Utility Maximization in Hidden Regime-Switching Markets with Default Risk

Transcription

1 Utility Maximization in Hidden Regime-Switching Markets with Default Risk José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis pages.wustl.edu/figueroa Department of Mathematics North Carolina State University Feb. 8, 2019 (joint work with Agostino Capponi from Columbia and Andrea Pascucci from Bologna)

2 Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

3 The Problem Motivation Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

4 The Problem Motivation Portfolio Optimization with Regime Switching 1 Why regime switching? Market and credit factors exhibit different behavior depending on the overall state of economy. The latter can be measured by, e.g., a macro-economic index X t and/or other type of hidden underlying macro-economic factor. 2 Portfolio Optimization has largely focused on default-free markets: Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor & Cadenillas (2009); 3 However, defaultable securities are a significant portion of the market. 4 Portfolio optimization problems with defaultable securities has focused on Brownian driven risky factors: Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang (2008), Jeanblanc & Runggaldier (2010);

5 The Problem Motivation Portfolio Optimization with Regime Switching 1 Why regime switching? Market and credit factors exhibit different behavior depending on the overall state of economy. The latter can be measured by, e.g., a macro-economic index X t and/or other type of hidden underlying macro-economic factor. 2 Portfolio Optimization has largely focused on default-free markets: Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor & Cadenillas (2009); 3 However, defaultable securities are a significant portion of the market. 4 Portfolio optimization problems with defaultable securities has focused on Brownian driven risky factors: Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang (2008), Jeanblanc & Runggaldier (2010);

6 The Problem Motivation Portfolio Optimization with Regime Switching 1 Why regime switching? Market and credit factors exhibit different behavior depending on the overall state of economy. The latter can be measured by, e.g., a macro-economic index X t and/or other type of hidden underlying macro-economic factor. 2 Portfolio Optimization has largely focused on default-free markets: Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor & Cadenillas (2009); 3 However, defaultable securities are a significant portion of the market. 4 Portfolio optimization problems with defaultable securities has focused on Brownian driven risky factors: Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang (2008), Jeanblanc & Runggaldier (2010);

7 The Problem Motivation Portfolio Optimization with Regime Switching 1 Why regime switching? Market and credit factors exhibit different behavior depending on the overall state of economy. The latter can be measured by, e.g., a macro-economic index X t and/or other type of hidden underlying macro-economic factor. 2 Portfolio Optimization has largely focused on default-free markets: Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor & Cadenillas (2009); 3 However, defaultable securities are a significant portion of the market. 4 Portfolio optimization problems with defaultable securities has focused on Brownian driven risky factors: Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang (2008), Jeanblanc & Runggaldier (2010);

8 The Problem Motivation Portfolio Optimization with Regime Switching 1 Why regime switching? Market and credit factors exhibit different behavior depending on the overall state of economy. The latter can be measured by, e.g., a macro-economic index X t and/or other type of hidden underlying macro-economic factor. 2 Portfolio Optimization has largely focused on default-free markets: Guo et al. (2005); Nagai & Runggaldier (2008); Sotomayor & Cadenillas (2009); 3 However, defaultable securities are a significant portion of the market. 4 Portfolio optimization problems with defaultable securities has focused on Brownian driven risky factors: Bielecki & Jang (2006), Bo et al. (2010), Lakner & Liang (2008), Jeanblanc & Runggaldier (2010);

9 The Problem Motivation Overview of Main Results [Capponi, F-L, & Pascucci, Power Utility Maximization in Hidden Regime-Switching Markets with Default Risk, Finance & Stochastic, 2015] 1 Solved Merton s utility maximization problem from wealth (without consumption) in finite-horizon with (1) a risk-free asset, (2) a risky asset, and (3) a defaultable security, under a Markovian model driven by a hidden regime-switching factor; 2 Established the Hamilton-Jacobi-Bellman (HJB) PDE Equations for the optimal value function of the problem under a power utility function U(v) = 1 γ v γ, γ (0, 1); 3 Found explicit representations for the optimal allocation positions. 4 Proved existence and uniqueness of classical solutions to the HJB equations; 5 Obtained Verification Theorems for the HJB equations;

10 The Problem Motivation Overview of Main Results [Capponi, F-L, & Pascucci, Power Utility Maximization in Hidden Regime-Switching Markets with Default Risk, Finance & Stochastic, 2015] 1 Solved Merton s utility maximization problem from wealth (without consumption) in finite-horizon with (1) a risk-free asset, (2) a risky asset, and (3) a defaultable security, under a Markovian model driven by a hidden regime-switching factor; 2 Established the Hamilton-Jacobi-Bellman (HJB) PDE Equations for the optimal value function of the problem under a power utility function U(v) = 1 γ v γ, γ (0, 1); 3 Found explicit representations for the optimal allocation positions. 4 Proved existence and uniqueness of classical solutions to the HJB equations; 5 Obtained Verification Theorems for the HJB equations;

11 The Problem Motivation Overview of Main Results [Capponi, F-L, & Pascucci, Power Utility Maximization in Hidden Regime-Switching Markets with Default Risk, Finance & Stochastic, 2015] 1 Solved Merton s utility maximization problem from wealth (without consumption) in finite-horizon with (1) a risk-free asset, (2) a risky asset, and (3) a defaultable security, under a Markovian model driven by a hidden regime-switching factor; 2 Established the Hamilton-Jacobi-Bellman (HJB) PDE Equations for the optimal value function of the problem under a power utility function U(v) = 1 γ v γ, γ (0, 1); 3 Found explicit representations for the optimal allocation positions. 4 Proved existence and uniqueness of classical solutions to the HJB equations; 5 Obtained Verification Theorems for the HJB equations;

