A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond

Transcription

1 A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond Bogdan IFTIMIE "Simion Stoilow" Institute of Mathematics of the Romanian Academy October 7, 2011, IMAR, Bucharest Conferinţa internaţională: Creştere economică şi sustenabilitate socială. Provocări şi perspective europene. Contractul de finanţare nr. POSDRU/89/1.5/S/62988 CERBUN "Cercetarea Ştiinţifică Economică, Suport al Bunăstării şi Dezvoltării Umane în Context European" Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond1 / 261/2

2 Introduction My interest arises in the study of a portfolio optimization problem in a financial market, with investment options in: 1 a money market account; 2 a stock; 3 a corporative bond (which may default at some random time τ). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond2 / 262/2

3 Introduction My interest arises in the study of a portfolio optimization problem in a financial market, with investment options in: 1 a money market account; 2 a stock; 3 a corporative bond (which may default at some random time τ). Main idea: to decompose the the original optimization problem (which is posed in an incomplete market, due to the jump of the wealth process at default) into two subproblems in complete markets: a pre-default problem (considered till the default time); a post-default problem (after the post-default time). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond2 / 262/2

4 Introduction My interest arises in the study of a portfolio optimization problem in a financial market, with investment options in: 1 a money market account; 2 a stock; 3 a corporative bond (which may default at some random time τ). Main idea: to decompose the the original optimization problem (which is posed in an incomplete market, due to the jump of the wealth process at default) into two subproblems in complete markets: a pre-default problem (considered till the default time); a post-default problem (after the post-default time). The post-default problem is stated in the Merton framework (with no default) and thus the corresponding optimal portfolio should be given by the Merton optimal portfolio. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond2 / 262/2

5 Introduction My interest arises in the study of a portfolio optimization problem in a financial market, with investment options in: 1 a money market account; 2 a stock; 3 a corporative bond (which may default at some random time τ). Main idea: to decompose the the original optimization problem (which is posed in an incomplete market, due to the jump of the wealth process at default) into two subproblems in complete markets: a pre-default problem (considered till the default time); a post-default problem (after the post-default time). The post-default problem is stated in the Merton framework (with no default) and thus the corresponding optimal portfolio should be given by the Merton optimal portfolio. In what follows, we shall consider the cases of CRRA utility functions logarithmic utility U(x) = ln(x); power utility U(x) = xγ γ, for 0 < γ < 1. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond2 / 262/2

6 Introduction My interest arises in the study of a portfolio optimization problem in a financial market, with investment options in: 1 a money market account; 2 a stock; 3 a corporative bond (which may default at some random time τ). Main idea: to decompose the the original optimization problem (which is posed in an incomplete market, due to the jump of the wealth process at default) into two subproblems in complete markets: a pre-default problem (considered till the default time); a post-default problem (after the post-default time). The post-default problem is stated in the Merton framework (with no default) and thus the corresponding optimal portfolio should be given by the Merton optimal portfolio. In what follows, we shall consider the cases of CRRA utility functions logarithmic utility U(x) = ln(x); power utility U(x) = xγ γ, for 0 < γ < 1. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond2 / 262/2

7 The free-default market Consider a probability space (Ω, F, P) endowded a filtration (F t ), which is the default-free market filtration (it is also called the reference filtration); it stands for the natural filtration generated by a two-dimensional standard Brownian motion W(t) := (W 1 (t), W 2 (t)). Here 1 W 1 (t) stands for the source of randomness of the default-free market; 2 W 2 (t) stands for the source of randomness generated by the defaultable asset. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond3 / 263/2

8 The free-default market Consider a probability space (Ω, F, P) endowded a filtration (F t ), which is the default-free market filtration (it is also called the reference filtration); it stands for the natural filtration generated by a two-dimensional standard Brownian motion W(t) := (W 1 (t), W 2 (t)). Here 1 W 1 (t) stands for the source of randomness of the default-free market; 2 W 2 (t) stands for the source of randomness generated by the defaultable asset. The dynamics of the money market account are given by dr(t) = R(t)r(t)dt, (1) Assume that the short rate r(t) is stochastic and follows a Hull-White process dr(t) = (a 1 (t) b 1 (t)r(t))dt + σ 1 (t)dw 1 (t). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond3 / 263/2

