Static Hedging of Barrier Options under Zero-Drift CEV. Sergey Nadtochiy Joint work with Peter Carr ORFE Department Princeton University March 27, 2009 5th Oxford-Princeton Workshop on Financial Mathematics and Stochastic Analysis Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 1 / 22
Introduction and Notation Problem Formulation Definitions Up-and-Out Put (UOP) option on underlying X with maturity T , strike K and barrier U > K has the following payoff at time of maturity I { sup t ∈ [0 , T ] X t < U } · ( K − X T ) + Static Hedging strategy is given by function G : R + → R , such that European option with payoff G ( X T ) has the same price as UOP option, up until the barrier is hit . Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 2 / 22
Introduction and Notation Problem Formulation Black’s Model P.Carr and J.Bowie ”Static Simplicity”(1994) : � + x − U 2 G ( x ) = ( K − x ) + − K � U K Figure: Static Hedging of UOP in Black’s model. K ∗ = U 2 K Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 3 / 22
Introduction and Notation Method of Images PDE Approach In Complete Diffusion models price of a European-type claim satisfies 2 σ 2 ( x , t ) ∂ 2 1 ∂ x 2 u ( x , t ) + µ ( x , t ) ∂ ∂ x u ( x , t ) − r ( t ) u ( x , t ) + ∂ ∂ t u ( x , t ) = 0 u ( x , T ) = h ( x ) where h is the payoff function. For options with upper barrier define domain and add boundary condition: x ∈ (0 , U ) , u ( U , t ) = 0 Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 4 / 22
Introduction and Notation Method of Images PDE Approach In Complete Diffusion models price of a European-type claim satisfies 2 σ 2 ( x , t ) ∂ 2 1 ∂ x 2 u ( x , t ) + µ ( x , t ) ∂ ∂ x u ( x , t ) − r ( t ) u ( x , t ) + ∂ ∂ t u ( x , t ) = 0 u ( x , T ) = h ( x ) where h is the payoff function. For options with upper barrier define domain and add boundary condition: x ∈ (0 , U ) , u ( U , t ) = 0 Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 4 / 22
Introduction and Notation Method of Images Method of Images Want to get rid of the boundary condition in pricing PDE for Barrier Option - make it ”vanilla”. In case of upper barrier, payoff h always has support in [0 , U ]. Problem: Find new payoff function g with support in [ U , ∞ ), such that u h ( U , t ) = u g ( U , t ) , t ∈ [0 , T ] , where u h , u g are solutions of pricing PDE with terminal conditions h and g respectively ( without boundary condition at x = U ! ). Then G = h − g , since function u h − g = u h − u g satisfies pricing PDE in domain ( x , t ) ∈ (0 , U ) × (0 , T ), with terminal condition h and zero boundary condition at x = U ). Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 5 / 22
Introduction and Notation Method of Images Method of Images Want to get rid of the boundary condition in pricing PDE for Barrier Option - make it ”vanilla”. In case of upper barrier, payoff h always has support in [0 , U ]. Problem: Find new payoff function g with support in [ U , ∞ ), such that u h ( U , t ) = u g ( U , t ) , t ∈ [0 , T ] , where u h , u g are solutions of pricing PDE with terminal conditions h and g respectively ( without boundary condition at x = U ! ). Then G = h − g , since function u h − g = u h − u g satisfies pricing PDE in domain ( x , t ) ∈ (0 , U ) × (0 , T ), with terminal condition h and zero boundary condition at x = U ). Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 5 / 22
Introduction and Notation Method of Images Method of Images Figure: Solutions to pricing PDE - u h (blue) and u g (green) - along the line x = U Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 6 / 22
Introduction and Notation Method of Images General Result C.Bardos, R.Douady, A.Fursikov, ”Static Hedging of Barrier Options with a smile” : treats this problem for general parabolic PDE, and proves the existence of approximate solutions g ε , such that � � � u h ( U , t ) − u g ε ( U , t ) sup � < ε � � t ∈ [0 , T ] They show that exact solution doesn’t exist in general... Proof is not constructive - finding the approximate solutions is a separate problem. Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 7 / 22