12 The Problem Motivation Overview of Main Results [Capponi, F-L, & Pascucci, Power Utility Maximization in Hidden Regime-Switching Markets with Default Risk, Finance & Stochastic, 2015] 1 Solved Merton s utility maximization problem from wealth (without consumption) in finite-horizon with (1) a risk-free asset, (2) a risky asset, and (3) a defaultable security, under a Markovian model driven by a hidden regime-switching factor; 2 Established the Hamilton-Jacobi-Bellman (HJB) PDE Equations for the optimal value function of the problem under a power utility function U(v) = 1 γ v γ, γ (0, 1); 3 Found explicit representations for the optimal allocation positions. 4 Proved existence and uniqueness of classical solutions to the HJB equations; 5 Obtained Verification Theorems for the HJB equations;

13 The Problem The Model Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

14 The Problem The Model Probabilitic Framework 1 (Ω, G, P): complete probability space; 2 P: real world or statistical probability measure; 3 Ft I : flow of information generated by the tradable security assets, driven by a 2-d Wiener process W := {(W (1) t, W (2) t ) } t 0 ; 4 H t : flow of information generated by the default process H t := 1 t τ, with τ denoting the default time of a vulnerable security; 5 G I = {G I t } t 0 = {F I t H t } t 0 : investor s available information; 6 F X := {F X t } t 0 : information generated by a hidden risk factor X = {X t } t 0 ; 7 G = {G t } t 0 = {G I t F X t } t 0 : all information;

15 The Problem The Model Regime Switching Risky Factor The different regimes of the market are modeled by a hidden continuous-time Markov process X = {X t } t 0 with finite state-space S; WLOG, S = {e 1,..., e N }, where e i = (0,..., 1,..., 0) is the i th -unit vector of R N. A(t) := [ϖ i,j (t)] N i,j=1 denotes the transition rates or generator of X : 1 ϖ i,j (t) = lim δ 0 δ P(X t+δ = e j X t = e i ), ϖ i,i (t) = j i ϖ i,j (t). (i j); Equivalently, the process ϕ t := X t X 0 is a P-martingale. t 0 A (s)x s ds,

16 The Problem The Model Regime Switching Market Model 1 Risk-free asset: db t = rb t dt, B 0 = 1; 2 Risky asset: ds t := µ t S t dt + σs t dw (1) t, S 0 = s, µ t := µ(x t ) := µ, X t = µ i, if X t = e i, for some pre-specified potential appreciation rates µ := (µ 1,..., µ N ) and a known constant volatility σ. 3 Vulnerable or Defaultable asset: [Linetsky (2006)] dp t := a(t, X t )P t dt + υp t dw (2) t, for t < τ, P t := 0, for t τ. for some deterministic function a : [0, ) S R +, a specified default time τ, and a known constant volatility υ.

17 The Problem The Model Regime Switching Market Model 1 Risk-free asset: db t = rb t dt, B 0 = 1; 2 Risky asset: ds t := µ t S t dt + σs t dw (1) t, S 0 = s, µ t := µ(x t ) := µ, X t = µ i, if X t = e i, for some pre-specified potential appreciation rates µ := (µ 1,..., µ N ) and a known constant volatility σ. 3 Vulnerable or Defaultable asset: [Linetsky (2006)] dp t := a(t, X t )P t dt + υp t dw (2) t, for t < τ, P t := 0, for t τ. for some deterministic function a : [0, ) S R +, a specified default time τ, and a known constant volatility υ.

18 The Problem The Model Regime Switching Market Model 1 Risk-free asset: db t = rb t dt, B 0 = 1; 2 Risky asset: ds t := µ t S t dt + σs t dw (1) t, S 0 = s, µ t := µ(x t ) := µ, X t = µ i, if X t = e i, for some pre-specified potential appreciation rates µ := (µ 1,..., µ N ) and a known constant volatility σ. 3 Vulnerable or Defaultable asset: [Linetsky (2006)] dp t := a(t, X t )P t dt + υp t dw (2) t, for t < τ, P t := 0, for t τ. for some deterministic function a : [0, ) S R +, a specified default time τ, and a known constant volatility υ.

19 The Problem The Model The Default Framework 1 We adopt the double-stochastic framework to default, where τ := inf { t R + : t 0 h u du χ }, for {h t } t 0 independent of W and χ exp(1) indep. of X and W. 2 Interpretation: The so-called hazard rate process h = {h t } t 0 is such that P (τ t + δ τ > t, F t ) = h t δ + o(δ), (δ 0). 3 Assume that h = {h t } t 0 is also determined by the state of the economy: h t := h(x t ) := h, Xt = h i, if X t = e i, for some pre-specified potential default rates h := (h 1,..., h N ). 4 A useful result: The processes H t := 1 t τ and h t are such that ξ t := H t t 0 1 u<τ h u du is a P-martingale.

20 The Problem The Model The Default Framework 1 We adopt the double-stochastic framework to default, where τ := inf { t R + : t 0 h u du χ }, for {h t } t 0 independent of W and χ exp(1) indep. of X and W. 2 Interpretation: The so-called hazard rate process h = {h t } t 0 is such that P (τ t + δ τ > t, F t ) = h t δ + o(δ), (δ 0). 3 Assume that h = {h t } t 0 is also determined by the state of the economy: h t := h(x t ) := h, Xt = h i, if X t = e i, for some pre-specified potential default rates h := (h 1,..., h N ). 4 A useful result: The processes H t := 1 t τ and h t are such that ξ t := H t t 0 1 u<τ h u du is a P-martingale.

