A PDE pricing framework for cross-currency interest rate derivatives Duy Minh Dang Department of Computer Science University of Toronto, Toronto, Canada dmdang@cs.toronto.edu Joint work with Christina Christara, Ken Jackson and Asif Lakhany Workshop on Computational Finance and Business Intelligence International Conference on Computational Science 2010 (ICCS 2010) Amsterdam, May 30–June 2, 2010 1 / 22
Outline Power Reverse Dual Currency (PRDC) swaps 1 The model and the associated PDE 2 Numerical methods 3 Numerical results 4 Summary and future work 5 2 / 22
Outline Power Reverse Dual Currency (PRDC) swaps 1 The model and the associated PDE 2 Numerical methods 3 Numerical results 4 Summary and future work 5
b b b Power Reverse Dual Currency (PRDC) swaps PRDC swaps: dynamics • Long-dated cross-currency swaps ( ≥ 30 years); • Two currencies (domestic and foreign) and their foreign exchange (FX) rate • FX-linked PRDC coupon amounts in exchange for LIBOR payments, ν β − 1 L d ( T β − 2 , T β − 1 ) N d ν 1 L d ( T 0 , T 1 ) N d ν 2 L d ( T 1 , T 2 ) N d T β − 1 T β T 0 T 1 T 2 ν β − 1 C β − 1 N d ν 1 C 1 N d ν 2 C 2 N d � � � � s ( T α ) • C α = min max F (0 , T α ) − c d , b f c f , b c ◦ s ( T α ) : the spot FX-rate at time T α ◦ F (0 , T α ) = P f (0 , T α ) P d (0 , T α ) s (0) , the forward FX rate ◦ c d , c f : domestic and foreign coupon rates; b f , b c : a cap and a floor • When b f = 0, b c = ∞ , C α is a call option on the spot FX rate h α = c f , k α = f α c d C α = h α max( s ( T α ) − k α , 0) , f α c f 4 / 22
Power Reverse Dual Currency (PRDC) swaps PRDC swaps: issues in modeling and pricing • Essentially, a PRDC swap are long dated portfolio of FX options ◦ effects of FX skew (log-normal vs. local vol/stochastic vol.) √ ◦ interest rate risk (Vega ( ≈ T ) vs. Rho ( ≈ T )) ⇒ high dimensional model, calibration difficulties • Moreover, the swap usually contains some optionality: ◦ knockout ◦ FX-Target Redemption (FX-TARN) ◦ Bermudan cancelable This talk is about • Pricing framework for cross-currency interest rate derivatives via a PDE approach using a three-factor model • Bermudan cancelable feature • Local volatility function • Analysis of pricing results and effects of FX volatility skew 5 / 22
Power Reverse Dual Currency (PRDC) swaps Bermudan cancelable PRDC swaps The issuer has the right to cancel the swap at any of the times { T α } β − 1 α =1 after the occurrence of any exchange of fund flows scheduled on that date. • Observations : terminating a swap at T α is the same as i. continuing the underlying swap, and ii. entering into the offsetting swap at T α ⇒ the issuer has a long position in an associated offsetting Bermudan swaption • Pricing framework : ◦ Over each period: dividing the pricing of a Bermudan cancelable PRDC swap into i. the pricing of the underlying PRDC swap (a “vanilla” PRDC swap), and ii. the pricing of the associated offsetting Bermudan swaption ◦ Across each date: apply jump conditions and exchange information ◦ Computation: 2 model-dependent PDE to solve over each period, one for the PRDC coupon, one for the “option” in the swaption 6 / 22
Outline Power Reverse Dual Currency (PRDC) swaps 1 The model and the associated PDE 2 Numerical methods 3 Numerical results 4 Summary and future work 5
The model and the associated PDE The pricing model Consider the following model under domestic risk neutral measure ds ( t ) s ( t ) =( r d ( t ) − r f ( t )) dt + γ ( t , s ( t )) dW s ( t ) , dr d ( t )=( θ d ( t ) − κ d ( t ) r d ( t )) dt + σ d ( t ) dW d ( t ) , dr f ( t )=( θ f ( t ) − κ f ( t ) r f ( t ) − ρ fs ( t ) σ f ( t ) γ ( t , s ( t ))) dt + σ f ( t ) dW f ( t ) , • r i ( t ) , i = d , f : domestic and foreign interest rates with mean reversion rate and volatility functions κ i ( t ) and σ i ( t ) • s ( t ): the spot FX rate (units domestic currency per one unit foreign currency) • W d ( t ) , W f ( t ) , and W s ( t ) are correlated Brownian motions with dW d ( t ) dW s ( t ) = ρ ds dt , dW f ( t ) dW s ( t ) = ρ fs dt , dW d ( t ) dW f ( t ) = ρ df dt � s ( t ) � ς ( t ) − 1 • Local volatility function γ ( t , s ( t )) = ξ ( t ) L ( t ) - ξ ( t ): relative volatility function - ς ( t ): constant elasticity of variance (CEV) parameter - L ( t ): scaling constant (e.g. the forward FX rate F (0 , t )) 8 / 22
