A Term Structure Model for Dividends and Interest Rates

Damir Filipovi´c Joint work with Sander Willems

Ecole´ Polytechnique F´ed´eralede Lausanne Swiss Finance Institute

Consortium for Data Analytics in Risk Seminar UC Berkeley, July 31, 2018 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Introduction 2/34 A new market for dividend derivatives

How can we trade dividends?

I Synthetic replication. I Dividend swaps (OTC) or dividend futures (on exchange). I Latest innovations: single names, options, dividend-rates hybrids, . . . Dividend derivative pricing.

I Buehler et al. (2010), Buehler (2015), Kragt et al. (2016), Tunaru (2017). Asset pricing: term structure of equity risk premium.

I Lettau and Wachter (2007), Binsbergen et al. (2012), Binsbergen et al. (2013), Binsbergen and Koijen (2017). Interest rates: hybrid products, long maturity dividend claims.

Introduction 3/34 Notional Outstanding Dividend Swaps and Futures

Figure: Total notional outstanding as of June 2015. Source: Mixon and Onur (2016)

Introduction 4/34 Notional Outstanding Dividend Swaps and Futures

Figure: Notional outstanding per expiry as of June 2015. Source: Mixon and Onur (2016)

Introduction 5/34 Contribution of this paper

Term-structure model for dividends and interest rates with Closed-form prices for dividend futures/swaps, bonds, and dividend paying stocks. Moment-based approximations for a broad class of exotic payoﬀs. Positive dividends and possible seasonal behaviour. Flexible correlation between dividends and interest rates.

Introduction 6/34 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Polynomial Framework 7/34 Factor process

Filtered probability space (Ω, F, Ft , Q), with Q risk-neutral pricing measure. d Multivariate factor process Xt on E ⊆ R

dXt = κ(θ − Xt )dt + dMt ,

d×d d for κ ∈ R , θ ∈ R , and martingale Mt such that Xt is a polynomial jump-diﬀusion, cfr. Filipovi´cand Larsson (2017). Generator G maps polynomials to polynomials:

GPoln(E) ⊆ Poln(E), ∀n ∈ N,

with Poln(E) space of polynomials on E of degree n or less.

Polynomial Framework 8/34 PJD Moment Formula

Fix polynomial basis for Poln(E):

> Hn(x) = (h1(x),..., hNn (x)) , n+d with Nn = dim(Poln(E)) ≤ n . G restricts to a linear operator on Poln(E)

GHn(x) = GnHn(x).

First order linear ODE for s 7→ Et [Hn(Xs )] Z T Et [Hn(XT )] = Hn(Xt ) + Gn Et [Hn(Xs )] ds t Solving ODE gives for all t ≤ T

Gn(T −t) Et [Hn(XT )] = e Hn(Xt ).

Polynomial Framework 9/34 Dividend Futures

Instantaneous dividend rate:

> Dt = p H1(Xt ),

d+1 > for p ∈ R such that p H1(x) ≥ 0 for all x ∈ E. Linear dividend futures price:

Z T2 Dfut (t, T1, T2) = Et Ds ds T1 Z T2 > G1(s−t) = p e ds H1(Xt ). T1

> > E.g., if H1(x) = (1, x ) , then

0 0 G = . 1 κθ −κ

Polynomial Framework 10/34 Interest Rates

Risk-neutral discount factor:

R t − rs ds ζt = ζ0e 0 , t ≥ 0,

where rt denotes the short rate.

Directly specify ζt : −γt > ζt := e q H1(Xt ), d+1 for γ ∈ R and q ∈ R such that ζt is a positive and absolutely continuous process. Implied short rate: > q G1H1(Xt ) rt = γ − > . q H1(Xt ) Cfr. Filipovi´cet al. (2017), Ackerer and Filipovi´c(2016).

Polynomial Framework 11/34 Bond Prices

Time-t price of zero-coupon bond maturing at T ≥ t:

> G1(T −t) 1 −γ(T −t) q e H1(Xt ) P(t, T ) = Et [ζT ] = e > . ζt q H1(Xt )

Linear discounted bond price ζt P(t, T ). If <(eig(κ)) > 0: log(P(t, T )) lim − = γ. T →∞ T − t

Polynomial Framework 12/34 Dividend Paying Stock

Fundamental stock price Z ∞ ∗ 1 St = Et ζs Ds ds . ζt t

If <(eig(G2)) < γ, then

> −1 ∗ ~v (γ Id − G2) H2(Xt ) St = > < ∞, q H1(Xt )

∗ Quadratic discounted fundamental stock price ζt St . Deﬁne arbitrage-free stock price as

Lt ∗ St = + St , ζt

for some nonnegative (local) martingale Lt , cfr. Buehler (2015), Jarrow et al. (2007, 2010). Polynomial Framework 13/34 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Option Pricing 14/34 Maximum Entropy Moment Matching