21 The Problem The Model The Default Framework 1 We adopt the double-stochastic framework to default, where τ := inf { t R + : t 0 h u du χ }, for {h t } t 0 independent of W and χ exp(1) indep. of X and W. 2 Interpretation: The so-called hazard rate process h = {h t } t 0 is such that P (τ t + δ τ > t, F t ) = h t δ + o(δ), (δ 0). 3 Assume that h = {h t } t 0 is also determined by the state of the economy: h t := h(x t ) := h, Xt = h i, if X t = e i, for some pre-specified potential default rates h := (h 1,..., h N ). 4 A useful result: The processes H t := 1 t τ and h t are such that ξ t := H t t 0 1 u<τ h u du is a P-martingale.

22 The Problem The Model The Default Framework 1 We adopt the double-stochastic framework to default, where τ := inf { t R + : t 0 h u du χ }, for {h t } t 0 independent of W and χ exp(1) indep. of X and W. 2 Interpretation: The so-called hazard rate process h = {h t } t 0 is such that P (τ t + δ τ > t, F t ) = h t δ + o(δ), (δ 0). 3 Assume that h = {h t } t 0 is also determined by the state of the economy: h t := h(x t ) := h, Xt = h i, if X t = e i, for some pre-specified potential default rates h := (h 1,..., h N ). 4 A useful result: The processes H t := 1 t τ and h t are such that ξ t := H t t 0 1 u<τ h u du is a P-martingale.

23 The Problem Portfolio Optimization Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

24 The Problem Portfolio Optimization The problem Investor wants to maximize her expected final utility from wealth, E P (U(V T )), at a finite horizon T, with power utility U(v) = v γ, γ (0, 1), γ given an initial fixed endowment V 0 = v. Investor can dynamically allocate her financial wealth into the risk-free bank account, the stock, and the defaultable security. Investor does not have intermediate consumption nor external capital influx to support purchase of assets (self-financiability condition).

25 The Problem Portfolio Optimization The problem Investor wants to maximize her expected final utility from wealth, E P (U(V T )), at a finite horizon T, with power utility U(v) = v γ, γ (0, 1), γ given an initial fixed endowment V 0 = v. Investor can dynamically allocate her financial wealth into the risk-free bank account, the stock, and the defaultable security. Investor does not have intermediate consumption nor external capital influx to support purchase of assets (self-financiability condition).

26 The Problem Portfolio Optimization The problem Investor wants to maximize her expected final utility from wealth, E P (U(V T )), at a finite horizon T, with power utility U(v) = v γ, γ (0, 1), γ given an initial fixed endowment V 0 = v. Investor can dynamically allocate her financial wealth into the risk-free bank account, the stock, and the defaultable security. Investor does not have intermediate consumption nor external capital influx to support purchase of assets (self-financiability condition).

27 The Problem Portfolio Optimization Merton s Portfolio Optimization Problem The Wealth Dynamics: Let π t := (π B t, π S t, π P t ) represent the fractions of wealth invested in the bank account, the stock, and the defaultable security at time t. Then, the dynamics of the wealth process V π t where dv π,v t V π,v 0 = v, = V π,v t {π B t db t B t + π S t ds t S t + π P t := V π,v t dp t P t is }, t > 0, π B t + π P t + π S t = 1, π P t = 0, for t τ, (with the convention that 0/0 = 0).

28 The Problem Portfolio Optimization Merton s Portfolio Optimization Problem Optimization Problem with horizon T : ϕ T 0 (v) := sup E P [ U(V π,v T )] π A for a suitable class A of admissible strategies π = {π t } t T such that π t G I t := σ(s u, P u, H u : u t). 1 Issue: Incomplete information problem! 2 If X t were observable so that π t is allowed to depend on X t, the optimal solution would be π,s t = µ(x t) r σ 2 (1 γ), π,p t = a(t, X t) r υ 2 (1 γ) 1 {t<τ}; i.e., directly proportional to the Sharpe ratio of each asset (a measure of performance) and inversely proportional to the volatility; (Merton s Solution)

29 The Problem Portfolio Optimization Merton s Portfolio Optimization Problem Optimization Problem with horizon T : ϕ T 0 (v) := sup E P [ U(V π,v T )] π A for a suitable class A of admissible strategies π = {π t } t T such that π t G I t := σ(s u, P u, H u : u t). 1 Issue: Incomplete information problem! 2 If X t were observable so that π t is allowed to depend on X t, the optimal solution would be π,s t = µ(x t) r σ 2 (1 γ), π,p t = a(t, X t) r υ 2 (1 γ) 1 {t<τ}; i.e., directly proportional to the Sharpe ratio of each asset (a measure of performance) and inversely proportional to the volatility; (Merton s Solution)

30 The Problem Portfolio Optimization Merton s Portfolio Optimization Problem Optimization Problem with horizon T : ϕ T 0 (v) := sup E P [ U(V π,v T )] π A for a suitable class A of admissible strategies π = {π t } t T such that π t G I t := σ(s u, P u, H u : u t). 1 Issue: Incomplete information problem! 2 If X t were observable so that π t is allowed to depend on X t, the optimal solution would be π,s t = µ(x t) r σ 2 (1 γ), π,p t = a(t, X t) r υ 2 (1 γ) 1 {t<τ}; i.e., directly proportional to the Sharpe ratio of each asset (a measure of performance) and inversely proportional to the volatility; (Merton s Solution)

