF * using the * component stocks Index Stock 3
Stock 2 Dispersion Trading: Reconstruct an index option using options on the Stock 1 component stocks Main U.S. indices and sectors
• Major Indices : SPX, DJX, NDX SPY, DIA, QQQQ (Exchange-Traded Funds)
• Sector Indices : Semiconductors: SMH, SOX Biotech: BBH, BTK Pharmaceuticals: PPH, DRG Financials: BKX, XBD, XLF, RKH Oil & Gas: XNG, XOI, OSX High Tech, WWW, Boxes: MSH, HHH, XBD, XCI Retail: RTH n = = Intuition… I ∑ wi Si wi number of shares in index i=1
n n dI = 1 = wi Si dS i ∑ widS i ∑ I I i=1 i=1 I Si n = dS i = wi Si ∑ pi , pi i=1 Si I
n σ 2 = dI = dS i I Var Var ∑ pi I i=1 Si dS = dS i j ∑ pi p jCov ij Si S j Fair value relation for σ 2 = σ σ ρ I ∑ pi p j i j ij volatilities assuming a ij given correlation matrix The trade in pictures
Index Sell index call
Stock 1 Stock 2
Buy calls on different stocks.
Delta-hedge using index and stocks Profit-loss scenarios for a dispersion trade in a single day
Scenario 1 Scenario 2
1.5 2.5 2 1 1.5 0.5 1 0.5 0 0
-0.5 -0.5 -1 standardmove standardmove -1 -1.5 -2 -1.5 -2.5 -2 -3 1 2 3 4 5 6 7 8 9 101112131415 1 2 3 4 5 6 7 8 9 101112131415 stock # stock #
Stock P/L: - 2.30 Stock P/L: +9.41 Index P/L: - 0.01 Index P/L: - 0.22 Total P/L: - 2.41 Total P/L: +9.18 First approximation to the dispersion package: ``Intrinsic Value Hedge’’
M = = I ∑wiSi wi number of shares, scaled by ``divisor' ' i=1
M IVH: use index = K ∑wi Ki ⇒ weights for option j=1 hedge
M ()()− ≤ − IVH: max I K 0, ∑wimax Si Ki 0, premium from index = j 1 is less than premium from components M “Super-replication” ()()≤ CI I, K,T ∑wiCi Si , Ki ,T j=1 Makes sense for deep- -in-the-money options Intrinsic-Value Hedging is `exact’ only if stocks are perfectly correlated
1 M M σ N − σ 2T ()()= = i i 2 i I T ∑wi Si T ∑wi Fie i=1 i=1 ρ ≡ ≡ = ij 1 ⇒ Ni N standardiz ed normal
Similar to M σ −1σ 2 i X i T Jamshidian (1989) = 2 Solve for X in : K ∑wi Fie for pricing bond = i 1 options in 1-factor 1 σ X − σ 2T model = i 2 i Set : Ki Fie ∴ M ()()()− = ()− ∀ max I T K 0, ∑wi max Si T Ki 0, T i=1 IVH : Hedge with ``equal-delta’’ options
σ −1σ 2 i X T i T 1 K 1 K = Fe 2 ∴ X = ln i + σ T i i σ i i T Fi 2 1 F 1 − X = ln i − σ T = d σ i 2 i T Ki 2 ()= N d2 constant log - moneyness ≈ constant Del tas ≈ constant What happens after you enter an option trade ?