9 The free-default market Consider a probability space (Ω, F, P) endowded a filtration (F t ), which is the default-free market filtration (it is also called the reference filtration); it stands for the natural filtration generated by a two-dimensional standard Brownian motion W(t) := (W 1 (t), W 2 (t)). Here 1 W 1 (t) stands for the source of randomness of the default-free market; 2 W 2 (t) stands for the source of randomness generated by the defaultable asset. The dynamics of the money market account are given by dr(t) = R(t)r(t)dt, (1) Assume that the short rate r(t) is stochastic and follows a Hull-White process dr(t) = (a 1 (t) b 1 (t)r(t))dt + σ 1 (t)dw 1 (t). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond3 / 263/2

10 The setting The stock price process is a geometric Brownian motion with time-dependent coefficients ds(t) = S(t)(µ(t)dt + σ(t)dw 1 (t)); S(0) = S 0. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond4 / 264/2

11 The setting The stock price process is a geometric Brownian motion with time-dependent coefficients ds(t) = S(t)(µ(t)dt + σ(t)dw 1 (t)); S(0) = S 0. We adopt the reduced form approach for the defaultable asset, i.e. the bond may default at some random time τ which is not a stopping time with respect to the default-free market filtration (F t ). It satisfies P(τ = 0) = 0 (the default cannot arrive at the initial time); For any 0 < t < T, P(τ > t) > 0 (default can arrive at any time till maturity). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond4 / 264/2

12 The setting The stock price process is a geometric Brownian motion with time-dependent coefficients ds(t) = S(t)(µ(t)dt + σ(t)dw 1 (t)); S(0) = S 0. We adopt the reduced form approach for the defaultable asset, i.e. the bond may default at some random time τ which is not a stopping time with respect to the default-free market filtration (F t ). It satisfies P(τ = 0) = 0 (the default cannot arrive at the initial time); For any 0 < t < T, P(τ > t) > 0 (default can arrive at any time till maturity). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond4 / 264/2

13 The defaultable asset Define the default indicator process H t := 1 (τ t) ; the filtration generated by the default indicator process, H t := σ(h s ; 0 s t) the minimal filtration with respect to which τ is a stopping time; the enlarged filtration (called also the full filtration) G t := F t H t. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond5 / 265/2

14 The defaultable asset Define the default indicator process H t := 1 (τ t) ; the filtration generated by the default indicator process, H t := σ(h s ; 0 s t) the minimal filtration with respect to which τ is a stopping time; the enlarged filtration (called also the full filtration) G t := F t H t. Remark Clearly τ is a stopping time with respect to the enlarged filtration! Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond5 / 265/2

15 The defaultable asset Define the default indicator process H t := 1 (τ t) ; the filtration generated by the default indicator process, H t := σ(h s ; 0 s t) the minimal filtration with respect to which τ is a stopping time; the enlarged filtration (called also the full filtration) G t := F t H t. Remark Clearly τ is a stopping time with respect to the enlarged filtration! Let Q be a risk-neutral probability (it will be specified later). Set F t := Q(τ t F t ). Then F t is clearly a bounded non-negative F t - submartingale. According to the Doob-Meyer decomposition it can be written as the sum of a martingale and an increasing process. Assume that the martingale part is 0. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond5 / 265/2

16 The defaultable asset If F t is absolutely continuous with respect to the Lebesque measure, then F t = Q(τ t F t ) = where (f t ) is a non-negative F t -adapted process. t 0 f s ds, Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond6 / 266/2

17 The defaultable asset If F t is absolutely continuous with respect to the Lebesque measure, then F t = Q(τ t F t ) = t 0 f s ds, where (f t ) is a non-negative F t -adapted process. Define Γ t := ln(1 F t ). Γ t is called the hazard process of τ under Q, conditionally on F t. Since F t is increasing, then Γ t is also increasing and Γ t = t 0 λ sds. λ t is called the conditional hazard rate of τ given the free default filtration. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond6 / 266/2

