lorenzo bergomi - École polytechnique€¦ · 2 are periodic functions with period d = 1 day: r s...
TRANSCRIPT
![Page 1: Lorenzo Bergomi - École Polytechnique€¦ · 2 are periodic functions with period D = 1 day: r S = 1 D R t+D 1d t rs1s 2ds q 1 D R t+D t s 2 1 ds q 1 D R t+D d t d s 2 2 ds r A](https://reader035.vdocument.in/reader035/viewer/2022071215/6045c7dfe93745630764d1e2/html5/thumbnails/1.jpg)
MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Correlations in Asynchronous Markets
Lorenzo Bergomi
Global Markets Quantitative Research
Paris, January 2011
Lorenzo Bergomi Correlations in Asynchronous Markets
![Page 2: Lorenzo Bergomi - École Polytechnique€¦ · 2 are periodic functions with period D = 1 day: r S = 1 D R t+D 1d t rs1s 2ds q 1 D R t+D t s 2 1 ds q 1 D R t+D d t d s 2 2 ds r A](https://reader035.vdocument.in/reader035/viewer/2022071215/6045c7dfe93745630764d1e2/html5/thumbnails/2.jpg)
MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Outline
Motivation
Estimating correlations and volatilities in asynchronous markets
Historical correlations: Stoxx50 —S&P500 —Nikkei
Comparison with other heuristic estimators —options and correlation swaps
Correlations larger than 1
Conclusion
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Motivation
Equity derivatives generally involve baskets of stocks/indices traded indifferent geographical areas
Operating hours of: Asian and European exchanges, Asian and Americanexchanges, usually have no overlap
Standard methodology on equity derivatives desks:
Use standard multi-asset model based on assumption of continuously tradedsecurities
Compute/trade deltas at the close of each market, using stale values forsecurities not trading at that time
Likewise, valuation is done using stale values for securities whose marketsare closed
B How should we estimate volatility and correlation parameters ?
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Correlation estimation in asynchronous markets
Consider following situation:
Stoxx50
Nikkei
11 −ir
ir1
12 −ir i
r2
1−it
it
1+it
δ−−1it δ−
+1itδ−
it
12 +ir
Valuation of the option is done at the close of the Stoxx50Deltas are computed and traded on the market close of each security
Daily P&L:
P&L = − [f (ti+1, S1,i+1, S2,i+1) − f (ti , S1,i , S2,i )]
+dfdS1
(ti , S1,i , S2,i ) (S1,i+1 − S1,i )
+dfdS2
(ti − δ, S1,i−1, S2,i ) (S2,i+1 − S2,i )
Note that arguments of dfdS2
are different
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Rewrite delta on S2 so that arguments are same as f and dfdS1
dfdS2
(ti − δ, S1,i−1, S2,i ) =dfdS2
(ti , S1,i , S2,i ) −d 2f
dS1dS2(ti , S1,i , S2,i ) (S1,i − S1,i−1)
Other correction terms contribute at higher order in ∆
P&L now reads:
P&L = − [f (ti+1, S1,i+1, S2,i+1) − f (ti , S1,i , S2,i )]
+dfdS1
(S1,i+1 − S1,i ) +[dfdS2− d 2fdS1dS2
(S1,i − S1,i−1)](S2,i+1 − S2,i )
Expanding at 2nd order in δS1, δS2:
P&L = − dfdt
∆ −[12d 2fdS 21
δS 21+ +12d 2fdS 21
δS 22+ +d 2f
dS1dS2δS1+δS2+
]− d 2fdS1dS2
δS1−δS2+
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Assume f is given by a Black-Scholes equation:
dfdt
+σ212S 21d 2fdS 21
+σ222S 22d 2fdS 22
+ ρσ1σ2S1S2d 2f
dS1dS2= 0
P&L now reads:
P&L = − 12S 21d 2fdS 21
[(δS+1S1
)2− σ21∆
]− 12S 22d 2fdS 22
[(δS+2S2
)2− σ22∆
]
−S1S2d 2f
dS1dS2
[(δS−1S1
+δS+1S1
)δS+2S2
− ρσ1σ2∆]
B Prescription for estimating volatilities & correlations so that P&L vanisheson average:
σ?21 =1∆
⟨(δS+1S1
)2⟩σ?21 =
1∆
⟨(δS+2S2
)2⟩ρ?σ?1σ?2 =
1∆
⟨(δS−1S1
+δS+1S1
)δS+2S2
⟩