31 The Problem Portfolio Optimization Merton s Portfolio Optimization Problem Optimization Problem with horizon T : ϕ T 0 (v) := sup E P [ U(V π,v T )] π A for a suitable class A of admissible strategies π = {π t } t T such that π t G I t := σ(s u, P u, H u : u t). 1 Issue: Incomplete information problem! 2 If X t were observable so that π t is allowed to depend on X t, the optimal solution would be π,s t = µ(x t) r σ 2 (1 γ), π,p t = a(t, X t) r υ 2 (1 γ) 1 {t<τ}; i.e., directly proportional to the Sharpe ratio of each asset (a measure of performance) and inversely proportional to the volatility; (Merton s Solution)

32 Solution Reduction to a Complete Observation Problem Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

33 Solution Reduction to a Complete Observation Problem Ingredient 1: Change of Probability Measure 1 Σ Y := ( σ 0 0 υ ) [ ], ϑ(t, X t ) := µ(x t ) σ2 2, a(t, X t) υ Define a probability ˆP on (Ω, G, G) such that ˆP (A) = E P (ρ t 1 A ), A G t EˆP (Z) = E P (ρ t Z), Z G t, where ρ t := e t 0 ϑ(s,xs) Σ 1 Y dws 1 T 2 0 ϑ (Σ Y Σ Y ) 1 ϑ(s,x s)ds h 1 {τ t} e t τ τ 0 (1 h u)du, 3 What is special about ˆP? d[log S t, log P t ] = Σ Y dŵ t, where t t τ Ŵ t := W t + Σ 1 Y ϑ(s, X s)ds, ˆξt := H t du 0 0 are a ˆP-Wiener process and a ˆP-martingale, respectively, and, furthermore, both are independent of {X t } t 0, under ˆP.

34 Solution Reduction to a Complete Observation Problem Ingredient 1: Change of Probability Measure 1 Σ Y := ( σ 0 0 υ ) [ ], ϑ(t, X t ) := µ(x t ) σ2 2, a(t, X t) υ Define a probability ˆP on (Ω, G, G) such that ˆP (A) = E P (ρ t 1 A ), A G t EˆP (Z) = E P (ρ t Z), Z G t, where ρ t := e t 0 ϑ(s,xs) Σ 1 Y dws 1 T 2 0 ϑ (Σ Y Σ Y ) 1 ϑ(s,x s)ds h 1 {τ t} e t τ τ 0 (1 h u)du, 3 What is special about ˆP? d[log S t, log P t ] = Σ Y dŵ t, where t t τ Ŵ t := W t + Σ 1 Y ϑ(s, X s)ds, ˆξt := H t du 0 0 are a ˆP-Wiener process and a ˆP-martingale, respectively, and, furthermore, both are independent of {X t } t 0, under ˆP.

35 Solution Reduction to a Complete Observation Problem Ingredient 1: Change of Probability Measure 1 Σ Y := ( σ 0 0 υ ) [ ], ϑ(t, X t ) := µ(x t ) σ2 2, a(t, X t) υ Define a probability ˆP on (Ω, G, G) such that ˆP (A) = E P (ρ t 1 A ), A G t EˆP (Z) = E P (ρ t Z), Z G t, where ρ t := e t 0 ϑ(s,xs) Σ 1 Y dws 1 T 2 0 ϑ (Σ Y Σ Y ) 1 ϑ(s,x s)ds h 1 {τ t} e t τ τ 0 (1 h u)du, 3 What is special about ˆP? d[log S t, log P t ] = Σ Y dŵ t, where t t τ Ŵ t := W t + Σ 1 Y ϑ(s, X s)ds, ˆξt := H t du 0 0 are a ˆP-Wiener process and a ˆP-martingale, respectively, and, furthermore, both are independent of {X t } t 0, under ˆP.

36 Solution Reduction to a Complete Observation Problem Ingredient 2: Filter Probabilities 1 Recall that Gt I = σ(s u, P u, H u : u t) is the investor s available information up to time t; 2 The filter probabilities p t := (pt 1,..., pt N ) are defined as ( ) pt i := P X t = e i Gt I, i = 1,..., N. 3 Useful notation: Given a function g : D 1 {e 1,..., e N } D 2 R, with D 1 and D 2 arbitrary, possibly empty, domains, and a vector α = (α 1,..., α N ) R N, [ ĝ(y, p t, z) := E P g(y, X t, z) ] G t I = [ ˆα(p t ) := E P α, X t ] G t I = N g(y, e i, z)pt, i i=1 N α i pt. i i=1

37 Solution Reduction to a Complete Observation Problem Ingredient 2: Filter Probabilities 1 Recall that Gt I = σ(s u, P u, H u : u t) is the investor s available information up to time t; 2 The filter probabilities p t := (pt 1,..., pt N ) are defined as ( ) pt i := P X t = e i Gt I, i = 1,..., N. 3 Useful notation: Given a function g : D 1 {e 1,..., e N } D 2 R, with D 1 and D 2 arbitrary, possibly empty, domains, and a vector α = (α 1,..., α N ) R N, [ ĝ(y, p t, z) := E P g(y, X t, z) ] G t I = [ ˆα(p t ) := E P α, X t ] G t I = N g(y, e i, z)pt, i i=1 N α i pt. i i=1

38 Solution Reduction to a Complete Observation Problem Ingredient 2: Filter Probabilities 1 Recall that Gt I = σ(s u, P u, H u : u t) is the investor s available information up to time t; 2 The filter probabilities p t := (pt 1,..., pt N ) are defined as ( ) pt i := P X t = e i Gt I, i = 1,..., N. 3 Useful notation: Given a function g : D 1 {e 1,..., e N } D 2 R, with D 1 and D 2 arbitrary, possibly empty, domains, and a vector α = (α 1,..., α N ) R N, [ ĝ(y, p t, z) := E P g(y, X t, z) ] G t I = [ ˆα(p t ) := E P α, X t ] G t I = N g(y, e i, z)pt, i i=1 N α i pt. i i=1

