Approximate pricing of European and Barrier claims in a - PowerPoint PPT Presentation

Approximate pricing of European and Barrier claims in a local-stochastic volatility setting Weston Barger Based on work with Matthew Lorig Department of Applied Mathematics, University of Washington

Problem statement We are interested in computing the price of a barrier-style claim V = ( V t ) 0 ≤ t ≤ T (option) written on an asset S = ( S t ) 0 ≤ t ≤ T (asset) whose payoff at the maturity date T is given by ∈ � 1 { τ>T } � ϕ ( S T ) , τ = inf { t ≥ 0 : S t / I } . (payoff) where � I is an interval in R . • The option becomes worthless if S leaves � I at any time t ≤ T . • These types of options are known as knock-out options.

Problem statement Examples: • � I = ( L, U ) - double-barrier knock-out • � I = ( L, ∞ ) - single-barrier option with lower barrier • � I = ( −∞ , U ) - single-barrier option with upper barrier • � I = ( −∞ , ∞ ) - European option

Problem statement Examples: • � I = ( L, U ) - double-barrier knock-out • � I = ( L, ∞ ) - single-barrier option with lower barrier • � I = ( −∞ , U ) - single-barrier option with upper barrier • � I = ( −∞ , ∞ ) - European option We can price knock-in options by pricing European and knock-out options using knock-in knock-out parity V ( knock-in ) + V ( knock-out ) = V ( European ) , � � I I where the payoff of a knock-in option is given by 1 { τ ≤ T } � ϕ ( S T ) .

Asset model For an asset S , we consider models of in a general local-stochastic volatility setting S t = e X t , d X t = µ ( X t , Y t )d t + σ ( X t , Y t )d W t , d Y t = c ( X t , Y t )d t + g ( X t , Y t )d B t , d � W, B � t = ρ d t, where W and B are correlated Brownian motions under the pricing probability measure P .

Risk-neutral price Let • r = 0 , • I = log � I , ϕ (e x ) = � • ϕ ( x ) = � ϕ ( s ) . To avoid arbitrage, all traded assets must be martingales under the pricing measure P . The value V t of the claim with the payoff 1 { τ>T } ϕ ( X T ) , τ = inf { t ≥ 0 : X t / ∈ I } (payoff) at time t ≤ T is given by V t = 1 { τ>t } u ( t, X t , Y t ) , where � � u ( t, x, y ) := E 1 { τ>T } ϕ ( X T ) | X t = x, Y t = y, τ > t .

Possible Approaches How might one solve the pricing problem? • Simulation • Ex: Monte Carlo • Limitation: Simulation gives you the price for one ( X 0 , Y 0 ) and parameter choice. • Limitation: Low degree of precision

Possible Approaches How might one solve the pricing problem? • Simulation • Ex: Monte Carlo • Limitation: Simulation gives you the price for one ( X 0 , Y 0 ) and parameter choice. • Limitation: Low degree of precision • Numerical PDE solver • Ex: Solve PDE using finite difference or finite element • Limitation: Numerical solvers suffer from the “curse of dimensionality.” • Limitation: Discretized solution

Possible Approaches How might one solve the pricing problem? • Simulation • Ex: Monte Carlo • Limitation: Simulation gives you the price for one ( X 0 , Y 0 ) and parameter choice. • Limitation: Low degree of precision • Numerical PDE solver • Ex: Solve PDE using finite difference or finite element • Limitation: Numerical solvers suffer from the “curse of dimensionality.” • Limitation: Discretized solution • Analytical techniques on the PDE • Ex: perturbation theory • Advantage: Fast evaluation at higher dimension • Advantage: Ease of implementation

Pricing PDE The function u � � u ( t, x, y ) = E 1 { τ>T } ϕ ( X T ) | X t = x, Y t = y, τ > t , is the unique classical solution of the Kolmogorov Backward equation 0 = ( ∂ t + A ) u, u ( T, · ) = ϕ, where A , the generator of ( X, Y ) , is given explicitly by A = 1 x + ρσ ( x, y ) g ( x, y ) ∂ x ∂ y + 1 2 σ 2 ( x, y ) ∂ 2 2 g 2 ( x, y ) ∂ 2 y + µ ( x, y ) ∂ x + c ( x, y ) ∂ y , and the domain of A is given by dom ( A ) := { g ∈ C 2 : lim x → ∂I g ( x, y ) = 0 } .

