Deterministic Income under a Stochastic Interest Rate

Transcription

1 Deterministic Income under a Stochastic Interest Rate Julia Eisenberg, TU Vienna Scientic Day, 1

2 Agenda 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted Consumption under a Deterministic Income 2

3 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model The Surplus Process We consider an insurance company whose wealth is modelled by a Brownian motion with drift: X t = x + µt + σw t, t 0. x: the initial capital µ R: the drift parameter σ: the diusion parameter W : a standard Brownian motion on (Ω, F, {F t }, P), adapted to {F t }. 3

4 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model Dividends A dividend strategy {C t } models the accumulated dividend payments up to time t. The Ex-dividend process is and time to ruin under C X C t = X t C t, t 0 τ C := inf{t : X C t 0}. A dividend strategy C is admissible if C is adapted, right-continuous and non-decreasing; X C t 0 for all t τ C. 4

5 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model Discounting Let δ > 0 denote a xed discounting rate. The target is to maximise the expected discounted dividend payments until ruin: [ τ C ] max E e δt dc t, C 0 to nd the[ optimal strategy C and the corresponding value function ] τ C V (x) := E e δt dc 0 t. 5

6 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model The Optimal Strategy It turns out that the optimal strategy is a barrier strategy. Assume C t = t 0 c s ds with c s [0, ξ] and ξ > 0. Then, there is an x 0 R + such that the optimal strategy is given by c t = ξ1i [X C t >x 0 ]. If we allow for lump sum payments, the optimal strategy becomes C t = max{ sup 0 s τ C X s x 0, 0}. 6

7 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model Constant Discount Rate? + A constant discount rate simplies the calculations; The economic growth is unlikely to be constant over long time horizons. A constant discount rate assumption does not adequately account for uncertainty. 7

8 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted Consumption under a Deterministic Income Geometric Brownian Motion as an Interest Rate Process Discounting by the Price of a Pure-Discount Bond at Time Zero Ornstein-Uhlenbeck Process as an Interest Rate 8

9 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted Consumption under a Deterministic Income Geometric Brownian Motion as an Interest Rate Process Discounting by the Price of a Pure-Discount Bond at Time Zero Ornstein-Uhlenbeck Process as an Interest Rate 9

10 Consider an individual or household, whose wealth is modelled by a deterministic process: Dene further X t = x + µt t 0. r t = r + mt + σw t, where {W t } is a standard Brownian motion and m > σ2 2. We target to maximize the discounted consumption and dene the return function corresponding to a strategy C = {c s } and the value function to be [ V C (r, x) = E 0 ] e rs c s ds r 0 = r V (r, x) = sup V C (r, x) (r, x) R R +. C (r, x) R R +, 10

11 Maximizing Discounted Consumption under a Deterministic Income, Geometric Brownian Motion as an Interest Rate Process HJB Equation A strategy C = {c s } is called admissible if c s [0, ξ], ξ > 0, Xt C 0 for all t R +, C is adapted and cadlag. The HJB equation corresponding to the problem is } µv x + mv r + σ2 2 V rr + sup c {e r V x = 0. 0 c ξ 11

12 Maximizing Discounted Consumption under a Deterministic Income, Geometric Brownian Motion as an Interest Rate Process The Optimal Strategy Consider the strategy Ĉ = {ĉ s } for ξ > µ ĉ s = { ξ 0 s x ξ µ µ s > x ξ µ. The corresponding return function is given by V Ĉ (r, x) = ξ 0 x ξ µ σ2 r (m e 2 )s σ2 r (m ds + µ e 2 )s ds. x ξ µ and V Ĉ x (r, x) = e r (m σ 2 ) x 2 ξ µ. Obviously, this function solves the HJB equation. 12

13 Maximizing Discounted Consumption under a Deterministic Income, Geometric Brownian Motion as an Interest Rate Process Unrestricted Payments If we allow for lump sum payments, the optimal strategy is to pay out the initial capital immediately and to pay on the rate µ up to an innite time horizon. The value function is then given by V (r, x) = e r x + e r µ m σ

14 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted Consumption under a Deterministic Income Geometric Brownian Motion as an Interest Rate Process Discounting by the Price of a Pure-Discount Bond at Time Zero Ornstein-Uhlenbeck Process as an Interest Rate 14

15 Let {r s } be an Ornstein-Uhlenbeck process. I.e. {r s } fulls the following integral equation t r t = re at + b(1 e at ) + σe at e as dw s, where r is the initial value of the process, a, σ > 0, b R are constants and {W s } is a standard Brownian motion. Further U r s = s 0 0 r u du if r 0 = r. Here, E [ e U r s ] denotes the price at zero of a zero-coupon bond (or pure-discount bond) with maturity s. 15

16 Maximizing Discounted Consumption under a Deterministic Income, Discounting by the Price of a Pure-Discount Bond at Time Zero HJB Equation The return function corresponding to an admissible consumption rate strategy C = {c s } (c s [0, ξ] and Xt C 0 for t [0, T ]) is dened as T [ [ ] V C (t, x) = E e U s ]c r s ds + X CT E e U r T, t We target to maximize the value of discounted consumption. V (t, x) = sup V C (t, x). C The HJB equation corresponding to the problem is given by HJB { [ ] } V t + µv x + sup c E e U r t V x = 0. 0 c ξ 16

17 Maximizing Discounted Consumption under a Deterministic Income, Discounting by the Price of a Pure-Discount Bond at Time Zero Pure-Discount Bond Let σ := σ 2a and b := b σ2 2a 2. Then, Let E[e U r s ] = exp { bs r b ( ) 1 e as σ2 ( ) } 1 e as 2. a a 2 f (s) := bs r b ( ) 1 as e σ2 Then, the HJB equation becomes a a 2 (1 e as ) 2. V t + µv x + sup { c e f (t) } V x = 0. 0 c ξ 17

18 Depending on the parameter choice, the function f (s) will have dierent properties. 18

19 Maximizing Discounted Consumption under a Deterministic Income, Discounting by the Price of a Pure-Discount Bond at Time Zero The Value Function The idea is to establish a backward algorithm by cutting the function f (t) in dierent areas. Starting at t = T, we nd the optimal strategy until the next change point in the behaviour of the function f. Here, one can also consider an arbitrary drift function µ(t) with just nitely many zeros in the interval [0, T ]. Then, one has to take into consideration the behaviour of the function µ(t). This algorithm can be applied to an arbitrary deterministic discounting function, for example on sin(t). The case of unrestricted payments is very easy. Basically, one has to wait until a local maximum and pay out everything there. 19

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21 Maximizing Discounted Consumption under a Deterministic Income, Discounting by the Price of a Pure-Discount Bond at Time Zero Example 21

22 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted Consumption under a Deterministic Income Geometric Brownian Motion as an Interest Rate Process Discounting by the Price of a Pure-Discount Bond at Time Zero Ornstein-Uhlenbeck Process as an Interest Rate 22

23 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate r t = re at + b(1 e at ) + σe at t 0 e as dw s, Here, we assume that the long-term mean b of the process {r s } fulls: b > σ2 2a 2. The return function corresponding to a strategy C = {c s } and the value function are given by [ V C (r, x) = E 0 ] e U s r cs ds X 0 = x, (r, x) R R +, V (r, x) = sup V C (r, x), (r, x) R R +. C 23

24 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Realisations of exp{ U r s }, r = 1, b = 4, b = 0, b = 1 24

25 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate The HJB Equation HJB } µv x + a( b r)v r + σ2 2 V rr rv + sup c {1 V x = 0. 0 c ξ 25

26 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Restricted Rates with ξ µ The return function V ξ corresponding to the constant strategy c s ξ is: [ V ξ ] (r, x) = ξe e U s r ds = ξ 0 0 e f (r,s) ds. Note that V ξ does not depend on x in this case. In particular: 1 V ξ x (r, x) = 1. And it is an easy exercise to prove that V ξ solves the ODE a( b r)v r + σ2 2 v rr rv + ξ = 0. V ξ is the value function. 26

27 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Restricted rates with ξ > µ. I The return function corresponding to the strategy { ξ 0 s x ξ µ ĉ s = µ s > x ξ µ is given by [ V Ĉ (r, x) = E ξ = ξ 0 0 x ξ µ x ξ µ e U r s e f (r,s) ds + µ ] ds + µ e U s r ds x ξ µ x ξ µ e f (r,s) ds. 27

28 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Restricted rates with ξ > µ. II The derivative of V Ĉ with respect to x is given by ( ) V Ĉ x (r, x) = e f x r, ξ µ. If r < 0 and s > 0, then for every xed r R the function f (r, s) is at rst increasing and then decreasing in s. Further, since f (r, 0) = 0 for all r R and lim f (r, s) = the curve s α(s) := a { bs + b } 1 e as a (1 e as ) σ2 2a 2 (1 e as ) 2 is unique with f ( α(s), s ) 0. V ξ is not the value function. 28

29 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Restricted rates with ξ > µ. III Let now τ := inf{t 0 : r t = 0, r 0 = r < 0} ϱ := inf{t 0 : r t = 0, r 0 = r > 0} and [ ] G(r, x) := E e U τ r (x + µτ + C) [ F (r, x) := E ξ 0 + e U r ϱ x ξ µ ϱ e U s r e U r s ξ µ ϱ ϱ ds + µ x ( ( x ) x + (µ ξ) ξ µ ϱ + C ds )] 29

30 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Restricted rates with ξ > µ. IV Then: G(0, x) = F (0, x) and G x (0, x) = F x (0, x) = 1 for all x 0. [ G x (r, x) = E F x (r, x) = E e U r τ [e U r x ] > 1 for r < 0, x R + and ξ µ ϱ ] < 1 for r > 0, x R +. 30

31 Maximizing Discounted Consumption under a Deterministic Income, Ornstein-Uhlenbeck Process as an Interest Rate Unrestricted Consumption HJB } max {µv x + a( b r)v r + σ2 2 V rr rv, 1 V x = 0. Using the same notation like above, we let (x ] G(r, x) = E[ + µτ + G(0, 0))e Uτ r for r 0, [ ϱ ] F (r, x) = x + E µ e U s r ds + G(0, r 0)e U ϱ for r 0. 0 Obviously, G x (r, x) = E [ e U r τ ] > 1 for r < 0 and Fx (r, x) = 1. Then, it is clear G(0, x) = F (0, x) and G x (0, x) = F x (0, x). 31

32 Literature Asmussen, S. and Taksar, M. (1997): Controlled diusion models for optimal dividend pay-out. Insurance: Mathematics and Economics 20, pp Borodin, A.N. and Salminen, P. (1998): Handbook of Brownian Motion Facts and Formulae. Birkhäuser Verlag, Basel. Brigo, D. and Mercurio, F. (2006): Interest Rate Models - Theory and Practice. Springer, Heidelberg. 2 edition Shreve, S.E., Lehoczky, J.P. and Gaver, D.P.: (1984). Optimal consumption for general diusions with absorbing and reecting barriers. SIAM J. Control and Optimization 22,