Pricing problem: h i πt = Et F g(XT ) ,

with g ∈ Poln(E) and F : R → R. Goal: Approximate density of g(XT ) based on moments. Maximize Boltzmann-Shannon entropy:

Z max − f (x) ln f (x) dx f Z n s.t. x f (x) dx = Mn, n = 0,..., N

Unique solution:

N ! X i f (x) = exp − λi x i=0 Option Pricing 15/34 Option Pricing

Swaptions:

+ swpt 1 swap πt = Et ζT πT ζt " n !+# 1 X = Et ζT − ζT P(T , Tn) − δK ζT P(T , Tk ) ζt k=1 Stock options

stock 1 + πt = Et (ζT ST − ζT K) ζt 1 ∗ + = Et (LT + ζT ST − ζT K) ζt

Option Pricing 16/34 Option Pricing

Dividend options

" +# Z T1 div πt = Et Ds ds − K T0 + = Et (IT1 − IT0 − K) ,

R T with IT = 0 Ds ds. Augment factor process: (It , Xt ) is PJD. n Compute moments Et [(IT1 − IT0 ) ] using law of iterated expectations.

Option Pricing 17/34 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Linear Jump-Diﬀusion Model 18/34 Linear Jump-Diﬀusion

Specify martingale part dMt as

dXt = κ(θ − Xt ) dt + diag(Xt−) (Σ dBt + dJt )

d×d Bt : standard d-dimensional Brownian motion, Σ ∈ R lower triangular with Σii > 0

Jt : compensated compound Poisson process, jump intensity ξ and Z i.i.d. jump amplitudes e − 1, Z ∼ N (µJ , ΣJ ).

Unique positive solution if κθ ≥ 0 and if κij ≤ 0 for i 6= j. Allows for ﬂexible instant. correlation between factors through Σ. Moments in closed-form (PJD).

Linear Jump-Diﬀusion Model 19/34 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Calibration 20/34 Model Speciﬁcation

I I I D D > Five factor model Xt = (X0t , X1t , X2t , X1t , X2t ) I I I I > Rate factors Xt = (X0t , X1t , X2t ) :

 I I   I I    κ0 −κ0 0 θ − X0t 0 0 I I I I I I I dB1t dXt =  0 κ1 −κ1 θ − X1t  dt + diag(Xt ) Σ11 0  , I I I I I dB2t 0 0 κ2 θ − X2t Σ21 Σ22

−γt I I with ζt = e X0t , θ = 1, and γ = 4.2%. D D D > Dividend factors Xt = (X1t , X2t ) :

D D D D D D κ1 −κ1 θ − X1t D Σ11 0 dB3t dJ1t dXt = D D D dt + diag(Xt−) D D + , 0 κ2 θ − X2t Σ21 Σ22 dB4t dJ2t

D with Dt = X1t . ∗ Explicit restrictions on parameters s.t. St < ∞.

Calibration 21/34 Model Speciﬁcation

Jump-diﬀusive stock price bubble:

L L L dLt = Lt−(σ dBt + dJt )

Assume Lt independent of Xt

stock 1 ∗ + πt = Et (LT + ζT ST − ζT K) ζt 1 h M ˜ i = Et Ct (Lt , K(XT )) , ζt

M ˜ where Ct (Lt , K(XT )) is the Merton (1976) option price with spot Lt ˜ ∗ and strike K(XT ) = ζT K − ζT ST . Dependence between Lt and Xt is possible as long as (Xt , Lt ) remains jointly a PJD (at the cost of an increased dimension).

Calibration 22/34 Data

Calibration date: December 21 2015. Euribor swaps (7) Tenors: 1, 2, 3, 5, 7, 10, 15y. Euribor swaptions (12) Expiries: 3m, 6m, 1y Tenors: 3, 5, 10, 15y Moneyness: ATM Eurostoxx 50 index dividend futures (10) Expiry: 1-10 y Eurostoxx 50 index dividend options (21) Expiry: 2, 3, 4y Moneyness: 0.9, 0.95, 0.975, 1, 1.025, 1.05, 1.1 Eurostoxx 50 index options (24) Expiry: 3m, 6m, 1y Moneyness: 0.8, 0.9, 0.95, 0.975, 1, 1.025, 1.05, 1.1

Calibration 23/34 Swaps and Dividend Futures (21/12/2015)

1.6 150 Model Model 1.4 Market 21/12/2015 Market 21/12/2015

1.2

1 100

0.8

0.6 Futures price Swap rate (%) 0.4 50

0.2

-0.2 0 1 2 3 5 7 10 15 Dec16 Dec17 Dec18 Dec19 Dec20 Dec21 Dec22 Dec23 Dec24 Dec25 Tenor (yrs) Maturity (a) Euribor swap rates (b) Eurostoxx 50 dividend futures

