Credit Risk, I. Summer School in Financial Mathematics September Ljubljana

Transcription

1 Credit Risk, I. Summer School in Financial Mathematics 7-20 September 2009 Ljubljana 1

2 I.Hazard Process Approach of Credit Risk: A Toy Model 1. The Model 2. Toy Model and Martingales 3. Valuation and Trading Defaultable Claims 2

3 The Model 3

4 The Market We begin with the case where a riskless asset, with deterministic interest rate (r(s; s 0 is the only asset available in the default-free market. ( t R(t =exp r(sds 0 4

5 The Market We begin with the case where a riskless asset, with deterministic interest rate (r(s; s 0 is the only asset available in the default-free market. ( t R(t =exp r(sds 0 The time-t price B(t, T of a risk-free zero-coupon bond with maturity T is B(t, T def = exp ( T t r(sds. 5

6 Default occurs at time τ, where τ is assumed to be a positive random variable with density f, constructed on a probability space (Ω, G, P. F (t =P(τ t = t 0 f(sds. We assume that F (t < 1, t 6

7 Defaultable Zero-coupon with Payment at Maturity A defaultable zero-coupon bond (DZC in short- or a corporate bond- with maturity T and rebate δ paid at maturity, consists of The payment of one monetary unit at time T if default has not occurred before time T, Apaymentofδ monetary units, made at maturity, if τ<t, where 0 δ<1. 7

8 Value of the defaultable zero-coupon bond The value of the defaultable zero-coupon bond is defined as D (δ,t (0,T = E ( B(0,T(1 {T<τ} + δ 1 {τ T } = B(0,T(1 (1 δf (T. 8

9 Value of the defaultable zero-coupon bond The value of the defaultable zero-coupon bond is defined as D (δ,t (0,T = E ( B(0,T(1 {T<τ} + δ 1 {τ T } = B(0,T(1 (1 δf (T. The value D (δ,t (t, T of the DZC is the conditional expectation of the discounted payoff B(t, T [1 {T<τ} + δ 1 {τ T } ] given the information: D (δ,t (t, T = 1 {τ t} B(t, T δ + 1 {t<τ} D(δ,T (t, T 9

10 Value of the defaultable zero-coupon bond The value of the defaultable zero-coupon bond is defined as D (δ,t (0,T = E ( B(0,T(1 {T<τ} + δ 1 {τ T } = B(0,T(1 (1 δf (T. The value D (δ,t (t, T of the DZC is the conditional expectation of the discounted payoff B(t, T [1 {T<τ} + δ 1 {τ T } ] given the information: D (δ,t (t, T = 1 {τ t} B(t, T δ + 1 {t<τ} D(δ,T (t, T where the predefault value D (δ,t (t, T is defined as D(δ,T (t, T = E ( B(t, T (1{T<τ} + δ 1 {τ T } t<τ 10

11 D(δ,T (t, T = E ( B(t, T (1{T<τ} + δ 1 {τ T } t<τ B(t, T ( 1 (1 δp(τ T t<τ ( P(t <τ T B(t, T 1 (1 δ P(t <τ ( F (T F (t B(t, T 1 (1 δ 1 F (t 11

12 D(δ,T (t, T = E ( B(t, T (1{T<τ} + δ 1 {τ T } t<τ = B(t, T ( 1 (1 δp(τ T t<τ ( P(t <τ T B(t, T 1 (1 δ P(t <τ ( F (T F (t B(t, T 1 (1 δ 1 F (t 12

13 D(δ,T (t, T = E ( B(t, T (1{T<τ} + δ 1 {τ T } t<τ = B(t, T ( 1 (1 δp(τ T t<τ ( P(t <τ T = B(t, T 1 (1 δ P(t <τ 13

14 D(δ,T (t, T = E ( B(t, T (1{T<τ} + δ 1 {τ T } t<τ = B(t, T ( 1 (1 δp(τ T t<τ ( P(t <τ T = B(t, T 1 (1 δ P(t <τ ( F (T F (t = B(t, T 1 (1 δ 1 F (t 14

15 The formula D(δ,T P(t <τ T (t, T =B(t, T B(t, T (1 δ P(t <τ can be read as D(δ,T (t, T =B(t, T EDLGD DP 15

16 The formula D(δ,T P(t <τ T (t, T =B(t, T B(t, T (1 δ P(t <τ can be read as D(δ,T (t, T =B(t, T EDLGD DP where the Expected Discounted Loss Given Default (EDLGD is defined as B(t, T (1 δ andthedefault Probability (DP is DP = P(t <τ T P(t <τ = P(τ T t <τ. 16

