(RP13) Efficient numerical methods on high-performance computing platforms for the underlying financial models: Series Solution and Option Pricing

Transcription

1 (RP13) Efficient numerical methods on high-performance computing platforms for the underlying financial models: Series Solution and Option Pricing Jun Hu Tampere University of Technology Final conference 15 March 2016, London

2 Formula Please check out solutions for at European option for stochastic volatility models (Heston, GARCH and 3/2). American option for the Black-Scholes model.

3 Results Publication: Code: Asymptotic expansion of European options with mean-reverting stochastic volatility dynamics, Finance Research Letters, 14, 1 10 Limit order book models and market phenomenology, Project report Doctoral dissertation European option for stochastic volatility models American option for the Black-Scholes model

4 Series solution to ODE Consider the ordinary differential equation d V (x) = V (x), dx (1) V (0) = 1. (2) The solution is V (x) = e x. Suppose we do not know the answer and try to obtain the series solution V (x) = a i x i, (3) d dx V (x) = ia i x i 1 = (i + 1)a i+1 x i. (4)

5 Then the ODE becomes Equivalently (i + 1)a i+1 x i = a i x i, (5) a 0 = 1. (6) a i+1 = 1 i + 1 a i, (7) a 0 = 1. (8) The result is a i = 1 i! V (x) = 1 i! xi = e x. (9) Exactly the same result as we expect. Series solutions has the potential to be exact.

6 Series solution and pricing PDE (simplified) The pricing PDE usually looks like BV = plv, (10) V (p = 0) = S N( ) Ke rt N( ). (11) In comparison V (x) = d V (x), dx (12) V (0) = 1. (13) Similarly, we propose V (p) = V i (x, t,...)p i. (14)

7 The original PDE BV = plv, (15) V (p = 0) = S N( ) Ke rt N( ), (16) becomes BV i p i = pl V i p i = LV i 1 p i, (17) i=1 V 0 = S N( ) Ke rt N( ), (18) equivalently BV 0 = 0, (19) BV i = LV i 1 V i = B 1 LV i 1, (20) V 0 = S N( ) Ke rt N( ). (21)

8 Difference Iteration for ODE leads to a i+1 = 1 i + 1 a i, (22) a 0 = 1, (23) a i = 1 i!. (24) Iteration for PDE V i = B 1 LV i 1, (25) V 0 = S N( ) Ke rt N( ), (26) is hard to generalize, because the iteration usually involves complicated integration, e.g. s ) V 1 = e ay+bs 1 (x y)2 dt dx exp ( ax bt 0 2πvt 2v(s t) [ v S N( ) Ke rt N( ) ]. (27)

9 Better choice of basis The option price is bounded 0 < C(p) < S, for q (0, ). (28) However p i is not, therefore always blows up at p. C N V i p i (29) p i C(p) p

10 The solution: basis function with better asymptotic behavior ( ) p i, (30) 1 + p because ( ) p i ( ) p i lim = p i, lim = 1. (31) p p p 1 + p C C(p) p i p C(p) = V i p i = ( ) p i V i (32) 1 + p

11 Best expansion (so far) European option under stochastic volatility models looks like Error is bounded lim C = η,v i,j=0 lim η,v i,j=0 ( ) η i ( ) v θ j u ij. (33) 1 + η 1 + v θ i,j=0 N ( η u ij 1 + η ) i ( ) v θ j = 1 + v θ N u ij <, (34) i,j=0 N u ij η i (v θ) j. (35)

12 Scale invariance Scale invariance for the Heston model ( ) t V (t, x, r, v, θ, ρ, η, κ) = V, x, λr, λv, λθ, ρ, λη, λκ, (36) λ because λ t V λv ( ) 2 2 x x V λr ( x 1) V 1 ρληλv x λ vv 1 2 λ2 η 2 λv 1 λ 2 2 vv λκλ(v θ) 1 λ vv (37) ( = λ t V v ( ) 2 2 x x V r ( x 1) V ρηv x v V 1 ) 2 η2 v vv 2 κ(v θ) v V = 0. (38)

13 Scale invariance for the Heston model ( ) t V (t, x, r, v, θ, ρ, η, κ) = V, x, λr, λv, λθ, ρ, λη, λκ λ Scale invariance for the GARCH model ( t V (t, x, r, v, θ, ρ, η, κ) = V λ, x, λr, λv, λθ, ρ, ) λη, λκ (39) (40) Scale invariance for the 3/2 model V (t, x, r, v, θ, ρ, η, κ) = V ( ) t, x, λr, λv, λθ, ρ, η, κ λ (41)

14 Breaking of symmetry (Scale invariance) Scale invariance for the Heston model is ( ) t V (t, x, r, v, θ, ρ, η, κ) = V, x, λr, λv, λθ, ρ, λη, λκ. (42) λ Symmetry is broken possibly (in fact it is not!) for Because N u ij η i (v θ) j. (43) i,j=0 λη 1+λη λ η 1+η, symmetry is broken definitely for N ( ) η i ( ) v θ j u ij. (44) 1 + η 1 + v θ i,j=0 We have a new degree of freedom to manipulate convergence.

15 Comparison v = η = 0 (η, v) (η + 1, v + 1) FFT

16 American option PDE (after front-fixing) t V v ( ) 2 2 x x V r( x 1)V = d (t) x V, (45) defined on (t, x) (0, ) (0, ), with V (0, x) = 0, (46) V (t, 0) = 1 e d(t), (47) x V (t, 0) = e d(t). (48) Difficulties d(t) and V (t, x) are coupled and should be solved simultaneously d x V is non-linear No obvious starting point p = 0 Integration B 1 V

17 Special functions We denote the set of following functions Σ 1 ( ) x e ax+bt erfc A ij x i ) t j + exp (ax + bt x2 B ij x i t j, 2vt 2vt i,j i,j (49) the set of following functions Σ 2 i C i t. (50) i If we propose the expansion V = V i (t, x)p i, d = d i (t)p i. (51) Then iteration can go on V i (t, x) Σ 1, d i (t) Σ 2. (52)

18 Advanced models The PDE looks like t V v ( ) 2 2 x x V r( x 1)V = d (t) x V + OV, (53) defined on (t, x) (0, ) (0, ), with Actually, we are solving t V θ 2 V (0, x) = 0, (54) V (t, 0) = 1 e d(t), (55) x V (t, 0) = e d(t). (56) ( 2 x x ) V r( x 1)V = d (t) x V + v θ 2 It has better convergence when r v, when we set θ = r. ( 2 x x ) V. (57)

19 Numerical results Time Boundary Price Tree Series Price Tree Series Price Tree Series Price Tree Series Price Tree Series

20 HPC There are two types of calculations involved in expansion methods. Calculation of formula. It is very computationally intensive, therefore HPC is needed. The computation is parallelizable, e.g. B 1 (V 1 + V 2 ) = B 1 V 1 + B 1 V 2. The formula can be calculated and published. In research, Mathematica is used, however more efficient languages can be used. Calculation of price. It is within the power of a PC, and HPC is not needed.

21 Summary Expansion methods have the potential to solve complicated option under complicated models, which is dimensionally more difficult for finite-difference methods. European American Black-Scholes trivial Beyond Black-Scholes Expansion methods are easy to implement. HPC is not needed for end-users. Greeks are straight-forward to calculate. Pricing option families is efficient.

22 Please try the code. Thank you! Questions?