Estimating Pricing Kernel via Series Methods Maria Grith Wolfgang Karl Härdle Melanie Schienle Ladislaus von Bortkiewicz Chair of Statistics Chair of Econometrics C.A.S.E. Center for Applied Statistics and Economics Humboldt Universität zu Berlin http://lvb.wiwi.hu-berlin.de
().5 -.5 - - -.5.5 g Motivation - The Financial Market In an arbitrage-free market, the European call price is given by C t (X ) = e rt,τ τ (S T X ) + q t (S T ) ds T. () S t the underlying asset price at time t X the strike price τ the time to maturity T = t + τ the epiration date r t,τ deterministic risk free interest rate q t (S T ) RND of S T conditional on information at t
().5 -.5 - - -.5.5 g Motivation -2 The Financial Market Under the subjective measure P t, s.t. dp t (S T ) = p t (S T )ds T C t (X ) = e rt,τ τ (S T X ) + q t(s T ) p t (S T ) p t(s T ) ds T (2) = e rt,τ τ (S T X ) + m t (S T )p t (S T ) ds T m t (S T ) pricing kernel at time t for discounting payoffs occuring at T
().5 -.5 - - -.5.5 g Motivation -3 Empirical Pricing Kernel (EPK) estimation of pricing kernel crucially depends on underlying distributional assumptions and data generating process of the underlying asset in practice, parametric stock price specifications do not hold (e.g. GBM in Black-Scholes model, Heston model) employ nonparametric methods
().5 -.5 - - -.5.5 g Motivation -4 Empirical Pricing Kernel () estimate p and q separately, see e.g. Aït-Sahalia and Lo (2), Brown and Jackwerth (24), Grith et al. (29) ˆm t (S T ) = ˆq t(s T ) ˆp t (S T ) (3) q nonparametric second derivative of C w.r.t. X based on intraday option prices (kernel, local polynomial, splines, basis epansion) p simpler methods (e.g. kernel density estimator) based on daily stock prices
().5 -.5 - - -.5.5 g Motivation -5 Empirical Pricing Kernel (2) one step estimation of m in Engle and Rosenberg (22) by series epansion m t (S T ) L α tl g l (S T ). (4) l= where {g l } L l= are known basis functions and α t = (α t,, α tl ) are time varying coefficients vectors better interpretability of the curves in dynamics
().5 -.5 - - -.5.5 g Motivation -6 Outline. Motivation 2. The Model 3. Estimation 4. Statistical Properties 5. Empirical Study 6. Final remarks 7. Bibliography
().5 -.5 - - -.5.5 g The Model 2- The Model () For the i.i.d. call/strike data {Y i, X i } n i= observed at time t consider a nonparametric regression model Y i = C(X i ) + ε i, E[ε i X i ] = (5) where we assume that the call price C(X ) : R R is a function in X only given by (2) C(X ) = e rt,τ τ (S T X ) + m(s T )p t (S T ) ds T We are interested in estimating the function m(s T ) : R R
().5 -.5 - - -.5.5 g The Model 2-2 The Model (2) Rewrite (5) using the series approimation for m in (4) Y i = C(X i ) + u i, where u i = (C(X i ) C(X i )) + ε i and L C(X ) = e rt,τ τ (S T X ) + α l g l (S T )p t (S T )ds T (6) = l= L { } α l e rt,τ τ (S T X ) + g l (S T )p t (S T )ds T l=
().5 -.5 - - -.5.5 g Estimation 3- Estimation of α For known basis functions and fied L, the vector α = (α,..., α L ) is estimated using the following linear least square criteria In (7), define ˆα = arg min α n i= { Y i C(X i )} 2 (7) ψ il = e rτ (S T X i ) + g l (S T )p t (S T )ds T. (8) s.t. C i (X ) = L l= α lψ il.
().5 -.5 - - -.5.5 g Estimation 3-2 Estimation of α, C and m Then ˆα = (Ψ Ψ) Ψ Y, (9) where Ψ (n L) = (ψ il ) and Y = (Y,, Y n ), ˆ C(X ) = ψ L (X ) ˆα, () where ψ L (X ) = (ψ (X ),..., ψ L (X )) and where g L (S T ) = (g (S T ),..., g L (S T )). ˆm(S T ) = g L (S T ) ˆα, ()
().5 -.5 - - -.5.5 g Estimation 3-3 Simulation of S T In practice the integral in (8) is replaced by the sum where {S s T }J s= J J (ST s X i) + g l (ST s ). (2) s= is a simulated sample from the historical returns r t s,τ = log(s t s /S t (s+) ). so that S s T = S te r t s,τ.
