Remarks on American Options for Jump Diffusions

Transcription

1 Remarks on American Options for Jump Diffusions Hao Xing University of Michigan joint work with Erhan Bayraktar, University of Michigan Department of Systems Engineering & Engineering Management Chinese University of Hong Kong, Feb. 5, 2009

2 Why jump models? P market (K,T) = P BS (S 0,K,T,σ imp ). 70 Implied Volatility of S&P 500 index put option Jan. 16, 09 Dec. 30, Implied Volatility % Moneyness = strike / spot index

3 Crash Fear! D. S. Bates [00] studied the index option data for the 87 crash: After the crash out of the money (OTM) put options that provide explicit portfolio insurance against substantial downward movements in the market have been trading at high prices (as measured by implicit volatilities)...even more overpriced relative to OTM calls that will pay off only if the markets rises substantially. Crash Fear

4 Crash Fear! D. S. Bates [00] studied the index option data for the 87 crash: After the crash out of the money (OTM) put options that provide explicit portfolio insurance against substantial downward movements in the market have been trading at high prices (as measured by implicit volatilities)...even more overpriced relative to OTM calls that will pay off only if the markets rises substantially. Crash Fear Comparing to models with stochastic volatilities, models with jumps can better explain this phenomenon.

5 Jump diffusions Assume the stock price S follows jump diffusions (under a risk neutral measure calibrated from the market) N t ( ) ds t = bs t dt + σ(s t,t)s t dw t + S t d e Y i 1, in which i=1 N t is a Poisson process with rate λ independent of W, Y i are iid. random variables with distribution ν(dy), b r λ(ξ 1), we assume ξ E[e Y i] < + so that {e rt S t } t 0 is a martingale. We assume σ(s,t) ǫ > 0 for some positive constant ǫ.

6 Jump diffusions Assume the stock price S follows jump diffusions (under a risk neutral measure calibrated from the market) N t ( ) ds t = bs t dt + σ(s t,t)s t dw t + S t d e Y i 1, in which i=1 N t is a Poisson process with rate λ independent of W, Y i are iid. random variables with distribution ν(dy), b r λ(ξ 1), we assume ξ E[e Y i] < + so that {e rt S t } t 0 is a martingale. We assume σ(s,t) ǫ > 0 for some positive constant ǫ. Examples: Merton s model: ν(dy) is the dist. of normal r.v. Kou s model: ν(dy) is the dist. of double exponential r.v.

7 The American put option The value function of the American put option is given by [ ] V (S,t) sup E e r(τ t) g(s τ ) S t = S, τ T t,t where g(s) = (K S) +.

8 The American put option The value function of the American put option is given by [ ] V (S,t) sup E e r(τ t) g(s τ ) S t = S, τ T t,t where g(s) = (K S) +. The domain R n [0,T) can be divided into the continuation region and the stopping region: C {(S, t) R + [0, T) : V(S, t) > g(s)} and D {(S, t) R + [0, T) : V(S, t) = g(s)}.

9 The American put option The value function of the American put option is given by [ ] V (S,t) sup E e r(τ t) g(s τ ) S t = S, τ T t,t where g(s) = (K S) +. The domain R n [0,T) can be divided into the continuation region and the stopping region: C {(S, t) R + [0, T) : V(S, t) > g(s)} and D {(S, t) R + [0, T) : V(S, t) = g(s)}. There exists an optimal exercise boundary s(t), such that C = {S > s(t)} and D = {S s(t)}.

10 The American put option The value function of the American put option is given by [ ] V (S,t) sup E e r(τ t) g(s τ ) S t = S, τ T t,t where g(s) = (K S) +. The domain R n [0,T) can be divided into the continuation region and the stopping region: C {(S, t) R + [0, T) : V(S, t) > g(s)} and D {(S, t) R + [0, T) : V(S, t) = g(s)}. There exists an optimal exercise boundary s(t), such that C = {S > s(t)} and D = {S s(t)}. One of the optimal exercising time τ D is the hitting time of the stopping region D. V (S t,t) is a super-martingale, V (S t τd,t τ D ) is a martingale.