17 In case the payment is a function of the default time, say δ(τ, the value of this defaultable zero-coupon is D (δ,t (0,T = E ( B(0,T 1 {T<τ} + B(0,Tδ(τ 1 {τ T } [ = B(0,T P(T <τ+ T 0 ] δ(sf(sds. 17

18 In case the payment is a function of the default time, say δ(τ, the value of this defaultable zero-coupon is D (δ,t (0,T = E ( B(0,T 1 {T<τ} + B(0,Tδ(τ 1 {τ T } [ = B(0,T P(T <τ+ T 0 ] δ(sf(sds. The predefault price D (δ,t (t, T is D(δ,T (t, T = B(t, T E( 1{T<τ} + δ(τ 1 {τ T } t<τ [ ] T P(T <τ = B(t, T P(t <τ + 1 δ(sf(sds P(t <τ t. 18

19 We introduce the increasing hazard function Γ defined by Γ(t = ln(1 F (t and its derivative γ(t = f(t 1 F (t where f(t =F (t, i.e., 1 F (t =e Γ(t =exp ( t γ(sds = P(τ >t. 0 19

20 We introduce the increasing hazard function Γ defined by Γ(t = ln(1 F (t and its derivative γ(t = f(t 1 F (t where f(t =F (t, i.e., 1 F (t =e Γ(t =exp ( t γ(sds = P(τ >t. 0 The quantity γ(t called the hazard rate is the probability that the default occurs in a small interval dt given that the default has not occured before time t 1 γ(t = lim P (τ t + h τ >t. h 0 h 20

21 For δ =0, D(t, T =exp ( T t (r + γ(sds in other terms, the spot rate has to be adjusted by means of a spread (γ in order to evaluate DZCs. 21

22 Defaultable Zero-coupon with Payment at Hit Here, a defaultable zero-coupon bond with maturity T consists of The payment of one monetary unit at time T if default has not yet occurred, A payment of δ(τ monetary units, where δ is a deterministic function, made at time τ if τ<t. Here, we do not assume that F is differentiable. 22

23 Value of the defaultable zero-coupon The value of this defaultable zero-coupon bond is D (δ (0,T = E(B(0,T 1 {T<τ} + B(0,τδ(τ 1 {τ T } = G(T B(0,T T 0 B(0,sδ(sdG(s, where G(t = 1 F (t = P(t < τis the survival probability. 23

24 For t<t, D (δ (t, T = 1t<τ D (δ (t, T where D (δ (t, T is called the predefault price defined by B(0,t D (δ (t, T = E(B(0,T 1 {T<τ} + B(0,τδ(τ 1 {τ T } t<τ = P(T <τ P(t <τ B(0,T+ 1 P(t <τ T t B(0,sδ(sdF (s. Hence, B(0,tG(t D (δ (t, T =G(T B(0,T T t B(0,sδ(sdG(s. 24

25 In terms of the hazard function, the time-t value D (δ (t, T satisfies: B(0,te Γ(t D(δ (t, T =e Γ(T B(0,T+ T t B(0,se Γ(s δ(sdγ(s. 25

26 A particular case If F is differentiable, the function γ =Γ satisfies f(t =γ(te Γ(t. Then, Rd(t D (δ (t, T =Rd(T + T t Rd(sγ(sδ(sds with ( t Rd(t =exp (r(s+γ(s ds 0 The defaultable interest rate is r + γ and is, as expected, greater than r (the value of a DZC with δ = 0 is smaller than the value of a default-free zero-coupon. 26

27 The dynamics of D (δ (t, T are d D (δ (t, T =(r(t+γ(t D (δ (t, T dt δ(tγ(tdt. The dynamics of D (δ includes a jump at time τ. 27

28 Spreads A term structure of credit spreads associated with the zero-coupon bonds S(t, T is defined as S(t, T = 1 D(t, T ln T t B(t, T. In our setting, on the set {τ >t} S(t, T = 1 ln Q(τ >T τ >t, T t whereas S(t, T = on the set {τ t}. 28

29 Toy Model and Martingales We denote by (Ht,t 0 the right-continuous increasing process Ht = 1 {t τ} and by (Ht its natural filtration. Any integrable Ht-measurable r.v. H is of the form H = h(τ t =h(τ 1 {τ t} + h(t 1 {t<τ} where h is a Borel function. 29