().5 -.5 - - -.5.5 g Estimation 3-4 Choice of the Tuning Parameter L () Optimal selection L: the resulting MISE equals the smallest possible integrated square error Li and Racine (27) Mallows s C L (or C p ), Mallows (973) C L = n n i= { Y i ˆ C(X i )} 2 + 2σ 2 (L/n) where σ 2 is the variance of u i. One can estimate σ 2 by ˆσ 2 = n n ûi 2, with û i = Y i ˆ C(X i ). i=
().5 -.5 - - -.5.5 g Estimation 3-5 Choice of the Tuning Parameter L (2) Generalized cross-validation, Craven and Wahba (979) CV G L n { n = i= Y i ˆ C(X } 2 i ) { (L/n)} 2. Leave one out cross-validation, Stone (974) CV L = n i= { Y i ˆ C } 2 i (X i ) where ˆ C i (X i ) is the leave one estimate of C(X i ) obtained by removing (X i, Y i ) from the sample.
().5 -.5 - - -.5.5 g Statistical Properties 4- Assumptions: Newey (997) Assumption. {X i, Y i } is i.i.d. as (X, Y ), var(y ) is bounded on S, the compact support of X. Assumption 2. For every L there is a nonsinguar matri of constants B such that, for G L (S T ) = g L (S T ). (i) The smallest eigenvalue of E[G L (S s T )G L (S s T ) ] is bounded away from zero uniformly in L.
().5 -.5 - - -.5.5 g Statistical Properties 4-2 Assumptions: Newey (997) (ii) There eists a sequence of constants ξ (L) that satisfy the condition sup S G L (S T ) ξ (L), where L = L(n) such that ξ (L) 2 /n as n. (iii) As n, L and L/n Assumption 3. There eists θ >, such that sup m(s T ) g L (S T ) α = O(L θ ) as L. (3) θ
().5 -.5 - - -.5.5 g Statistical Properties 4-3 Convergence of ˆm Under Assumptions, 2 and 3 one can show that { ˆm(S T ) m(s T )} 2 dp(s T ) = = {g L (S T ) (ˆα α) + g L (S T ) α m(s T )} 2 dp(s T ) ˆα α 2 + {g L (S T ) α m(s T )} 2 dp(s T ) = O p (L/n + L 2θ ) + O(L 2θ ) = O p (L/n + L 2θ ).
().5 -.5 - - -.5.5 g Empirical Study 5- Data Source: Reseach Data Center (RDC) http://sfb649.wiwi.hu-berlin.de Datastream DAX 3 Price Inde; 5 overlapping monthly returns EUREX European Option Data; tick observations; Cross-sectional data: 242
().5 -.5 - - -.5.5 g Empirical Study 5-2 Series Functions Use Laguerre polynomials, with the first two polynomials g () = () = Recurrence relation for l = 2,, L g l+ () = l ((2l )g l () (l )g l 2 ()).
().5 -.5 - - -.5.5 g Empirical Study 5-3 g.5 g l () -.5 - - -.5.5 Figure : First five terms of the Laguerre polynomials
().5 -.5 - - -.5.5 g Empirical Study 5-4 3 2 EPK 3 35 4 45 5 Stock price at T Figure 2: EPK on 242 using Laguerre polynomials as basis functions and L = 5 given by CV ; S t = 433
().5 -.5 - - -.5.5 g Empirical Study 5-5.8.6 Residuals' density.4.2-3 -.5.5 3 Residuals Figure 3: Kernel density estimators for the residuals of the fitted call curves (h by plug-in method) (blue curve) against a simulated normal density (red curve)
().5 -.5 - - -.5.5 g Empirical Study 5-6 2.5-3 RND/PDF.5.5 3 35 4 45 5 Stock price at T Figure 4: Simulated PDF for S T (green curve) and estimated RND (magenta curve)
().5 -.5 - - -.5.5 g Final Remarks 6- Further Improvements individual estimation of PK curves underutilize the available information FDA methods that assume a common curve structure seem to perform better e.g. FPCA, DSFM a structural model for the coefficients can improve estimation functional time series
().5 -.5 - - -.5.5 g Bibligraphy 7- Bibliography Aït-Sahalia, Y. and Lo, A. W. (2) Nonparametric risk management and implied risk aversion. Journal of Econometrics 94: 9-5 Black, F. and Scholes, M. (973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy 8: 637-654. Brown, D.P. and Jackwerth, J. C. (24) The pricing kernel puzzle: Reconciling inde option data and economic theory. Manuscript
().5 -.5 - - -.5.5 g Bibligraphy 7-2 Bibliography Engle, R. F. and Rosenberg, J. V. (22) Empirical pricing kernels. Journal of Financial Economics 64: 34-372 Grith, M., Härdle, W. and Park, J. (29) Shape invariant modelling pricing kernels and risk aversion, Submitted to Journal of Financial Econometrics on June 29 Heston, S. (993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327-343
().5 -.5 - - -.5.5 g Bibligraphy 7-3 Bibliography Li, Q. and J. S. Racine (27) Nonparametric econometrics: Theory and practice. Princeton University Press Newey, W.K. (997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79: 47-68