Unhedged call option Hedged option
€ 35 € 7
€ 30 € 6
€ 25 € 5
€ 20 € 4
€ 15 € 3
€ 10 € 2
€ 1 € 5 € 0 € 0 € 70 € 75 € 80 € 85 € 90 € 95 € 100 € 105 € 110 € 115 € 120 € 125 € 130 € 70 € 75 € 80 € 85 € 90 € 95 € 100 € 105 € 110 € 115 € 120 € 125 € 130 -€ 1
Profit-loss for a hedged single option position (Black –Scholes)
dσ P / L ≈ θ ⋅ (n2 −1) + NV ⋅ σ ∆S ∂C θ = time - decay (dollars), n = , NV = normalized Vega = σ Sσ ∆t ∂σ n ~ standardized move Gamma P/L for an Index Option
Assume dσ = 0
= θ ( 2 − ) Index Gamma P/L I nI 1
M pσ w S n = i i n p = i i I ∑ σ i i M i=1 I ∑w j S j j=1 M σ 2 = σ σ ρ I ∑ pi p j i j ij ij =1
M p2σ 2 p p σ σ Index P/L = θ i i ()n2 −1 +θ i j i j ()n n − ρ I ∑ σ 2 i I ∑ σ 2 i j ij i=1 I i≠ j I
Gamma P/L for Dispersion Trade
th ≈θ ⋅( 2 − ) i stock P/L i ni 1
M p2σ 2 p p σ σ Dispersion Trade P/L ≈ θ + i i θ ()n2 −1 +θ i j i j ()n n − ρ ∑ i σ 2 I i I ∑ σ 2 i j ij i=1 I i≠ j I
diagonal term: off-diagonal term: realized single-stock realized cross-market movements vs. movements vs. implied volatilities implied correlation Dispersion Statistic
N ∆ ∆ 2 = ()− 2 = Si = I D ∑ pi X i Y X i , Y i=1 Si I
N 2 = σ 2 2 −σ 2 2 D ∑ pi i ni I nI i=1
N = θ ()()2 − +θ 2 − P/L ∑ i ni 1 I nI 1 i=1 N N = θ 2 +θ 2 − Θ Θ ≡ θ +θ ∑ ini I nI ∑ i I i=1 i=1 N θ N θ N = θ n2 + I pσ 2n2 − I pσ 2n2 +θ n2 − Θ ∑ i i σ 2 ∑ i i i σ 2 ∑ i i i I I i=1 I i=1 I i=1 N θ pσ 2n2 θ = I i i i +θ n2 − I D2 − Θ ∑ σ 2 i i σ 2 i=1 I I Summary of Gamma P/L for Dispersion Trade
N θ pσ 2n2 θ Gamma P/L = I i i i +θ n2 − I D2 − Θ ∑ σ 2 i i σ 2 i=1 I I
“Idiosyncratic” Dispersion Time-Decay Gamma Gamma
Example: ``Pure long dispersion” (zero idiosyncratic Gamma): 2 σ 2 σ 2 ∑ pi i ∑ pi i pσ θ = −θ i i Θ = θ i −1 ≥ θ i −1 > 0 i I σ 2 I σ 2 I σ 2 I I I 25 20 15 10 5 0 70 30 25 20 15 75 10 5 0 80
85 70
90 75
95 80
100 85 90 105 95 110 100
115 105
120 110 115 125 120 130 125 130 70 70 08 80 90 100 90 110 120 10 0 130 110 120 130 IVH as a position function of Value function (B&S) the for stock prices (2 stocks) Payoff function for a trade In general: short index IVHIngeneral: index short options (IVH),options 2stocks short index/longwith `rnvra’ moves``transversal’’ diagonal, long-Gamma for short-Gammais along the Gamma Risk: Negative exposure for ‘parallel’ shifts, positive ‘exposure’ to transverse shifts
+10.84 10.31 130 -2.29
125
120
115
110 σ = 1 30 % 105 σ = 2 40 % 100 ρ = 12 5. 95
90
5.80 85 20.49 80
75
-6.80 70 70 75 80 85 90 95 100 105 110 115 120 125 130+7.88 D/(Y*Y)= Normalized Dispersion D=cross-sectional move, Dispersion, or Gamma-Risk for Baskets for Gamma-Risk normalized normalized dispersion -2.E+05 1.E+06 -6.E+05 -4.E+05 8.E+05 -1.E+06 -8.E+05 4.E+05 6.E+05 -1.E+06 0.E+00 2.E+05
1.21 0.3
0.07
0.012
0 0.13
0.06
-0.01 -0.08 -0.15
ind ex D D X i / = Y = Fromrealistic portfolio ∑ i 2 N = ∆ 1 S = S i p i ∑ i i