Our approach The full pricing PDE 0 = ∂ t u + 1 2 σ 2 ( x, y ) ∂ 2 x u + ρσ ( x, y ) g ( x, y ) ∂ x ∂ y u + 1 2 g 2 ( x, y ) ∂ 2 y u + µ ( x, y ) ∂ x u + c ( x, y ) ∂ y u, u ( T, · , · ) = ϕ, is not generally solvable in closed form.

Our approach The full pricing PDE 0 = ∂ t u + 1 2 σ 2 ( x, y ) ∂ 2 x u + ρσ ( x, y ) g ( x, y ) ∂ x ∂ y u + 1 2 g 2 ( x, y ) ∂ 2 y u + µ ( x, y ) ∂ x u + c ( x, y ) ∂ y u, u ( T, · , · ) = ϕ, is not generally solvable in closed form. If σ, g, µ, c were constant and ρ = 0 , the pricing PDE would be ∂ t u + 1 x u + 1 2 σ 2 ∂ 2 2 g 2 ∂ 2 y u + µ∂ x u + c∂ y u = 0 , which is solvable. This suggests a perturbation expansion...

Perturbation framework Let f ∈ { 1 2 σ 2 , σg, 1 2 g 2 , µ, c } and (¯ x, ¯ y ) ∈ I × R . We introduce ε ∈ [0 , 1] and define f ε ( x, y ) := f (¯ x + ε ( x − ¯ x ) , ¯ y + ε ( y − ¯ y )) . Note that f ε ( x, y ) | ε =1 = f ( x, y ) and f ε ( x, y ) | ε =0 = f (¯ x, ¯ y ) .

Perturbation framework Let f ∈ { 1 2 σ 2 , σg, 1 2 g 2 , µ, c } and (¯ x, ¯ y ) ∈ I × R . We introduce ε ∈ [0 , 1] and define f ε ( x, y ) := f (¯ x + ε ( x − ¯ x ) , ¯ y + ε ( y − ¯ y )) . Note that f ε ( x, y ) | ε =1 = f ( x, y ) and f ε ( x, y ) | ε =0 = f (¯ x, ¯ y ) . Taylor expanding f ε about the point ε = 0 yields f ε = f 0 + εf 1 + ε 2 f 2 + · · · , where n � ∂ n − i ∂ i y f (¯ x, ¯ y ) x x ) n − i ( y − ¯ y ) i . f n ( x, y ) = ( x − ¯ i !( n − i )! i =0

Perturbation framework Recall A = 1 x + ρσ ( x, y ) g ( x, y ) ∂ x ∂ y + 1 2 σ 2 ( x, y ) ∂ 2 2 g 2 ( x, y ) ∂ 2 y + µ ( x, y ) ∂ x + c ( x, y ) ∂ y , 2 g 2 , µ, c } with f ε in A and expanding Replacing f ∈ { 1 2 σ 2 , σg, 1 yields ∞ � A ε,ρ = ε n ( A n, 0 + ρ A n, 1 ) , n =0 where 2 σ 2 ) n ∂ 2 2 g 2 ) n ∂ 2 A n, 0 := ( 1 x + ( 1 y + µ n ∂ x + c n ∂ y A n, 1 := ( σg ) n ∂ x ∂ y .

Perturbation framework We try to solve ( ∂ t + A ε,ρ ) u ε,ρ = 0 , u ε,ρ ( T, · , · ) = ϕ by expanding u ε,ρ in powers of ε and ρ as follows ∞ n � � u ε,ρ = ε n − i ρ i u n − i,i . n =0 i =0 An approximation to the solution of the original pricing PDE ( ∂ t + A ) u = 0 , u ( T, · , · ) = ϕ will be obtained by setting ε = 1 in u ε,ρ .

