Path-dependent options and randomization techniques

Transcription

1 Path-dependent options and randomization techniques Daniele Marazzina Joint work with G. Fusai and M. Marena Dipartimento di Matematica F. Brioschi Politecnico di Milano April 8, 2009

2 The Talk In this talk, we will show how we can price discretely monitored options Arithmetic Asian Barrier European Lookback Benchmark combining maturity randomization and Fourier transform techniques Numerical approach Numerical solution of a set of independent integral equations in the Lévy world

3 Discrete knock-and-out barrier option: as you hit you die Payoff at maturity (T = 1) if still alive Path-Dependent

4 Methods for pricing path-dependent options Barrier and Lookback Options Airoldi, Quantitative Finance, 2005 Perturbative Series Expansion Approximation of a generic probability function around another simpler pdf by matching all moments up to a generic order Chiarella, El-Hassan & Kucera, Computational Methods in Financial Engineering, 2008 Path integral approach Backward recursion functional equation solved by expanding the pricing functions in Fourier-Hermite series

5 Methods for pricing path-dependent options Barrier and Lookback Options - 2 Feng & Linetsky, Mathematical Finance, 2008 Fast Hilbert Transform Approach Apply the Hilbert transform and a discretization procedure using the Whittaker cardinal series Backward induction in the Fourier space, rather than in the state space Jackson, Jaimungal & Surkov, Journal of Computational Finance, 2008 Fourier Space Time-stepping method Apply the Fourier transform to the pricing PIDE to obtain a linear system of easily solvable ordinary differential equations Lord, Fang, Bervoets & Oosterlee, SIAM Journal on Scientific Computing, 2008 Convolution method Based on Fourier transforms combined with quadrature rules

6 Methods for pricing path-dependent options Asian Options Benhamou, Journal of Computational Finance, 2002 Convolution method Albrecher & Predota, Journal of Computational and Applied Mathematics, 2004 Moments approximation Approximations of the arithmetic option price based on the moments of the average Fusai & Meucci, Journal of Banking and Finance, 2008 Recursive integration Cerny & Kyriakou, to appear Backward price convolution They provide analytical upper bound for the pricing error due to truncations

7 CONTRACTS

8 The setup 1. Risk-free interest rate r 2. Asset having log-price x, i.e., x t = log(s t) 3. Derivative contract having maturity T N monitoring dates = T the time-distance between monitoring dates N strike K

9 Barrier and European Options For European and Barrier options, the pricing problem can be written in a recursive way (see Hull, 2003) C (x, j) = e r f (x, ξ; ) C (ξ, j + 1) dξ j = N 1,, 0 Ω C (x, N) = φ (x) where Ω depends on the contract f (x, ξ; ) the transition density φ (x) is the payoff C(x, j) is the option price at the j th monitoring date

10 1: Choice of Ω European Options: options that can only be exercised at the end of their lives Ω = R Barrier Options: options die/start to live if a lower or upper barrier is hit Ω = (ln(l), + ) for lower barrier Ω = (, ln(u)) for upper barrier Ω = (ln(l), ln(u)) for double barriers

11 2: Choice of the transition density: Lévy World Assumption The underlying asset evolves according to a generic Lévy process Thus the increments are i.i.d. the transition density f (x, ξ; ) from ξ at time t to x at time t + is of convolution type, i.e. (with an abuse of notation) f (x, ξ; ) = f (x ξ; ) Why Lévy Process? The volatility surface shows the inconsistence of the Black-Scholes model In general the transition density f is known only through its Fourier transform it is a computational advantage that f is of convolution type

12 The considered Lévy processes are G (Gaussian) CGMY (Carr-Geman-Madan-Yor) DE (Double Exponential) Lévy World 0 JD (Jump Diffusion or Merton) log return G CGMY Figure: Density of the log-returns For the CGMY, the DE and the JD processes tails are fatter than in the Gaussian case: we better model the occurrence of large (negative) returns Model (Parameters) Characteristic Exponent ψ (γ) G(σ) 1 2 σ2 γ 2 ) CGMY (C, G, M, Y ) C Γ ( Y ) ((M iγ) Y M Y + (G + iγ) Y G Y ( ) DE(σ, λ, p, η 1, η 2) 1 2 σ2 γ 2 + λ (1 p) η2 + p η 1 1 η 2 iγ η 1 +iγ ( ) JD(σ, α, λ, δ) 1 2 σ2 γ 2 + λ e iγα 1 2 γ2 δ 2 1 DE JD