B Volatility estimators are the usual ones, involving daily returnsB The correlation estimator involves daily returns as well
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Define ri =δS+iSi. At lowest order in ∆, δS−i
Si' δS−i
Si−1
σ?21 =1∆⟨r 21i⟩
σ?22 =1∆⟨r 22i⟩
ρ? =〈(r1i−1 + r1i ) r2i 〉√⟨
r 21i⟩ ⟨r 22i⟩
Stoxx50
Nikkei
11 −ir
ir1
12 −ir i
r2
1−it
it
1+it
δ−−1it δ−
+1itδ−
it
12 +ir
Had we chosen the close of the Nikkei for valuing the option: symmetricalestimator:
σ?21 =1∆⟨r 21i⟩
σ?22 =1∆⟨r 22i⟩
ρ? =〈r1i (r2i + r2i+1)〉√⟨
r 21i⟩ ⟨r 22i⟩
If returns are time-homogeneous 〈r1i−1r2i 〉 = 〈r1i r2i+1〉In practice 1N ∑N1 (r1i−1 + r1i ) r2i − r1i (r2i + r2i+1) = 1
N (r10r21 − r1N r2N+1)
B Difference between two estimators of ρ?: finite size effect of order 1N
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
In conclusion, in asynchronous markets: 2 correlations ρS , ρA :
ρS =〈r1i r2i 〉√⟨r 21i⟩ ⟨r 22i⟩ ρA =
〈r1i r2i+1〉√⟨r 21i⟩ ⟨r 22i⟩ S
A
S
A
and derivatives should be priced with ρ?:
ρ? = ρS + ρA
B Does ρ? depend on the particular delta strategy used in derivation ?
B Is ρ? in [−1, 1] ?B How does ρ? compare to standard correlations estimators evaluated with3-day, 5-day, n-day returns ?
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
What if had computed deltas differently — for example "predicting" thevalue of the stock not trading at the time of computation ?
B Option delta-hedged one way minus option delta-hedged the other way.Final P&L is:
∑(∆at − ∆bt )(St+∆ − St )B Price of pure delta strategy is zero: correlation estimator is independenton delta strategy used in derivation
Imagine processes are continuous yet observations are asynchronous:assume that ρσ1σ2, σ21, σ22 are periodic functions with period ∆ = 1 day:
ρS =1∆
∫ t+∆−δt ρσ1σ2 ds√
1∆
∫ t+∆t σ21ds
√1∆
∫ t+∆−δt−δ σ22ds
ρA =1∆
∫ t+∆t+∆−δ ρσ1σ2 ds√
1∆
∫ t+∆t σ21ds
√1∆
∫ t+∆−δt−δ σ22ds
ρ? = ρS + ρAS
A
S
A
B Recovers value of "synchronous correlation": no bias
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Historical correlations
ρS (blue), ρA (pink), ρ? = ρS + ρA (green) — 6-month EWMA
Stoxx50 / SP500
0%
20%
40%
60%
80%
100%
4-Dec-99 3-Dec-01 3-Dec-03 2-Dec-05 2-Dec-07 1-Dec-09
Nikkei / Stoxx50
0%
20%
40%
60%
80%
100%
4-Dec-99 3-Dec-01 3-Dec-03 2-Dec-05 2-Dec-07 1-Dec-09
Nikkei / SP500
0%
20%
40%
60%
80%
100%
4-Dec-99 3-Dec-01 3-Dec-03 2-Dec-05 2-Dec-07 1-Dec-09
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Is the signal for ρA in the Stoxx50/S&P500 case real ?
Switch time series of Stoxx50 and S&P500 and redo computation:
-20%
0%
20%
40%
60%
80%
100%
4-Dec-99 3-Dec-01 3-Dec-03 2-Dec-05 2-Dec-07 1-Dec-09
Stoxx50 / SP500 - normal
-20%
0%
20%
40%
60%
80%
100%
4-Dec-99 3-Dec-01 3-Dec-03 2-Dec-05 2-Dec-07 1-Dec-09
Stoxx50 / SP500 - reversed
B In the reversed situation, ρA hovers around 0.
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
ρS , ρA seem to move antithetically
Imagine σ1(s) = σ1λ(s), σ2(s) = σ2λ(s), ρ constant, with λ(s) such that1∆
∫ ∆0 λ2(s)ds = 1. Then:
ρS = ρ1∆
∫ ∆−δ
0λ2 (s) ds
ρA = ρ1∆
∫ ∆
∆−δλ2 (s) ds
and ρ? is given by:ρ? = ρS + ρA = ρ
By changing λ(s) we can change ρS , ρA , while ρ? stays fixed.
B The relative sizes of ρS , ρA are given by the intra-day distribution of therealized covariance.
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Comparison with heuristic estimators
Trading desks have long ago realized that merely using ρS is inadequateStandard fix: compute standard correlation using 3-day, 5-day, you-name-it,rather than daily returnsHow do these estimators differ from ρ? ?
