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Early exercise boundary regularity close to expiry in indifference setting Early exercise boundary regularity close to Teitur Arnarson KTH, Stockholm expiry in indifference setting American options History and background The blow-up Teitur Arnarson technique KTH, Stockholm Application to indifference pricing Workshop and Mid-Term Conference on Advanced Mathematical Methods for Finance September, 17th-22nd, 2007

Early exercise American option boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options History and background ◮ A contract on one or several underlying assets that can The blow-up technique be exercised during some predetermined period [ t , T ]. Application to indifference pricing

Early exercise American option boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options History and background ◮ A contract on one or several underlying assets that can The blow-up technique be exercised during some predetermined period [ t , T ]. Application to indifference pricing ◮ Payoff g : R n → R at exercise τ ∈ [ t , T ].

Early exercise Example: American put option boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm Gives you the right, but not the obligation, to sell the underlying stock X s for a predetermined price K any time American options s ∈ [ t , T ]. History and background The blow-up technique Application to indifference pricing

Early exercise Example: American put option boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm Gives you the right, but not the obligation, to sell the underlying stock X s for a predetermined price K any time American options s ∈ [ t , T ]. History and background The blow-up technique At exercise τ the payoff is g ( X τ ) = max( K − X τ , 0). Application to indifference pricing g ( x ) x K

Early exercise Complete markets boundary regularity close to expiry in indifference setting Teitur Arnarson The market consists of KTH, Stockholm American options ◮ non-risky asset History and background = ρ B s ds dB s The blow-up technique B t = B . Application to indifference pricing ◮ traded asset dX s = µ X s ds + σ X s dW s = X t x W s is Brownian motion.

Early exercise Option price boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options The price h of an American option with payoff g is given by History and background The blow-up Theorem (Risk-neutral valuation formula) technique Application to indifference pricing e − ρ ( τ − t ) E ( g ( X τ ) | X t = x ) . h ( x , t ) = sup τ ∈ [ t , T ]

Early exercise Variational inequality boundary regularity close to expiry in indifference setting h solves the following linear variational inequality Teitur Arnarson KTH, Stockholm − h t − 1 � 2 σ 2 x 2 h xx − ρ xh x + ρ h , min American options History and � background h ( x , t ) − g ( x ) = 0 in R × [0 , T ) The blow-up technique h ( x , T ) = g ( x ) in [0 , T ) Application to indifference pricing

Early exercise Variational inequality boundary regularity close to expiry in indifference setting h solves the following linear variational inequality Teitur Arnarson KTH, Stockholm − h t − 1 � 2 σ 2 x 2 h xx − ρ xh x + ρ h , min American options History and � background h ( x , t ) − g ( x ) = 0 in R × [0 , T ) The blow-up technique h ( x , T ) = g ( x ) in [0 , T ) Application to indifference pricing A free boundary Γ separates the sets {− h t − 1 2 σ 2 x 2 h xx − ρ xh x + ρ h = 0 } C = = { h − g = 0 } . E

Early exercise History boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options History and Independent results for the American put. background The blow-up technique ◮ Kuske & Keller (1998) Application to indifference pricing ◮ Bunch & Johnsson (2000) ◮ Stamicar, Sevcovic & Chadam (1999)

Early exercise Chen, Chadam: Reformulation boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options In dimensionless variables the price function ˜ h ( x , t ) solves History and background The blow-up h t − ˜ ˜ h xx − ( k − 1)˜ h x + k ˜ for x > ˜ h = 0 β ( t ) technique ˜ for x < ˜ 1 − e x = β ( t ) Application to h indifference pricing ˜ (1 − e x ) + , h (0 , x ) = where x = ˜ β ( t ) is a parameterization of the free boundary Γ.