13 Lookback Options Their settlement is based on the minimum or the maximum value of the underlying asset as registered during the lifetime of the options Consider fixed strike lookback put options, i.e., the payoff is ( ) + (K M N ) + = e k e m N where M N := min j=0,,n S j m N := min j=0,,n x j x j is the asset log-price at the j th monitoring date, i.e., x j = log(s j ) k is the logarithm of the strike K

14 Assuming x 0 > k, the lookback put with fixed strike has price k e rt e u P(m N u) du Problem: P(m N u) =???

15 P(m N u) = 1 v(x 0 u, N) where v(, ) satisfies the following recursion v(z, j) = + 0 v(z, 0) = 1 (z>0) f (z ξ; )v(ξ, j 1)dξ j = 1,, N Solving the recursion, we obtain the distribution of the minimum Once this distribution is computed, we have to perform a numerical integration to obtain the option price

16 Asian Options The payoff of an arithmetic Asian option depends on the arithmetic average of the values of the underlying asset at the monitoring dates Payoff of a Call Option ( ) + Fixed-Strike Asian Options: AN K N+1 ( Floating-Strike Asian Options: S N A N N+1 where A N N j=0 S j ) +

17 Fixed-Strike Asian Options We define a random variable Z such that A j+1 = A j + S j e Z j+1, j = 0,, N 1 Letting L N e Z N and defining recursively the following quantities we obtain L j e Z j (1 + L j+1 ), j = N 1,, 1 A N S 0 (1 + L 1) The density of L 1 turns out to be the key variable in pricing fixed strike Asian options Option Price e rt + where f B1 is the density of B 1 log(l 1) ( ) + S0 T + 1 (1 + ex ) K f B1 (x) dx

18 Recursion Problem: How to compute f B1? with a recursion The density of f Bn satisfies the recursion f Bj (x) = + f (x log (e y + 1)) f Bj+1 (y) dy, j = N 1,, 1 with initial condition f BN f, i.e., the transition density

19 Floating-Strike Asian Options The following recursion for the option price holds for j = N 1,, 0 C (S N, A N ; N) = ( S N A ) + N N C (S j, A j ; j) = e r f (s) C (S j e s, A j + S j e s ; j + 1) ds

20 Since the payoff is homogeneous of degree one in the stock price, we can write ( C (S j, A j ; j) = S j C 1, A ) j ; j S j If we set then the following recursion holds + g j (x) = e r f with initial condition 1 g j (x) C (1, x; j) ( log g N (x) = ( x )) y 1 ( 1 x ) + N + 1 x (y 1) 2 g j+1 (y) dy The floating strike option price is S 0g 0 (1)

21 Maturity Randomization

22 All the recursion can be written as v (x, j) = K (x, ξ; ) v (ξ, j 1) dξ Ω v (x, 0) = ψ (x) j = 1,, M for suitable choices of K, ψ, M and Ω In practice, to solve the pricing problem, we have to compute M nested integrals

23 The strategy The problem: recursive computation of an integral 1. Idea: let us randomize the option expiry according to a geometric distribution 2. Randomization transforms the recursive pricing problem into a set of integral equations 3. Let us look for numerical solutions of the integral equations 4. De-randomize the expiry and get the option price

24 Randomize the option expiry Let us consider a coin tossing game where q is the probability of getting tails If we get heads for the first time after j tails, we receive an option having j monitoring dates Therefore the expected gain W (x, q) is W (x, q) = (1 q)q j v (x, j) j=0

25 Randomize the option expiry Let us consider a coin tossing game where q is the probability of getting tails If we get heads for the first time after j tails, we receive an option having j monitoring dates Therefore the expected gain W (x, q) is W (x, q) = (1 q)q j v (x, j) j=0 The recursive equation becomes an integral equation W (x, q) = q K (x, ξ; ) W (ξ, q) dξ + (1 q)ψ (x) Ω