39 Solution Reduction to a Complete Observation Problem Re-formulation of the problem 1 Changing the probability measure, E P [U (V T )] can be written as 1 γ EP [(V T ) γ ] = 1 [ γ EP (V T ) γ ρ 1 T ρ ] 1 T = EˆP [ V γ ] 1 γ T ρ 1 T =: EˆP [L T ] γ 2 By Itô formula, L t := (V t ) γ ρ 1 follows the dynamics dl t = L t t ( Q(t, X t, π t ) Σ Y dŵ t + (h(x t ) 1)d ˆξ t γη(t, X t, π t )dt ), where Q(s, e i, π s ) := ( 1 σ 2 ( 1 υ 2 ) µ i σ2 + γπs S 2 a(t, e i ) υ2 2 ) + γπ P s, η(t, e i, π t ) := r + πt S (r µ i ) + πt P (r a(t, e i )) + 1 γ π 2 tσ Y Σ Y π t

40 Solution Reduction to a Complete Observation Problem Re-formulation of the problem 1 Changing the probability measure, E P [U (V T )] can be written as 1 γ EP [(V T ) γ ] = 1 [ γ EP (V T ) γ ρ 1 T ρ ] 1 T = EˆP [ V γ ] 1 γ T ρ 1 T =: EˆP [L T ] γ 2 By Itô formula, L t := (V t ) γ ρ 1 follows the dynamics dl t = L t t ( Q(t, X t, π t ) Σ Y dŵ t + (h(x t ) 1)d ˆξ t γη(t, X t, π t )dt ), where Q(s, e i, π s ) := ( 1 σ 2 ( 1 υ 2 ) µ i σ2 + γπs S 2 a(t, e i ) υ2 2 ) + γπ P s, η(t, e i, π t ) := r + πt S (r µ i ) + πt P (r a(t, e i )) + 1 γ π 2 tσ Y Σ Y π t

41 Solution Reduction to a Complete Observation Problem Re-formulation of the problem Theorem (Capponi, F-L, & Pascussi (2015)) For any π adapted to G I = (G I t ) t (G I t = σ(s u, P u, H u : u t)), 1 [ˆL ] γ EP [(V T ) γ ] = EˆP [L T ] = EˆP T, where ( ) dl t = L t Q(t, X t, π t ) Σ Y dŵ t + (h(x t ) 1)d ˆξ t γη(t, X t, π t )dt, d ˆL t = ˆL ( t ˆQ(t, pt, π t ) Σ Y dŵt + (ĥ(p t ) 1)d ˆξ ) t γ ˆη(t, p t, π t )dt. Remark: The process ˆL only depends on the observable filter probabilities p i t = P ( X t = e i G I t ).

42 Solution Reduction to a Risk-Sensitive Control Problem Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

43 Solution Reduction to a Risk-Sensitive Control Problem Reduction to a Risk-Sensitive Control Problem 1 By Itô formula, one can decompose ˆL t as where ζ t := ζ (1) t ζ (2) t ˆL t = ζ t e γ t 0 ˆη(s,ps,πs)ds with ζ (1) and ζ (2) satisfying the SDEs: dζ (1) t = ζ (1) t ˆQ (t, p t, π t ) Σ Y dŵ t, dζ (2) t = ζ (2) t 2 Define a probability P on (Ω, G, G) such that (ĥ(pt ) 1) d ˆξ t. P (A) = EˆP (1 A ζ t ), A G t E P (Z) = EˆP (Zζ t ), Z G t. 3 In particular, EˆP [ˆL T ] = EˆP [ ζ t e γ t 0 ˆη(v,pv,πv )dv] = E P [ e γ t 0 ˆη(v,pv,πv )dv]. 4 To wit, sup π A E P [ [ U(V π,v T )] = sup E P e γ T 0 ˆη(v,pv,πv )dv], π A

44 Solution Reduction to a Risk-Sensitive Control Problem Reduction to a Risk-Sensitive Control Problem 1 By Itô formula, one can decompose ˆL t as where ζ t := ζ (1) t ζ (2) t ˆL t = ζ t e γ t 0 ˆη(s,ps,πs)ds with ζ (1) and ζ (2) satisfying the SDEs: dζ (1) t = ζ (1) t ˆQ (t, p t, π t ) Σ Y dŵ t, dζ (2) t = ζ (2) t 2 Define a probability P on (Ω, G, G) such that (ĥ(pt ) 1) d ˆξ t. P (A) = EˆP (1 A ζ t ), A G t E P (Z) = EˆP (Zζ t ), Z G t. 3 In particular, EˆP [ˆL T ] = EˆP [ ζ t e γ t 0 ˆη(v,pv,πv )dv] = E P [ e γ t 0 ˆη(v,pv,πv )dv]. 4 To wit, sup π A E P [ [ U(V π,v T )] = sup E P e γ T 0 ˆη(v,pv,πv )dv], π A

45 Solution Reduction to a Risk-Sensitive Control Problem Reduction to a Risk-Sensitive Control Problem 1 By Itô formula, one can decompose ˆL t as where ζ t := ζ (1) t ζ (2) t ˆL t = ζ t e γ t 0 ˆη(s,ps,πs)ds with ζ (1) and ζ (2) satisfying the SDEs: dζ (1) t = ζ (1) t ˆQ (t, p t, π t ) Σ Y dŵ t, dζ (2) t = ζ (2) t 2 Define a probability P on (Ω, G, G) such that (ĥ(pt ) 1) d ˆξ t. P (A) = EˆP (1 A ζ t ), A G t E P (Z) = EˆP (Zζ t ), Z G t. 3 In particular, EˆP [ˆL T ] = EˆP [ ζ t e γ t 0 ˆη(v,pv,πv )dv] = E P [ e γ t 0 ˆη(v,pv,πv )dv]. 4 To wit, sup π A E P [ [ U(V π,v T )] = sup E P e γ T 0 ˆη(v,pv,πv )dv], π A