11 Parabolic integro-differential equations (PIDE) Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08]) The value function V (S,t) is the unique classical solution of ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) = 0, S > s(t), V(S, t) = K S, S s(t), V(S, T) = (K S) +, in which L D 1 2 σ2 S 2 SS 2 + b S S. Moreover, S V(s(t), t) = 1, and ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) 0. R R

12 Parabolic integro-differential equations (PIDE) Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08]) The value function V (S,t) is the unique classical solution of ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) = 0, S > s(t), V(S, t) = K S, S s(t), V(S, T) = (K S) +, in which L D 1 2 σ2 S 2 SS 2 + b S S. Moreover, S V(s(t), t) = 1, and ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) 0. R R Remark Because of the nonlocal integral term, the classical finite difference method cannot be applied directly to solve the PIDE numerically.

13 Numerical algorithms Various numerical algorithms were studied in the literature: [Zhang 97] [Andersen & Andreasen 00] [Kou & Wang 04] [Hirsa & Madan 04] [Cont & Voltchkova 05] [d Halluin et al. 05]

14 Numerical algorithms Various numerical algorithms were studied in the literature: [Zhang 97] [Andersen & Andreasen 00] [Kou & Wang 04] [Hirsa & Madan 04] [Cont & Voltchkova 05] [d Halluin et al. 05] We propose yet another numerical algorithm which 1. A slight modification of classical finite difference schemes 2. Shows the connection between jump diffusion problems and diffusion problems

15 The functional operator J Instead of the jump diffusions, we consider the GBM: ds 0 t = b S 0 t dt + σ S 0 t dw t.

16 The functional operator J Instead of the jump diffusions, we consider the GBM: ds 0 t = b S 0 t dt + σ S 0 t dw t. We introduce a functional operator J, for any f : R + [0, T] R +, [ τ J f (S,t) sup E e (r+λ)(u t) λ P f ( Su 0,u) du + τ T t,t t e (r+λ)(τ t) ( K Sτ 0 ) ] + S 0 t = S, in which P f (S,u) f (e y S,u)ν(dy). R

17 The functional operator J Instead of the jump diffusions, we consider the GBM: ds 0 t = b S 0 t dt + σ S 0 t dw t. We introduce a functional operator J, for any f : R + [0, T] R +, [ τ J f (S,t) sup E e (r+λ)(u t) λ P f ( Su 0,u) du + τ T t,t t e (r+λ)(τ t) ( K Sτ 0 ) ] + S 0 t = S, in which P f (S,u) R f (e y S,u)ν(dy). Remark J f is the value function for an optimal stopping problem for the diffusion S 0.

18 The approximating sequence Let us iteratively define v 0 (S,t) = (K S) +, v n+1 (S,t) = J v n (S,t), n 0, for (S,t) R + [0,T].

19 The approximating sequence Let us iteratively define v 0 (S,t) = (K S) +, v n+1 (S,t) = J v n (S,t), n 0, for (S,t) R + [0,T]. Theorem For each n 1, v n is the unique classical solution of the following free boundary PDE: ( t + L D (r + λ)) v n (S,t) = λ (P v n 1 ) (S,t), S > s n (t), v n (S,t) = K S, S s n (t), v n (S,t) > (K S) +, S > s n (t), S v n (s n (t),t) = 1, v n (S,T) = (K S) +.

20 The approximating sequence Let us iteratively define v 0 (S,t) = (K S) +, v n+1 (S,t) = J v n (S,t), n 0, for (S,t) R + [0,T]. Theorem For each n 1, v n is the unique classical solution of the following free boundary PDE: ( t + L D (r + λ)) v n (S,t) = λ (P v n 1 ) (S,t), S > s n (t), v n (S,t) = K S, S s n (t), v n (S,t) > (K S) +, S > s n (t), S v n (s n (t),t) = 1, v n (S,T) = (K S) +. Remark [ v n (S,t) = sup E e r(τ σn t) (K S τ σn ) + ] St = S, τ T t,t where σ n is the n-th jump time of the Poisson process N t.

21 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K.

22 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K. Proof. J v 0 = v 1 (K S) + = v 0, J maps positive functions to positive functions. The statement follows from the induction.

23 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K. Proof. J v 0 = v 1 (K S) + = v 0, J maps positive functions to positive functions. The statement follows from the induction. There exists a limit v (S,t) lim n + v n (S,t).

24 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K. Proof. J v 0 = v 1 (K S) + = v 0, J maps positive functions to positive functions. The statement follows from the induction. There exists a limit v (S,t) lim n + v n (S,t). Moreover, v is the fixed point of J = v is the unique classical solution of the PIDE.