26 De-Randomization How to go back from W to v Complex inversion integral v (x, M) = 1 2πρ M 2π 0 ( W x, ρe ) 1u e 1Mu du Numerical approximation { v h (x, M)= 1 M 1 W (x, ρ)+( 1) M W (x, ρ)+2 ( 1) j Re 2Mρ M Solution of M + 1 independent integral equations j=1 ( ( W x, ρe )) } 1jπ M

27 The Algorithm 1. Compute the transition density f using FFT 2. For j = 0,, M, set q (j) = ρ exp ( 1jπ/M ), ρ = 10 γ/2m, γ = 8 3. Discretization of the integral equation Truncate the domain Ω to [a, b], if necessary Choose a quadrature rule and define the m nodes x i We obtain the linear systems (I m q(j)k md m) W m = Ψ m j = 0,, M where I m is the identity matrix of size m K m is the m m square matrix with elements K ij = K(x i, x j ; ) D m is the m m matrix which contains the weights associated to the quadrature formula Ψ i = (1 q)ψ (x i ) 4. Solve the M + 1 linear systems 5. Using the M + 1 solutions of the integral equation parametrized by q (j), reconstruct v h (x, M)

28 Computational Cost

29 How to improve performances? 1.EULER ACCELERATION 2.PRECONDITIONING TECHNIQUES 3.GRID COMPUTING

30 1.Euler Acceleration

31 Euler Acceleration Euler Summation is a convergence-acceleration technique well suited for evaluating alternating series Numerical Inversion Formula { v h (x, M)= 1 M 1 W (x, ρ)+( 1) M W (x, ρ)+2 ( 1) j Re 2Mρ M This formula can be rewritten as v h (x, M) = 1 ρ M j=0 j=1 ( ( W x, ρe M ( ( 1) j a j Re W (x, ρe )) 1jπ/M )) } 1jπ M

32 The idea of the Euler acceleration technique is to approximate the above sum with the following one ( ) v h (x, M) 1 m e m b 2 me ρ M ne +j j where b k = j=0 j=0 k ( ( 1) j a j Re W (x, ρe )) 1jπ/M with n e and m e suitably chosen, such that n e + m e < M Thus the number of linear systems to be solved is min{m, n e + m e} + 1 We set n e = 12 and m e = 10

33 2.Preconditioning Technique Barrier and Lookback Options K(, ) is of convolution type we deal with Toeplitz type matrices [Joint work with M. Ng]

34 Barrier and Lookback Options We consider Composite Rectangle, Trapezoidal and Simpson s formula (I qkd) W = Φ All the linear systems are solved with the GMRES iterative method Composite Rectangle K is a Toeplitz matrix D is the identity matrix I qkd is a Toeplitz matrix Composite Trapezoidal and Simpson s they are higher-order formulas K is a Toeplitz matrix I qkd is not a Toeplitz matrix Toeplitz Matrix Each descending diagonal from left to right is constant Multiplication of a m m Toeplitz matrix to a vector requires O(m log m) operations m m Toeplitz matrix store only 2m 1 elements

35 Preconditioner A preconditioner P of a matrix A is a matrix such that P 1 A has a smaller condition number than A Preconditioners are useful when using an iterative method to solve a large linear system Ax = b, since the rate of convergence for most iterative linear solvers degrades as the condition number of a matrix increases: instead of solving the original linear system, one solve P 1 Ax = P 1 b Lin, Ng & Chan Preconditioner Based on Circulant Matrix Already Inverted Lin, Ng & Chan

36 GMRES iterations GMRES Iterations & Execution Time (ETime - in sec): m = 3001 No Preconditioner CR CT CS M Iterations ETime Iterations ETime Iterations ETime Preconditioner CR CT CS M Iterations ETime Iterations ETime Iterations ETime Computational Cost: O(min{M, n e + m e}m log m)

37 2.Preconditioning Technique Asian Options K(, ) is no more of convolution type Reichel algorithm

