Genetics and/of basket options

Transcription

1 Genetics and/of basket options Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics Humboldt-Universität zu Berlin

2 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 is i (t). Then payoffs of some basket options: Basket call: {B(T ) K B } [ + ] + Rainbow (best-of-n): max {S i(t )} K B 1 i N Atlas (Mountain range): + N N 1 2 S j (T ) N (N 1 + N 2 ) S j (0) K B 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.

3 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?

4 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

5 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.

6 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: db(t) N B(t) = (r q i=1 B)dt + w is i (t)σ i dw i (t) N i=1 w = (3) is 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: w i = a i S i (t) N l=1 a ls l (t) (4)

7 Basket dynamics in the Black-Scholes framework 2-3 Dynamics of correlated basket constituents Let ρ 11 ρ 1N Σ =..... ρ N1 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: ds i (t) S i (t) = (r q)dt + σ i N m i,l dw l (t) (5) l=1

8 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 ]: S i (t 2 ) = S i (t 1 ) exp { (r d σ2 i 2 ) t + σ i } N m i,l tgl l=1 (6) where g l N(0, 1), i.i.d.

9 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 ρ ij = λk {X i (t k) X i (t)}{x j (t k) X j (t)} T k=0 λk {X i (t k) X i (t)} 2 T k=0 λk {X j (t k) X j (t)} 2 to obtain the historical correlation matrix 1 ρ 12 ρ 1N ρ 12 1 ρ 2N ρ 1N ρ N2 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).

10 Estimating correlation matrix: historical (time series) correlation 3-2 Equicorrelation matrix Basket variance σ 2 Basket = N i=1 w 2 i σ 2 i + 2 N N i=1 j=i+1 w i w j σ i σ j ρ ij (7) 1 ρ 12 ρ 1N 1 ρ ρ ρ 21 1 ρ 2N replace with ρ 1 ρ......, ρ N1 ρ N2 1 ρ ρ 1 then ρ = σ2 Basket N i=1 w i 2σ2 i 2 N N i=1 j=i+1 w (8) iw j σ i σ j is the average basket correlation. Nice property: for {1/(N 1)} < ρ < 1 - positive definite (see Mardia et al, 1979).

11 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: ρ(κ, τ) = σ2 Basket (κ, τ) N i=1 w i 2 σ i 2 (κ, τ) 2 N N i=1 j=i+1 w iw j σ i (κ, τ) σ j (κ, τ) (9)

12 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

13 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) Y t,j = L Z t,l m l (X t,j ) + ε t,j = Zt m(x t,j ) + ε t,j (10) l=1 where Z t = (Z t1,..., Z tl ) - unobservable L-dimensional process m - L-tuple (m 1,..., m L ) of unknown real-valued functions X t,j,..., X T,JT and ε t,j,..., ε T,JT 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.

14 Estimating correlation matrix: implied correlation 4-4 Dynamic modeling of correlation surfaces Y t,j = L K Z t,l l=1 k=1 a l,k ψ k (X t,j ) + ε t,j = Z t AΨ t + ε t (11) 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.

15 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)

16 Estimating correlation matrix: implied correlation 4-6 Choice of space basis 1. estimate µ(u)(µ(v)): T J {Y tj a t=1 j=1 2 ( ) b c (u c Xtj)} c 2 Xtj u K c=1 2. estimate φ(u, v): T 2 2 {Y tj Y tk a 0 b1(u c c Xtj) c b2(v c c Xtk c )}2 t=1 1 j k J t c=1 c=1 ( ) ( ) Xtj u Xtj v K K h φ 3. compute ψ(u, v) = φ(u, v) µ(u) µ(v) and take K eigenfunctions corresponding to the largest eigenvalues h φ h µ

17 Estimating correlation matrix: implied correlation 4-7 Basis functions 1st eigenfunction 2nd eigenfunction time to maturity moneyness time to maturity moneyness Figure 1: Eigenfunctions as basis functions estimated on 10x10 grid

18 Estimating correlation matrix: implied correlation 4-8 Estimated time series of factors Ẑt1, Ẑt2 Z

19 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 ρ k+1 ρ 1 ρ ρ k ρ k ρ k 1 ρ k ρ k ρ k ρ k 1

20 From equicorrelation to block correlation 5-2 Correlation matrix for 2 groups of assets (3 blocks) 1 ρ 1 ρ 1 ρ 1 1 ρ ρ 1 ρ 1 1 ρ 3 ρ 3 1 ρ 2 ρ 2 ρ 2 1 ρ ρ 2 ρ 2 1

21 From equicorrelation to block correlation 5-3 Block implied correlation, 3 blocks +2 σ 2 Basket (K, τ) = N M M i=1 j=i+1 N M i=1 M N M j=i+1 N i=1 j=m+1 i=1 w 2 i σ 2 i (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 1 (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 2 (K, τ)+ w i w j σ i (K, τ)σ j (K, τ)ρ 3 (K, τ) where M - number of assets in the 1-st block.

22 From equicorrelation to block correlation 5-4 Challenges Moving to high-dimensional portfolios (N ) with block structure of covariance matrix: need well-conditioned estimate of covariance matrix (Ledoit and Wolf (2003), Bickel and Levina (2008)) need to define the grouping procedure and way of finding the optimal block size (Hautsch, Kyj and Oomen (2009)) Improving correlation surface modeling: need to expand the time effect in a series model Z t as a sum of basis functions ( Song Härdle and Ritov (2010))

23 Genetics and/of basket options Wolfgang Karl Härdle Elena Silyakova Ladislaus von Bortkiewicz Chair of Statistics Humboldt-Universität zu Berlin

24 Bibliography 6-1 Bibliography Alexander, C. Market Models, A Guide to Financial Data Analysis John Wiley & Sons (2001) Bai, Z.D. Methodologies in Spectral Analysis of Large Dimensional Random Matrices, A Review Statistica Sinica, (1999) Efron, B. Bootstrap Methods: Another Look at the Jackknife Annals of Statistics, (1979)

25 Bibliography 6-2 Bibliography Fengler, M. R., Pilz K.F. and P. Schwendner Basket Volatility and Correlation Volatility As An Asset Class, Risk Publications (2007) Fengler, M. R. and P. Schwendner Quoting multiasset equity options in the presence of errors from estimating correlations Journal of Derivatives, (2004) Hall, P., Müller, H. G. and Wang J. Properties of principal component methods for functional and longitudinal data analysis Ann. Statist., 34(3): , (2006)

26 Bibliography 6-3 Bibliography Laloux, L., et al. Random Matrix Theory and Financial Correlations International Journal of Theoretical and Applied Finance, (2000) Ledoit, O., and M. Wolf Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection Journal of Empirical Finance 105, (2003) Mardia, K. V., Kent, J. T. and Bibby, J. M. Multivariate Analysis Academic Press,Duluth, London, (1979) Plerou, V., et al. Genetics Random and/ofmatrix basket options Approach - COMPSTAT to Cross 2010 Correlations in Financial Data