30 Key Lemma If X is any integrable, G-measurable r.v. E(X 1 {t<τ} E(X Ht 1 {t<τ} = 1 {t<τ} P(t <τ. 30

31 Key Lemma If X is any integrable, G-measurable r.v. E(X 1 {t<τ} E(X Ht 1 {t<τ} = 1 {t<τ} P(t <τ. Let Y = h(τ be a H-measurable random variable. Then E(Y Ht = 1 {τ t} h(τ+ 1 {t<τ} h(ue Γ(t Γ(u dγ(u t 31

32 An important Martingale The process (Mt,t 0 defined as Mt = Ht τ t 0 df (s 1 F (s = H t is a H-martingale. 32 t 0 df (s (1 Hs 1 F (s

33 Hazard Function The hazard function is Γ(t = ln(1 F (t = t 0 df (s 1 F (s In particular, if F is differentiable, the process Mt = Ht τ t 0 γ(sds = Ht t 0 γ(s(1 Hsds is a martingale, where γ(s = f(s 1 F (s function, called the intensity of τ. is a deterministic non-negative 33

34 The Doob-Meyer decomposition of the submartingale H is Ht = Mt +Γ(t τ The predictable process At = Γt τ is called the compensator of H. 34

35 The process is a H-martingale. Lt ( t def = 1 {τ>t} exp γ(sds 0 35

36 Proof: We shall give 3 different arguments, each of which constitutes a proof. a Since the function γ is deterministic, for t>s ( t E(Lt Hs =exp γ(udu E( 1 {t<τ} Hs. 0 From the Key Lemma E( 1 {t<τ} Hs = 1 {τ>s} 1 F (t 1 F (s = 1 {τ>s} exp ( Γ(t+Γ(s. Hence, ( s E(Lt Hs = 1 {τ>s} exp γ(udu = Ls. 0 36

37 b Another method is to apply integration by parts formula to the ( t process Lt =(1 Htexp γ(sds If U and V are two finite 0 variation processes, Stieltjes integration by parts formula can be written as follows U(tV (t = U(0V (0 + ]0,t] V (s du(s+ ]0,t] U(s dv (s + s t ΔU(sΔV (s. ( t ( t dlt = dht exp γ(sds + γ(texp γ(sds (1 Htdt ( t = exp γ(sds dmt

38 c A third (sophisticated method is to note that L is the exponential martingale of M, i.e., the solution of the SDE dlt = Lt dmt,l0 =1. 38

39 In the case where N is an inhomogeneous Poisson process with deterministic intensity λ and τ is the first time when N jumps, let Ht = Nt τ.itiswellknownthatnt t 0 λ(sds is a martingale. Therefore, the process stopped at time τ is also a martingale, i.e., Ht t τ 0 λ(sds is a martingale. 39

40 Change of probability Let P be a probability equivalent to P on the space (Ω, H where H = H is the σ-algebra generated by τ. Then, dp = h(τ dp where h is a strictly positive fonction, such that EP(h(τ = 1. Let Γ (t = ln P (τ>t. If Γ is continuous, Γ is continuous and dγ (t = h(t g(t dγ(t where g(t =e Γ(t EP( 1t<τ h(τ. 40

41 Proof: P (τ>t=ep( 1t>τ h(τ = Hence t h(udf (u =e Γ (t e Γ (t dγ (t =h(tdf (t =h(te Γ(t dγ(t Therefore EP( 1t<τ h(τdγ (t =h(te Γ(t dγ(t It follows that dγ (t = h(t dγ(t =h(t e Γ(t EP( 1t<τ h(τ g(t dγ(t 41

42 Exercices: Let ηt = EP (h(τ Ht. Prove that ηt = t 0 h(sdhs +(1 Htg(t Prove that the martingale η admits a representation in terms of M as ηt = 1+ t 0 ηu ( h(t g(t 1dM u Note that γ (t =γ(t(1 ( h(u g(u 1 42

43 Incompleteness of the Toy model If the market consists only of the risk-free zero-coupon bond, there exists infinitely many e.m.m s. The discounted asset prices are constant, hence the set Q of equivalent martingale measures is the set of probabilities equivalent to the historical one. For any Q Q, we denote by FQ the cumulative function of τ under Q, i.e., FQ(t =Q(τ t. 43

44 The range of prices is defined as the set of prices which do not induce arbitrage opportunities. For a DZC with a constant rebate δ paid at maturity, the range of prices is equal to the set {EQ ( B(0,T( 1{T<τ} + δ 1 {τ<t}, Q Q}. This set is exactly the interval ]δrt,rt [. 44