38 Asian Options We consider Gauss-Chebyshew quadrature rules for the following linear system (I qkd) W = Ψ In order to speed up the solution of the above linear system, we consider an algorithm due to Reichel, who proposes a fast solution method for the one-dimensional Fredholm integral equation Reichel proves that the matrix K can be well approximated by a matrix K of rank much smaller than m Thus he proposes an iterative algorithm with a suitable preconditioner (based on K ) having a computational cost of O(m 2 ) arithmetic operations

39 Fixed Strike Asian Option Price. M = 50 Reichel GMRes GE m Price CPU Price CPU Price CPU Benchmark price: Cérny and Kyriakou

40 How to improve performances? 3.Grid Computing

41 Grid Solving independent integral equations Grid Computing Solve linear systems on different computers Grid Computing is a special type of parallel computing which relies on complete computers connected to a network Our grid consists of six Intel Pentium systems, each equipped with 4GB of RAM and Intel Core 2 Quad Q6600 (2400MHz) processor Floating Strike Asian Option - CPU Time (in seconds) N m = 1000 m = 2000 m = 3000 m = 4000 Serial Grid

42 NUMERICAL RESULTS

43 Barrier Options Double Barrier Options and Double Exponential Distribution: N = 252 Logarithmic Scales Benchmark Price: Feng & Linetsky, Rectangle Trapezoidal Simpson 10 3 Pointwise Error m

44 Lookback Options Lookback Options and DE Model - Prices N m = 1001 m = 2001 m = 3001 m = 4001 CI Composite Rectangle Composite Trapezoidal Composite Simpson s Parameters: r = , T = 1, S 0 = 100, and K = 100 DE: σ = , λ = , p = , η 1 = and η 2 = m >> 4001

45 Fixed Strike Asian Options Fixed Strike Asian Options: a comparison with Cérny & Kyriakou Model Parameters m Price Benchmark G σ = CGMY C = G = M = Y = NIG δ = α = β = Parameters: r = 0.04, T = 1, S 0 = 100, M = 50 and K = 100

46 Floating Strike Asian Options Floating Strike Asian Options: a comparison with Lim (2002) Model m Price Benchmark G Parameters: σ = 0.2, r = 0.1, T = 182, S0 = 100, M = 91 and K =

47 Conclusions We have presented a method for pricing options intuitive easy to be implemented accurate can be used with different products and processes

48 Conclusions We have presented a method for pricing options intuitive easy to be implemented accurate can be used with different products and processes References G.Fusai, D.M., M.Marena, Option Pricing, Maturity Randomization and Grid Computing, IPDPS2008 Proceedings G.Fusai, D.M., M.Marena, M.Ng, Randomization and Preconditioning Techniques for Option Pricing, submitted G.Fusai, D.M., M.Marena, Pricing Discretely Monitored Asian Options Using Maturity Randomization, in preparation

49 Discrete and continuous monitoring Path-Dependent Options One of the most important characteristic is the monitoring frequency of the underlying asset With discrete monitoring, the updating conditions occur only at prefixed dates, e.g. the end of the day/week/month The available analytical formula in the Black-Scholes setting assume continuous monitoring Approximations for the discrete monitoring case are given in Broadie, Glasserman & Kou (1999) Back

50 Lookback Options and Composite Rectangle Formula: Prices m DE CGMY ETime N = 50 N = 100 N = 50 N = 100 N = 50 N = CT CT CS CS Back

51 Lin, Ng & Chan Preconditioner We have to solve the linear system where: I is the identity matrix K is a complex Toeplitz matrix (I + KD)W = Φ D is a diagonal matrix - for example, for the trapezoidal rule D = diag([ 1 2, 1, 1, 1, 1,, 1, 1 2 ])

52 We consider the equivalent preconditioned linear system P 1 (I + KD)W = P 1 Φ Let C be a circulant preconditioners for K, then (I + C) 1 = I + P where the eigenvalues of the circulant matrix P are given by λ 1 + λ and λ are the eigenvalues of C Hence, the preconditioned linear system becomes (I + PD)(I + KD)W = (I + PD)Φ P 1 = I + PD can be computed easily (the preconditioner is already inverted) Back