46 Solution Reduction to a Risk-Sensitive Control Problem Reduction to a Risk-Sensitive Control Problem 1 By Itô formula, one can decompose ˆL t as where ζ t := ζ (1) t ζ (2) t ˆL t = ζ t e γ t 0 ˆη(s,ps,πs)ds with ζ (1) and ζ (2) satisfying the SDEs: dζ (1) t = ζ (1) t ˆQ (t, p t, π t ) Σ Y dŵ t, dζ (2) t = ζ (2) t 2 Define a probability P on (Ω, G, G) such that (ĥ(pt ) 1) d ˆξ t. P (A) = EˆP (1 A ζ t ), A G t E P (Z) = EˆP (Zζ t ), Z G t. 3 In particular, EˆP [ˆL T ] = EˆP [ ζ t e γ t 0 ˆη(v,pv,πv )dv] = E P [ e γ t 0 ˆη(v,pv,πv )dv]. 4 To wit, sup π A E P [ [ U(V π,v T )] = sup E P e γ T 0 ˆη(v,pv,πv )dv], π A

47 Solution Hamilton-Jacobi-Bellman Equations Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

48 Solution Hamilton-Jacobi-Bellman Equations Dynamic Optimization Problem 1 Since pt pt N = 1, it is more convenient to express the problem in terms of p t := (pt 1,..., pt N 1 ) =: ( p t 1,..., p t N 1 ) : [ sup E P e γ T 0 η(u, pv,πv )dv], where η(v, p v, π v ) := ˆη(v, ˆp v, π v ). π A 2 Feedback or Markov controls: π t = (π S t, π P t ) such that π S t := π S (t, p t, H t ) and π P t := π P (t, p t, H t ) 3 For each t [0, T ) and z {0, 1}, consider the function w(t, p, z) such that [ e w(t, p,z) = sup E P e γ T ] t η(u, p v,π v )dv p t = p, H t = z ; π A t 4 By Markov s property, [ e w(t, pt,ht) = sup E P t e γ ] T t η(v, p v,π v )dv, π A t ( [ ]) E P t [ ] := E P Gt I

49 Solution Hamilton-Jacobi-Bellman Equations Dynamic Optimization Problem 1 Since pt pt N = 1, it is more convenient to express the problem in terms of p t := (pt 1,..., pt N 1 ) =: ( p t 1,..., p t N 1 ) : [ sup E P e γ T 0 η(u, pv,πv )dv], where η(v, p v, π v ) := ˆη(v, ˆp v, π v ). π A 2 Feedback or Markov controls: π t = (π S t, π P t ) such that π S t := π S (t, p t, H t ) and π P t := π P (t, p t, H t ) 3 For each t [0, T ) and z {0, 1}, consider the function w(t, p, z) such that [ e w(t, p,z) = sup E P e γ T ] t η(u, p v,π v )dv p t = p, H t = z ; π A t 4 By Markov s property, [ e w(t, pt,ht) = sup E P t e γ ] T t η(v, p v,π v )dv, π A t ( [ ]) E P t [ ] := E P Gt I

50 Solution Hamilton-Jacobi-Bellman Equations Dynamic Optimization Problem 1 Since pt pt N = 1, it is more convenient to express the problem in terms of p t := (pt 1,..., pt N 1 ) =: ( p t 1,..., p t N 1 ) : [ sup E P e γ T 0 η(u, pv,πv )dv], where η(v, p v, π v ) := ˆη(v, ˆp v, π v ). π A 2 Feedback or Markov controls: π t = (π S t, π P t ) such that π S t := π S (t, p t, H t ) and π P t := π P (t, p t, H t ) 3 For each t [0, T ) and z {0, 1}, consider the function w(t, p, z) such that [ e w(t, p,z) = sup E P e γ T ] t η(u, p v,π v )dv p t = p, H t = z ; π A t 4 By Markov s property, [ e w(t, pt,ht) = sup E P t e γ ] T t η(v, p v,π v )dv, π A t ( [ ]) E P t [ ] := E P Gt I

51 Solution Hamilton-Jacobi-Bellman Equations Dynamic Optimization Problem 1 Since pt pt N = 1, it is more convenient to express the problem in terms of p t := (pt 1,..., pt N 1 ) =: ( p t 1,..., p t N 1 ) : [ sup E P e γ T 0 η(u, pv,πv )dv], where η(v, p v, π v ) := ˆη(v, ˆp v, π v ). π A 2 Feedback or Markov controls: π t = (π S t, π P t ) such that π S t := π S (t, p t, H t ) and π P t := π P (t, p t, H t ) 3 For each t [0, T ) and z {0, 1}, consider the function w(t, p, z) such that [ e w(t, p,z) = sup E P e γ T ] t η(u, p v,π v )dv p t = p, H t = z ; π A t 4 By Markov s property, [ e w(t, pt,ht) = sup E P t e γ ] T t η(v, p v,π v )dv, π A t ( [ ]) E P t [ ] := E P Gt I

52 Solution Hamilton-Jacobi-Bellman Equations Dynamic Programming Principle 1 Bellman s Optimality Principle An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision 2 In our case, we can break the problem as follows, for any u [t, T ), [ sup E P t e γ ] T t η(v, p v,π v )dv π A t [ = sup E P t e γ u t η(v, p v,π v )dv e γ T u η(v, pv,πv )dv] π A t [ = sup E P t e γ [ u t η(v, p v,π v )dv E P u e γ T u η(v, pv,πv )dv]] π A t = sup π A t E P t [ e γ [ u t η(v, p v,π v )dv sup E P u e γ T u η(v, pv,πv )dv]] π A u

