Efficient Valuation Methods for Contracts in Finance and Insurance - PowerPoint PPT Presentation

Efficient Valuation Methods for Contracts in Finance and Insurance Kees Oosterlee 1 , 2 1 CWI, Center for Mathematics and Computer Science, Amsterdam, 2 Delft University of Technology, Delft. Joint Work with Fang Fang, Lech Grzelak, Stefan Singor Eindhoven, August 29th, 2011 C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 1 / 59

Contents Option pricing method, based on Fourier-cosine expansions ◮ Focus on European options and calibration Generalize to hybrid products ◮ Models with stochastic interest rate; stochastic volatility C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 2 / 59

Financial industry; Banks at Work Pricing approach: 1. Define some financial product 2. Model asset prices involved (SDEs) 3. Calibrate the model to market data (Numerics, Optimization) 4. Model product price correspondingly (PDE, Integral) 5. Price the product of interest (Numerics, MC) 6. Set up hedge to remove the risk related to the product (Optimization) C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 3 / 59

Pricing: Feynman-Kac Theorem Given the final condition problem  2 σ 2 S 2 ∂ 2 v  ∂ v 1 ∂ S 2 + rS ∂ v + ∂ S − rv = 0 , ∂ t  v ( S , T ) = given Then the value, v ( S ( t ) , t ), is the unique solution of 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. and S satisfies the system of stochastic differential equations: dS t = rS t dt + σ S t dW Q t , Similar relations hold for other SDEs in Finance C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 4 / 59

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: ◮ Speed of solution methods (for example, for calibration) ◮ Early exercise feature (P(I)DE → free boundary problem) ◮ The problem’s dimensionality (not treated here) C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 5 / 59

Motivation Fourier Methods Derive pricing methods that ◮ are computationally fast ◮ are not restricted to Gaussian-based models ◮ should work as long as we have the characteristic function, Z ∞ Z ∞ e iux f ( x ) dx ; f ( x ) = 1 “ e iuX ” Re ( φ ( u ) e − iux ) du φ ( u ) = E = π 0 −∞ (available for L´ evy processes and also for Heston’s model). ◮ In probability theory a characteristic function of a continuous random variable X , equals the Fourier transform of the density of X . Generalize basic method w.r.t. SDEs, contracts, applications C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 6 / 59

Class of Affine Jump Diffusion (AJD) processes Duffie, Pan, Singleton (2000): The following system of SDEs: d X t = µ ( X t ) dt + σ ( X t ) d W t + d Z t , is of the affine form, if the drift, volatility, jump intensity and interest rate satisfy: a 0 + a 1 X t for ( a 0 , a 1 ) ∈ R n × R n × n , µ ( X t ) = b 0 + b T 1 X t , for ( b 0 , b 1 ) ∈ R × R n , λ ( X t ) = ij X t , ( c 0 , c 1 ) ∈ R n × n × R n × n × n , σ ( X t ) σ ( X t ) T ( c 0 ) ij + ( c 1 ) T = r 0 + r T 1 X t , for ( r 0 , r 1 ) ∈ R × R n . r ( X t ) = The discounted characteristic function then has the following form: φ ( u , X t , t , T ) = e A ( u , t , T )+ B ( u , t , T ) T X t , The coefficients A ( u , t , T ) and B ( u , t , T ) T satisfy a system of Riccati-type ODEs. C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 7 / 59

The COS option pricing method, based on Fourier Cosine Expansions C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 8 / 59

Series Coefficients of the Density and the Ch.F. Fourier-Cosine expansion of a 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 F n has a direct relation to ch.f., φ ( u ) := � R f ( x ) e iux dx ( � 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 na π = φ exp . b − a Re b − a b − a C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 9 / 59

Recovering Densities Replace F n by A n , and truncate the summation: � � n π � � �� � � � ′ N − 1 in π − a n π x − a 2 f ( x ) ≈ φ exp cos , n =0 Re b − a 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 . Similar behaviour for other L´ evy processes. C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 10 / 59

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) - Eurandom Workshop 29/8-2011 11 / 59

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 k π y − a 2 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) - Eurandom Workshop 29/8-2011 12 / 59

Heston model The Heston stochastic volatility model can be expressed by the following 2D system of SDEs � r t S t dt + √ ν t S t dW S dS t = t , − κ ( ν t − ν ) dt + γ √ ν t dW ν d ν t = t , With x t = log S t this system is in the affine form. ⇒ Itˆ o’s Lemma: multi-D partial differential equation C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 13 / 59

Characteristic Functions Heston Model For L´ evy and Heston models, the ChF can be represented by ϕ levy ( u ) · e iu x φ ( u ; x ) = with ϕ levy ( u ) := φ ( u ; 0) , ϕ hes ( u ; ν 0 ) · e iu x , φ ( u ; x , ν 0 ) = The ChF of the log-asset price for Heston’s model: � � 1 − e − D ∆ t � � iur ∆ t + ν 0 ( κ − i ργ u − D ) · ϕ hes ( u ; ν 0 ) = exp γ 2 1 − Ge − D ∆ t � κ ¯ � �� ∆ t ( κ − i ργ u − D ) − 2 log(1 − Ge − D ∆ t ν exp ) , γ 2 1 − G � ( κ − i ργ u ) 2 + ( u 2 + iu ) γ 2 G = κ − i ργ u − D with D = and κ − i ργ u + D . C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 14 / 59

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 − 1 � n π � � 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 ( u ) := φ ( u ; 0) C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 15 / 59

Numerical Results Pricing 21 strikes K = 50 , 55 , 60 , · · · , 150 simultaneously under Heston’s model. Other parameters: S 0 = 100 , r = 0 , q = 0 , T = 1 , κ = 1 . 5768 , γ = 0 . 5751 , ¯ ν = 0 . 0398 , ν 0 = 0 . 0175 , ρ = − 0 . 5711 . N 96 128 160 COS (msec.) 2.039 2.641 3 . 220 max. abs. err. 4.52e-04 2.61e-05 4 . 40 e − 06 Error analysis for the COS method is provided in the paper. C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 16 / 59

Numerical Results within Calibration Calibration for Heston’s model: Around 10 times faster than Carr-Madan. C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 17 / 59

What do we do with the COS method? Generalizations: ◮ Early-exercise options (Bermudan, barrier, American) ◮ Context of CDS pricing (with Wim Schouten, Henrik J¨ onsson) ◮ Swing options (commodity market) ◮ Stochastic control problems, economic decision making (dikes, climate) ◮ Asian options ◮ Multi-asset options Generalize to hybrid products (Rabobank, Ortec Finance) ◮ Models with stochastic interest rate; stochastic volatility ◮ Heston Hull-White, Heston SV-LMM C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 18 / 59

An exotic contract: A hybrid product Based on sets of assets with different expected returns and risk levels. Proper construction may give reduced risk and an expected return greater than that of the least risky asset. A simple example is a portfolio with a stock with a high risk and return and a bond with a low risk and return. Example: � � �� 0 , 1 S T + 1 B T R T 0 r s ds max e − V ( S , t 0 ) = E Q 2 2 S 0 B 0 C.W.Oosterlee (CWI) - Eurandom Workshop 29/8-2011 19 / 59