Regularity of the Optimal Exercise Boundary of American Options for Jump Diffusions

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1 Regularity of the Optimal Exercise Boundary of American Options for Jump Diffusions Hao Xing University of Michigan joint work with Erhan Bayraktar, University of Michigan SIAM Conference on Financial Mathematics & Engineering New Brunswick, Nov. 22, 2008

2 American options in Black-Scholes Model Let us consider the underlying dynamics ds t = (r q)s t dt + σs t dw t, q 0 is the dividend.

3 American options in Black-Scholes Model Let us consider the underlying dynamics 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 E [ e rτ (K S τ ) + S 0 = S ]. τ S 0,T t

4 American options in Black-Scholes Model Let us consider the underlying dynamics 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 E [ e rτ (K S τ ) + S 0 = S ]. τ S 0,T t V (S, t) satisfies the following free boundary problem t V σ2 S 2 SS 2 V + (r q)s SV rv = 0, V (S, t) = K s(t), S V (s(t), t) = 1 V (S, T ) = (K S) +. S > s(t) S s(t)

5 Regularity of the value function S It is well known that (Karatzas & Shreve 1998, Peskir 2005) 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 ).

6 Regularity of the value function S It is well known that (Karatzas & Shreve 1998, Peskir 2005) 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?

7 Regularity of the value function S It is well known that (Karatzas & Shreve 1998, Peskir 2005) 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 [2007]).

8 American options for jump diffusions Assume the underlying process follows jump diffusions (under the risk neutral measure calibrated from the market) ds t = µs t dt + σ(s t, t)s t dw t + S t R (e z 1)N(dt, dz), σ(s, t) is continuously differentiable in S and t, 0 < δ σ, for some positive constants δ and. N(dt, dz) is a Poisson random measure with the mean measure λdt ν(dz). ν(dz) is a finite measure on R. At the time of jump, S t S t e Z, Z has distribution ν(dz), (jump up for Z > 0, down for Z < 0 ). We assume E[e Z ] < +, so that (e rt S t ) t 0 is a martingale.

9 American options for jump diffusions Assume the underlying process follows jump diffusions (under the risk neutral measure calibrated from the market) ds t = µs t dt + σ(s t, t)s t dw t + S t R (e z 1)N(dt, dz), σ(s, t) is continuously differentiable in S and t, 0 < δ σ, for some positive constants δ and. N(dt, dz) is a Poisson random measure with the mean measure λdt ν(dz). ν(dz) is a finite measure on R. At the time of jump, S t S t e Z, Z has distribution ν(dz), (jump up for Z > 0, down for Z < 0 ). We assume E[e Z ] < +, so that (e rt S t ) t 0 is a martingale. Examples: Merton s model: Z is normal; Kou s model: Z is double exponential.

10 Why jump diffusions? Proposition (Pham 1995, Yang et al. 2006, Bayraktar 2007) The value function V (S, t) is the unique classical solution of L D V + λ V (Se z, t) ν(dz) = 0, S > s(t), R V (S, t) = K S, S s(t), V (S, T ) = (K S) +, in which L D t σ2 S 2 2 SS + µs S (r + λ). Moreover, S V (s(t), t) = 1 and L D V + λ R V (Sez, t) ν(dz) 0.

11 Why jump diffusions? Proposition (Pham 1995, Yang et al. 2006, Bayraktar 2007) The value function V (S, t) is the unique classical solution of L D V + λ V (Se z, t) ν(dz) = 0, S > s(t), R V (S, t) = K S, S s(t), V (S, T ) = (K S) +, in which L D t σ2 S 2 2 SS + µs S (r + λ). Moreover, S V (s(t), t) = 1 and L D V + λ R V (Sez, t) ν(dz) 0. Proposition (Friedman 1975, Yang et al. 2006) Probabilitistic argument? lim tv (S, t) = 0, t [0, T ). S s(t)

12 Literature on American options in jump models In the Lévy models with infinite activity jumps, V is not expected to be a classical solution in general. Pham [1998]: viscosity sense, Lamberton and Mikou [2008]: distribution sense, Achdou [2008]: Sobolev sense. We need V to be a classical solution.

13 Literature on American options in jump models In the Lévy models with infinite activity jumps, V is not expected to be a classical solution in general. Pham [1998]: viscosity sense, Lamberton and Mikou [2008]: distribution sense, Achdou [2008]: Sobolev sense. We need V to be a classical solution. Continuity of the free boundary: Pham [1995], Yang et al. [2006], Lamberton and Mikou [2008].