25 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K. Proof. J v 0 = v 1 (K S) + = v 0, J maps positive functions to positive functions. The statement follows from the induction. There exists a limit v (S,t) lim n + v n (S,t). Moreover, v is the fixed point of J = v is the unique classical solution of the PIDE. The verification argument = v = V.

26 The convergence of the sequence Proposition {v n } n 0 is an monotone increasing sequence, moreover v n K. Proof. J v 0 = v 1 (K S) + = v 0, J maps positive functions to positive functions. The statement follows from the induction. There exists a limit v (S,t) lim n + v n (S,t). Moreover, v is the fixed point of J = v is the unique classical solution of the PIDE. The verification argument = v = V. Proposition v n (S,t) V (S,t) v n (S,t) +K ( ( ) 1 e (r+λ)(t t)) n λ n. λ + r

27 Numerical scheme Defining x = log S and u n (x,t) = v n (S,t) for n 1, we solves the PDE satisfied by u n inductively. We discretize the equation by Crank-Nicolson scheme. For fixed t and x such that M t = T and L x = x max x min, we denote ũn l,m as the solution of the difference equation. P u n 1 = R u n 1(x + y,t)ρ(y)dy is a convolution integral. We evaluate it by FFT: P u n 1 = ( u n 1 g ). We use Brennan-Schwartz algorithm (UI decomposition) or PSOR method to solve the free boundary problem.

28 Numerical scheme Defining x = log S and u n (x,t) = v n (S,t) for n 1, we solves the PDE satisfied by u n inductively. We discretize the equation by Crank-Nicolson scheme. For fixed t and x such that M t = T and L x = x max x min, we denote ũn l,m as the solution of the difference equation. P u n 1 = R u n 1(x + y,t)ρ(y)dy is a convolution integral. We evaluate it by FFT: P u n 1 = ( u n 1 g ). We use Brennan-Schwartz algorithm (UI decomposition) or PSOR method to solve the free boundary problem. Complexity: Let the grid point in x and t be O(N), Iterating Brennan-Schwartz: O(N 2 log N), Iterating PSOR: O(N 5/2 ).

29 Convergence of the numerical solutions Proposition 1. ũ n converges to ũ uniformly, moreover ( max ũ l,m l,m ) ( ũn l,m 1 η M) ( n λ λ + r ) n K, 1 where η = 1+(λ+r) t (0,1). 2. ũ l,m u(x l,m t) 0, as x, t, y 0. The convergence is quadratic. 3. The numerical scheme is stable.

30 Numerical experiments Table: American options in Kou s model η 1 = η 2 = 25, p = 0.6, S 0 = 100. The accuracy of Amin price is up tp a penny. KPW 5EXP price is from [Kou et al. 05], B-S stands for the Brennan-Schwartz. The computations are done on the same kind of computer and run time is in seconds. Parameter Values Amin KPW 5EXP Proposed Algorithm K T σ λ Price Value Error Time Value Error B-S PSOR Time Time

31 Numerical experiments cont. Table: American options in Merton model K=100, T=0.25, r=0.05, σ = 0.15, λ = 0.1. Stock price has lognormal jump distribution with µ = 0.9 and σ = dflv comes from [D Halluin et al. 05]. B-S stands for the Brennan-Schwartz. S(0) dflv Proposed Algorithm Value Error LU(B-S) Time PSOR Time

32 Numerical experiments cont. Figure: Iteration of the price functions: v n (S, 0) V(S, 0), S v 3 ( S, 0 ) 7 Option Price v i ( S, 0 ), i = 4, 5, 6 3 ( K S ) + 2 v 1 ( S, 0 ) 1 v 2 ( S, 0 ) Stock Price ( S )

33 Numerical experiments cont. Figure: Iteration of the Exercise Boundary: s n (t) s(t), t [0, T). Both s n (t) and s(t) will converge to S < K as t T. K = S * approx = Continuation Region 95 Stock Price ( S ) s 1 ( t ) 90 s 2 ( t ) s i ( t ), i = 4, 5, 6 s 3 ( t ) t ( in years ) Stopping Region

34 Numerical experiments cont. Figure: Iteration of the Exercise Boundary: s n (t) s(t), t [0, T). Both s n (t) and s(t) will converge to S < K as t T. K = S * approx = Continuation Region 95 Stock Price ( S ) s 1 ( t ) 90 s 2 ( t ) s i ( t ), i = 4, 5, 6 s 3 ( t ) t ( in years ) Stopping Region Coefficients satisfies r < λ 0 (e y 1) ν(dy).