Introduction and Notation Method of Images General Result C.Bardos, R.Douady, A.Fursikov, ”Static Hedging of Barrier Options with a smile” : treats this problem for general parabolic PDE, and proves the existence of approximate solutions g ε , such that � � � u h ( U , t ) − u g ε ( U , t ) sup � < ε � � t ∈ [0 , T ] They show that exact solution doesn’t exist in general... Proof is not constructive - finding the approximate solutions is a separate problem. Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 7 / 22
Introduction and Notation Method of Images General Result C.Bardos, R.Douady, A.Fursikov, ”Static Hedging of Barrier Options with a smile” : treats this problem for general parabolic PDE, and proves the existence of approximate solutions g ε , such that � � � u h ( U , t ) − u g ε ( U , t ) sup � < ε � � t ∈ [0 , T ] They show that exact solution doesn’t exist in general... Proof is not constructive - finding the approximate solutions is a separate problem. Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 7 / 22
Solution Setup Specification We provide an explicit (analytic) solution to the static hedging problem in the following setting: Risk-neutral dynamics of the underlying are given by Constant Elasticity Volatility (CEV) model with zero drift ( r ≡ q , or, equivalently, X is a forward price) dX t = δ X β +1 dB t t Interest rate is positive and constant. We restrict ourselves to β < 0. Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 8 / 22
Solution Setup Back to PDE Problem: find g : [0 , ∞ ) → R with support in ( U , ∞ ), such that there exists u : R + × R + → R which satisfies Pricing PDE δ 2 2 x 2 β +2 ∂ 2 ∂ x 2 u ( x , τ ) − ru ( x , τ ) − ∂ ∂τ u ( x , τ ) = 0 u ( x , 0) = g ( x ) and u ( U , t ) = P ( U , t , K ) for all t ∈ [0 , T ]. Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 9 / 22
Solution Numerical Approach Least - Squares optimization Choose some strike values ( U < ) κ 1 < κ 2 , ... < κ N Approximate solution: N � ψ j · ( x − κ j ) + g ( x ) = j =1 Choose partitioning t 1 , ..., t M of the interval [0 , T ], and solve a simple quadratic optimization problem 2 M N � � min ψ j C ( U , t k , κ j ) − P ( U , t k , K ) ψ k =1 j =1 (plus some term to penalize for non-smooth solutions) Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 10 / 22
Solution Numerical Approach Results of numerical approach: β = − 0 . 5 , K = 1 , U = 1 . 5 Figure: 1. Payoff ’g’ Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 11 / 22
Solution Matching Prices in Laplace-Carson space Analytic Solution We will provide analytic solution. Look for payoff g in the form � ∞ ψ ( κ )( x − κ ) + d κ, g ( x ) = 0 for some generalized function ψ . Problem: find ψ such that � ∞ ψ ( κ ) C ( U , t , κ ) d κ = P ( U , t , K ) , ∀ t ∈ [0 , T ] , 0 and ψ ( κ ) = 0 for κ ≤ U . Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 12 / 22
Solution Matching Prices in Laplace-Carson space Laplace-Carson transform Let’s work in Laplace-Carson space: � ∞ ˜ 0 e − λτ C ( S , τ, K ) d τ C ( x , λ, K ) := λ Solving the pricing ODE, obtain √ � � 2( λ + r ) 2( λ + r ) ˜ x − β ) K ν ( K − β ) , C ( x , λ, K ) = ˜ c xKI ν ( δ | β | δ | β | for x ∈ [0 , K ]. Where I ν and K ν are the Modified Bessel functions . Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 13 / 22
Solution Matching Prices in Laplace-Carson space New Problem formulation √ 2( λ + r ) Introduce new variable z = δ | β | Problem: find ψ such that � ∞ √ K I ν ( zK − β ) K ν ( zU − β ) √ κψ ( κ ) K ν ( z κ − β ) d κ = (1) I ν ( zU − β ) 0 √ 2 r holds for all z > δ | β | . Sergey Nadtochiy (Princeton University) Static Hedging under Zero-Drift CEV Oxford-Princeton Workshop 14 / 22
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