14 Literature on American options in jump models In the Lévy models with infinite activity jumps, V is not expected to be a classical solution in general. Pham [1998]: viscosity sense, Lamberton and Mikou [2008]: distribution sense, Achdou [2008]: Sobolev sense. We need V to be a classical solution. Continuity of the free boundary: Pham [1995], Yang et al. [2006], Lamberton and Mikou [2008]. Differentiability of the free boundary: Yang et al. [2006] proved s(t) C 1 [0, T ) under the condition r q + λ (e z 1)ν(dz). (C) R +

15 Behavior of the free boundary Levendorskii [2004], Yang et al. [2006], Lamberton and Mikou [2008] showed that { K (C) holds, s(t ) lim s(t) = min{k, S 0 } = t T S 0 (C) fails, where S 0 is the unique solution of the integral equation J 0 (S) qs rk + λ (Se z K) + ν(dz) = 0. R

16 Behavior of the free boundary Levendorskii [2004], Yang et al. [2006], Lamberton and Mikou [2008] showed that { K (C) holds, s(t ) lim s(t) = min{k, S 0 } = t T S 0 (C) fails, where S 0 is the unique solution of the integral equation J 0 (S) qs rk + λ (Se z K) + ν(dz) = 0. Our goals: Extend regularity properties of the free boundary s(t) to the case where (C) fails. Our approach treats both cases simultaneously. R

17 Main results Let us change the variables 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.

18 Main results Let us change the variables 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 For any ɛ > 0, there exists δ > 0, if ɛ t 1 t 2 T and 0 < t 2 t 1 < δ, b(t 1 ) b(t 2 ) C ɛ (t 2 t 1 ) 5 8, in which C ɛ is a positive constant independent with t 1 and t 2. This means b(t) H 5 8 ([ɛ.t 0 ]).

19 Main results Let us change the variables 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 For any ɛ > 0, there exists δ > 0, if ɛ t 1 t 2 T and 0 < t 2 t 1 < δ, b(t 1 ) b(t 2 ) C ɛ (t 2 t 1 ) 5 8, in which C ɛ is a positive constant independent with t 1 and t 2. This means b(t) H 5 8 ([ɛ.t 0 ]). Theorem b(t) C 1 (0, T ].

20 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 ([ɛ, T ]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ɛ, T ]).

21 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 ([ɛ, T ]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ɛ, 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 ].

22 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 ([ɛ, T ]). If ρ(z) H l 1+2α (R ) for l 1, then b(t) H l 2 +α ([ɛ, 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.

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

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

25 Denominator Let us introduce the auxiliary function on R [0, T ]: J(x, t) qe x [ rk + λ u(x + z, t) + e x+z K ] ν(dz). Lu(x, t) = J(x, t), for x < b(t), t (0, T ], R lim x J(x, t) = rk and lim x + J(x, t) = +, x J(x, t) is strictly increasing and t J(x, t) is non-decreasing.

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

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

28 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 J(x, t) is strictly increasing, we have J(b(t), t) < 0, for t (0, T ]. On the other hand, 0 = lim Lu(x, t) x b(t) = 1 2 σ2 lim x b(t) 2 xxu(x, t) 1 2 σ2 e b(t) j Z h i ff qe b(t) rk + λ u(b(t) + z, t) + e b(t)+z K ν(dz) R = σ2 lim x b(t) x u(x, t) σ2 e b(t) J(b(t), t). The statement follows from J(b(t), t) < 0.

29 Numerator Lemma 2 xtu(b(t)+, t) is continuous in t [0, T ).

30 Numerator Lemma 2 xtu(b(t)+, t) is continuous in t [0, T ). Proof. Setp 1: Use the Maximum Principle of the differential integral operator L and a test function used in Friedman & Shen (2002) to show b(t) H 5 8 ([ɛ, T 0 ]). (Key step to drop the condition (C).)

31 Numerator Lemma 2 xtu(b(t)+, t) is continuous in t [0, T ). Proof. Setp 1: Use the Maximum Principle of the differential integral operator L and a test function used in Friedman & Shen (2002) to show b(t) H 5 8 ([ɛ, 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. [1974] gives us 2 xtu(b(t)+, t) lim x b(t) 2 xtu(x, t) is a continuous function with respect to t.

32 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. )

33 Future Research Consider the regularity of the value function and the optimal exercise boundary with the underlying process Jump diffusions with infinite activity jump, Pure jump Lévy processes (Caffarelli & Silvestre s work).

34 Thanks for your attention!

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

36 Appendix 1: The free boundary is Hölder continuous Lemma For any ɛ > 0, if there exists δ > 0 such that for any t 1, t 2 satisfying ɛ t 1 < t 2 T 0 and t 2 t 1 δ u(b(t 1 ), t) u(b(t 1 ), t 1 )) C ɛ (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 ɛ(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.

37 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 ɛ t 1 t 2 δ T 0 t χ(x) = { [ Cɛ (t 2 t 1 ) α 2 + β (x b(t1 ))] + } 2, b(t 2 ) x b(t 1 ). We have χ(x) > 0 when x > b(t 1 ) Cɛ 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 Cɛ x > b(t 1 ) β (t 2 t 1 ) α 2 = b(t 1 ) b(t 2 ) Cɛ β (t 2 t 1 ) α 2.

38 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 J(x, t) is strictly increasing, we have J(x, t 0 ) > 0 for all x (B(t 0 ), b(t 0 )). It implies Lu(x, t 0 ) = J(x, t 0 ) < 0, for any x (B(t 0 ), b(t 0 )), which contradicts with Lu(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 L(u g)(x, t) = J(x, t) > 0, when x > B(t). x L D (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

ANALYSIS OF THE OPTION PRICES IN JUMP DIFFUSION MODELS

ANALYSIS OF THE OPTION PRICES IN JUMP DIFFUSION MODELS by Hao Xing A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University