N = ) ( 1 X p i i − ) ( Y X Y = i / 2 ∆ Y I I −
1 2 Vega Risk
Sensitivity to volatility: perturb all single-stock implied volatilities by the same percent amount
M = ∆σ + ∆σ Vega P/L ∑Vega j j Vega I I j=1 M ∆σ ∆σ = ()NV j + ()NV I ∑ j σ I σ j=1 j I M ∆σ = ()()NV + NV ∑ j I σ j=1
∂V NV = normalized vega = σ ∂σ Market/Volatility Risk
130 20 125 19 18 120 17 115 16 15 110 14 13 105 12 11 100 10 95 9 8
90 marketlevel 7 6 85 5 80 4 3 75 2 1 70 0 Vol % multipler 130 125 70% 80% 90% 130% 120 115 95 110 90 100% 110% 120% 130% 105 115% 85 100 80 75 70 100% Market level 85%
vol % multiplier 70%
Short Gamma on a perfectly correlated move Monotone-increasing dependence on volatility (IVH) ``Rega’’: Sensitivity to correlation
ρ → ρ + ∆ρ ≠ ij ij i j
M σ 2 → σ σ ρ + σ σ ∆ρ I ∑ pi p j i j ij ∑ pi p j i j ij =1 i≠ j
M M ∆σ 2 = []()()σ )1( 2 − σ )0( 2 ∆ρ σ )1( = σ σ )0( = 2σ 2 I I I , I ∑ p j j , I ∑ p j j j=1 j=1
2 2 ∆σ 1 ()()σ )1( − σ )0( I = I I ∆ρ σ σ 2 I 2 I
2 2 2 2 1 ()()σ )1( − σ )0( 1 ()σ ()1 − ()σ ()0 Correlatio n P/L = ()NV I I ∆ρ R ega = I I × ()NV I σ 2 σ 2 I 2 I 2 I Market/Correlation Sensitivity
130 125 5.1 4.8 120 4.5 4.2 115 3.9 110 3.6 3.3 105 3 2.7 100 market level 2.4 2.1 95 1.8 90 1.5 1.2 85 0.9 130 0.6 80 0.3 110 0 75 90 market level 70 -0.3 0 0 -0.2 0.1 0.2 0.3 -0.1 70 -0.3 -0.2 -0.1 0.1 0.2 0.3 corr change corr change
Short Gamma on a perfectly correlated move Monotone-decreasing dependence on correlation A model for dispersion trading signals (taking into account volatility skews)
• Given an index (DJX, SPX, NDX) construct a proxy for the index with small residual.
m dI = β dS k + ε ∑ k (multiple regression ) I k=1 Sk
• Alternatively, truncate at a given capitalization level and keep the original weights, modeling the remainder as a stock w/o options.
• Build a Weighted Monte Carlo simulation for the dynamics of the m stocks and value the index options with the model
• Compare the model values with the bid/offer values for the index options traded in the market. Morgan Stanley High-Technology 35 Index (MSH)
ADP JDSU AMAT JNPR AMZN LU AOL MOT 35 Underlying Stocks BRCM MSFT CA MU Equal-dollar weighted index, adjusted CPQ NT CSCO ORCL annually DELL PALM EDS PMTC EMC PSFT Each stock has typically O(30) options ERTS SLR over a 1yr horizon FDC STM HWP SUNW IBM TLAB INTC TXN INTU XLNX YHOO Test problem: 35 tech stocks
Price options on basket of 35 stocks underlying the MSH index
Number of constraints : 876
Number of paths : 10,000 to 30,000 paths
