On Highly Efficient Methods for Pricing Options with and without - PowerPoint PPT Presentation

On Highly Efficient Methods for Pricing Options with and without Early Exercise Cornelis W. Oosterlee 1 , 2 , Fang Fang 2 1 CWI, Center for Mathematics and Computer Science, Amsterdam, 2 Delft University of Technology, Delft. Linz, Semester on Finance, November 2008 C.W.Oosterlee (CWI) The COS Method Linz 1 / 42

Contents Brief overview of derivative pricing Our contribution: The COS method: ◮ Efficient way to recover the density function; ◮ Efficient alternative for FFT-based methods for calibration; ◮ Focus on L´ evy processes and Heston stochastic volatility COS method for European options Bermudan and discretely-monitored barrier options Credit Default Swaps C.W.Oosterlee (CWI) The COS Method Linz 2 / 42

Multi-D asset prices Asset price, S i , can be modeled by geometric Brownian motion: dS i ( t ) = µ i S i ( t ) dt + σ i S i dW i ( t ) , with W i ( t ) Wiener process, µ i drift, σ i volatility. ⇒ Itˆ o’s Lemma: multi-D Black-Scholes equation: (for a European option) d d ∂ 2 V ∂ V ∂ t + 1 ∂ V � � [ σ i σ j ρ i , j S i S j ] + [ rS i ] − rV = 0 . 2 ∂ S i ∂ S j ∂ S i i , j =1 i =1 Correlation between a pair of assets, S i and S j , is ρ i , j . C.W.Oosterlee (CWI) The COS Method Linz 3 / 42

Pricing: Feynman-Kac Theorem Given the system of stochastic differential equations: dS i ( t ) = rS i ( t ) dt + σ i S i dW i ( t ) with E { dW i ( t ) dW j ( t ) } = ρ ij dt and an option, V , such that V ( S , t ) = e − r ( T − t ) E Q { V ( S ( T ) , T ) | S ( t ) } with the sum of the first derivatives of the option square integrable. Then the value, V ( S ( t ) , t ), is the unique solution of the final condition problem  i , j =1 [ σ i σ j ρ i , j S i S j ∂ 2 V ∂ V 1 i =1 [ rS i ∂ V � d ∂ S i ∂ S j ] + � d + ∂ S i ] − rV = 0 ,  2 ∂ t V ( S , T ) = given  C.W.Oosterlee (CWI) The COS Method Linz 4 / 42

Numerical Pricing Approach One can apply several numerical techniques to calculate the option price: ◮ Numerical integration, ◮ Monte Carlo simulation, ◮ Numerical solution of the partial-(integro) differential equation (P(I)DE) Each of these methods has its merits and demerits. Numerical challenges: ◮ The problem’s dimensionality ◮ Speed of solution methods ◮ Early exercise feature (P(I)DE → free boundary problem) C.W.Oosterlee (CWI) The COS Method Linz 5 / 42

L´ evy Processes Use Heston’s model, or a L´ evy process with jumps, to better fit market data, and allow for smile effects A L´ evy process is a stochastic process that starts at 0 and has independent and stationary increments. The L´ evy processes of our interest here include ◮ The CGMY model (generalized VG model; driven by four parameters); ◮ The Normal Inverse Gaussian (NIG) model (a variance-mean mixture of a Gaussian distribution with an inverse Gaussian; driven by four parameters). C.W.Oosterlee (CWI) The COS Method Linz 6 / 42

Motivation Our motivation: To derive pricing methods that ◮ are computationally fast ◮ are not restricted to Gaussian-based models ◮ should work as long as we have a characteristic function, Z ∞ Z ∞ e i ω x f ( x ) dx ; f ( x ) = 1 Re ( φ ( ω ) e − i ω x ) d ω φ ( ω ) = π 0 −∞ ◮ Preferably faster than approaches based on the FFT The characteristic function of a L´ evy process equals: exp ( t ( i µω − 1 � 2 σ 2 ω 2 + ( e i ω x − 1 − i ω x 1 [ | x | < 1] ν ( dx ))) , φ ( ω ) = I R the celebrated L´ evy-Khinchine formula. C.W.Oosterlee (CWI) The COS Method Linz 7 / 42