The model and the associated PDE The 3-D pricing PDE Over each period of the tenor structure, we need to solve two PDEs of the form ∂ u ∂ t + L u ≡ ∂ u ∂ t +( r d − r f ) s ∂ u ∂ s � � ∂ u � � ∂ u + θ d ( t ) − κ d ( t ) r d + θ f ( t ) − κ f ( t ) r f − ρ fS σ f ( t ) γ ( t , s ( t )) ∂ r d ∂ r f 2 γ 2 ( t , s ( t )) s 2 ∂ 2 u d ( t ) ∂ 2 u f ( t ) ∂ 2 u + 1 ∂ s 2 + 1 + 1 2 σ 2 2 σ 2 ∂ r 2 ∂ r 2 d f + ρ dS σ d ( t ) γ ( t , s ( t )) s ∂ 2 u ∂ r d ∂ s + ρ fS σ f ( t ) γ ( t , s ( t )) s ∂ 2 u ∂ r f ∂ s + ρ df σ d ( t ) σ f ( t ) ∂ 2 u − r d u = 0 ∂ r d ∂ r f • Derivation: multi-dimensional Itˆ o’s formula • Boundary conditions: Dirichlet-type “stopped process” boundary conditions • Backward PDE: solved from T α to T α − 1 via change of variable τ = T α − t • Difficulties: high-dimensionality, cross-derivative terms 9 / 22
Outline Power Reverse Dual Currency (PRDC) swaps 1 The model and the associated PDE 2 Numerical methods 3 Numerical results 4 Summary and future work 5
Numerical methods Discretization • Space: Second-order central finite differences on uniform mesh • Time: - Alternating Direction Implicit - Crank-Nicolson : (ADI) : solving a system of the form A m u m = b m − 1 by preconditioned ¯ solving several tri-diagonal systems A m is GMRES, where ¯ for each space dimension block-tridiagonal 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 nz = 8713 11 / 22
Numerical methods GMRES with a preconditioner solved by FFT techniques A m u m = b m − 1 with nonsymmetric ¯ • Applicable to ¯ A m • Starting from an initial guess update the approximation at the i -th iteration by by linear combination of orthonormal basis of the i -th Krylov’s subspace • Problem : slow converge (greatly depends on the spectrum of ¯ A m ) • Solution : preconditioning - find a matrix P such that A m u m = P − 1 b m − 1 converges faster i. GMRES method applied to P − 1 ¯ ii. P can be solved fast • Our choice : ◦ P = ∂ 2 u ∂ s 2 + ∂ 2 u d + ∂ 2 u f + u ∂ r 2 ∂ r 2 ◦ P is solved by Fast Sine Transforms (FST) ◦ Complexity: O ( npq log( npq )) flops 12 / 22
Numerical methods ADI Timestepping scheme from time t m − 1 to time t m : Phase 1: v 0 = u m − 1 + ∆ τ ( A m − 1 u m − 1 + g m − 1 ) , ( I − 1 i ) v i = v i − 1 − 1 u m − 1 + 1 2∆ τ A m − 1 i − g m − 1 2∆ τ A m 2∆ τ ( g m ) , i = 1 , 2 , 3 , i i Phase 2: v 0 = v 0 + 1 2∆ τ ( A m v 3 − A m − 1 u m − 1 ) + 1 2∆ τ ( g m − g m − 1 ) , � ( I − 1 v i − 1 − 1 2∆ τ A m 2∆ τ A m i ) � v i = � i v 3 , i = 1 , 2 , 3 , u m = � v 3 . • u m : the vector of approximate values • A m 0 : matrix of all mixed derivatives terms; A m i , i = 1 , . . . , 3: matrices of the second-order spatial derivative in the s -, r d -, and r s - directions, respectively • g m i , i = 0 , . . . , 3 : vectors obtained from the boundary conditions • A m = � 3 i ; g m = � 3 i =0 A m i =0 g m i 13 / 22
Outline Power Reverse Dual Currency (PRDC) swaps 1 The model and the associated PDE 2 Numerical methods 3 Numerical results 4 Summary and future work 5
Numerical results Market Data • Two economies: Japan (domestic) and US (foreign) • s (0) = 105, r d (0) = 0 . 02 and r f (0) = 0 . 05 • Interest rate curves, volatility parameters, correlations: ρ df = 25% P d (0 , T ) = exp( − 0 . 02 × T ) σ d ( t ) = 0 . 7% κ d ( t ) = 0 . 0% ρ dS = − 15% P f (0 , T ) = exp( − 0 . 05 × T ) σ f ( t ) = 1 . 2% κ f ( t ) = 5 . 0% ρ fS = − 15% • Local volatility function: period period (years) ( ξ ( t )) ( ς ( t )) (years) ( ξ ( t )) ( ς ( t )) (0 0.5] 9.03% -200% (7 10] 13.30% -24% (0.5 1] 8.87% -172% (10 15] 18.18% 10% (1 3] 8.42% -115% (15 20] 16.73% 38% (3 5] 8.99% -65% (20 25] 13.51% 38% (5 7] 10.18% -50% (25 30] 13.51% 38% • Truncated computational domain: { ( s , r d , r f ) ∈ [0 , S ] × [0 , R d ] × [0 , R f ] } ≡ { [0 , 305] × [0 , 0 . 06] × [0 , 0 . 15] } 15 / 22
Numerical results Specification Bermudan cancelable PRDC swaps • Principal: N d (JPY); Settlement/Maturity dates: 1 Jun. 2010/1 Jun. 2040 • Details: paying annual PRDC coupon, receiving JPY LIBOR Year coupon funding (FX options) leg s (1) 1 max( c f F (0 , 1) − c d , 0) N d L d (0 , 1) N d . . . . . . . . . s (29) 29 max( c f F (0 , 29) − c d , 0) N d L d (28 , 29) N d • Leverage level level low medium high 4.5% 6.25% 9.00% c f 2.25% 4.36% 8.10% c d • The payer has the right to cancel the swap on each of { T α } β − 1 α =1 , β = 30 (years) 16 / 22
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