Optimization technique : Quasi-Newton method (explicit gradient) OptionNameStockTickerExpDate Strike Type Intrinsic Bid Ask Volume OpenInterestStockPrice QuoteDate ZQN AC-E AMZN 1/20/01 15 Call 0 4.125 4.375 13 3058 16.6875 12/20/00 ZQN AT-E AMZN 1/20/01 16.75 Call 0 3.125 3.375 0 1312 16.6875 12/20/00 ZQN AO-E AMZN 1/20/01 17.5 Call 0 2.875 3.25 20 10 16.6875 12/20/00 ZQN AU-E AMZN 1/20/01 18.375 Call 0 2.625 2.875 10 338 16.6875 12/20/00 ZQN AD-E AMZN 1/20/01 20 Call 0 1.9375 2.125 223 5568 16.6875 12/20/00 ZQN BC-E AMZN 2/17/01 15 Call 0 5.125 5.625 30 1022 16.6875 12/20/00 ZQN BO-E AMZN 2/17/01 17.5 Call 0 4 4.375 0 0 16.6875 12/20/00 ZQN BD-E AMZN 2/17/01 20 Call 0 3.125 3.5 10 150 16.6875 12/20/00 ZQN DC-E AMZN 4/21/01 15 Call 0 5.875 6.375 0 639 16.6875 12/20/00 ZQN DO-E AMZN 4/21/01 17.5 Call 0 5 5.375 0 168 16.6875 12/20/00 ZQN DD-E AMZN 4/21/01 20 Call 0 3.875 4.125 5 1877 16.6875 12/20/00 ZQN DS-E AMZN 4/21/01 22.5 Call 0 3.125 3.375 20 341 16.6875 12/20/00 ZQN GC-E AMZN 7/21/01 15 Call 0 6.875 7.375 0 134 16.6875 12/20/00 ZQN GO-E AMZN 7/21/01 17.5 Call 0 5.625 6.125 0 63 16.6875 12/20/00 ZQN GD-E AMZN 7/21/01 20 Call 0 4.875 5.25 5 125 16.6875 12/20/00 ZQN GS-E AMZN 7/21/01 22.5 Call 0 4.125 4.5 0 180 16.6875 12/20/00 ZQN GE-E AMZN 7/21/01 25 Call 0 3.5 3.875 65 79 16.6875 12/20/00 AOE AZ-E AOL 1/20/01 32.5 Call 0 6.6 7 20 1972 37.25 12/20/00 AOE AO-E AOL 1/20/01 33.75 Call 0 5.6 6 0 596 37.25 12/20/00 AOE AG-E AOL 1/20/01 35 Call 0 4.7 5.1 153 5733 37.25 12/20/00 AOE AU-E AOL 1/20/01 37.5 Call 0 3.4 3.7 131 3862 37.25 12/20/00 AOE AH-E AOL 1/20/01 40 Call 0 2.5 2.7 1229 19951 37.25 12/20/00 AOE AR-E AOL 1/20/01 41.25 Call 0 2 2.3 6 1271 37.25 12/20/00 AOE AV-E AOL 1/20/01 42.5 Call 0 1.65 1.85 219 4423 37.25 12/20/00 AOE AS-E AOL 1/20/01 43.75 Call 0 1.3 1.5 44 3692 37.25 12/20/00 AOE AI-E AOL 1/20/01 45 Call 0 1.2 1.25 817 11232 37.25 12/20/00 AOE BZ-E AOL 2/17/01 32.5 Call 0 7 7.4 0 0 37.25 12/20/00 AOE BG-E AOL 2/17/01 35 Call 0 5.4 5.8 31 4 37.25 12/20/00 AOE BU-E AOL 2/17/01 37.5 Call 0 4.1 4.5 0 0 37.25 12/20/00 AOE BH-E AOL 2/17/01 40 Call 0 3.1Fragment 3.4 299 of data 48 for 37.25 12/20/00 AOE BV-E AOL 2/17/01 42.5 Call 0 2.15calibration 2.45 with 191 876 266 constraints 37.25 12/20/00 AOE BI-E AOL 2/17/01 45 Call 0 1.55 1.75 235 1385 37.25 12/20/00 AOE DZ-E AOL 4/21/01 32.5 Call 0 8.4 8.8 16 10 37.25 12/20/00 AOE DG-E AOL 4/21/01 35 Call 0 6.9 7.3 32 179 37.25 12/20/00 AOE DU-E AOL 4/21/01 37.5 Call 0 5.5 5.9 36 200 37.25 12/20/00 AOE DH-E AOL 4/21/01 40 Call 0 4.5 4.9 264 2164 37.25 12/20/00 AOE DV-E AOL 4/21/01 42.5 Call 0 3.6 3.9 209 632 37.25 12/20/00 AOE DI-E AOL 4/21/01 45 Call 0 2.9 3.1 415 3384 37.25 12/20/00 AOE DW-E AOL 4/21/01 47.5 Call 0 2.15 2.45 37 1174 37.25 12/20/00 AOO DJ-E AOL 4/21/01 50 Call 0 1.75 1.95 224 7856 37.25 12/20/00 AOE GZ-E AOL 7/21/01 32.5 Call 0 9.4 9.8 0 0 37.25 12/20/00 Near-month options (Pricing Date: Dec 2000)
MSH Basket option: model vs. market Front Month
80 78 76 74 model 72 midmarket 70 68 bid impliedvol 66 offer 64 62 60 600 610 620 630 640 650 660 670 680 690 700 strike Second-month options
Basket option: model vs. market
70 68 66 64 model 62 midmarket 60 bid 58 impliedvol offer 56 54 52 50 600 620 640 660 680 700 strike Third-month options
