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Free boundary regularity close to initial state and applications to finance Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Teitur Arnarson KTH, Stockholm PDE Methods in Finance 2007 October 15-16, University of Marne-la-Vallée Application to indifference pricing Outline Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options ? ? ? ? Problem and setup History The blow-up technique Application to indifference pricing History and background The blow-up technique Application to indifference pricing American option Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background I A contract on one or several underlying assets that can be exercised during some predetermined period [t, T ]. The blow-up technique Application to indifference pricing American option Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background I I A contract on one or several underlying assets that can be exercised during some predetermined period [t, T ]. Payoff g : Rn → R at exercise τ ∈ T[t,T ] . The blow-up technique Application to indifference pricing Example: American put option Gives you the right, but not the obligation, to sell the underlying stock Xs for a predetermined price K any time s ∈ [t, T ]. Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Example: American put option Gives you the right, but not the obligation, to sell the underlying stock Xs for a predetermined price K any time s ∈ [t, T ]. Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique At exercise τ the payoff is g (Xτ ) = max(K − Xτ , 0). g (x) K x Application to indifference pricing Free boundary regularity close to initial state and applications to finance Complete markets Teitur Arnarson KTH, Stockholm The market consists of American options I non-risky asset History and background dBs Bt I = ρBs ds = B. traded asset dXs Xt = µXs ds + σXs dWs = x Ws is Brownian motion. The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Option price Teitur Arnarson KTH, Stockholm American options The price h of an American option with payoff g is given by Theorem (Risk-neutral valuation formula) History and background The blow-up technique Application to indifference pricing h(x, t) = sup E (e −ρ(τ −t) g (Xτ )|Xt = x). τ ∈[t,T ] Free boundary regularity close to initial state and applications to finance Variational inequality h solves the following linear variational inequality 1 min − ht − σ 2 x 2 hxx − ρxhx + ρh, 2 h(x, t) − g (x) = 0 Teitur Arnarson KTH, Stockholm American options in R × [0, T ) h(x, T ) = g (x) in [0, T ) History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Variational inequality h solves the following linear variational inequality 1 min − ht − σ 2 x 2 hxx − ρxhx + ρh, 2 h(x, t) − g (x) = 0 Teitur Arnarson KTH, Stockholm American options in R × [0, T ) h(x, T ) = g (x) in [0, T ) A free boundary Γ separates the sets C E 1 = {−ht − σ 2 x 2 hxx − ρxhx + ρh = 0} 2 = {h − g = 0}. History and background The blow-up technique Application to indifference pricing History Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options Results for the American put. History and background The blow-up technique I Kuske & Keller (1998) I Bunch & Johnsson (2000) I Stamicar, Sevcovic & Chadam (1999) Application to indifference pricing Free boundary regularity close to initial state and applications to finance Chen, Chadam: Reformulation Teitur Arnarson KTH, Stockholm American options In dimensionless variables the price function h̃(x, t) solves h̃t − h̃xx − (k − 1)h̃x + k h̃ = 0 for x > β̃(t) h̃ = 1 − e x for x < β̃(t) h̃(0, x) = (1 − e x )+ , where x = β̃(t) is a parameterization of the free boundary Γ. History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Fundamental solution Teitur Arnarson KTH, Stockholm Find the fundamental solution for the PDE 1 (x + (k − 1)t)2 √ exp − Φ(x, t) = 4t 2 πt and get the following integral representation Z 0 (1 − e y )Φ(x − y , t)dy h̃(x, t) = −∞ Z tZ β̃(t−θ) Φ(x − y , θ)dydθ. +k 0 −∞ American options History and background The blow-up technique Application to indifference pricing ODE for the free boundary Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm Derive an ODE for the free boundary 2Φx (β̃(t), t) −2 β̃˙ = − k Z 0 t ˙ − θ)dθ. Φx (β̃(t) − β̃(t − θ), θ)β̃(t American options History and background The blow-up technique Application to indifference pricing ODE for the free boundary Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm Derive an ODE for the free boundary 2Φx (β̃(t), t) −2 β̃˙ = − k Z t ˙ − θ)dθ. Φx (β̃(t) − β̃(t − θ), θ)β̃(t 0 Asymptotic expansion β̃ 2 1 1 17 = −ξ − + + + ... 