A Term-Structure Model for Dividends and Interest Rates

Transcription

1 A Term-Structure Model for Dividends and Interest Rates Sander Willems Joint work with Damir Filipović School and Workshop on Dynamical Models in Finance May 24th 2017 Sander Willems A TSM for Dividends and Interest Rates May 24th / 29

2 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

3 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

4 A new market for dividend derivatives How can we trade dividends? Synthetic replication. Dividend swaps (OTC) or dividend futures (on exchange). Latest innovations: single names, options, dividend-rates hybrids,... Asset pricing: term structure of equity risk premium. Lettau and Wachter (2007), Binsbergen et al. (2012), Binsbergen et al. (2013), Binsbergen and Koijen (2017). Dividend derivative pricing. Buehler et al. (2010), Tunaru (2014), Buehler (2015), Kragt et al. (2016). Derivative pricing with dividend paying stock Deterministic dividends: Bos and Vandermark (2002), Bos et al. (2003), Vellekoop and Nieuwenhuis (2006). Proportional dividends: Merton (1973), Korn and Rogers (2005). Interest rates: hybrid products, long maturity dividend claims. Sander Willems A TSM for Dividends and Interest Rates May 24th / 29

5 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

6 State Process Filtered probability space (Ω, F, F t, Q), with Q the risk-neutral pricing measure. Multivariate state process X t in E R d with linear drift: dx t = κ(θ X t )dt + dm t, for κ R d d, θ R d, and some d-dimensional martingale M t. First moment is linear in the state: [( )] ( ) ( ) 1 E t = e A(T t) 0 0, A =, t T. X T 1Xt κθ κ Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

7 Dividend Futures Instantaneous dividend rate: D t = p + q X t, for p R, q R d such that p + q x 0 for all x E. Dividend futures price: If κ is invertible: D fut (t, T 1, T 2 ) = E t [ T2 D fut (t, T 1, T 2 ) =(T 2 T 1 ) T 1 ] D s ds = ( p q ) T 2 T 1 ( ) p + q θ ( e A(s t) 1 ds X t ). q κ 1 ( e κ(t 2 t) e κ(t 1 t) ) (X t θ). Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

8 Dividend Seasonality DVP Index points Time Figure: Monthly dividend payments by Eurostoxx 50 constituents (in index points) from October 2009 until October Source: Eurostoxx 50 DVP index, Bloomberg. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

9 Dividend Seasonality Standard choice to model annual cycles: δ(t) = ρ 0 + ρ Γ(t), Γ(t) = Remark, Γ(t) is the solution of a linear ODE: ) dγ(t) = blkdiag,..., (( 0 2π 2π 0 We can add Γ to the state vector! For example: sin(2πt) cos(2πt). sin(2πkt) cos(2πkt) ( 0 2πK 2πK 0 dx t = κ(δ(t) X t )dt + dm t. )) Γ(t)dt. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

10 Interest Rates Risk-neutral discount factor: where r t denotes the short rate. Directly specify dynamics for ζ t : ζ t = ζ 0 e t 0 rs ds, t 0, ζ t := e γt Y t, dy t = λ(φ + ψ X t Y t )dt, for φ, λ, γ R and ψ R d such that Y t > 0. Cfr. Filipović et al. (2017), Ackerer and Filipović (2016). Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

11 Bond Prices Time-t price of zero-coupon bond maturing at T : P(t, T ) = 1 ζ t E t [ζ T ] = Implied short rate: e γ(t t) e t) d+2 eb(t Y t X t, B = κθ κ 0 Y t λφ λψ λ r t = γ + λ λ φ + ψ X t Y t. If all eigenvalues of κ have positive real parts and λ > 0: T )) lim log(p(t, = γ. T T t. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

12 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

13 GARCH Diffusion Specify martingale part M t as follows dx t = κ(θ X t ) dt + diag(x 1t,..., X dt )Σ db t, (1) with B t a standard d-dimensional Brownian motion and Σ R d d lower triangular with Σ ii > 0. Used before for stochastic volatility (Nelson (1990), Barone-Adesi et al. (2005)), energy markets (Pilipović (1997)), interest rates (Brennan and Schwartz (1979)), and Asian option pricing (Linetsky (2004)). Attractive features: Unique positive solution. Flexible correlation structure. Moments in closed-form. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

14 GARCH Diffusion Proposition Consider the following system of SDEs: { dxt = κ(θ X t ) dt + diag(x 1t,..., X dt )Σ db t, If (X 0, Y 0 ) R d+1 +, and dy t = λ(φ + ψ X t Y t )dt (κθ) i, ψ i 0, λ, φ 0, κ ij 0 for i j, then the system has a unique strong solution in R d+1 +. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