Perturbation framework We now have the parameterized set of PDEs ( ∂ t + A ε,ρ ) u ε,ρ = 0 , u ε,ρ ( T, · , · ) = ϕ. Inserting A ε,ρ and u ε,ρ and collecting powers of ε and ρ gives O ( ε 0 ρ 0 ) : ( ∂ t + A 0 , 0 ) u 0 , 0 = 0 , u 0 , 0 ( T, · , · ) = ϕ, O ( ε n ρ k ) : ( ∂ t + A 0 , 0 ) u n,k + F n,k = 0 , u n,k ( T, · , · ) = 0 , where n k � � F n,k = (1 − δ i + j, 0 ) A i,j u n − i,k − j . i =0 j =0 • O ( ε 0 ρ 0 ) is a constant coefficient heat equation. • O ( ε n ρ k ) is a constant coefficient heat equation with a forcing term.

N th order approximation Definition Let u be the unique classical solution of PDE problem (1). ( ∂ t + A ) u = 0 , u ( T, · , · ) = ϕ, (1) u ρ We define ¯ N , the N th order approximation of u , as � � N � i � u ρ ε j ρ i − j u j,i − j ( t, x, y ) ¯ N ( t, x, y ) := � y,ε )=( x,y, 1) , (¯ x, ¯ i =0 j =0 where u 0 , 0 satisfies (2) and u n,k satisfies (3) for ( n, k ) � = (0 , 0) . ( ∂ t + A 0 , 0 ) u 0 , 0 = 0 , u 0 , 0 ( T, · , · ) = ϕ, (2) ( ∂ t + A 0 , 0 ) u n,k + F n,k = 0 , u n,k ( T, · , · ) = 0 . (3)

Duhamel’s principal Duhamel’s principle states that the the unique classical solution to ( ∂ t + A 0 , 0 ) u + F = 0 , u ( T, · , · ) = h, is given by � T u ( t, x, y ) = P 0 , 0 ( t, T ) h ( x, y ) + d s P 0 , 0 ( t, s ) F ( s, x, y ) , t where we have introduced P 0 , 0 the semigroup generated by A 0 , 0 , which is defined as follows � � P 0 , 0 ( t, s ) h ( x, y ) = d ξ d η Γ 0 , 0 ( t, x, y ; s, ξ, η ) h ( ξ, η ) , I R where 0 ≤ t ≤ s ≤ T , and Γ 0 , 0 is the solution of 0 = ( ∂ t + A 0 , 0 )Γ 0 , 0 ( · , · , · ; T, ξ, η ) , Γ 0 , 0 ( T, · , · ; T, ξ, η ) = δ ξ,η .

Formula for u n,k Proposition The function u 0 , 0 is given by u 0 , 0 ( t ) = P 0 , 0 ( t, T ) ϕ, and for ( n, k ) � = (0 , 0) , we have � T � T � T n + k � � u n,k ( t ) = d s 1 d s 2 · · · d s j t s 1 s j − 1 j =1 I n,k,j P 0 , 0 ( t, s 1 ) A n 1 ,k 1 · · · P 0 , 0 ( s j − 1 , s j ) A n j ,k j P 0 , 0 ( s j , T ) ϕ, with I n,k,j given by  �  � n 1 + · · · + n j = n,  � n 1 , · · · , n j �   �  � ∈ Z 2 × j I n,k,j = k 1 + · · · + k j = k, . � + k 1 , · · · , k j  �    � 1 ≤ n i + k i , for all 1 ≤ i ≤ j

Asymptotic accuracy for European claims Let I = R (European option), and let h − 1 be the number of Lipschitz continuous derivatives of ϕ . Then under certain regularity assumptions on the coefficients ( µ, σ, g, c ) , the approximate solution satisfies the following: