Global Markets Quantitative Research lorenzo.bergomi@sgcib.com Paris, January 011
Outline Motivation Motivation Estimating correlations and volatilities in asynchronous markets : Stoxx50 S&P500 Nikkei Comparison with other heuristic estimators options and correlation swaps Conclusion
Motivation Equity derivatives generally involve baskets of stocks/indices traded in different geographical areas Operating hours of: Asian and European exchanges, Asian and American exchanges, usually have no overlap Standard methodology on equity derivatives desks: Use standard multi-asset model based on assumption of continuously traded securities Compute/trade deltas at the close of each market, using stale values for securities not trading at that time Likewise, valuation is done using stale values for securities whose markets are closed How should we estimate volatility and correlation parameters?
Consider following situation: Stoxx50 Nikkei ti 1 δ ti 1 ri 1 r1i 1 δ t i r i r 1 i ti t i+ 1 t i+1 δ r i+1 Valuation of the option is done at the close of the Stoxx50 Deltas are computed and traded on the market close of each security Daily P&L: P&L = [f (t i+1, S 1,i+1, S,i+1 ) f (t i, S 1,i, S,i )] + df (t ds i, S 1,i, S,i ) (S 1,i+1 S 1,i ) 1 Note that arguments of df ds + df ds (t i δ, S 1,i 1, S,i ) (S,i+1 S,i ) are different
Rewrite delta on S so that arguments are same as f and df ds 1 df (t i δ, S 1,i 1, S,i ) = df (t i, S 1,i, S,i ) ds ds d f (t i, S 1,i, S,i ) (S 1,i S 1,i 1 ) ds 1 ds Other correction terms contribute at higher order in P&L now reads: P&L = [f (t i+1, S 1,i+1, S,i+1 ) f (t i, S 1,i, S,i )] + df [ df (S 1,i+1 S 1,i ) + d ] f (S 1,i S 1,i 1 ) (S,i+1 S,i ) ds 1 ds ds 1 ds Expanding at nd order in δs 1, δs : P&L = df dt [ 1 d f ds1 δs1+ + 1 d f ds1 δs+ + d ] f δs 1+ δs + ds 1 ds d f δs 1 δs + ds 1 ds
Assume f is given by a Black-Scholes equation: df dt + σ 1 S 1 d f ds1 + σ S d f ds + ρσ 1 σ S 1 S d f ds 1 ds = 0 P&L now reads: [ ( P&L = 1 S 1 d f δs + 1 ds1 S 1 d f S 1 S ds 1 ds ) ] σ 1 [( δs 1 S 1 + δs + 1 S 1 [ ( 1 S d f δs + ds S ) δs + ] ρσ 1 σ S ) ] σ Prescription for estimating volatilities & correlations so that P&L vanishes on average: σ 1 = 1 ( δs + 1 S 1 ) σ 1 = 1 ( δs + S ) ρ σ 1σ = 1 ( δs 1 S 1 + δs + 1 S 1 ) δs + S Volatility estimators are the usual ones, involving daily returns The correlation estimator involves daily returns as well
Define r i = δs + i S i σ1 = 1 r 1i. At lowest order in, δs i S i σ = 1 r i δs i S i 1 ρ = (r 1i 1 + r 1i ) r i r 1i r i Stoxx50 ti 1 r1i 1 r 1 i ti t i+ 1 Nikkei ti 1 δ ri 1 δ t i r i t i+1 δ Had we chosen the close of the Nikkei for valuing the option: symmetrical estimator: σ1 = 1 r 1i σ = 1 r i ρ = r 1i (r i + r i+1 ) r 1i r i r i+1 If returns are time-homogeneous r 1i 1 r i = r 1i r i+1 In practice 1 N N 1 (r 1i 1 + r 1i ) r i r 1i (r i + r i+1 ) = 1 N (r 10r 1 r 1N r N +1 ) Difference between two estimators of ρ : finite size effect of order 1 N
In conclusion, in asynchronous markets: correlations ρ S, ρ A : ρ S = r 1i r i r ρ A = r 1i r i+1 1i r r i 1i r i S A and derivatives should be priced with ρ : ρ = ρ S + ρ A Does ρ depend on the particular delta strategy used in derivation? Is ρ in [ 1, 1]? How does ρ compare to standard correlations estimators evaluated with 3-day, 5-day, n-day returns?
