Genetics and/of basket options Wolfgang Karl Hrdle Elena Silyakova - PowerPoint PPT Presentation

Genetics and/of basket options Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics Humboldt-Universität zu Berlin http://lvb.wiwi.hu-berlin.de

Motivation 1-1 Basket derivatives Let us consider a basket of N assets with value at time t defined by B ( t ) = � N i = 1 a i S i ( t ) . Then payoffs of some basket options: ⊡ Basket call: { B ( T ) − K B } + � � + ⊡ Rainbow (best-of-N): 1 ≤ i ≤ N { S i ( T ) } − K B max ⊡ Atlas (Mountain range):   +   N − N 2 � 1 S j ( T ) S j ( 0 ) − K B  N − ( N 1 + N 2 )  j = 1 + N 1 where S i ( t ) - price of the i -th basket constituent at time t , a i - quantity of the i -th asset, K B - exercise price (strike) of a basket option, T - time of the option’s expiry, N 1 , N 2 - number of best and worst performing stocks. Genetics and/of basket options - COMPSTAT 2010

Motivation 1-2 Research questions 1. Which pricing model is suitable for multiasset options? 2. How to estimate dependence (correlation) between assets in the basket? 3. How to estimate correlations in large dimensional baskets? Genetics and/of basket options - COMPSTAT 2010

Outline 1. Motivation � 2. Basket dynamics in the Black-Scholes framework 3. Estimating correlation matrix ◮ Historical (time series) correlation ◮ Implied correlation 4. From equicorrelation to block correlation 5. Conclusion

Basket dynamics in the Black-Scholes framework 2-1 Price dynamics of basket constituents The price dynamic of the i -th stock in a basket is given by: dS i ( t ) S i ( t ) = ( r − q i ) dt + σ i dW i ( t ) (1) ρ ij dt = dW i ( t ) dW j ( t ) (2) where r - interest rate, q i - dividend yield of a stock i , σ i - constant volatility of the i -th stock, ρ ij - constant correlation between the i -th and the j -th stock, W - Brownian motion. Genetics and/of basket options - COMPSTAT 2010

Basket dynamics in the Black-Scholes framework 2-2 Dynamics of the basket’s value The dynamics of the basket’s value is then given by: � N dB ( t ) i = 1 w i S i ( t ) σ i dW i ( t ) B ( t ) = ( r − q B ) dt + = (3) � N i = 1 w i S i ( t ) = ( r − q B ) dt + dZ ( t ) where q B is the dividend yield of the basket and the relative weight w i of the i -th constituent varies over time and is given by: a i S i ( t ) w i = (4) � N l = 1 a l S l ( t ) Genetics and/of basket options - COMPSTAT 2010

Basket dynamics in the Black-Scholes framework 2-3 Dynamics of correlated basket constituents Let   ρ 11 · · · ρ 1 N  . .  ... . . Σ =   . . ρ N 1 · · · ρ NN the correlation matrix of a basket. By Cholesky decomposition Σ = MM ⊤ we obtain M = ( m i , j ) 1 ≤ i ≤ N , 1 ≤ j ≤ N , a lower triangular matrix, a ”square root” of Σ . The process for every individual asset S i is then defined by: N � dS i ( t ) S i ( t ) = ( r − q ) dt + σ i m i , l dW l ( t ) (5) l = 1 Genetics and/of basket options - COMPSTAT 2010