18 The defaultable asset If F t is absolutely continuous with respect to the Lebesque measure, then F t = Q(τ t F t ) = t 0 f s ds, where (f t ) is a non-negative F t -adapted process. Define Γ t := ln(1 F t ). Γ t is called the hazard process of τ under Q, conditionally on F t. Since F t is increasing, then Γ t is also increasing and Γ t = t 0 λ sds. λ t is called the conditional hazard rate of τ given the free default filtration. We have the following formula relating λ t to f t λ t = 1 F t f t. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond6 / 266/2

19 The defaultable asset If F t is absolutely continuous with respect to the Lebesque measure, then F t = Q(τ t F t ) = t 0 f s ds, where (f t ) is a non-negative F t -adapted process. Define Γ t := ln(1 F t ). Γ t is called the hazard process of τ under Q, conditionally on F t. Since F t is increasing, then Γ t is also increasing and Γ t = t 0 λ sds. λ t is called the conditional hazard rate of τ given the free default filtration. We have the following formula relating λ t to f t λ t = 1 F t f t. We assume that λ(t) follows a Hull-White process dλ t = (a 2 (t) b 2 (t)λ t )dt + σ 1 (t)dw 1 (t) + σ 2 (t)dw 2 (t). (2) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond6 / 266/2

20 A Remark Remark It is usually assumed that market risk and default risk are correlated, so it would be convenient to represent f (t) as dλ t = (a 2 (t) b 2 (t)λ t )dt + σ 1 (t)db(t), (3) where the Brownian motions W 1 (t) and B(t) are correlated and let ρ(t) := E(W 1 (t)b(t)) = W 1, B t their cross variation process which it is assumed deterministic in the subsequent. Notice that this leads to the same formulation, since if we define W 1 := W 1 (t) and W 2 (t) := t 0 1 t 1 ρ 2 (s) db(s) ρ(s) 1 ρ 2 (s) dw 1(s), 0 Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond7 / 267/2

21 A Remark Remark It is usually assumed that market risk and default risk are correlated, so it would be convenient to represent f (t) as dλ t = (a 2 (t) b 2 (t)λ t )dt + σ 1 (t)db(t), (3) where the Brownian motions W 1 (t) and B(t) are correlated and let ρ(t) := E(W 1 (t)b(t)) = W 1, B t their cross variation process which it is assumed deterministic in the subsequent. Notice that this leads to the same formulation, since if we define W 1 := W 1 (t) and W 2 (t) := t 0 1 t 1 ρ 2 (s) db(s) ρ(s) 1 ρ 2 (s) dw 1(s), 0 Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond7 / 267/2

22 Recovery at Default then W 1 (t) and W 2 (t) are independent standard Brownian motions. We would get then for λ(t) a similar dynamics similar to (3). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond8 / 268/2

23 Recovery at Default then W 1 (t) and W 2 (t) are independent standard Brownian motions. We would get then for λ(t) a similar dynamics similar to (3). We adopt the recovery rate at default given by the recovery of the market value at default RMV (see Duffie and Singleton (1999)). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond8 / 268/2

24 Recovery at Default then W 1 (t) and W 2 (t) are independent standard Brownian motions. We would get then for λ(t) a similar dynamics similar to (3). We adopt the recovery rate at default given by the recovery of the market value at default RMV (see Duffie and Singleton (1999)). At the time of occurence of the default (if it occurs) the bond cesses to exist and the holder of the bond receives a compensation z(t) given by a proportion of the pre-default value of the bond where z(t) = (1 L(t))D(t, T), D(t, T) stands for the value of the bond at time t (it has a jump at the default time τ) and D(τ ) stands for the value prior to default; L(t) stands for the loss-rate. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond8 / 268/2

25 Price of the defaultable bond The price at time t of a defaultable zero-coupon bond with maturity T and recovery z(t) is given by (see Bielecki and Rutkovski (2004)) ( D(t, T) = E Q 1 (τ>t) e T t r udu X + 1 (t<τ T) e ) τ t r uds Gt z τ ( = 1 (τ>t) E Q e T T t (r u+λ u)du X + e s t ) t (ru+λu)du z s λ s ds F t, where E Q stands for the expectation with respect to the probability measure Q. (4) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond9 / 269/2

26 Price of the defaultable bond The price at time t of a defaultable zero-coupon bond with maturity T and recovery z(t) is given by (see Bielecki and Rutkovski (2004)) ( D(t, T) = E Q 1 (τ>t) e T t r udu X + 1 (t<τ T) e ) τ t r uds Gt z τ ( = 1 (τ>t) E Q e T T t (r u+λ u)du X + e s t ) t (ru+λu)du z s λ s ds F t, where E Q stands for the expectation with respect to the probability measure Q. In the case of recovery of the market value at default D(t, T) = 1 (τ>t) E Q ( e T t (r s+λ sl s)ds) X F t ) := 1 (τ>t) B(t, T), (5) where B(t, T) can be viewed as the pre-default value of the bond and may be seen as the value of a non-defaultable bond with default-risk adjusted interest rate ˆr t := r t + λ t L t ; credit spread given by ˆr t r t = λ t L t. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond9 / 269/2 (4)

27 The wealth process Consider an investor who can invest in the assets specified from above. We denote by N R (t), N S (t) and N D (t) the quantity of each asset (money market, stock respectively defaultable bond) detained by the investor at time t. N R (t) and N S (t) are assumed (F t ) predictable processes and N D (t) a (G t ) predictable process. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 10 / 2610

28 The wealth process Consider an investor who can invest in the assets specified from above. We denote by N R (t), N S (t) and N D (t) the quantity of each asset (money market, stock respectively defaultable bond) detained by the investor at time t. N R (t) and N S (t) are assumed (F t ) predictable processes and N D (t) a (G t ) predictable process. The wealth process is given by X(t) = N R (t)r(t) + N S (t)s(t) + N D (t)d(t, T) and is assumed self-financed, which means that dx(t) = N R (t)dr(t) + N S (t)ds(t) + N D (t)dd(t, T). (6) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 10 / 2610

29 The wealth process Consider an investor who can invest in the assets specified from above. We denote by N R (t), N S (t) and N D (t) the quantity of each asset (money market, stock respectively defaultable bond) detained by the investor at time t. N R (t) and N S (t) are assumed (F t ) predictable processes and N D (t) a (G t ) predictable process. The wealth process is given by X(t) = N R (t)r(t) + N S (t)s(t) + N D (t)d(t, T) and is assumed self-financed, which means that dx(t) = N R (t)dr(t) + N S (t)ds(t) + N D (t)dd(t, T). (6) Set π R (t), π S (t) and π D (t) the corresponding fractions of wealth, i.e. π R (t) := N R(t)R(t), π S (t) := N S(t)S(t) X(t ) X(t ), π D(t) := N D(t)D(t, T). X(t ) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 10 / 2610

30 Decomposition of the investment strategy The self-financing condition imposed on the portfolio reads ( dx π (t) = X π (t ) π R (t) dr(t) R(t) + π S(t) ds(t) ) S(t) + π dd(t, T) D(t). (7) D(t, T) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 11 / 2611

31 Decomposition of the investment strategy The self-financing condition imposed on the portfolio reads ( dx π (t) = X π (t ) π R (t) dr(t) R(t) + π S(t) ds(t) ) S(t) + π dd(t, T) D(t). (7) D(t, T) The strategy π(t) adopted by the investor can be decomposed in a pre-default strategy π(t) (for t < τ) and a post-default strategy π(t) (for t > τ), according to which the wealth process evolves as Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 11 / 2611

32 Decomposition of the investment strategy The self-financing condition imposed on the portfolio reads ( dx π (t) = X π (t ) π R (t) dr(t) R(t) + π S(t) ds(t) ) S(t) + π dd(t, T) D(t). (7) D(t, T) The strategy π(t) adopted by the investor can be decomposed in a pre-default strategy π(t) (for t < τ) and a post-default strategy π(t) (for t > τ), according to which the wealth process evolves as ( dx π (t) = X π (t) (1 π S (t) π D (t))r(t)dt + π S (t)µ(t)dt 1 ) (8) + π S (t)σ(t)dw 1 (t) + π D (t) B(t, T) db(t, T), for t < τ, Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 11 / 2611

33 Decomposition of the investment strategy The self-financing condition imposed on the portfolio reads ( dx π (t) = X π (t ) π R (t) dr(t) R(t) + π S(t) ds(t) ) S(t) + π dd(t, T) D(t). (7) D(t, T) The strategy π(t) adopted by the investor can be decomposed in a pre-default strategy π(t) (for t < τ) and a post-default strategy π(t) (for t > τ), according to which the wealth process evolves as ( dx π (t) = X π (t) (1 π S (t) π D (t))r(t)dt + π S (t)µ(t)dt 1 ) (8) + π S (t)σ(t)dw 1 (t) + π D (t) B(t, T) db(t, T), for t < τ, respectively dx π (t) = X π (t) [ (1 π S (t))r(t)dt + π S (t)µ(t)dt + π S (t)σ(t)dw 1 (t) ], for t τ. (9) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 11 / 2611

34 Decomposition of the investment strategy The self-financing condition imposed on the portfolio reads ( dx π (t) = X π (t ) π R (t) dr(t) R(t) + π S(t) ds(t) ) S(t) + π dd(t, T) D(t). (7) D(t, T) The strategy π(t) adopted by the investor can be decomposed in a pre-default strategy π(t) (for t < τ) and a post-default strategy π(t) (for t > τ), according to which the wealth process evolves as ( dx π (t) = X π (t) (1 π S (t) π D (t))r(t)dt + π S (t)µ(t)dt 1 ) (8) + π S (t)σ(t)dw 1 (t) + π D (t) B(t, T) db(t, T), for t < τ, respectively dx π (t) = X π (t) [ (1 π S (t))r(t)dt + π S (t)µ(t)dt + π S (t)σ(t)dw 1 (t) ], for t τ. (9) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 11 / 2611

35 Admissible portfolios The jump of the wealth process in the default time τ is X π (τ) = X π (τ) X π (τ ) = N D (τ)z(τ) N D (τ )B(τ, T) = N D (τ)l(τ)b(τ, T) = X π (τ )π D (τ)l(τ), by the left-continuity of π D (t) and the continuity of B(t, T). Then X π (τ) = X π (τ) + X π (τ ) = X π (τ ) (1 π D (τ)l(τ)). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 12 / 2612

36 Admissible portfolios The jump of the wealth process in the default time τ is X π (τ) = X π (τ) X π (τ ) = N D (τ)z(τ) N D (τ )B(τ, T) = N D (τ)l(τ)b(τ, T) = X π (τ )π D (τ)l(τ), by the left-continuity of π D (t) and the continuity of B(t, T). Then X π (τ) = X π (τ) + X π (τ ) = X π (τ ) (1 π D (τ)l(τ)). The set A(x) of the admissible portfolios is determined by the bounded and left-continuous portfolio processes (π(t)) 0 t T such that all the integrals appearing in the formulas (8) and (9) are well defined; the initial endownement is given by the positive amount x, X π (0) = x; the wealth remains positive during the investment process, i.e. for each t [0, T], X π (t) 0, P a.s. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 12 / 2612

37 The optimization problem Our interest to maximize the expected utility (under the historical probability P) of the investor from the final wealth over the class A(x) of admissible portfolios. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 13 / 2613

38 The optimization problem Our interest to maximize the expected utility (under the historical probability P) of the investor from the final wealth over the class A(x) of admissible portfolios. The optimization problem is stated as V(x) := sup E[U(X π (T))] = sup J(π), (10) π A(x) π A(x) where the utility function U : (0, ) R describing the preferences of the investor is a strictly increasing, strictly concave and continuously differentiable function; satisfies the Inada conditions: lim x 0 U (x) = + and lim x U (x) = 0. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 13 / 2613

39 The optimization problem Our interest to maximize the expected utility (under the historical probability P) of the investor from the final wealth over the class A(x) of admissible portfolios. The optimization problem is stated as V(x) := sup E[U(X π (T))] = sup J(π), (10) π A(x) π A(x) where the utility function U : (0, ) R describing the preferences of the investor is a strictly increasing, strictly concave and continuously differentiable function; satisfies the Inada conditions: lim x 0 U (x) = + and lim x U (x) = 0. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 13 / 2613

40 Related citations Continuous-time portfolio optimization problems were studied starting with the papers of Merton ((1969), (1971), (1973)). Other significant contributions in the case of complete financial markets: Karatzas, Lehoczky and Shreve (1987); Korn and Kraft (2001) - where the interest rate is stochastic; Blanchet-Scaillet, El Karoui, Jeanblanc and Martellini (2008) - where the time-horizon is random. in the case of incomplete financial markets: Kramkov and Schachermayer (1999), Jiao and Pham (2010) and Lim and Quenez (2010) - in a market with counterparty default risk. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 14 / 2614

41 Related citations Continuous-time portfolio optimization problems were studied starting with the papers of Merton ((1969), (1971), (1973)). Other significant contributions in the case of complete financial markets: Karatzas, Lehoczky and Shreve (1987); Korn and Kraft (2001) - where the interest rate is stochastic; Blanchet-Scaillet, El Karoui, Jeanblanc and Martellini (2008) - where the time-horizon is random. in the case of incomplete financial markets: Kramkov and Schachermayer (1999), Jiao and Pham (2010) and Lim and Quenez (2010) - in a market with counterparty default risk. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 14 / 2614

42 Approaches There are two possible approaches 1 the dynamic programming approach (as a tool of stochastic control theory), leading to, in the case of complete markets to some nonlinear PDE the Hamilton-Jacobi-Bellman equation (which generally is not easy to solve); incomplete markets to some BSDE; 2 the martingale approach using convex duality arguments (using the properties of the convex dual of U). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 15 / 2615

43 The convex dual of U In this spirit of the martingale approach, define the convex dual function of U U (x) := sup(u(y) xy). (11) y>0 Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 16 / 2616

44 The convex dual of U In this spirit of the martingale approach, define the convex dual function of U Remark U (x) := sup(u(y) xy). (11) y>0 The function U (x) stands for the Legendre transform of the function U( y). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 16 / 2616

45 The convex dual of U In this spirit of the martingale approach, define the convex dual function of U Remark U (x) := sup(u(y) xy). (11) y>0 The function U (x) stands for the Legendre transform of the function U( y). Under the assumptions imposed on U, U is invertible; if I := (U ) 1 then (U ) = I; the supremum in the formula (11) is attained for y = I(x), which leads to for any x, y > 0. U(y) xy U(I(x)) xi(x), Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 16 / 2616

46 A result of Kramkov and Schachermayer Using Theorem 2.2 from Kramkov and Schachermayer (1999), we know that the optimization problem (10) admitts a solution under the assumptions (i) The asymptotic elasticity of the utility function U(x) satisfies xu (x) AE(U) := lim sup x U(x) < 1; (ii) There exist at least an equivalent local martingale measure, i.e. a probability measure Q equivalent with P under which the discounted wealth process is a (local) martingale; (iii) for some x > 0 the value function V(x) is finite. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 17 / 2617

47 A result of Kramkov and Schachermayer Using Theorem 2.2 from Kramkov and Schachermayer (1999), we know that the optimization problem (10) admitts a solution under the assumptions (i) The asymptotic elasticity of the utility function U(x) satisfies xu (x) AE(U) := lim sup x U(x) < 1; (ii) There exist at least an equivalent local martingale measure, i.e. a probability measure Q equivalent with P under which the discounted wealth process is a (local) martingale; (iii) for some x > 0 the value function V(x) is finite. The assumption (i) is obviously satisfied for our choices of CARA utility functions (power utility and logarithmic utility). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 17 / 2617

48 An existence result Next, it can be shown that the price of the defaultable bond satisfies t D(t, T) = H t B(t, T) = D(t, T)(r t λ t (1 L t ))dt + H t e 0 ˆr(s)ds p(t)dw(t) where H t := 1 (τ>t) = 1 H t ; recall that ˆr t := r t + λ t L t ; B(t, T)dM t, (12) p(t) appears when we apply the Representation of Brownian Martingales Theorem for the process m t := E Q [e T 0 ˆr(s)ds X F t ], i.e. dm t = p(t)dw(t); The process M t := H t t τ 0 λ s ds is a G martingale. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 18 / 2618