Fourier-Cosine Expansion The COS method: ◮ Exponential convergence; ◮ Greeks are obtained at no additional cost. ◮ For discretely-monitored barrier and Bermudan options as well; The basic idea: ◮ Replace the density by its Fourier-cosine series expansion; ◮ Series coefficients have simple relation with characteristic function. C.W.Oosterlee (CWI) The COS Method Linz 8 / 42

Series Coefficients of the Density and the Ch.F. Fourier-Cosine expansion of density function on interval [ a , b ]: � ′∞ � � n π x − a f ( x ) = n =0 F n cos , b − a with x ∈ [ a , b ] ⊂ R and the coefficients defined as � b 2 � n π x − a � F n := f ( x ) cos dx . b − a b − a a R f ( x ) e i ω x dx ( � F n has direct relation to ch.f., φ ( ω ) := � R \ [ a , b ] f ( x ) ≈ 0), 2 � n π x − a � � F n ≈ A n := f ( x ) cos dx b − a b − a R � n π 2 � � � − i ka π �� = φ exp . b − a Re b − a b − a C.W.Oosterlee (CWI) The COS Method Linz 9 / 42

Recovering Densities Replace F n by A n , and truncate the summation: � n π � ′ N − 1 � � � �� � � 2 in π − a n π x − a f ( x ) ≈ φ b − a ; x exp cos , n =0 Re b − a b − a b − a 2 x 2 , [ a , b ] = [ − 10 , 10] and x = {− 5 , − 4 , · · · , 4 , 5 } . 2 π e − 1 1 Example: f ( x ) = √ N 4 8 16 32 64 error 0.2538 0.1075 0.0072 4.04e-07 3.33e-16 cpu time (sec.) 0.0025 0.0028 0.0025 0.0031 0.0032 Exponential error convergence in N . C.W.Oosterlee (CWI) The COS Method Linz 10 / 42

Pricing European Options Start from the risk-neutral valuation formula: � v ( x , t 0 ) = e − r ∆ t E Q [ v ( y , T ) | x ] = e − r ∆ t v ( y , T ) f ( y | x ) dy . R Truncate the integration range: � v ( x , t 0 ) = e − r ∆ t v ( y , T ) f ( y | x ) dy + ε. [ a , b ] Replace the density by the COS approximation, and interchange summation and integration: � n π v ( x , t 0 ) = e − r ∆ t � ′ N − 1 � � � a e − in π ˆ φ b − a ; x V n , n =0 Re b − a where the series coefficients of the payoff, V n , are analytic. C.W.Oosterlee (CWI) The COS Method Linz 11 / 42

Pricing European Options Log-asset prices: x := ln( S 0 / K ) and y := ln( S T / K ) , The payoff for European options reads v ( y , T ) ≡ [ α · K ( e y − 1)] + . For a call option, we obtain � b � � 2 k π y − a K ( e y − 1) cos V call = dy k b − a b − a 0 2 = b − aK ( χ k (0 , b ) − ψ k (0 , b )) , For a vanilla put, we find 2 V put = b − aK ( − χ k ( a , 0) + ψ k ( a , 0)) . k C.W.Oosterlee (CWI) The COS Method Linz 12 / 42