4t 2ξ 8ξ 2 24ξ 3 √ where ξ = 4πk 2 t. American options History and background The blow-up technique Application to indifference pricing Summary of the expansion method Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options Advantage I Good precision History and background The blow-up technique Application to indifference pricing Drawback I One-dimensional, linear setting. Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options The blow-up technique (only 4 slides) History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance A general obstacle problem Teitur Arnarson KTH, Stockholm American options Obstacle problem with a non-linear, n + 1-dimensional, parabolic operator min(Dt u − F (D 2 u, Du, u, x, t), u − g ) = 0 in B1 × (0, 1) u(x, 0) = g (x) where B1 is the unit ball in Rn . in B1 History and background The blow-up technique Application to indifference pricing Scaling in the point (0, 0) Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm For simplicity assume: u(0, 0) = g (0) = 0. American options Scaled function History and background u(r x, r 2 t) ur (x, t) = αr Scaled operator Fr (D 2 u, Du, u, x, t) = F (D 2 u, r Du, r 2 u, r x, r 2 t). The blow-up technique Application to indifference pricing Scaling in the point (0, 0) Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm For simplicity assume: u(0, 0) = g (0) = 0. American options Scaled function History and background u(r x, r 2 t) ur (x, t) = αr Scaled operator Fr (D 2 u, Du, u, x, t) = F (D 2 u, r Du, r 2 u, r x, r 2 t). Choose αr so that 0 < limr →0 ur < ∞ . The blow-up technique Application to indifference pricing Scaled obstacle problem Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options Under standard assumptions on F the scaled function ur solves min(Dt ur − Fr (D 2 ur , Dur , ur , x, t), 1 ur − gr ) = 0 in B1/r × (0, 2 ) r ur (x, 0) = gr (x) in B1/r . History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Blow-up limit Teitur Arnarson KTH, Stockholm American options Take the so called blow-up limit by letting r → 0. History and background If we have the right growth and continuity of u the limit function u0 = limr →0 ur will solve Application to indifference pricing 2 min(Dt u0 − F0 (D u0 , Du0 , u0 , x, t), u0 − g0 ) = 0 The blow-up technique in R × R+ u0 (x, 0) = g0 (x) in R. Free boundary regularity close to initial state and applications to finance Blow-up limit Teitur Arnarson KTH, Stockholm American options Take the so called blow-up limit by letting r → 0. History and background If we have the right growth and continuity of u the limit function u0 = limr →0 ur will solve Application to indifference pricing 2 min(Dt u0 − F (D u0 , 0, 0, 0, 0), u0 − g0 ) = 0 The blow-up technique in R × R+ u0 (x, 0) = g0 (x) in R. Free boundary regularity close to initial state and applications to finance Free boundary regularity Teitur Arnarson KTH, Stockholm Assume we have a free boundary. American options t u=g History and background Dt u − Fu = 0 The blow-up technique Γ Application to indifference pricing x Free boundary regularity close to initial state and applications to finance Free boundary regularity Teitur Arnarson KTH, Stockholm Assume that the free boundary stays above t = cx 2 . American options t = cx 2 t u=g History and background Dt u − Fu = 0 The blow-up technique Γ Application to indifference pricing x Free boundary regularity close to initial state and applications to finance Free boundary regularity Pick a sequence X1 , X2 . . . ∈ {t = cx 2 },where Xj = (xj , tj ). Teitur Arnarson KTH, Stockholm American options t = cx 2 t u=g X1 History and background Dt u − Fu = 0 The blow-up technique Γ Application to indifference pricing X2 X3 Xi x Free boundary regularity close to initial state and applications to finance Free boundary regularity Teitur Arnarson KTH, Stockholm Set rj = |Xj |. . . American options t = cx 2 t u=g History and background Dt u − Fu = 0 The blow-up technique Γ Application to indifference pricing Xi Brj × (0, rj2 ) x Free boundary regularity close to initial state and applications to finance Free boundary regularity . . . and scale the problem by rj . X̃j = (xj /rj , tj /rj2 ). Teitur Arnarson KTH, Stockholm American options t = cx 2 History and background t urj = grj X̃i Dt urj − Frj urj = 0 Γ Application to indifference pricing B1 × (0, 1) Xi The blow-up technique Brj × (0, rj2 ) x Free boundary regularity close to initial state and applications to finance Free boundary regularity Teitur Arnarson KTH, Stockholm Take the limit as j → ∞. Note |X̃∞ | = 1. American options t = cx 2 t u0 = g0 X̃∞ History and background Dt u0 − F0 u0 = 0 Γ The blow-up technique Application to indifference pricing B1 × (0, 1) x The blow-up limit problem I I For the limit problem no lower order terms occur in the PDE. The limit obstacle g0 is possibly simpler than the original g . Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing The blow-up limit problem I I For the limit problem no lower order terms occur in the PDE. The limit obstacle g0 is possibly simpler than the original g . Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing ⇓ When analysing the limit problem we might arrive at different conclusios: I We might find an analytic solution. I We might be able to contradict the picture above. The obstacle is a strict subsolution g0 is a strict subsolution if 2 −F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1). Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing The obstacle is a strict subsolution g0 is a strict subsolution if 2 −F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1). Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options Dt u0 − F (u0 , 0, 0, 0, 0) ≥ 0 in B1 × (0, 1) and the maximum principle ⇓ u0 > g0 in B1 × (0, 1). History and background The blow-up technique Application to indifference pricing The obstacle is a strict subsolution g0 is a strict subsolution if 2 −F (D g0 , 0, 0, 0, 0) < 0 in B1 × (0, 1). Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options Dt u0 − F (u0 , 0, 0, 0, 0) ≥ 0 in B1 × (0, 1) and the maximum principle ⇓ u0 > g0 in B1 × (0, 1). ⇓ No free boundary exists for the limit problem, i.e. Γ ∈ {t < x 2 · σ(x)} for some modulus of continuity σ(x). History and background The blow-up technique Application to indifference pricing Incomplete markets: Market components The market consists of Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm I non-risky asset (zero interest rate for simplicity) American options Bs I The blow-up technique traded asset dXs Xt I = B. = µXs ds + σXs dWs = x non-traded asset dYs Yt History and background = b(Ys , s)ds + a(Ys , s)dWs0 = y Ws and Ws0 are correlated with correlation ρ ∈ (−1, 1). Application to indifference pricing Aim Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background Define the indifference price h of a call option written on the non-traded asset Ys . The blow-up technique Application to indifference pricing Investment alternatives Alternative 1: Invest in stock Xs and bond Bs Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options I Allocation in traded stock Xs : πs Allocation in bond: πs0 History and background I Wealth: Zs = πs0 + πs . The blow-up technique dZs Zt = πs µds + πs σdWs = z. Application to indifference pricing Investment alternatives Alternative 1: Invest in stock Xs and bond Bs Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options I Allocation in traded stock Xs : πs Allocation in bond: πs0 History and background I Wealth: Zs = πs0 + πs . The blow-up technique dZs Zt = πs µds + πs σdWs = z. Alternative 2: Invest in stock Xs , bond Bs and buy a call option on non-traded asset Ys at time t for price h I American call payoff: g (y ) = (y − K )+ . Application to indifference pricing Indifference pricing I Alternative 1 (Stock and bond only) Initial wealth: z Terminal wealth: ZT Value function: V1 (z, t) = sup E (U(ZT )|Zt = z). π where U(z) = −e −γz . Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Indifference pricing I Alternative 1 (Stock and bond only) Initial wealth: z Terminal wealth: ZT Value function: V1 (z, t) = sup E (U(ZT )|Zt = z). π −e −γz . I where U(z) = Alternative 2 (Stock, bond and call option) Initial wealth: z −h Wealth at exercise time τ : Zτ + g (Yτ ) Value function: V2 (z, y , t) = sup E (V1 (Zτ + g (Yτ ), τ )|Zτ = z, Yτ = y ) π,τ Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Indifference pricing I Alternative 1 (Stock and bond only) Initial wealth: z Terminal wealth: ZT Value function: V1 (z, t) = sup E (U(ZT )|Zt = z). π −e −γz . I where U(z) = Alternative 2 (Stock, bond and call option) Initial wealth: z −h Wealth at exercise time τ : Zτ + g (Yτ ) Value function: V2 (z, y , t) = sup E (V1 (Zτ + g (Yτ ), τ )|Zτ = z, Yτ = y ) π,τ I Definition: The indifference price h satisfies V1 (z, t) = V2 (z − h, y , t) Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Variational inequality Teitur Arnarson KTH, Stockholm American options min(Hh, h − g ) = 0 h(y , T ) = g (y ) in R × [0, T ) in R where µ 1 Hu = Dt u − a2 (y , t)Dy2 u − b(y , t) − ρ a(y , t) Dy u 2 σ 1 + γ(1 − ρ2 )a2 (y , t)(Dy u)2 . 