Calibration 24/34 Euribor Swaptions (21/12/2015)

100 100 Model Model 90 Market 21/12/2015 90 Market 21/12/2015

80 80

70 70

60 60

50 50

40 40

30 30 Normal implied vol (bps) Normal implied vol (bps)

20 20

10 10

0 0 3 5 10 15 3 5 10 15 Tenor (yrs) Tenor (yrs) (a) 3 month maturity (b) 6 month maturity

100 Model 90 Market 21/12/2015

30 Normal implied vol (bps)

0 3 5 10 15 Tenor (yrs) (c) 1 year maturity Calibration 25/34 Eurostoxx 50 Dividend Futures Options (21/12/2015)

25 25 Model Model Market 21/12/2015 Market 21/12/2015

20 20

15 15

Black IV (%) 10 Black IV (%) 10

5 5

0 0 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 Strike/Futures Strike/Futures (a) Dec 2016-2017 (b) Dec 2017-2018

25 Model Market 21/12/2015

Black IV (%) 10

0 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 Strike/Futures (c) Dec 2018-2019 Calibration 26/34 Index Options (21/12/2015)

40 40 Model Model 35 Market 21/12/2015 35 Market 21/12/2015

30 30

25 25

20 20

15 15 Black-Scholes IV (%) Black-Scholes IV (%) 10 10

5 5

0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Strike/Spot Strike/Spot (a) 3 month maturity (b) 6 month maturity

40 Model 35 Market 21/12/2015

15 Black-Scholes IV (%) 10

0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Strike/Spot (c) 1 year maturity Calibration 27/34 Moments and Option Prices

63 22 Maximal entropy Maximal entropy 62.5 MC MC MC 95% CI 21.5 MC 95% CI 62

61.5 21 61

60.5 20.5

60 Normal IV (bps) 20 59.5 Black-Scholes IV (%)

59 19.5 58.5

58 19 2 4 6 8 10 2 4 6 8 10 Number of moments matched Number of moments matched (a) Swaption, 3m expiry (b) Index option, 3m expiry

14 Maximal entropy 13.5 MC MC 95% CI

12.5

Black IV (%) 11.5

10.5

10 2 4 6 8 10 Number of moments matched (c) Dividend option, 2y

Calibration expiry 28/34 Overview

1 Introduction

2 Polynomial Framework

3 Option Pricing

4 Linear Jump-Diﬀusion Model

5 Calibration

6 Extensions

Extensions 29/34 Dividend Seasonality

30 DVP Index points

0 2009 2010 2011 2012 2013 2014 2015 2016 2017 Time

Figure: Monthly dividend payments by Eurostoxx 50 constituents (in index points) from October 2009 until October 2016. Source: Eurostoxx 50 DVP index, Bloomberg.

Extensions 30/34 Dividend Seasonality

Standard choice to model annual cycles:

 sin(2πt)  cos(2πt) >  .  δ(t) = ρ0 + ρ Γ(t), Γ(t) =  .  . sin(2πKt) cos(2πKt)

Remark, Γ(t) is the solution of a linear ODE:

h 0 2πi h 0 2πKi dΓ(t) = blkdiag −2π 0 ,..., −2πK 0 Γ(t)dt.

→ We can add Γ to the state vector! For example:

dXt = κ(δ(t) − Xt )dt + dMt

Extensions 31/34 Dividend Swaps

Dividend swap/forward price:

1 1 Z T2 Dswap(t, T1, T2) = Et ζT2 Ds ds P(t, T2) ζt T1 h i Cov ζ , R T2 D ds t T2 T1 s = Dfut (t, T1, T2) + . P(t, T2)ζt In polynomial framework:

w(t, T , T )>H (X ) D (t, T , T ) = 1 2 2 t , swap 1 2 > G (T −t) q e 1 2 H1(Xt )

T with w(t, T , T ) = R 2 q>eG1(T2−s) Q eG2(s−t) ds and 1 2 T1 > QH2(x) = H1(x)H1(x) p.

Extensions 32/34 Conclusion

Joint term-structure model for dividends and interest rates. Explicit prices for dividend futures/swaps, bonds, and dividend paying stock. Moment-based approximations for (path dependent) option prices using principle of maximum entropy. Future work: ∗ I Time-series estimation of St . I SVJ model for bubble component. I Single-stock framework with credit risk.

https://ssrn.com/abstract=3016310

Extensions 33/34 Thank you for your attention! ReferencesI

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Jarrow, R. A., P. Protter, and K. Shimbo (2010). Asset price bubbles in incomplete markets. Mathematical Finance 20(2), 145–185. 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), 55–92. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3(1-2), 125–144. Mixon, S. and E. Onur (2016). Dividend swaps and dividend futures: State of play. Journal of Alternative Investments, Forthcoming. Tunaru, R. (2017). Dividend derivatives. Quantitative Finance, 1–19.