Characteristic Functions Heston Model The characteristic function of the log-asset price for Heston’s model: � 1 − e − D ∆ t � � � i ωµ ∆ t + u 0 ϕ hes ( ω ; u 0 ) = exp ( λ − i ρηω − D ) · η 2 1 − Ge − D ∆ t ∆ t ( λ − i ρηω − D ) − 2 log(1 − Ge − D ∆ t � λ ¯ � �� u exp ) , η 2 1 − G ( λ − i ρηω ) 2 + ( ω 2 + i ω ) η 2 G = λ − i ρηω − D � with D = and λ − i ρηω + D . For L´ evy and Heston models, the ChF can be represented by ϕ levy ( ω ) · e i ω x φ ( ω ; x ) = with ϕ levy ( ω ) := φ ( ω ; 0) , ϕ hes ( ω ; u 0 ) · e i ω x , φ ( ω ; x , u 0 ) = C.W.Oosterlee (CWI) The COS Method Linz 13 / 42

Characteristic Functions L´ evy Processes For the CGMY/KoBol model: exp ( i ω ( r − q )∆ t − 1 2 ω 2 σ 2 ∆ t ) · ϕ levy ( ω ) = exp (∆ tC Γ( − Y )[( M − i ω ) Y − M Y + ( G + i ω ) Y − G Y ]) , where Γ( · ) represents the gamma function. The parameters should satisfy C ≥ 0 , G ≥ 0 , M ≥ 0 and Y < 2. The characteristic function of the log-asset price for NIG: � � α 2 − β 2 − α 2 − ( β + i ω ) 2 ) � � ϕ NIG ( ω ) = exp i ωµ + δ ( with α, δ > 0 , β ∈ ( − α, α − 1) C.W.Oosterlee (CWI) The COS Method Linz 14 / 42

Heston Model We can present the V k as V k = U k K , where 2 � b − a ( χ k (0 , b ) − ψ k (0 , b )) for a call U k = 2 b − a ( − χ k ( a , 0) + ψ k ( a , 0)) for a put . The pricing formula simplifies for Heston and L´ evy processes: � n π �� ′ N − 1 � � v ( x , t 0 ) ≈ K e − r ∆ t · Re U n · e in π x − a n =0 ϕ , b − a b − a where ϕ ( ω ) := φ ( ω ; 0) C.W.Oosterlee (CWI) The COS Method Linz 15 / 42

Numerical Results Pricing for 21 strikes K = 50 , 55 , 60 , · · · , 150 under Heston’s model. Other parameters: S 0 = 100 , r = 0 , q = 0 , T = 1 , λ = 1 . 5768 , η = 0 . 5751 , ¯ u = 0 . 0398 , u 0 = 0 . 0175 , ρ = − 0 . 5711 . 96 128 160 N COS (msec.) 2.039 2.641 3 . 220 max. abs. err. 4.52e-04 2.61e-05 4 . 40 e − 06 N 2048 4096 8192 Carr-Madan (msec.) 20.36 37 . 69 76.02 max. abs. error 2.61e-01 2 . 15 e − 03 2.08e-07 Error analysis for the COS method is provided in the paper. C.W.Oosterlee (CWI) The COS Method Linz 16 / 42

Numerical Results within Calibration Calibration for Heston’s model: Around 10 times faster than Carr-Madan. C.W.Oosterlee (CWI) The COS Method Linz 17 / 42

Pricing Bermudan Options 0 m m+1 M s s 0 K � � � � 0 T t ������� ������� The pricing formulae e − r ∆ t � � c ( x , t m ) = R v ( y , t m +1 ) f ( y | x ) dy v ( x , t m ) = max ( g ( x , t m ) , c ( x , t m )) and v ( x , t 0 ) = e − r ∆ t � R v ( y , t 1 ) f ( y | x ) dy . ◮ Use Newton’s method to locate the early exercise point x ∗ m , which is the root of g ( x , t m ) − c ( x , t m ) = 0 . ◮ Recover V n ( t 1 ) recursively from V n ( t M ), V n ( t M − 1 ) , · · · , V n ( t 2 ). ◮ Use the COS formula for v ( x , t 0 ). C.W.Oosterlee (CWI) The COS Method Linz 18 / 42