2 History and background The blow-up technique Application to indifference pricing Free boundary at expiry I I I Parameterization of free boundary: Γ = (β(t), t) Location at expiry: β0 = limt→0 β(t) call A(y , t) = −Hg = b − ρ σµ a − 1 2 γ(1 − ρ2 )a2 Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Free boundary at expiry I I I Parameterization of free boundary: Γ = (β(t), t) Location at expiry: β0 = limt→0 β(t) call A(y , t) = −Hg = b − ρ σµ a − Lemma 1 If A(y0 , 0) = 0 and A(y0 + δ, 0)A(y0 − δ, 0) < 0 for all small δ then either no free boundary exists or 1 2 γ(1 − Teitur Arnarson KTH, Stockholm ρ2 )a2 American options t History and background The blow-up technique Application to indifference pricing β0 = y0 . K A>0 y0 A<0 x Free boundary regularity close to initial state and applications to finance Free boundary at expiry I I I Parameterization of free boundary: Γ = (β(t), t) Location at expiry: β0 = limt→0 β(t) call A(y , t) = −Hg = b − ρ σµ a − Lemma 1 If A(y0 , 0) = 0 and A(y0 + δ, 0)A(y0 − δ, 0) < 0 for all small δ then either no free boundary exists or 1 2 γ(1 − Teitur Arnarson KTH, Stockholm ρ2 )a2 American options t History and background The blow-up technique Application to indifference pricing β0 = y0 . K A>0 y0 A<0 x t Lemma 2 If A(y , 0) < −ε for some ε > 0 and all y ∈ {g > 0} then β0 = K . K A<0 x Free boundary regularity: β0 6= K Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm Theorem 1 There exists ξ0 and r > 0 such that for ξ1 < ξ0−2 < ξ2 and t < r (β(t), t) ∈ {(y , t) : ξ1 (y − β0 )2 ≤ t ≤ ξ2 (y − β0 )2 }. American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Free boundary regularity: β0 6= K Teitur Arnarson KTH, Stockholm Theorem 1 There exists ξ0 and r > 0 such that for ξ1 < ξ0−2 < ξ2 and t < r (β(t), t) ∈ {(y , t) : ξ1 (y − β0 )2 ≤ t ≤ ξ2 (y − β0 )2 }. ξ0 solve u(ξ0 ) − ξ0 u 0 (ξ0 ) u(ξ) = ξ(6a2 (β0 , 0) + ξ 2 ) ξ −∞ The blow-up technique Application to indifference pricing = 0 where Z American options History and background exp (6a2 (β0 −x 2 0 ,0) dx. , 0) + x 2 )2 x 2 4a2 (β Proof ? Rewrite equation Ĥu = A(y , t)χ{u>0} where Ĥ = H + γ(1 − ρ2 )a2 gy Dy . Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Proof ? Rewrite equation Ĥu = A(y , t)χ{u>0} where Ĥ = H + γ(1 − ρ2 )a2 gy Dy . ? Scale by r 3 u(ry + β0 , r 2 t) ur (y , t) = r3 and take the limit r → 0 1 Dt u0 − a02 Dy2 u0 = A0 y χ{u0 >0} . 2 Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Proof ? Rewrite equation Teitur Arnarson KTH, Stockholm Ĥu = A(y , t)χ{u>0} where Ĥ = H + γ(1 − ρ2 )a2 gy Dy . ? Scale by r 3 u(ry + β0 , r 2 t) ur (y , t) = r3 and take the limit r → 0 1 Dt u0 − a02 Dy2 u0 = A0 y χ{u0 >0} . 2 √ ? Self-similar solution in the variable ξ = −y / t. ũ(ξ) = u(y , t). −ũ 00 − 1 3 ξũ 0 + 2 = −A0 ξ 2 2a0 2a0 in {ũ > 0} American options History and background The blow-up technique Application to indifference pricing Free boundary regularity: β0 = K Free boundary regularity close to initial state and applications to finance Teitur Arnarson KTH, Stockholm American options History and background Theorem 2 There exists a modulus of continuity σ(r ) such that 2 (β(t), t) ∈ {(y , t) : t < (y − K ) σ(y − K )}. The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Proof ? Scale by r hr (y , t) = h(ry + K , r 2 t) r and take limit r → 0 1 min(Dt h0 − a02 Dy2 h0 , h0 − g0 ) = 0 2 h0 (y , 0) = g0 (y ) Teitur Arnarson KTH, Stockholm American options History and background The blow-up technique Application to indifference pricing Free boundary regularity close to initial state and applications to finance Proof ? Scale by r hr (y , t) = h(ry + K , r 2 t) r Teitur Arnarson KTH, Stockholm American options and take limit r → 0 1 min(Dt h0 − a02 Dy2 h0 , h0 − g0 ) = 0 2 h0 (y , 0) = g0 (y ) ? g0 = y + is a strict subsolution to the limit PDE. ⇓ The limit problem does not have a free boundary. ⇓ (β(t), t) ∈ {t < (y − K )2 σ(y − K )}. History and background The blow-up technique Application to indifference pricing