45 Risk Neutral Probability Measures It is usual to interpret the absence of arbitrage opportunities as the existence of an e.m.m.. If DZCs are traded, their prices are given by the market, and the equivalent martingale measure Q, chosen by the market, is such that, on the set {t <τ}, D(t, T =B(t, T EQ( [ 1 T<τ + δ 1t<τ T ] t<τ. Therefore, we can characterize the cumulative function of τ under Q from the market prices of the DZC as follows. 45

46 Zero Recovery If a DZC with zero recovery of maturity T is traded at apriced(t, T which belongs to the interval ]0,R T t [, then, under any risk-neutral probability Q, the process R(tD(t, T is a martingale, the following equality holds ( D(t, T B(0,t=EQ(B(0,T 1 {T<τ} Ht =B(0,T 1 {t<τ} exp T t γ Q (sds where γ Q (s = df Q(s/ds 1 FQ(s. The process γq is the Q-intensity of τ. Therefore the unique risk-neutral intensity can be obtained from the prices of DZCs as r(t+γ Q (t = T ln D(t, T T =t 46

47 Fixed Payment at maturity If the prices of DZCs with different maturities are known, then B(0,T D(0,T B(0,T(1 δ = FQ(T where FQ(t =Q(τ t, so that the law of τ is known under the e.m.m.. 47

48 Payment at hit In this case, denoting by T D the derivative of the value of the DZC at time 0 with respect to the maturity, we obtain T D(0,T=g(T B(0,T G(T B(0,Tr(T δ(t g(t B(0,T, where g(t =G (t. Therefore, solving this equation leads to [ t ] 1 Q(τ >t=g(t =Δ(t 1+ T D(0,s B(0,s(1 δ(s (Δ(s 1 ds ( t where Δ(t = exp 0 0 r(u 1 δ(u du., 48

49 Representation Theorem Let h be a (bounded Borel function. Then, the martingale M h t = E(h(τ Ht admits the representation E(h(τ Ht =E(h(τ t τ 0 ( h(s h(s dms, where Mt = Ht Γ(t τ and h(t = t h(udg(u G(t. 49

50 Representation Theorem Let h be a (bounded Borel function. Then, the martingale M h t = E(h(τ Ht admits the representation E(h(τ Ht =E(h(τ + t τ 0 (h(s h(s dms, where Mt = Ht Γ(t τ and h(t = t h(udg(u G(t. Note that h(t =M h t on t<τ. 50

51 Representation Theorem Let h be a (bounded Borel function. Then, the martingale M h t = E(h(τ Ht admits the representation E(h(τ Ht =E(h(τ t τ 0 ( h(s h(s dms, where Mt = Ht Γ(t τ and h(t = t h(udg(u G(t. Note that h(t =M h t on t<τ. In particular, any square integrable H-martingale (Xt,t 0 can be written as Xt = X0 + t 0 x sdms where (xt,t 0 is a predictable process. 51

52 Proof: A proof consists in computing the conditional expectation E(h(τ Ht =h(τht +(1 Hte Γ(t h(sdf (s t and to use integration by parts formula. 52

53 Partial information: Duffie and Lando s model Duffie and Lando study the case where τ =inf{t : Vt m} where V satisfies dvt = μ(t, Vtdt + σ(t, VtdWt. 53

54 Partial information: Duffie and Lando s model Duffie and Lando study the case where τ =inf{t : Vt m} where V satisfies dvt = μ(t, Vtdt + σ(t, VtdWt. Here the process W is a Brownian motion. If the information is the Brownian filtration, the time τ is a stopping time w.r.t. a Brownian filtration, therefore is predictable and admits no intensity. 54

55 Partial information: Duffie and Lando s model Duffie and Lando study the case where τ =inf{t : Vt m} where V satisfies dvt = μ(t, Vtdt + σ(t, VtdWt. Here the process W is a Brownian motion. If the information is the Brownian filtration, the time τ is a stopping time w.r.t. a Brownian filtration, therefore is predictable and admits no intensity. If the agent does not know the behavior of V, but only the minimal information Ht, i.e. he knows when the default appears, the price of a zero-coupon is, in ( the case where the default is not yet occurred, exp T t γ(sds where γ(s = f(s G(s and G(s =P(τ >s,f = G,assoonasthe cumulative function of τ is differentiable. 55