49 An existence result Next, it can be shown that the price of the defaultable bond satisfies t D(t, T) = H t B(t, T) = D(t, T)(r t λ t (1 L t ))dt + H t e 0 ˆr(s)ds p(t)dw(t) where H t := 1 (τ>t) = 1 H t ; recall that ˆr t := r t + λ t L t ; B(t, T)dM t, (12) p(t) appears when we apply the Representation of Brownian Martingales Theorem for the process m t := E Q [e T 0 ˆr(s)ds X F t ], i.e. dm t = p(t)dw(t); The process M t := H t t τ 0 λ s ds is a G martingale. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 18 / 2618

50 An existence result With a properly defined random variable F T measurable random variable Z (via a stochastic exponential), the probability Q which is absolutely continuous with respect to the historical probability P, having the Radon Nikodym density Z is such that, under Q, the discounted price of the defaultable asset, e t 0 r(s)ds D(t, T) becomes a local martingale. This leads clearly to (ii). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 19 / 2619

51 An existence result With a properly defined random variable F T measurable random variable Z (via a stochastic exponential), the probability Q which is absolutely continuous with respect to the historical probability P, having the Radon Nikodym density Z is such that, under Q, the discounted price of the defaultable asset, e t 0 r(s)ds D(t, T) becomes a local martingale. This leads clearly to (ii). Next, according to Theorem 2.2 from Kramkov and Schachermayer (1999), V(x) is finite for some positive x if the conjugate function of the value function V, denoted V, is finite at y = V (x). A sufficient condition for the last assertion to hold is which we assume in the subsequent. E[U (yz)] <, for some y > 0, (13) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 19 / 2619

52 An existence result With a properly defined random variable F T measurable random variable Z (via a stochastic exponential), the probability Q which is absolutely continuous with respect to the historical probability P, having the Radon Nikodym density Z is such that, under Q, the discounted price of the defaultable asset, e t 0 r(s)ds D(t, T) becomes a local martingale. This leads clearly to (ii). Next, according to Theorem 2.2 from Kramkov and Schachermayer (1999), V(x) is finite for some positive x if the conjugate function of the value function V, denoted V, is finite at y = V (x). A sufficient condition for the last assertion to hold is which we assume in the subsequent. E[U (yz)] <, for some y > 0, (13) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 19 / 2619

53 An existence result We are now in position to state an existence result Theorem Under our standing assumptions, the optimization problem (10) has a solution. Remark Theorem 2.2 from Kramkov and Schachermayer (1999) allows us to provide a dual characterization of the value function in (10) and the associated optimal portfolio but not to obtain explicit formulas! Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 20 / 2620

54 An existence result We are now in position to state an existence result Theorem Under our standing assumptions, the optimization problem (10) has a solution. Remark Theorem 2.2 from Kramkov and Schachermayer (1999) allows us to provide a dual characterization of the value function in (10) and the associated optimal portfolio but not to obtain explicit formulas! Our next goal is to characterize the optimal portfolio for our choices of utility functions. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 20 / 2620

55 Characterization of the optimal portfolio The problem (10) is equivalent with the problem V(x) = sup π E[V(τ, (X π (τ))] = sup E[V(τ, X π (τ )(1 π D (τ)l(τ)))]. (14) π Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 21 / 2621

56 Characterization of the optimal portfolio The problem (10) is equivalent with the problem V(x) = sup π E[V(τ, (X π (τ))] = sup E[V(τ, X π (τ )(1 π D (τ)l(τ)))]. (14) π Remark We thus have to solve first the post-default optimization problem and the solution of the pre-default problem will depend on the solution on the former. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 21 / 2621

57 Characterization of the optimal portfolio The problem (10) is equivalent with the problem V(x) = sup π E[V(τ, (X π (τ))] = sup E[V(τ, X π (τ )(1 π D (τ)l(τ)))]. (14) π Remark We thus have to solve first the post-default optimization problem and the solution of the pre-default problem will depend on the solution on the former. Recall that the post-default value process satisfies dx π (t) = X π (t) [r(t) + π S (t)(µ(t) r(t))] dt + π S (t)σ(t)dw 1 (t), (15) for t > τ. Then ( t X π (t) = X π ( (τ) exp r(s) + πs (s)(µ(s) r(s)) 1 τ 2 π2 S(s)σ 2 (s) ) ) ds ( t ) exp π S (s)σ(s)dw 1 (s). τ (16) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 21 / 2621