53 Solution Hamilton-Jacobi-Bellman Equations Dynamic Programming Principle 1 Bellman s Optimality Principle An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision 2 In our case, we can break the problem as follows, for any u [t, T ), [ sup E P t e γ ] T t η(v, p v,π v )dv π A t [ = sup E P t e γ u t η(v, p v,π v )dv e γ T u η(v, pv,πv )dv] π A t [ = sup E P t e γ [ u t η(v, p v,π v )dv E P u e γ T u η(v, pv,πv )dv]] π A t = sup π A t E P t [ e γ [ u t η(v, p v,π v )dv sup E P u e γ T u η(v, pv,πv )dv]] π A u

54 Solution Hamilton-Jacobi-Bellman Equations HJB Equations 1 Therefore, we expect that [ e w(t, pt,ht) = sup E P t e w(u, pu,hu) γ ] u t η(v, p v,π v )dv, π A t 2 Next, applying Itô s Lemma to E(u, p u, H u ) := e w(u, pu,hu) (starting at t) and denoting the generator of (u, p u, H u ) is denoted by L π, 0 = sup E P t [E(t, p t, H t ) (e γ ) u t η(v, p v,π v )dv 1 π +e γ u t u η(v, p v,π v )dv t ] L π E(v, p v, H v )dv, 3 Dividing by u t and making u t, we get the HJB equations: sup [(L π γ η(t, p, π)) E(t, p, z)] = 0. π

55 Solution Hamilton-Jacobi-Bellman Equations HJB Equations 1 Therefore, we expect that [ e w(t, pt,ht) = sup E P t e w(u, pu,hu) γ ] u t η(v, p v,π v )dv, π A t 2 Next, applying Itô s Lemma to E(u, p u, H u ) := e w(u, pu,hu) (starting at t) and denoting the generator of (u, p u, H u ) is denoted by L π, 0 = sup E P t [E(t, p t, H t ) (e γ ) u t η(v, p v,π v )dv 1 π +e γ u t u η(v, p v,π v )dv t ] L π E(v, p v, H v )dv, 3 Dividing by u t and making u t, we get the HJB equations: sup [(L π γ η(t, p, π)) E(t, p, z)] = 0. π

56 Solution Hamilton-Jacobi-Bellman Equations HJB Equations 1 Therefore, we expect that [ e w(t, pt,ht) = sup E P t e w(u, pu,hu) γ ] u t η(v, p v,π v )dv, π A t 2 Next, applying Itô s Lemma to E(u, p u, H u ) := e w(u, pu,hu) (starting at t) and denoting the generator of (u, p u, H u ) is denoted by L π, 0 = sup E P t [E(t, p t, H t ) (e γ ) u t η(v, p v,π v )dv 1 π +e γ u t u η(v, p v,π v )dv t ] L π E(v, p v, H v )dv, 3 Dividing by u t and making u t, we get the HJB equations: sup [(L π γ η(t, p, π)) E(t, p, z)] = 0. π

57 Solution Post-Default and Pre-Default Analysis Agenda 1 The Problem Motivation The Model Optimal Portfolio Problem 2 Solution Reduction to a Complete Observation Problem Reduction to a Risk-Sensitive Control Problem Hamilton-Jacobi-Bellman Equations Post-Default and Pre-Default Analysis 3 Numerical example

58 Solution Post-Default and Pre-Default Analysis Post-Default Problem 1 The HJB equation for w(t, p) := w(t, p, 1) takes the form: w t tr(κ κ D 2 p w) + 1 2(1 γ) ( pw)κ κ ( p w) + ( p w)φ + Ψ = 0, w(t, p) = 0, where, fixing p N := 1 N 1 l=1 pl and µ( p) := N l=1 µ l p l, κ( p) := 1 σ Ψ( p) := γr + [ ] p 1 (µ 1 µ( p)),..., p N 1 (µ N 1 µ( p)) γ 2(1 γ) Φ(t, p) := β ϖ(t, p) + l=1 ( ) µ( p) r 2 σ γ µ( p) r κ( p), 1 γ σ [ N N 1 β ϖ(t, p) := ϖ l,1 (t) p l,..., ϖ l,n 1 (t) p l] 2 The optimal investment strategy in the stock is l=1 π S (s, p) := µ( p) r σ 2 (1 γ) + 1 pw(s, p)κ( p) σ(1 γ)

59 Solution Post-Default and Pre-Default Analysis Pre-Default Problem 1 The HJB equation for w(t, p) := w(t, p, 0) takes the form: 0 = w t tr( κ κ D 2 1 w) + 2(1 γ) ( p w) κ κ ( p w) + ( p w) Φ + Ψ + h( p) { ( ) } exp w t, p h( p) w(t, p), 0 = w(t, p), where h( p) := N l=1 h l p l, h( p) = [ h 1 / h( p),..., h N 1 / h( p) ], and is point-wise product. Above, ( σ κ(t, p) := 1 p 1 [µ 1 µ( p)],..., σ 1 p N 1 ) [µ N 1 µ( p)] υ 1 p 1 [a(t, e 1 ) ā(t, p)],..., υ 1 p N 1 [a(t, e N 1 ) ā(t, p)] Φ(t, p) := β ϖ(t, p) + γ ( ) µ( p) r ā(t, p) r, κ(t, p), 1 γ σ υ ( ) γ µ( p) r 2 ( ) γ ā(t, p) r 2 Ψ(t, p) := rγ + + h( p). 2(1 γ) σ 2(1 γ) σ 2 The optimal pre-default investment strategy is ( ) µ( p) r ( π S (s, p), π P ā(s, p) r (s, p)) :=, + 1 σ 2 (1 γ) υ 2 (1 γ) 1 γ Σ 1 Y p w(s, p) κ(s, p)