56 Valuation and Trading Defaultable Claims 56

57 We assume that the market has chosen a risk-neutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant. 57

58 We assume that the market has chosen a risk-neutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant. Price dynamics of a survival claim (X, 0,τ. Let (X, 0,τbeasurvival claim. The price of the payoff 1 {T<τ} X that settles at time T is Yt = e rt EQ( 1 {T<τ} e rt X Ht. 58

59 We assume that the market has chosen a risk-neutral probability Q and that M and γ are computed w.r.t. Q. We assume here that the interest rate r is constant. Price dynamics of a survival claim (X, 0,τ. Let (X, 0,τbeasurvival claim. The price of the payoff 1 {T<τ} X that settles at time T is Yt = e rt EQ( 1 {T<τ} e rt X Ht. The dynamics of the price process is dyt = ryt dt Yt dmt 59

60 Price dynamics of a recovery claim (0,Z,τ. The recovery Z is paid at the time of default. The ex-dividend price is St = e rt EQ( 1 {T τ>t} e rτ Z(τ Ht 60

61 Price dynamics of a recovery claim (0,Z,τ. The recovery Z is paid at the time of default. The ex-dividend price is St = e rt EQ( 1 {T τ>t} e rτ Z(τ Ht Hence dst =(rst Z(tγ(tdt +(Z(t St dmt Z(t(1 Htγ(tdt. 61

62 Price dynamics of a recovery claim (0,Z,τ. The recovery Z is paid at the time of default. The ex-dividend price is St = e rt EQ( 1 {T τ>t} e rτ Z(τ Ht Hence dst =(rst Z(tγ(tdt +(Z(t St dmt Z(t(1 Htγ(tdt. The cum-dividend price process Y of (0,Z,τis Yt = e rt EQ( 1 {T τ} e rτ Z(τ Ht, and dyt = ryt dt +(Z(t Yt dmt 62

63 Valuation of a Credit Default Swap A credit default swap (CDS is a contract between two counterparties A and B. Some maturity T is fixed. B agrees to pay, at default time τ, a default payment Z(τ toa if a default of the obligor C occurs before maturity. If there is no default until the maturity of the default swap, B pays nothing. A pays a fee for the default protection. The fee is paid till the maturity or till the default event, whichever occurs the first. A can not cancel the contract. Usually, the fee consists of Ci paid at time Ti (this is the fixed leg. However, here we shall consider a continuous payment κ ( i.e., κdt is paid during the time interval dt. The default payment is called the default leg. 63

64 Valuation of a Credit Default Swap A credit default swap (CDS is a contract between two counterparties A and B. Some maturity T is fixed. B agrees to pay, at default time τ, a default payment Z(τ toa if a default of the obligor C occurs before maturity. If there is no default until the maturity of the default swap, B pays nothing. A pays a fee for the default protection. The fee is paid till the maturity or till the default event, whichever occurs the first. A can not cancel the contract. Usually, the fee consists of Ci paid at time Ti (this is the fixed leg. However, here we shall consider a continuous payment κ ( i.e., κdt is paid during the time interval dt. The default payment is called the default leg. 64

65 For simplicity, we assume that the interest rate r = 0, so that the price of a savings account Bt =1foreveryt. Our results can be easily extended to the case of a constant r. 65

66 Ex-dividend Price of a CDS The ex-dividend price of a CDS maturing at T with spread κ is given by the formula ( St(κ =EQ δ(τ 1 {t<τ T } 1 {t<τ} κ ( (τ T t Ht. 66

67 Ex-dividend Price of a CDS The ex-dividend price of a CDS maturing at T with spread κ is given by the formula ( St(κ =EQ δ(τ 1 {t<τ T } 1 {t<τ} κ ( (τ T t Ht. The ex-dividend price at time t [s, T ] of a credit default swap with spread κ and recovery at default equals ( T 1 St(κ = 1 {t<τ} δ(u dg(u κ G(t t T t G(u du. 67

68 Proof: We have, on the set {t <τ}, dg(u St(κ κ = T δ(u t G(t ( = 1 G(t T t ( T u dg(u+tg(t t G(t δ(u dg(u κ ( TG(T tg(t t T t It remains to note that T G(u du = TG(T tg(t t T t udg(u, 68 udg(u.