What if had computed deltas differently for example "predicting" the value of the stock not trading at the time of computation? Option delta-hedged one way minus option delta-hedged the other way. Final P&L is: ( a t b t )(S t+ S t ) Price of pure delta strategy is zero: correlation estimator is independent on delta strategy used in derivation Imagine processes are continuous yet observations are asynchronous: assume that ρσ 1 σ, σ 1, σ are periodic functions with period = 1 day: ρ S = 1 1 t+ δ t ρσ 1 σ ds σ 1 ds 1 t+ t t+ δ t δ σ ds ρ A = 1 1 t+ t+ δ ρσ 1σ ds σ 1 ds 1 t+ t t+ δ t δ σ ds ρ = ρ S + ρ A S A Recovers value of "synchronous correlation": no bias
ρ S (blue), ρ A (pink), ρ = ρ S + ρ A (green) 6-month EWMA 10 Stoxx50 / SP500 10 Nikkei / Stoxx50 8 8 6 6 4 4 4-Dec-99 3-Dec-01 3-Dec-03 -Dec-05 -Dec-07 1-Dec-09 4-Dec-99 3-Dec-01 3-Dec-03 -Dec-05 -Dec-07 1-Dec-09 10 Nikkei / SP500 8 6 4 4-Dec-99 3-Dec-01 3-Dec-03 -Dec-05 -Dec-07 1-Dec-09
Is the signal for ρ A in the Stoxx50/S&P500 case real? Switch time series of Stoxx50 and S&P500 and redo computation: 10 Stoxx50 / SP500 - normal 10 Stoxx50 / SP500 - reversed 8 8 6 6 4 4 4-Dec-99 3-Dec-01 3-Dec-03 -Dec-05 -Dec-07 1-Dec-09 4-Dec-99 3-Dec-01 3-Dec-03 -Dec-05 -Dec-07 1-Dec-09 - - In the reversed situation, ρ A hovers around 0.
ρ S, ρ A seem to move antithetically Imagine σ 1 (s) = σ 1 λ(s), σ (s) = σ λ(s), ρ constant, with λ(s) such that 1 0 λ (s)ds = 1. Then: and ρ is given by: ρ S = ρ 1 δ λ (s) ds 0 ρ A = ρ 1 λ (s) ds δ ρ = ρ S + ρ A = ρ By changing λ(s) we can change ρ S, ρ A, while ρ stays fixed. The relative sizes of ρ S, ρ A are given by the intra-day distribution of the realized covariance.
Comparison with heuristic estimators Trading desks have long ago realized that merely using ρ S is inadequate Standard fix: compute standard correlation using 3-day, 5-day, you-name-it, rather than daily returns How do these estimators differ from ρ? Connected issue: how do we price an n-day correlation swap? S A An n day correlation swap should be priced with ρ n given by: ρ n = ρ S + n 1 ρ n A For n = 3, ρ 3 = ρ S + 3 ρ A If no serial correlation in historical sample, standard correlation estimator applied to n-day returns yields ρ n
Historical n-day correlations n-day correlations evaluated on 004-009 with: n-day returns (dark blue) using ρ S + n 1 n ρ A (light blue) compared to ρ (purple line) 10 Stoxx50 / SP500 10 Nikkei / Stoxx50 10 Nikkei / SP500 8 8 8 6 6 6 4 4 4 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 Common estimators ρ 3, ρ 5 underestimate ρ
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 the Stoxx50 at 5:30pm Paris time delta-hedge simultaneously both futures at say 4pm Paris time 1st case: use ρ, nd case: use standard correlation for synchronous securities are they different? ρ (light blue), standard sync. correlation (dark blue) 3-month EWMA 1 10 8 6 4 -Dec-05 -Dec-06 -Dec-07 1-Dec-08 1-Dec-09 Matches well, but not identical: difference stems from residual realized serial correlations.
Example of RBS/Citigroup correlations: ρ S (blue), ρ A (pink), ρ (green) 3-month EWMA 1 10 8 6 4 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 financial significance?
Consider a situation when no serial correlation is present. The global correlation matrix is positive, by construction. How large can ρ S + ρ A be? 0 0 0 S A S A S A S 0 0 0 Compute eigenvalues of full correlation matrix: assume both ladder uprights consist of N segments, with periodic boundary conditions assume eigenvalues have components e ikθ on higher upright, αe ikθ on lower upright express that λ is an eigenvalue: yields: αρ S + 1 + αe i θ = λ ρ S + α + e i θ ρ A = λα λ = 1 ± (ρ S + ρ A cos θ) + ρ A sin θ
Periodic boundary conditions impose θ = nπ N, where n = 0... N 1 λ (θ) extremal for θ = 0, π. For these values λ = 1 ± ρ S ± ρ A λ > 0 implies: 1 A 1 ρ S + ρ A 1 1 ρ S ρ A 1 1 S If no serial correlations ρ [ 1, 1] Instances when ρ > 1: evidence of serial correlations Impact of ρ > 1 on trading desk: price with the right realized volatilities, 10 correlation lose money!!
Example with basket option Sell 6-month basket option on basket of Japanese stock & French stock. ( ) + S1 Payoff is T S T S1 0S 1 0 Basket is lognormal with volatility given by σ = σ 1 + σ + ρσ 1σ Use following "historical" data: 10 100 80 Paris stock Tokyo stock 60 0 50 100 150 00 Realized vols are 1.8% for S 1, 3.6% for S. Realized correlations are ρ S = 63.3%, ρ A = 57.6%: ρ = 11%.
Backtest delta-hedging of option with: implied vols = realized vols different implied correlations Initial option price and final P&L: P&L / Price 8% 6% 4% Final P&L Initial option price % 6 8 10 1 14 -% -4% Correlation Final P&L vanishes when one prices and risk-manages option with an implied correlation ρ 15%.
Conclusion Motivation It is possible to price and risk-manage options on asynchronous securities using the standard synchronous framework, provided special correlation estimator is used. Correlation estimator quantifies correlation that is materialized as cross-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 short correlation position will lose money, even though one uses the right vols and 10 correlation.