15 Moment Formula (X t, Y t ) is a polynomial diffusions, cfr. Filipović and Larsson (2015). Fix basis for Pol n (R d R), n 1: H n (x, y) = (1, h 1 (x, y),..., h Nn (x, y)), with N n = ( ) n+d+1 n 1. There exists matrix G n such that for any z Pol n (R d R) E t [z(x T, Y T )] = z e Gn(T t) H n (X t, Y t ), where z is the coordinate representation of z with respect to the chosen basis. Efficient algorithms exist for z e Gn(T t), e.g. Al-Mohy and Higham (2011). Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

16 Dividend Paying Stock Absence of arbitrage: S t = 1 ζ t E t [ζ T S T ] + 1 ζ t E t [ T t ] ζ s D s ds, T t. If R(eig(G 2 )) < γ, then [ 1 ] E t ζ s D s ds = 1 v (γi G 2 ) 1 H 2 (X t, Y t ) <, ζ t Y t t with v the coordinate vector of (x, y) y(p + q x) wrt H 2 (x, y). Stock price representation: S t = L t ζ t + 1 ζ t E t [ t ] ζ s D s ds, t 0, for some non-negative martingale L t, cfr. Buehler (2010). Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

17 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

18 Data Eurostoxx 50 stock index Eurostoxx 50 dividend futures Underlying: sum of declared ordinary gross cash dividends (or cash equivalent) with ex-date during one calendar year, divided by index divisor on ex-date. Fixed maturity dates in Dec of each year, up to 10y. Example: Today we can trade in maturities Dec(17+k), k = 0,..., 9. Payoff of Dec(17+k) contract is sum of dividends in [Dec(17+(k 1)), Dec(17+k)]. We use 2nd, 3rd, 4th, 5th, 7th, and 10th contract. Euribor interest rate swaps Fixed leg pays annual, floating leg semi-annual. Fixed time to maturities: 1,2,3,5,7, and 10 years. Daily observations from October 2009 until October 2016, ( ) 1827 = 23, 751 observations in total. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

19 Summary Statistics Mean Median Std Min Max Swap rates (%) 1 yrs yrs yrs yrs yrs yrs Dividend futures 1-2 yrs yrs yrs yrs yrs yrs Eurostoxx 50 index 2, , , , Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

20 Model Specification We model X t = (Xt I, Xt D ) and set d = 2 2: dx1t I = κ I 1 (X 2t I X 1t I ) dt + X 1t I ΣI 1 dbp t dx2t I = κ I 2 (θi X2t I ) dt + X 2t I ΣI 2 dbp t dx 1t D = κ D 1 (X 2t D X 1t D) dt + X 1t DΣD 1 dbp t dx2t D = κ D 2 (θd X2t D) dt + X 2t DΣD 2 dbp t with B P t a 4-dimensional P-Brownian motion and Σ I 1 Σ I 11 Σ = Σ I 2 Σ D = 0 Σ I 22 1 Σ DI Σ D 11 0 Σ D Σ DI 22 0 Σ D 22, ψ = Constant market price of risk vector Λ R 4 : [ ] dq E P t = exp dp ( Λ B P t 1 2 Λ 2 t , q = ). Restrictions on parameters such that: (X t, Y t ) > 0, lim T E P [H 2 (X T, Y T )] <, and S t <, t > 0. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29,

21 Parameter Estimation Quasi-maximum likelihood + Kalman filtering. Transition equation: Z t = Φ 0 + Φ 1 Z t 1 + ε t, ε t N (0, q(z t 1 )). Measurement equation: M t = h(z t ) + ν t, ν t N (0, σm 2 Id). Dividend futures: linear. Swap rates and stock price: non-linear. Unscented Kalman filter. All observations are scaled by their sample mean. Three estimation steps: Estimate κ D 1, κ D 2, θ D, Σ D 11, ΣD 22 from dividend futures. Estimate κ I 1, κ I 2, θ I, Σ I 11, ΣI 22 from swap rates. Re-estimate all parameters using all instruments. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

22 Parameter Estimates ε I 1 ε I 2 ε D 1 ε D 2 Σ I 11 Σ I 22 Σ D (0.076) (0.001) (0.015) (0.002) (0.009) (0.004) (0.012) Σ D 22 Λ 1 Λ 2 Λ 3 Λ 4 Σ DI 1,1 Σ DI 2, (0.011) (0.040) (0.030) (0.082) (0.042) (0.011) (0.010) θ D γ λ σ M L (0.049) (0.002) (0.079) (0.000) Table: Quasi-maximum likelihood estimates with asymptotic standard deviations in parenthesis. Corresponding instantaneous correlation matrix: X1 I X2 I X1 D X2 D X 1 I 1.00 X2 I X1 D X2 D Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