35 Regularity of the optimal exercise boundary of American Options

36 American options in Black-Scholes model Let us assume that the stock price is governed by ds t = (r q)s t dt + σs t dw t, q 0 is the dividend.

37 American options in Black-Scholes model Let us assume that the stock price is governed by ds t = (r q)s t dt + σs t dw t, q 0 is the dividend. The value of the finite horizon American put option is defined by V (S,t) = sup τ S 0,T t E [ e rτ (K S τ ) + S0 = S ].

38 American options in Black-Scholes model Let us assume that the stock price is governed by ds t = (r q)s t dt + σs t dw t, q 0 is the dividend. The value of the finite horizon American put option is defined by V (S,t) = sup τ S 0,T t E [ e rτ (K S τ ) + S0 = S ]. V (S,t) satisfies the following free boundary problem t V σ2 S 2 2 SS V + (r q)s SV rv = 0, V (S,t) = K S, S V (s(t),t) = 1 V (S,T) = (K S) +. S > s(t) S s(t)

39 Regularity of the value function and the boundary S It is well known that ([Karatzas & Shreve 98], [Peskir 05]) l 0 s(t) C D T t lim S s(t) V (S,t) C 2,1 (C), V (S,t) C 2,1 (D), 1 2 σ2 S 2 SS 2 V rk qs(t) > 0, t < T, s(t ) = K,r q s(t ) = r q K,r < q, moreover, s(t) is continuous on[0,t).

40 Regularity of the value function and the boundary S It is well known that ([Karatzas & Shreve 98], [Peskir 05]) l 0 s(t) C D T t lim S s(t) V (S,t) C 2,1 (C), V (S,t) C 2,1 (D), 1 2 σ2 S 2 SS 2 V rk qs(t) > 0, t < T, s(t ) = K,r q s(t ) = r K,r < q, q moreover, s(t) is continuous on[0,t). Question: Is s(t) locally Lipschitz or even C 1?

41 Regularity of the value function and the boundary S It is well known that ([Karatzas & Shreve 98], [Peskir 05]) l 0 s(t) C D T t lim S s(t) V (S,t) C 2,1 (C), V (S,t) C 2,1 (D), 1 2 σ2 S 2 SS 2 V rk qs(t) > 0, t < T, s(t ) = K,r q s(t ) = r K,r < q, q moreover, s(t) is continuous on[0,t). Question: Is s(t) locally Lipschitz or even C 1? Answer: Yes! Moreover, s(t) is C [Chen & Chadam 07].

42 American options for jump diffusions The dynamics of stock price S t follows jump diffusions. Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08]) The value function V (S,t) is the unique classical solution of ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) = 0, S > s(t), V(S, t) = K S, S s(t), V(S, T) = (K S) +, in which L D 1 2 σ2 S 2 2 SS + b S S. Moreover, R S V(s(t), t) = 1, and ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) 0. R

43 American options for jump diffusions The dynamics of stock price S t follows jump diffusions. Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08]) The value function V (S,t) is the unique classical solution of ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) = 0, S > s(t), V(S, t) = K S, S s(t), V(S, T) = (K S) +, in which L D 1 2 σ2 S 2 2 SS + b S S. Moreover, S V(s(t), t) = 1, and ( t + L D (r + λ)) V + λ R R V(S e y, t)ν(dy) 0. Proposition ([Friedman 75], [Yang et al. 06]) lim tv (S,t) = 0, S s(t) t [0,T).

44 American options for jump diffusions The dynamics of stock price S t follows jump diffusions. Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08]) The value function V (S,t) is the unique classical solution of ( t + L D (r + λ)) V + λ V(S e y, t)ν(dy) = 0, S > s(t), V(S, t) = K S, S s(t), V(S, T) = (K S) +, in which L D 1 2 σ2 S 2 2 SS + b S S. Moreover, S V(s(t), t) = 1, and ( t + L D (r + λ)) V + λ R R V(S e y, t)ν(dy) 0. Proposition ([Friedman 75], [Yang et al. 06]) Probabilistic arguments? lim tv (S,t) = 0, S s(t) t [0,T).

