On the pricing equations in local / stochastic volatility models

On the pricing equations in local / stochastic volatility models Hao Xing Fields Institute/Boston University joint work with Erhan Bayraktar, University of Michigan Kostas Kardaras, Boston University Probability / Math Finance Seminar Carnegie Mellon, March 29, 2010

Local martingales in finance Throughout we assume r 0. FTAP-I: NFLVR Q P, s.t. S is a Q local martingale. Q is called the Equivalent Local Martingale Measure (ELMM).

Local martingales in finance Throughout we assume r 0. FTAP-I: NFLVR Q P, s.t. S is a Q local martingale. Q is called the Equivalent Local Martingale Measure (ELMM). Local martingale: {σ n } n N such that {S σ n t t 0} is a martingale for any n. Strict local martingale: local martingale that is not a martingale. Example: S = 1/X, where X is the 3-dim Bessel process. S is a strict local martingale. Nonnegative local martingales are supermartingales.

Local martingales in finance cont. Definition (Heston-Loewenstein-Willard 04, Cox-Hobson 05, Jarrow-Protter-Shimbo. 08) S has a bubble, if S is a strict local martingale under Q. E t [S T ] < S t, forward price is below the current price. Put-call parity fails. (Heston et al. 04, Cox-Hobson 05, etc.) European call price is not monotone in time to maturity. (Pal-Protter 08) It is not optimal to exercise American call at maturity. (Cox-Hobson 05, Bayraktar-Kardaras-X. 09) When S is a strict local martingale, the pricing equation of its options has multiple solutions.

Strict local martingale leads to multiple solutions Under Q, ds t = σ(s t )db t with σ(x) > 0 for x > 0 and σ(0) = 0.

Strict local martingale leads to multiple solutions Under Q, ds t = σ(s t )db t with σ(x) > 0 for x > 0 and σ(0) = 0. Consider a European option with payoff g of at most linear growth. The value function is u(x, T ) = E x [g(s T )]. The pricing equation is v T 1 2 σ2 (x)v xx = 0, (x, T ) R 2 ++, v(x, 0) = g(x), v(0, T ) = g(0). (1) v is a classical solution, if v C 2,1 (R 2 ++) C(R 2 +). Example (CEV model) ds t = σs α dw t, S is a strict local martingale when α > 1. v(x, T ) = x Γ(v,w) Γ(v,0) > 0 is of at most linear growth in x. v solves (1) with g 0. Clearly, ṽ 0 is another solution.

Martingale Uniqueness Condition on σ Uniqueness? Martingale? σ(x) C(1 + x) Yes S is a martingale (classical PDE)

Martingale Uniqueness Condition on σ Uniqueness? Martingale? σ(x) C(1 + x) Yes S is a martingale (classical PDE) σ(x) C x 2 No S is a strict local martingale

Martingale Uniqueness Condition on σ Uniqueness? Martingale? σ(x) C(1 + x) Yes S is a martingale (classical PDE) σ(x) C x 2 No S is a strict local martingale Theorem Assume that σ is locally 1/2-Hölder continuous on R +. When g has linear growth, T.F.A.E. 1. u is unique classical solution of (1) among functions of at most linear growth. 2. S, which satisfies ds t = σ(s t )db t, is a martingale. 3. x 1 dx =. (Delbaen-Shirakawa 02) σ 2 (x) When g has sublinear growth, the uniqueness always holds.

Strict local martingale property of S Proposition (Mijatovic-Urusov 09) Assume σ 2 L 1 loc (0, ). Then the following are equivalent: x 1 dx <. σ 2 (x) S is a strict local martingale.

Strict local martingale property of S Proposition (Mijatovic-Urusov 09) Assume σ 2 L 1 loc (0, ). Then the following are equivalent: x 1 dx <. σ 2 (x) S is a strict local martingale. S T is a strict local martingale for any T > 0. This property is not expected to hold for time inhomogeneous process. Example (Cox-Hobson 05) Let us consider ds u = S u T u dw u. Then S u is a martingale when u [0, t] for t < T, but S T = 0 a.s..

Main idea Define δ(x, T ) := x E x [S T ] 0.

Main idea Define δ(x, T ) := x E x [S T ] 0. Properties: S T is a strict local martingale δ(x, T ) > 0. δ is a classical solution to (1) with g 0.