69 The ex-dividend price of a CDS can also be represented as follows St(κ = 1 {t<τ} St(κ, t [0,T], where St(κ stands for the ex-dividend pre-default price of a CDS. 69

70 Price Dynamics of a CDS In what follows, we assume that ( t G(t =Q(τ >t=exp γ(u du 0 where the default intensity γ(t under Q is deterministic. We first focus on the dynamics of the ex-dividend price of a CDS with spread κ. 70

71 The dynamics of the ex-dividend price St(κ on [0,T] are dst(κ = St (κ dmt +(1 Ht(κ δ(tγ(tdt, where the H-martingale M under Q is given by the formula Mt = Ht t 0 (1 Huγ(u du, t R+. 71

72 Proof: It suffices to recall that St(κ = 1 {t<τ} St(κ =(1 Ht St(κ so that dst(κ =(1 Ht d St(κ St (κ dht. Using the explicit expression of St, we find easily that we have d St(κ =γ(t St(κ dt +(κ(s δ(tγ(t dt. The SDE for S follows. 72

73 Trading Strategies with a CDS A strategy φt =(φ 0 t,φ 1 t, t [0,T], is self-financing if the wealth process U(φ, defined as Ut(φ =φ 0 t + φ 1 t St(κ, satisfies dut(φ =φ 1 t dst(κ+φ 1 t ddt, where S(κ is the ex-dividend price of a CDS with the dividend stream D. Astrategyφ replicates a contingent claim Y if UT (φ =Y. 73

74 Hedging of a Contingent Claim in the CDS Market Our aim is to find a replicating strategy for the defaultable claim (X, 0,Z,τ, where X is a constant and Zt = z(t. 74

75 Hedging of a Contingent Claim in the CDS Market Our aim is to find a replicating strategy for the defaultable claim (X, 0,Z,τ, where X is a constant and Zt = z(t. Let ỹ and φ 1 be defined as ( ỹ(t = 1 XG(T G(t T t z(sdg(s φ 1 (t = z(t ỹ(t δ(t St(κ, 75

76 Hedging of a Contingent Claim in the CDS Market Our aim is to find a replicating strategy for the defaultable claim (X, 0,Z,τ, where X is a constant and Zt = z(t. Let ỹ and φ 1 be defined as ( ỹ(t = 1 XG(T G(t T t z(sdg(s φ 1 (t = z(t ỹ(t δ(t St(κ, Let φ 0 t = Vt(φ φ 1 (tst(κ, where Vt(φ =EQ(Y Ht and Y = 1 {T τ} z(τ+ 1 {T<τ} X 76

77 Hedging of a Contingent Claim in the CDS Market Our aim is to find a replicating strategy for the defaultable claim (X, 0,Z,τ, where X is a constant and Zt = z(t. Let ỹ and φ 1 be defined as ( ỹ(t = 1 XG(T G(t T t z(sdg(s φ 1 (t = z(t ỹ(t δ(t St(κ, Let φ 0 t = Vt(φ φ 1 (tst(κ, where Vt(φ =EQ(Y Ht and Y = 1 {T τ} z(τ+ 1 {T<τ} X Then the self-financing strategy φ =(φ 0,φ 1 based on the savings account and the CDS is a replicating strategy. 77

78 Proof: The terminal value of the wealth is Y = z(τ 1 {τ<t} + X 1 {T<τ} 78

79 Proof: The terminal value of the wealth is Y = z(τ 1 {τ<t} + X 1 {T<τ} On the one hand ( 1 E(Y Ht =Yt = z(τ 1 {τ t} + 1 {t<τ} XG(T G(t ( t 1 = z(sdhs +(1 Ht XG(T G(t 0 T t T t z(sdg(s z(sdg(s 79

80 Proof: The terminal value of the wealth is Y = z(τ 1 {τ<t} + X 1 {T<τ} On the one hand ( 1 E(Y Ht =Yt = z(τ 1 {τ t} + 1 {τ<t} XG(T G(t ( t 1 = z(sdhs +(1 Ht XG(T G(t 0 T t T t z(sdg(s z(sdg(s hence dyt = (z(t ỹ(t dmt with ỹ(t = 1 G(t (XG(T T t z(sdg(s. 80

81 Proof: The terminal value of the wealth is Y = z(τ 1 {τ<t} + X 1 {T<τ} On the one hand E(Y Ht =Yt = z(τ 1 {τ t} + 1 {t<τ} 1 G(t = t 0 1 z(sdhs +(1 Ht G(t ( XG(T + t ( XG(T + 0 t z(sdg(s 0 z(sdg(s hence dyt = (z(t ỹ(t dmt with ỹ(t = 1 G(t (XG(T T t z(sdg(s. On the other hand, dyt = φ 1 t (dst(κ κ(1 Htdt + δ(tdht =φ 1 t (δ(t St (κ dmt. 81