23 Error Analysis Maturities Swap rates All 1 y 2 y 3 y 5 y 7 y 10 y RMSE (bps) MAE (bps) Dividend futures All 1-2 y 2-3 y 3-4 y 4-5 y 6-7 y 9-10 y RMSRE (%) MARE (%) Eurostoxx 50 RMSRE (%) 5.38 MARE (%) 4.57 Table: The first five days of the sample are dropped when computing the error statistics to give the Kalman filter time to learn the current value of X t. The remaining sample period consists of 1,822 daily observations between October 8, 2009 and October 1, Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

24 Filtered State Y t X I 1t 1.6 X D 1t X D 2t 1.8 X I 2t Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 (a) Interest rate factors (b) Dividend factors Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

25 Filtered Swap Rates 4 Maturity 1 yrs 4 Maturity 2 yrs 4 Maturity 3 yrs 3 Filtered Observed 3 Filtered Observed 3 Filtered Observed Swap rate (%) 2 1 Swap rate (%) 2 1 Swap rate (%) Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time -1 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time -1 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 4 Maturity 5 yrs 4 Maturity 7 yrs 4 Maturity 10 yrs 3 Filtered Observed 3 Filtered Observed 3 Filtered Observed Swap rate (%) 2 1 Swap rate (%) 2 1 Swap rate (%) Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time -1 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time -1 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

26 Filtered Dividend Futures Prices 140 Maturity 1-2 yrs 140 Maturity 2-3 yrs 140 Maturity 3-4 yrs 130 Filtered Observed 130 Filtered Observed 130 Filtered Observed Futures price Futures price Futures price Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 60 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 60 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 140 Maturity 4-5 yrs 140 Maturity 6-7 yrs 140 Maturity 9-10 yrs 130 Filtered Observed 130 Filtered Observed 130 Filtered Observed Futures price Futures price Futures price Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 60 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time 60 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time Sander Willems A TSM for Dividends and Interest Rates May 24th / 29

27 Filtered Index Level Filtered Observed Index level Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Time Sander Willems A TSM for Dividends and Interest Rates May 24th / 29

28 Risk Premium Analysis Mean (%) Std (%) Sharpe β Dividend spot 2 yrs yrs yrs yrs yrs yrs yrs Bonds 2 yrs yrs yrs yrs yrs yrs yrs Index Table: Monthly returns in excess of the 1-month risk-free rate. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

29 Overview 1 Introduction 2 Dividend Futures and Bonds 3 Dividend Paying Stock 4 Empirical Analysis 5 Derivative Pricing Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

30 Derivative Pricing with Polynomial Expansions Denote Z t = (X t, Y t ) and Z = (Z t1,..., Z tn ) for some finite time partition t 1 < < t n. Time-t price of a (path dependent) derivative: π t = E t [F (Z)], for some discounted payoff function F on E = (R d R) n. Denote by g(dz) the (unknown) conditional distribution of Z. Let w(dz) be an auxiliary distribution such that g(dz) w(dz) and g(dz) = l(z)w(dz). Define Hilbert space L 2 w (E) with norm and scalar product f 2 w = f (z) 2 w(dz), f, h w = f (z)h(z)w(dz). E E Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

31 Derivative Pricing with Polynomial Expansions Assumptions: 1 Pol(E) L 2 w, 2 l L 2 w, 3 F L 2 w, 4 g w. Let H = {H 0 (z) = 1, H 1 (z), H 2 (z),...} be an orthonormal set of polynomials spanning the closure Pol(E) in L 2 w Let F be the orthogonal projection of F onto Pol(E) in L 2 w. Elementary functional analysis now gives: π t = E[ F (Z)] = F, l w = k 0 F k l k, F k = F, H k w = F, H k w = l k = l, H k H = E t [H k (Z)]. E F (z)h k (z) w(dz), Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

32 Derivative Pricing with Polynomial Expansions Truncating the series for π, we get: π (K) t = K F k l k k=0 ( π t π t ) = 0 if Pol(E) = L 2 w. = π t + ( π t π t ) }{{} + ( π (K) t π t ). }{{} projection bias truncation error ( π (K) t π t ) 0 as K. Crucial question: how to choose the auxiliary distribution? Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

33 The Auxiliary Distribution We choose the multivariate log-normal distribution: Definition A random vector (X 1,..., X k ) R k +, k 1, is said to have a multivariate log-normal distribution LN (µ, Λ) if (log(x 1 ),..., log(x k )) N (µ, Λ), for some µ R k and some positive semi-definite Λ R k k. Very easy to simulate from a log-normal. Finite moments of any order: E[X α 1 1 X α k k ] = exp {α µ α Λα Assumption 1 is always satisfied Moment indeterminate projection bias. } <, α N k. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