Basket dynamics in the Black-Scholes framework 2-4 Finally applying Itô’s lemma we obtain the closed-form expression for simulation of the i -th stock process on a time interval ∆ t = [ t 1 , t 2 ] : � � N √ � ( r − d − σ 2 i S i ( t 2 ) = S i ( t 1 ) exp 2 )∆ t + σ i m i , l ∆ tg l (6) l = 1 where g l ∼ N ( 0 , 1 ) , i.i.d. Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: historical (time series) correlation 3-1 Historical correlation X i ( t ) = log S i ( t ) − log S i ( t − 1 ) , log returns: � T k = 0 λ k { X i ( t − k ) − ¯ X i ( t ) }{ X j ( t − k ) − ¯ X j ( t ) } ρ ij = �� T X i ( t ) } 2 � T k = 0 λ k { X i ( t − k ) − ¯ k = 0 λ k { X j ( t − k ) − ¯ X j ( t ) } 2 to obtain the historical correlation matrix   1 ρ 12 · · · ρ 1 N   ρ 12 1 · · · ρ 2 N    . . .  ... . . .   . . . ρ 1 N ρ N 2 · · · 1 Here ¯ X i ( t ) the arithmetic mean of the i -th log return calculated at time t , λ - decay parameter (RiskMetrics: λ = 0 . 94). Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: historical (time series) correlation 3-2 Equicorrelation matrix Basket variance N N N � � � σ 2 w 2 i σ 2 Basket = i + 2 w i w j σ i σ j ρ ij (7) i = 1 i = 1 j = i + 1     1 ρ 12 · · · ρ 1 N 1 ρ · · · ρ     ρ 21 1 · · · ρ 2 N ρ 1 · · · ρ     replace  with  ,  . . .   . . .  ... ... . . . . . .   . . . . . . ρ N 1 ρ N 2 · · · 1 ρ ρ · · · 1 then Basket − � N σ 2 i = 1 w 2 i σ 2 i ρ = (8) 2 � N � N j = i + 1 w i w j σ i σ j i = 1 is the average basket correlation. Nice property: for −{ 1 / ( N − 1 ) } < ρ < 1 - positive definite (see Mardia et al, 1979). Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-1 Implied correlation Using (8) map the implied volatility surfaces of a basket σ Basket ( κ, τ ) and N constituents � σ i ( κ, τ ) to � ρ ( τ, κ ) the average � implied correlation surface of a basket : Basket ( κ, τ ) − � N σ 2 i = 1 w 2 σ 2 � i � i ( κ, τ ) ρ ( κ, τ ) = (9) � 2 � N � N j = i + 1 w i w j � σ i ( κ, τ ) � σ j ( κ, τ ) i = 1 Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-2 Dynamic modeling of correlation surfaces Every t we observe ( X t , j , Y t , j ) , 1 ≤ j ≤ J t , 1 ≤ t ≤ T where ⊡ Y t , j - implied correlation ⊡ X t , j - two-dimensional vector of κ and τ ⊡ T - number of observed time periods (days) ⊡ J t - number of observations at day t Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-3 Dynamic modeling of correlation surfaces Including explanatory variables X t , j influencing the factor loadings m l , j rewrite (10) L � Z t , l m l ( X t , j ) + ε t , j = Z ⊤ Y t , j = t m ( X t , j ) + ε t , j (10) l = 1 where ⊡ Z t = ( Z t 1 , . . . , Z tL ) ⊤ - unobservable L -dimensional process ⊡ m - L -tuple ( m 1 , ..., m L ) of unknown real-valued functions ⊡ X t , j , . . . , X T , J T and ε t , j , . . . , ε T , J T are independent ⊡ ε t , j are i . i . d . with zero mean and finite second moment ⊡ In such setting the modelling of Y t can be simplified to modelling of Z t = ( Z t , 1 , ..., Z t , L ) , which is more feasible for L << J . Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-4 Dynamic modeling of correlation surfaces L K � � a l , k ψ k ( X t , j ) + ε t , j = Z ⊤ Y t , j = Z t , l t A Ψ t + ε t (11) l = 1 k = 1 where ⊡ A - L × K coefficient matrix ⊡ Ψ t = { ψ 1 ( X t ) , ..., ψ R ( X t ) } ⊤ - space basis, in Park et al. (2009) a tensor product of one dimensional B-spline basis. Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-5 Choice of space basis Estimate basis functions in a FPCA framework, motivated by Hall et. al (2006): Find eigenfunctions corresponding to the K largest eigenvalues of the smoothed operator ψ ( u , v ) = � � φ ( u , v ) − � µ ( u ) � µ ( v ) Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-6 Choice of space basis 1. estimate � µ ( u )( µ ( v )) : � X tj − u � T J 2 � � � b c ( u c − X c tj ) } 2 K { Y tj − a − h µ t = 1 j = 1 c = 1 2. estimate � φ ( u , v ) : T 2 2 � � � � 1 ( u c − X c 2 ( v c − X c b c b c tk ) } 2 { Y tj Y tk − a 0 − tj ) − t = 1 1 ≤ j � = k ≤ J t c = 1 c = 1 � X tj − u � � X tj − v � × K K h φ h φ 3. compute � ψ ( u , v ) = � φ ( u , v ) − � µ ( u ) � µ ( v ) and take K eigenfunctions corresponding to the largest eigenvalues Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-7 Basis functions 1st eigenfunction 2nd eigenfunction 0.4 0.2 0.3 0.1 0.2 0 −0.1 0.1 −0.2 0 −0.3 −0.1 −0.4 2 2 1.5 2 1.5 2 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 time to maturity time to maturity 0 0 moneyness 0 0 moneyness Figure 1: Eigenfunctions as basis functions estimated on 10x10 grid Genetics and/of basket options - COMPSTAT 2010

Estimating correlation matrix: implied correlation 4-8 Estimated time series of factors � Z t 1 , � Z t 2 Z 80 60 40 20 0 −20 −40 −60 0 50 100 150 200 250 Genetics and/of basket options - COMPSTAT 2010

From equicorrelation to block correlation 5-1 From equicorrelation to block correlation Group assets in the basket into k blocks, then     1 ρ 1 · · · ρ 1     ρ 1 1 · · · ρ 1     · · · ρ k + 1   . . .   ... . . .     . . .     ρ 1 ρ 1 · · · 1    . .  ... . .   . .       1 ρ k · · · ρ k       ρ k 1 · · · ρ k     ρ k + 1 · · ·     . . . ... . . .     . . . ρ k ρ k · · · 1 Genetics and/of basket options - COMPSTAT 2010