Early exercise Fundamental solution boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm Find the fundamental solution for the PDE American options − ( x + ( k − 1) t ) 2 � � 1 History and Φ( x , t ) = 2 √ π t exp background 4 t The blow-up technique and get the following integral representation Application to indifference pricing � 0 ˜ (1 − e y )Φ( x − y , t ) dy h ( x , t ) = −∞ � t � ˜ β ( t − θ ) + k Φ( x − y , θ ) dyd θ. 0 −∞

Early exercise ODE for the free boundary boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm Derive an ODE for the free boundary American options � t β = − 2Φ x (˜ History and β ( t ) , t ) ˙ β ( t − θ ) , θ ) ˙ ˜ Φ x (˜ β ( t ) − ˜ ˜ background − 2 β ( t − θ ) d θ. k The blow-up 0 technique Application to indifference pricing

Early exercise ODE for the free boundary boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm Derive an ODE for the free boundary American options � t β = − 2Φ x (˜ History and β ( t ) , t ) ˙ β ( t − θ ) , θ ) ˙ ˜ Φ x (˜ β ( t ) − ˜ ˜ background − 2 β ( t − θ ) d θ. k The blow-up 0 technique Application to indifference pricing Asymptotic expansion ˜ β 2 4 t = − ξ − 1 2 ξ + 1 8 ξ 2 + 17 24 ξ 3 + . . . √ 4 π k 2 t . where ξ =

Early exercise Summary of the expansion method boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options History and Advantage background The blow-up ◮ Good precision technique Application to indifference pricing Drawback ◮ One-dimensional, linear setting.

Early exercise A general obstacle problem boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options Obstacle problem with a non-linear, n + 1-dimensional, History and background parabolic operator The blow-up technique min( D t u − F ( D 2 u , Du , u , x , t ) , u − g ) = 0 in B 1 × (0 , 1) Application to indifference pricing u ( x , 0) = g ( x ) in B 1 where B 1 is the unit ball in R n .

Early exercise Scaling in the point (0 , 0) boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm For simplicity assume: u (0 , 0) = g (0) = 0. American options History and Scaled function background The blow-up u r ( x , t ) = u ( rx , r 2 t ) technique α r Application to indifference pricing Scaled operator F r ( D 2 u , Du , u , x , t ) = F ( D 2 u , rDu , r 2 u , rx , r 2 t ) .

Early exercise Scaling in the point (0 , 0) boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm For simplicity assume: u (0 , 0) = g (0) = 0. American options History and Scaled function background The blow-up u r ( x , t ) = u ( rx , r 2 t ) technique α r Application to indifference pricing Scaled operator F r ( D 2 u , Du , u , x , t ) = F ( D 2 u , rDu , r 2 u , rx , r 2 t ) . Choose α r so that 0 < lim r → 0 u r < ∞ .

Early exercise Scaled obstacle problem boundary regularity close to expiry in indifference setting Teitur Arnarson KTH, Stockholm American options Under standard assumptions on F the scaled function u r History and background solves The blow-up technique min( D t u r − F r ( D 2 u r , Du r , u r , x , t ) , Application to indifference pricing in B 1 / r × (0 , 1 u r − g r ) = 0 r 2 ) u r ( x , 0) = g r ( x ) in B 1 / r .

Early exercise Blow-up limit boundary regularity close to expiry in indifference setting 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 The blow-up technique function u 0 = lim r → 0 u r will solve Application to indifference pricing min( D t u 0 − F ( D 2 u 0 , 0 , 0 , 0 , 0) , u 0 − g 0 ) in R × R + = 0 u 0 ( x , 0) = g 0 ( x ) in R .

Early exercise Free boundary regularity boundary regularity close to expiry in indifference setting Teitur Arnarson Assume we have a free boundary. KTH, Stockholm American options History and background t u = g D t u − Fu = 0 The blow-up technique Γ Application to indifference pricing x

Early exercise Free boundary regularity boundary regularity close to expiry in indifference setting Assume that the free boundary stays above t = cx 2 . Teitur Arnarson KTH, Stockholm American options t = cx 2 History and background t u = g D t u − Fu = 0 The blow-up technique Γ Application to indifference pricing x