60 Solution Post-Default and Pre-Default Analysis Two-Regime Case: Post-Default Equation w(t, p) t where κ(t, p)2 2 w(t, p) w(t, p) w(t, p) p 2 + Φ(t, p) + Ψ(t, p) p 1 γ = 0 w(t, p) = 1 κ(t, p) = µ 1 µ 2 p(1 p) σ Φ(t, p) = ϖ 11 p ϖ 22 (1 p) + γ 1 γ Ψ(t, p) = γr + γ 2(1 γ) µ 1 p + (1 p)µ 2 r κ(t, p) σ ( ) µ1 p + (1 p)µ 2 r 2 σ

61 Solution Post-Default and Pre-Default Analysis Two-Regime Case: Pre-Default Equation w(t, p) + 1 t 2 κ(t, p) κ(t, p) 2 w(t, p) p 2 + Φ(t, p) w p + Ψ(t, p) ( ) + (h 2 + (h 1 h 2 )p)e w ph t, 1 h 2 +(h 1 h 2 )p w(t, p) γ = 0, 1 γ w(t, p) = 1, w(t, p) 1 γ where [ ] µ1 µ 2 a(t, e 1 ) a(t, e 2 ) κ(t, p) = p(1 p), σ υ Φ(t, p) = ϖ 21 + (ϖ 11 ϖ 21 )p + γ ( κ1 (t, p) (ˆµ(p) r) + κ ) 2(t, p) (â(t, p) r) 1 γ σ υ ( ) Ψ(t, p) = 1 γ (ˆµ(p) r) 2 (â(t, p) r)2 2 1 γ σ 2 + υ 2 + γr h 2 + (h 1 h 2 ) p.

62 Solution Post-Default and Pre-Default Analysis Important Remarks 1 The optimal pre-default value function depends on the optimal post-default value function; 2 It can be shown that both the post- and pre-default HJB PDE admit classical solutions: This is highly not trivial since the pre-default equation is nonlinear and only locally, not globally, Lipschitz continuous; The existence is obtained via a monotone iterative method. 3 Under the following mild conditions on the transition rates {ϖ i,j (t)} of the chain X and the appreciation rates {a(t, e i )} of the defaultable asset sup 0 t T max ϖ i,j (t) <, i,j T max a 2 (t, e i )dt <, i 0 we have verification theorems for both the post- and pre-default problems.

63 Numerical example Test scenario T = 10 years, σ = 0.4, υ = 0.5, and r = a(t, e 1 ) = r + h 1 and a(r, e 2 ) = r + h 2. Two regimes: Regime i = 1 Regime i = 2 µ i h i ϖ ii Table: Parameters associated to the two regimes. Remarks: Mean rate of return on the stock is higher under regime 1; Default rate is higher under regime 1; ϖ 11 = 0.5 = Time spent at regime 1 is exponential with mean 2; ϖ 22 = 1 = Time spent at regime 2 is exponential with mean 1;

64 Numerical example Optimal Strategies Optimal Pre Default Strategy of Stock =0.05 =0.1 =0.2 = p 1 Optimal Strategy of Defaultable Security =0.1 =0.2 =0.3 = p Figure: The top (resp., bottom) panels report the stock (resp., vulnerable security) investment strategy for a square root investor (γ = 1/2) in terms of p = P(X t = 1 G I t ). In the left panels, we set t = 0. Crank-Nicolson Method used.

65 Conclusions Conclusions Develop a framework for solving continuous time portfolio optimization problems in hidden regime switching defaultable markets, consisting of a risky asset, a defaultable security and a risk-free asset. Obtained feasible explicit optimal trading strategies for a CRRA investor. Proved verification theorems for pre-default and post-default utility maximization subproblems. Proved that both the pre- and post-default HJB equations admit classical solutions.

66 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

67 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

68 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

69 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

70 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

71 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

72 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

73 Conclusions Future Work 1 Allow Time Varying Volatility: Both risky assets are assumed to have constant volatility: ds t = µ(x t )S t dt + σs t dw (1) t, dp t := a(t, X t )P t dt + υp t dw (2) t. Why not to make σ t = σ(x t ) and υ t = υ(x t ) (regime driven)? The main reason is because in that case X won t be hidden anymore since volatility is, in principle, estimable in finite time horizon. However, it is natural to consider a stochastic volatility model (say, Heston), where the drift is driven by X : dσ 2 t = α(θ(x t ) σ 2 t ) + βσ t dw (3) t. 2 Extend the results to deal with multiple correlated defaultable securities. 3 Extend to jump-diffusion models. 4 Combine observable and hidden regime switching factors.

74 Further Details Dynamics of the filter probabilities Dynamics of the filter probabilities Under P, the dynamics of the filter probabilities pt i := P(X t = e i G t ) are such that ( N ) ( dpt i = ϖ l,i (t)pt l + γpt i ϑ(t, e i ) ˆϑ(t, p t ) ) π t dt l=1 + p i t ( ϑ(t, e i ) ˆϑ(t, p t ) ) Σ 1 Y d W t + pt i h i ĥ(p t ) d ξ t, ĥ(p t ) where ϑ(s, e i ) := [ µ i σ2 2 a(t, e i ) υ2 2 ].