45 Behavior of the free boundary Continuity of the free boundary: [Pham 95], [Yang et al. 06], [Lamberton & Mikou 08].

46 Behavior of the free boundary Continuity of the free boundary: [Pham 95], [Yang et al. 06], [Lamberton & Mikou 08]. Differentiability of the free boundary: [Yang et al. 06] proved s(t) C 1 [0,T) under the condition r q + λ (e z 1)ν(dz). R + (C)

47 Behavior of the free boundary Continuity of the free boundary: [Pham 95], [Yang et al. 06], [Lamberton & Mikou 08]. Differentiability of the free boundary: [Yang et al. 06] proved s(t) C 1 [0,T) under the condition r q + λ (e z 1)ν(dz). R + (C) [Levendorskii 04], [Yang et al. 06], [Lamberton & Mikou 08] showed that { K (C) holds, s(t ) lim s(t) = min{k,s 0 } = t T S (C) fails, where S is the unique solution of the integral equation I 0 (S) qs rk + λ (S e y K) + ν(dy) = 0. R

48 Main results Our goals: Extend regularity properties of the free boundary s(t) to the case where (C) fails. Our approach treats both cases simultaneously.

49 Main results Our goals: Extend regularity properties of the free boundary s(t) to the case where (C) fails. Our approach treats both cases simultaneously. Let us change back to the x variable x = log S, u(x,t) = V (S,T t) and b(t) = log s(t t). We will state our main results in these variables.

50 Main results Our goals: Extend regularity properties of the free boundary s(t) to the case where (C) fails. Our approach treats both cases simultaneously. Let us change back to the x variable x = log S, u(x,t) = V (S,T t) and b(t) = log s(t t). We will state our main results in these variables. Theorem b(t) H 5 8([ǫ.T 0 ]).

51 Main results Our goals: Extend regularity properties of the free boundary s(t) to the case where (C) fails. Our approach treats both cases simultaneously. Let us change back to the x variable x = log S, u(x,t) = V (S,T t) and b(t) = log s(t t). We will state our main results in these variables. Theorem b(t) H 5 8([ǫ.T 0 ]). Theorem b(t) C 1 (0,T].

52 Main resuts cont. Theorem Assume ν has a density, i.e. ν(dz) = ρ(z)dz. Let α (0, 1 2 ). If ρ(z) satisfies u ρ(z)dz H2α (R ), then b(t) H 3 2 +α ([ǫ,t]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ǫ,t]).

53 Main resuts cont. Theorem Assume ν has a density, i.e. ν(dz) = ρ(z)dz. Let α (0, 1 2 ). If ρ(z) satisfies u ρ(z)dz H2α (R ), then b(t) H 3 2 +α ([ǫ,t]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ǫ,t]). Corollary If ρ(z) C (R ) with dl ρ(z) bounded in R dz l for each l 1, then b(t) C (0,T].

54 Main resuts cont. Theorem Assume ν has a density, i.e. ν(dz) = ρ(z)dz. Let α (0, 1 2 ). If ρ(z) satisfies u ρ(z)dz H2α (R ), then b(t) H 3 2 +α ([ǫ,t]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ǫ,t]). Corollary If ρ(z) C (R ) with dl ρ(z) bounded in R dz l for each l 1, then b(t) C (0,T]. Remark Free boundaries of Merton s and Kou s models are C.

55 Smooth-fit property By the smooth-fit property, for any ǫ > 0, we have 1 [ x u(b(t),t) x u(b(t ǫ),t ǫ) + e b (t) e b(t ǫ)] = 0. ǫ This yields b(t) b(t ǫ) lim = ǫ 0 ǫ 2 xx xtu(b(t)+,t) 2. u(b(t)+,t) + eb(t)

56 Smooth-fit property By the smooth-fit property, for any ǫ > 0, we have 1 [ x u(b(t),t) x u(b(t ǫ),t ǫ) + e b (t) e b(t ǫ)] = 0. ǫ This yields b(t) b(t ǫ) lim = ǫ 0 ǫ 2 xx xtu(b(t)+,t) 2. u(b(t)+,t) + eb(t) Two critical points to remove the condition (C): for t [0,T) 1. xxu(b(t)+,t) 2 lim x b(t) xxu(x,t) 2 > e b(t), (i.e. lim S s(t) SS 2 V (S,t) > 0) 2. xt 2 u(b(t)+,t) lim x b(t) xt 2 u(x,t) exists in the classical sense and is continuous in t.