34 The Auxiliary Distribution Theorem Let m = (d + 1)n. Suppose that the random vector Z = (Z t1,..., Z tn ) admits a continuous density with support on R m +. Let w be the LN (µ, Λ) distribution with µ R m and pos. def. Λ R m m. Define the matrix M R n n as M = 1 σ 2 T 1 (I n 1 d+1 )Λ 1 (I n 1 d+1 ), where σ 2 = max {(ΣΣ ) ij i, j = 1,..., d}, and T = (t i t j ) 1 i,j n. If M is positive semi-definite, then assumption 2 is satisfied. Lemma Suppose that Σ has strictly positive diagonal elements, λψ is different from the zero vector, and (X 0, Y 0 ) R d+1 +. Then for any t > 0, the random vector Z t = (X t, Y t ) has an infinitely differentiable density. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

35 Examples of (discounted) derivative payoffs Swaption: ζ ( ) T0 + π swap e γt 0 ( + ζ T 0 = wswap H 1 (X T0, Y T0 )). t Y t Dividend option: Denote I t = t 0 D s ds, ζ T1 ζ t ( T1 T 0 Stock option: ) + D s ds K = e γ(t 1 t) Y T1 (I T1 I T0 K) +, Y t ζ T (S T K) + = e γ(t t) ( + v (γi G 2 ) 1 H 2 (X T, Y T ) Y T K). ζ t Y t Dividend-Rates hybrid: ζ T ζ t ( T T 1 D s ds S T L 1y T ) +. Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th / 29

36 Thank you for your attention!

37 References I Ackerer, D. and D. Filipović (2016). Linear credit risk models. Swiss Finance Institute Research Paper (16-34). Al-Mohy, A. H. and N. J. Higham (2011). Computing the action of the matrix exponential, with an application to exponential integrators. SIAM Journal on Scientific Computing 33(2), Barone-Adesi, G., H. Rasmussen, and C. Ravanelli (2005). An option pricing formula for the GARCH diffusion model. Computational Statistics & Data Analysis 49(2), Binsbergen, J. H. v., M. W. Brandt, and R. S. Koijen (2012). On the timing and pricing of dividends. American Economic Review 102(4), Binsbergen, J. H. v., W. Hueskes, R. S. Koijen, and E. B. Vrugt (2013). Equity yields. Journal of Financial Economics 110(3), Binsbergen, J. H. v. and R. S. Koijen (2017). The term structure of returns: Facts and theory. Journal of Financial Economics, Forthcoming. Bos, M., A. Shepeleva, and A. Gairat (2003). Dealing with discrete dividends. Risk, Bos, M. and S. Vandermark (2002). Finessing fixed dividends. Risk,

38 References II Brennan, M. J. and E. S. Schwartz (1979). A continuous time approach to the pricing of bonds. Journal of Banking & Finance 3(2), Buehler, H. (2010). Volatility and dividends-volatility modelling with cash dividends and simple credit risk. Working Paper. Buehler, H. (2015). Volatility and dividends II-Consistent cash dividends. Working Paper. Buehler, H., A. S. Dhouibi, and D. Sluys (2010). Stochastic proportional dividends. Working Paper. Filipović, D. and M. Larsson (2015). Polynomial preserving diffusions and applications in finance. Finance and Stochastics, Filipović, D., M. Larsson, and A. B. Trolle (2017). Linear-rational term structure models. Journal of Finance 72, Korn, R. and L. G. Rogers (2005). Stocks paying discrete dividends: modeling and option pricing. Journal of Derivatives 13(2), Kragt, J., F. De Jong, and J. Driessen (2016). The dividend term structure. Working Paper. Lettau, M. and J. A. Wachter (2007). Why is long-horizon equity less risky? A duration-based explanation of the value premium. Journal of Finance 62(1),

39 References III Linetsky, V. (2004). Spectral expansions for Asian (average price) options. Operations Research 52(6), Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics 4(1), Nelson, D. B. (1990). ARCH models as diffusion approximations. Journal of Econometrics 45(1), Pilipović, D. (1997). Energy risk: Valuing and managing energy derivatives. Tunaru, R. (2014). Dividend derivatives. Working Paper. Vellekoop, M. H. and J. W. Nieuwenhuis (2006). Efficient pricing of derivatives on assets with discrete dividends. Applied Mathematical Finance 13(3),

40 Open interest Daily volume Time to maturity Mean Median Mean Median 1-2 yrs 178, ,906 4,692 3, yrs 132, ,800 3,892 2, yrs 87,059 85,116 2,461 1, yrs 57,530 54,968 1,514 1, yrs 23,169 22, yrs 3,354 1, Table: Open Interest and Daily Volume of Dividend Futures