Main idea Define δ(x, T ) := x E x [S T ] 0. Properties: S T is a strict local martingale δ(x, T ) > 0. δ is a classical solution to (1) with g 0. Given u as a classical solution to (1) (Ekstrom-Tysk 09), u + δ is another solution. When uniqueness holds, S must be a martingale. Otherwise u and u + δ are two solutions.

Verification Let v be another classical solution and τ 0 = inf {t 0 : S t = 0} T, then v(s τ0, T τ 0 ) is a local martingale. Let {σ n } n N be a localizing sequence. Then

Verification Let v be another classical solution and τ 0 = inf {t 0 : S t = 0} T, then v(s τ0, T τ 0 ) is a local martingale. Let {σ n } n N be a localizing sequence. Then v(x, T ) = E x [v(s σ n τ 0, T σ n τ 0 )] C(1 + E[S σn τ0 ]). S is a martingale = {S σn τ0 } n N is uniformly integrable. Therefore, we can exchange limit and expectation in v(x, T ) = lim n E x [v(s σ n τ 0, T σ n τ 0 )] = E x [g(s T )] = u(x, T ). When the payoff is of sublinear growth, the uniqueness always holds among sublinear growth functions.

Verification Let v be another classical solution and τ 0 = inf {t 0 : S t = 0} T, then v(s τ0, T τ 0 ) is a local martingale. Let {σ n } n N be a localizing sequence. Then v(x, T ) = E x [v(s σ n τ 0, T σ n τ 0 )] C(1 + E[S σn τ 0 ]). S is a martingale = {S σn τ 0 } n N is uniformly integrable. Therefore, we can exchange limit and expectation in v(x, T ) = lim n E x [v(s σ n τ 0, T σ n τ 0 )] = E x [g(s T )] = u(x, T ). When the payoff is of sublinear growth, the uniqueness always holds among sublinear growth functions. {v(s σ n τ 0, T σ n τ 0 )} n N is uniformly integrable by de la Vallée criterion.

Stochastic volatility models Let us consider ds t = S t b(y t ) dw t, S 0 = x > 0, dy t = µ(y t ) dt + σ(y t ) db t, Y 0 = y > 0, in which W and B have constant correlation ρ ( 1, 1). These models are so called log-linear models. Assumption 1: µ(0) 0, σ(y) > 0 for y > 0, σ(0) = 0, b(y) > 0 for y > 0, and b(0) = 0. µ(y) + σ(y) C(1 + y) for y R +. b has at most polynomial growth. Assumption 2: µ is locally Lipschitz and σ is locally 1/2-Hölder on R +.

Martingale property of S Assumption 3: bσ is locally Lipschitz on R +. ( ) dỹt = µ(ỹt) + ρbσ(ỹt) dt + σ(ỹt)d W t. { } has a unique strong solution up to ζ = inf t 0 : Ỹt =. We denote by Q the law of the solution. The scale function s of Ỹ is s(x) := { x c exp } y 2(µ+ρbσ) c (z) dz σ 2 v(x) := x c s(x) s(y) s (y)σ 2 (y) dy. dy.

Examples and Remarks Example (Andersen-Piterbarg 07) Consider ds t = S(t) Y t dw t and dy t = (θ Y t )dt + Y p t db t. When p 1/2, S is a martingale. When 1/2 < p < 3/2, S is a martingale if and only if ρ 0. Remark: v( ) < S is a strict local martingale. However, it is not clear whether S T is strict for ALL T > 0. Proposition Under Assumptions 1-3, T.F.A.E. v( ) <. S T is a strict local martingale for any T > 0. Remark: This generalizes Theorem 2.4 in (Lions-Musiela 07).

Pricing equation for European options The payoff g is continuous with g(x, y) C(1 + x + y m ). When g growth faster than linear in x, E[g(S T )] = for large T (Andersen-Piterbarg 07). The value function of a European option is The pricing equation is u(x, y, T ) = E x,y [g(s T, Y T )]. T v Lv = 0, (x, y, T ) R 3 ++, v(x, y, 0) = g(x, y), (x, y) R 2 +, (2) in which L := 1 2 b2 (y)x 2 2 x + 1 2 σ2 (y) 2 y + ρσ(y)x 2 xy + µ(y) y.

The boundary condition at y = 0 From the Markov property u(s t, Y t, T t) = E x,y [g(s T, Y T ) F t ] is a martingale on [0, T ].

The boundary condition at y = 0 From the Markov property u(s t, Y t, T t) = E x,y [g(s T, Y T ) F t ] is a martingale on [0, T ]. Let τ 0 = inf {t 0 : Y t = 0}. (2) only ensures v(s t τ0, Y t τ0, T t τ 0 ) to be a local martingale. On {τ 0 < T }, a boundary condition at y = 0 seems needed. A rule of thumb (used in numerical computation): Formally send y to zero in (2), i.e., T u(x, 0, t) µ(0) y u(x, 0, t) = 0. It implicitly requires T u and y u exist on y = 0. When µ(0) = 0, u(x, 0, T ) = g(x, 0). (BC1) (BC2)

Do we really need (BC1)? Lemma When σ is locally 1/2-Hölder on R +, L t (0) 0 for any t 0. Notice that L (0) = µ(0) 0 I {0} (Y u ) du. When µ(0) > 0, the time Y spend on 0 has zero Lebesgue measure, i.e., the boundary point 0 is instantaneously reflecting. Therefore, the boundary condition (BC1) is NOT needed. The local martingale property of v(s, Y, T ) is ensured by (2) until the next hitting time of 0.

Classical solutions Definition When P(τ 0 = ) = 1 or P(τ 0 < ) > 0 with µ(0) = 0, a solution v to (2) (with (BC2)) is a classical solution if v C(R 3 +) C 2,2,1 (R 3 ++). When P(τ 0 < ) > 0 with µ(0) > 0, a solution to (2) is a classical solution if v C(R 3 +) C 2,2,1 (R 3 ++) and T v, y v continuously extend to y = 0.

Main result Theorem Suppose Assumptions 1-3 are satisfied. Then 1. When P(τ 0 = ) = 1, u is the smallest nonneg. classical soln to (2). 2. When P(τ 0 < ) > 0 with µ(0) = 0, u is the S.N.C.S. to (2) and (BC2). 3. When P(τ 0 < ) > 0 with µ(0) > 0, if T u and y u continuously extend to y = 0. Then u is the S.N.C.S. to (2). In all above cases, 1. When g is of sublinear in x and poly. in y, u is the unique solution. 2. When g is of linear in x and poly. in y, u is the unique solution S is a martingale.

What if the uniqueness fails?

What if the uniqueness fails? u(x, y, T ) = lim n u n(x, y, T ) = lim n E x,y [g(s T, Y T ) n]. Each u n is the unique soln of pricing equation with the payoff g n.

What if the uniqueness fails? u(x, y, T ) = lim n u n(x, y, T ) = lim n E x,y [g(s T, Y T ) n]. Each u n is the unique soln of pricing equation with the payoff g n. (BC1) can be understood as asymptotic behavior of u near y = 0. If it is satisfied, it helps numerical computation. (Ekstrom-Tysk 10) gives a sufficient condition such that T u n and y u n extend continuously to y = 0 and (BC1) is satisfied by u n. Example (Bayraktar-Kardaras-X.) Consider g(s) = (S K) +. Define σ n = inf {t 0 : S t n} v := lim n v n = lim n E[g(S T σn )]. When S is a strict local martingale, v is the American call value and v u = δ > 0.

Two technical points in the proof Proposition Under Assumptions 1-3, u C(R 3 +). The pricing equation is degenerate at y = 0. use results of nondeg. PDE to show continuity in the interior. use a prob. argument to show u continuously extends to y = 0. Proposition When µ(0) > 0, define σ n = inf { t 0 : (S t, Y t ) / [ 1 n, n] [0, n]}. Assume that v is a classical solution to (2) with T v and y v continuously extending to y = 0. Then for any t σ n, v (S t, Y t, T t) = v(x, y, T ) t 0 1 µ(0) ( T v(s u, 0, T u) µ(0) y v(s u, 0, T u)) dl u (0) + mart.. Since Y hits 0, cannot apply Itô s lemma to v(s, Y, T ). Use Y ɛ = (Y ɛ) + + ɛ to approximate Y.

Summary In local / stochastic volatility models, Uniqueness of pricing equation among at most linear growth functions. Martingale property of S. Analytic conditions from Feller s test.

Thanks for your attention!