57 Denominator Let us introduce the auxiliary function on R [0,T]: I(x,t) qe x [ rk + λ u(x + z,t) + e x+z K ] ν(dz). ( t L + (r + λ)) u(x,t) = I(x,t), for x < b(t),t (0,T], x I(x, t) is strictly increasing, R lim x I(x,t) = rk and lim x + I(x,t) = +.

58 Denominator Let us introduce the auxiliary function on R [0,T]: I(x,t) qe x [ rk + λ u(x + z,t) + e x+z K ] ν(dz). ( t L + (r + λ)) u(x,t) = I(x,t), for x < b(t),t (0,T], x I(x, t) is strictly increasing, R lim x I(x,t) = rk and lim x + I(x,t) = +. Let us consider the level curve B(t) {x : I(x,t) = 0,t [0,T]} B(t) C 1 (0,T] C[0,T], B(t) > b(t) for t (0,T].

59 Denominator cont. Corollary 2 xxu (b(t)+,t) > e b(t), for t (0,T].

60 Denominator cont. Corollary 2 xxu (b(t)+,t) > e b(t), for t (0,T]. Proof. On the one hand, since B(t) > b(t) and x I(x,t) is strictly increasing, we have I(b(t),t) < 0, for t (0,T]. On the other hand, 0 = lim ( t L + (r + λ)) u(x,t) x b(t) = σ2 lim x b(t) x u(x,t) σ2 e b(t) I(b(t),t). The statement follows from I(b(t),t) < 0.

61 Numerator Lemma xt 2 u(b(t)+,t) is continuous in t [0,T).

62 Numerator Lemma xt 2 u(b(t)+,t) is continuous in t [0,T). Proof. Setp 1: Use the Maximum Principle of the differential integral operator t L + (r + λ) and a test function used in Friedman & Shen 02] to show b(t) H 5 8([ǫ,T 0 ]). (Key step to drop the condition (C).)

63 Numerator Lemma xt 2 u(b(t)+,t) is continuous in t [0,T). Proof. Setp 1: Use the Maximum Principle of the differential integral operator t L + (r + λ) and a test function used in Friedman & Shen 02] to show b(t) H 5 8([ǫ,T 0 ]). (Key step to drop the condition (C).) Step 2: Since 5 8 > 1 2, a generalization of the result of [Cannon et al. 74] gives us 2 xtu(b(t)+,t) lim x b(t) 2 xtu(x,t) is a continuous function with respect to t.

64 Higher regularity Let ξ x b(t), h(ξ, t) λ R w(ξ + z, t)ν(dz) + σ tσ( 2 xxu x u). w(ξ,t) = t u(x,t) satisfies t w 1 2 σ2 2 ξξw (µ + b (t) 12 σ2 ) ξ w + (r + λ)w = h(ξ, t), 1 b 2 (t) = σ2 ξ w(0, t) (µ r λ)e b(t) + (r + λ)k λ u(b(t) + z, t)ν(dz). R Regularity bootstrapping scheme: w H 2α,α h H 2α,α h H 2α,α + b H α = w H 2α+2,α+1 = ξ w(0,t) H α+ 1 2 = b (t) H α+ 1 2 α get updated to α ( needs regularity assumption on the jump density ρ, because h is a nonlocal integral and high order derivative of w from both continuation and stopping regions may not agree on the free boundary. )

65 Conclusion and future research The iterative method were also applied to Hedging problem [Kirch & Ruggaldier 04] Optimal investment problem in illiquid market [Pham & Tankov 08] The illiquidity is introduced when the asset prices are observed only at random times. Inventory control [Bayraktar & Ludkovski 08]

66 Conclusion and future research cont. Non-degenerate diffusion processes dominate pure jump process. the value function is smooth, the free boundary is smooth.