Basket option: model vs. market
70
65
model 60 midmarket bid 55 impliedvol offer
50
45 600 640 680 720 760 strike Six-month options
Basket option: model vs. market
60
55
model 50 midmarket bid 45 impliedvol offer
40
35 600 640 680 720 760 840 strike ---- Bid Price Broad Market Index Options (OEX) ---- Ask Price Pricing Date: Oct 9, 2001 ---- Model Fair Value Skew Graph
40.00
35.00
30.00
25.00
20.00 Volatility
15.00
10.00
5.00
0.00 500 510 520 530 540 550 555 560 565 570 580
Strike Price Hedging
• Covering the ``wings’’ in every name implies an excess Vega risk. Intrinsic Value Hedge implies long Volatility
• Use the WMC sensitivity method (regressions) to determine the best single co-terminal option to use for each component.
• Implement a Theta-Neutral hedge using the most important names with the corresponding Betas. Simulation for OEX Group: $10MM/ Targeting 1% daily stdev
SIGNALSTRENGTH > threshold 1080 trades OEX 2001 2002 2003 2001-2003 turnover time 60 days annualized return $4,239,794 $3,029,015 $1,339,717 $2,966,986 percentage 42.40 30.29 13.40 29.67 Sharpe Ratio 2.83 2.02 0.89 1.98
Constant-VaR portfolio (1% stdev per day)
Capital is allocated evenly among signals
Transaction costs in options/ stock trading included Dispersion OEX (return on $100)
100
90
80
70
60
50
$-return 40
30
20 signal 10 realized 0
0 2 0 0 - l-01 - r-04 éc jui ût d o av févr-02 a mars-03 sept-03
Results of Back-testing Simulation for QQQ group $10MM with 1% target daily stdev
signal >threshold trades 296 QQQ 2001 2002 2003 2001-2003 turnover time 76 annualized return -$1,369,462 $1,078,541 $5,339,452 $1,533,241 percentage -13.69 10.79 53.39 15.33 Sharpe Ratio -0.91 0.72 3.56 1.02 QQQ, return on $100
120
100
80
60 signal realized
$-return 40
20
0 sept-01 déc-01 avr-02 juil-02 oct-02 janv-03 mai-033 nov-03 août-0 mars-04 -20 QQQ; number of signals
20 QQQ 18
16
14
12
10
8 signals perday 6
4
2
0
1 1 1 2 3 3 -0 0 0 0 02 0 0 i l-01 t- - -03 - t-02 t-02 -02 r-03 t- a p r in û c v in-03 ût- c m jui e avr- ju o éc a ju o o s nov janv-02 fév ao d févr-03 a Simulation for QQQ+OEX $10MM with 1% daily stdev
QQQ + OEX 2001 2002 2003 2001-2003 turnover time 65 annualized return $3 054 673 $2 878 561 $2 264 803 $2 672 645 percentage 30.5 28.8 22.6 26.7 Sharpe Ratio 1.9 1.8 1.4 1.7 OEX + QQQ, return on $100
120.00
100.00
80.00
60.00 $-return 40.00
signal 20.00 realized 0.00
1 2 3 0 02 0 0 t-01 - i- r- vr ct- a in-04 v ep a o m ju fé s nov-03
Includes T.C., in options and stock trading Dispersion Capacity Estimate
USD 10 MM ~ 100 OEX contracts per day
If we assume 1000 contracts to be a liquidity limit, capacity is 100 MM just for OEX
Capacity is probably around 200 MM if we use sectors and Europe
Dispersion has higher Sharpe Ratio: It is an arb strategy based on waiting for profit opportunities