58 Logarithmic utility Let U(x) = ln(x). Then U(X π (T)) = ln(x π (τ)) + T 1 2 π2 S(t)σ 2 (t) ) dt + M t, τ ( r(t) + πs (t)(µ(t) r(t)) (17) where (M t ) is a (local) martingale (is a stochastic integral). ( E (U(X π (T))) = E (ln(x π T (τ))) + E 1 2 π2 S(t)σ 2 (t) ) ) dt. τ ( r(t) + πs (t)(µ(t) r(t)) (18) The second term on the r.h.s. of the last formula attains its maximum for π µ(t) r(t) S(t) = σ 2, (t) which is exactly the Merton s optimal strategy. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 22 / 2622

59 Logarithmic utility Next, we need to maximize E (ln(x π (τ))) = E ( ln ( X π (τ)(1 π D (τ)l(τ)) )), where X(t) is the solution of ( d X π (t) = X π (t) (1 π S (t) π D (t))r(t)dt + π S (t)µ(t)dt 1 ) + π S (t)σ(t)dw 1 (t) + π D (t) B(t, T) db(t, T), for 0 t T. (19) Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 23 / 2623

60 Logarithmic utility Next, we need to maximize E (ln(x π (τ))) = E ( ln ( X π (τ)(1 π D (τ)l(τ)) )), where X(t) is the solution of ( d X π (t) = X π (t) (1 π S (t) π D (t))r(t)dt + π S (t)µ(t)dt 1 ) + π S (t)σ(t)dw 1 (t) + π D (t) B(t, T) db(t, T), for 0 t T. (19) This is a maximization problem with uncertain time-horizon, for which we may apply the results of Blanchet-Scaillet, El Karoui, Jeanblanc and Martellini (2008). Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 23 / 2623

61 Power utility We look now at the case U(x) = xγ γ, with 0 < γ < 1. We have U(X π (T)) = (Xπ (τ)) γ ( T exp γ (r(t) + π S (t)(µ(t) r(t)) 1 γ τ 2 π2 S(t)σ 2 (t) ) ) dt ( T ) exp γ π S (s)σ(s)dw 1 (s). τ (20) If the interest rate r(t) is deterministic, by the so called Change-of-Measure Device (see Theorem 4.1 in Korn and Seifried (2009)), we know how to compute the supremum of the first exponential term in the last formula, while for the first one we could still apply the results of Blanchet-Scaillet, El Karoui, Jeanblanc and Martellini (2008). In the case of a stochastic interest rate, we think that the Change-of-Measure Device for semimartingales (see Section 3 in Seifried ( 2010)) could be a usefull tool. Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 24 / 2624

62 References T. Bielecki, M. Rutkovski, Credit Risk: Modeling, Valuation and Hedging, Springer Finance, C. Blanchet-Scaillet, N. El Karoui, M. Jeanblanc, L. Martellini, Optimal investment decisions when time-horizon is uncertain, Journal of Mathematical Economics 44, , Y. Jiao, H. Pham, Optimal investment with counterparty risk: a default density model approach, Finance Stochastics, published online R. Korn, H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM J. Control Optim. 40, No. 4, , R. Korn, F. T. Seifried, A worst-case approach to continuous-time portfolio optimization, Radon Series Comp. Appl. Math. 8, 1 19, Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 25 / 2625

63 References D. Kramkov, W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, The Annals of Applied Probability 9, No. 3, , T. Lim, M.-C. Quenez, Portfolio Optimization in a default model under full/partial information, draft, F. T. Seifried, Optimal investment for worst case crash scenarios: A martingale approach, Mathematics of Operations Research 35, No. 3, Bogdan Iftimie (IMAR) A Portfolio Optimization Problem with Stochastic Interest Rate and a Defaultable Bond 26 / 26 26