67 Conclusion and future research cont. Non-degenerate diffusion processes dominate pure jump process. the value function is smooth, the free boundary is smooth. When the diffusion component degenerate or vanish the value function is not expected to be a classical solution, Smooth-fit principle may fail: [Boyarchenko & Levendorskii 02] and [Alili & Kyprianou 05] Study the regularity of the value function and the optimal exercise boundary with the pure jump Lévy processes ([Caffarelli & Silvestre 08]).

68 Thanks for your attention!

69 Appendix 1: The free boundary is Hölder continuous Lemma For any ǫ > 0, if there exists δ > 0 such that for any t 1, t 2 satisfying ǫ t 1 < t 2 T 0 and t 2 t 1 δ u(b(t 1 ),t) u(b(t 1 ),t 1 )) C ǫ (t 2 t 1 ) α, t 1 t t 2, 0 < α 1 then there exists δ (0,δ] b(t 1 ) b(t 2 ) C ǫ(t 2 t 1 ) α/2, 0 t 2 t 1 δ. x b(t 1 ) u(b(t 1 ), t) b(t) 0 ǫ t 1 δ t 2 δ T 0 t

70 Appendix 1: The free boundary is Hölder continuous Lemma For any ǫ > 0, if there exists δ > 0 such that for any t 1, t 2 satisfying ǫ t 1 < t 2 T 0 and t 2 t 1 δ u(b(t 1 ),t) u(b(t 1 ),t 1 )) C ǫ (t 2 t 1 ) α, t 1 t t 2, 0 < α 1 then there exists δ (0,δ] b(t 1 ) b(t 2 ) C ǫ(t 2 t 1 ) α/2, 0 t 2 t 1 δ. Use this lemma twice, one can show b(t) is Hölder continuous. 1. u(b(t 1 ),t) u(b(t 1 ),t 1 ) max s t u(b(t 1 ),s)(t 2 t 1 ) = δ 1, b(t 1 ) b(t 2 ) C 1 (t 2 t 1 ) 1 2, 0 t 2 t 1 δ As a result, for 0 t 2 t 1 δ 1, t u(b(t 1 ),t) t u(b(t),t) C b(t 1 ) b(t) 1 2 C 2 (t 2 t 1 ) 1 4. Estimate for t u(b(t 1 ),t) get updated, u(b(t 1 ),t) u(b(t 1 ),t 1 ) C 2 (t 2 t 1 ) 5 4. Then apply the lemma again.

71 Appendix 1 cont. : Proof of the lemma x χ u g b(t 1 ) D χ 0 = u g Lχ L(u g) Choose sufficiently small β and δ b(t) Consider the test function 0 ǫ t 1 t 2 δ T 0 t χ(x) = { [ Cǫ (t 2 t 1 ) α 2 + β (x b(t 1 ))] + } 2, b(t 2 ) x b(t 1 ). We have χ(x) > 0 when x > b(t 1 ) Cǫ β (t 2 t 1 ) α/2. We want to show χ(x) (u g)(x, t), (x, t) D, for suitably chosen positive constant β and δ (0,δ]. For any (x,t) D, since (u g)(x,t) > 0, we have x > b(t 1 ) Cǫ β (t 2 t 1 ) α 2 = b(t 1 ) b(t 2 ) Cǫ β (t 2 t 1 ) α 2.

72 Appendix 2: B(t) > b(t), t (0, T] Step 1: B(t) b(t) If these is a t 0 (0,T] such that B(t 0 ) < b(t 0 ), since x I(x,t) is strictly increasing, we have I(x,t 0 ) > 0 for all x (B(t 0 ),b(t 0 )). It implies ( t L D + (r + λ)) u(x,t 0 ) = I(x,t 0 ) < 0, for any x (B(t 0 ),b(t 0 ) which contradicts with ( t L D + (r + λ)) u(x,t) 0. Step 2: B(t) > b(t) If there is a t 0 (0,T] such that B(t 0 ) = b(t 0 ), we have ( t L D + (r + λ)) (u g)(x,t) = I(x,t) > 0, when x > B(t). x ( t L D + (r + λ)) (u g) = J(x, t) > 0 l u(x, t) g(x, t) > 0 x (u g)(b(t 0 ), t 0 ) > 0 by the Hopf Lemma B(t) b(t) u(b(t 0 ), t 0 ) g(t 0 ) = 0 x (u g)(b(t 0 ),t 0 ) > 0 contradicts with the smooth-fit property at (b(t 0 ),t 0 ). 0 t 0 T t