Connected issue: how do we price an n-day correlation swap ?
S
A
S
A
B An n−day correlation swap should be priced with ρn given by:
ρn = ρS +n− 1n
ρA
For n = 3, ρ3 = ρS +23 ρA
If no serial correlation in historical sample, standard correlation estimatorapplied to n-day returns yields ρn
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Historical n-day correlations
n-day correlations evaluated on 2004-2009 with:
n-day returns (dark blue)using ρS +
n−1n ρA (light blue)
compared to ρ? (purple line)
Stoxx50 / SP500
0%
20%
40%
60%
80%
100%
1 3 5 7 9
Nikkei / Stoxx50
0%
20%
40%
60%
80%
100%
1 3 5 7 9
Nikkei / SP500
0%
20%
40%
60%
80%
100%
1 3 5 7 9
Common estimators ρ3, ρ5 underestimate ρ?
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
The S&P500 and Stoxx50 as synchronous securities
European and American exchanges have some overlap. We can either:delta-hedge asynchronously the S&P500 at 4pm New York time and theStoxx50 at 5:30pm Paris timedelta-hedge simultaneously both futures at — say — 4pm Paris time
1st case: use ρ?, 2nd case: use standard correlation for synchronoussecurities — are they different ?
ρ? (light blue), standard sync. correlation (dark blue) — 3-month EWMA
0%
20%
40%
60%
80%
100%
120%
2-Dec-05 2-Dec-06 2-Dec-07 1-Dec-08 1-Dec-09
Matches well, but not identical: difference stems from residual realizedserial correlations.
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Correlations larger than 1
Example of RBS/Citigroup correlations: ρS (blue), ρA (pink), ρ? (green) —3-month EWMA
0%
20%
40%
60%
80%
100%
120%
1-Dec-04 1-Dec-05 1-Dec-06 1-Dec-07 30-Nov-08 30-Nov-09
Are instances when ρ? > 1 an artifact ? Do they have financialsignificance ?
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Consider a situation when no serial correlation is present. The globalcorrelation matrix is positive, by construction. How large can ρS + ρA be ?
0 00
A
S
S
S
S
A
A
0 0 0
Compute eigenvalues of full correlation matrix:
assume both ladder uprights consist of N segments, with periodic boundaryconditionsassume eigenvalues have components e ikθ on higher upright, αe ikθ on loweruprightexpress that λ is an eigenvalue:
αρS + 1+ αe iθ = λ
ρS + α+ e−iθρA = λα
yields:
λ = 1±√(ρS + ρA cos θ)2 + ρ2A sin
2 θ
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Periodic boundary conditions impose θ = 2nπN , where n = 0 . . .N − 1
λ (θ) extremal for θ = 0,π. For these values λ = 1 ± |ρS ± ρA |λ > 0 implies:
−1 ≤ ρS + ρA ≤ 1
−1 ≤ ρS − ρA ≤ 1
1
A
1
S
B If no serial correlations ρ? ∈ [−1, 1]
B Instances when ρ? > 1: evidence of serial correlations
B Impact of ρ?> 1 on trading desk: price with the right realizedvolatilities, 100% correlation → lose money !!
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Example with basket option
Sell 6-month basket option on basket of Japanese stock & French stock.
Payoff is(√
ST1 ST2
S 01 S02− 1)+
Basket is lognormal with volatility given by σ =√
σ21 + σ22 + 2ρσ1σ2Use following "historical" data:
60
80
100
120
0 50 100 150 200
Paris stock
Tokyo stock
Realized vols are 21.8% for S1, 23.6% for S2. Realized correlations areρS = 63.3%, ρA = 57.6%: ρ? = 121%.
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Backtest delta-hedging of option with:implied vols = realized volsdifferent implied correlations
Initial option price and final P&L:
-4%
-2%
0%
2%
4%
6%
8%
60% 80% 100% 120% 140%
Correlation
P&
L / P
ric
e
Final P&L
Initial option price
Final P&L vanishes when one prices and risk-manages option with animplied correlation ρ ≈ 125%.
Lorenzo Bergomi Correlations in Asynchronous Markets
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MotivationCorrelation estimation in asynchronous markets
Historical correlationsCorrelations larger than 1
Conclusion
It is possible to price and risk-manage options on asynchronous securitiesusing the standard synchronous framework, provided special correlationestimator is used.
Correlation estimator quantifies correlation that is materialized ascross-Gamma P&L.
Correlation swaps and options have to be priced with different correlations.
Serial correlations may push realized value of ρ? above 1: a shortcorrelation position will lose money, even though one uses the right volsand 100% correlation.
Lorenzo Bergomi Correlations in Asynchronous Markets