Pricing and Hedging of Commodity Derivatives using the Fast Fourier Transform

Transcription

1 Pricing and Hedging of Commodity Derivatives using the Fast Fourier Transform Vladimir Surkov Department of Statistical and Actuarial Sciences, University of Western Ontario The Fields Institute, University of Toronto December 9, /33

2 1 Generalized Model for Commodity Spot Prices 2 Fourier Space Time-stepping framework 3 Computing Option Greeks 4 Dynamic and Static Hedging 2 /33

3 1 Generalized Model for Commodity Spot Prices 2 Fourier Space Time-stepping framework 3 Computing Option Greeks 4 Dynamic and Static Hedging 3 /33

4 WTI Crude Oil and Henry Hub Gas 4 /33

5 Great Britain System Sell Prices 5 /33

6 Generalized Model for Commodity Spot Prices Commodity prices Exhibit high volatilities and spikes and prices Tend to revert to long run equilibrium prices 6 /33

7 Generalized Model for Commodity Spot Prices Commodity prices Exhibit high volatilities and spikes and prices Tend to revert to long run equilibrium prices The Model dx(t) = (Θ(t) κx(t))dt + dw(t) + dj(t) S(t) = S(0) exp{bx(t)} 6 /33

8 Generalized Model for Commodity Spot Prices Commodity prices Exhibit high volatilities and spikes and prices Tend to revert to long run equilibrium prices The Model dx(t) = (Θ(t) κx(t))dt + dw(t) + dj(t) S(t) = S(0) exp{bx(t)} Multi-factor, mean-reverting model with jumps Generalizes a number of well-known models, such as Gibson and Schwartz (1990), Clewlow and Strickland (2000), Hikspoors and Jaimungal (2007) Mean-reversion rate is time dependent - allows to incorporate seasonality 6 /33

9 One-Factor Mean-Reverting Process with Jumps 7 /33

10 Two-Factor MR Process with Different Decay Rates 8 /33

11 One-Factor MR Processes with Co-dependent Jumps 9 /33

12 Numerical Methods for Option Pricing Monte Carlo methods Tree methods Finite difference methods Alternating Direction Implicit-FFT - Andersen and Andreasen (2000) Implicit-Explicit (IMEX) - Cont and Tankov (2004) IMEX Runge-Kutta - Briani, Natalini, and Russo (2004) Fixed Point Iteration - d Halluin, Forsyth, and Vetzal (2005) Quadrature methods Reiner (2001) QUAD - Andricopoulos, Widdicks, Duck, and Newton (2003) Q-FFT - O Sullivan (2005) Transform-based methods Carr and Madan (1999) Raible (2000) Lewis (2001) Lord, Fang, Bervoets, and Oosterlee (2008) 10/33

13 1 Generalized Model for Commodity Spot Prices 2 Fourier Space Time-stepping framework 3 Computing Option Greeks 4 Dynamic and Static Hedging 11/33

14 Fourier Space Time-stepping Framework Overview The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces 12/33

15 Fourier Space Time-stepping Framework Overview The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces A framework for numerical pricing of financial derivatives Fast and precise pricing of a wide range of European and path-dependent, single- and multi-asset, vanilla and exotic derivatives Efficient handling of path-independent and discretely monitored derivatives Generic handling of different spot-price models and option payoffs 12/33

16 Pricing Framework in Real Space Option price at time t is the discounted expected future payoff V(t,S(t)) = e r(t t) E [ ϕ(s(t)) ] 13/33

17 Pricing Framework in Real Space Option price at time t is the discounted expected future payoff V(t,S(t)) = e r(t t) E [ ϕ(s(t)) ] The discount-adjusted and log-transformed price process v(t,x(t)) e r(t t) V(t,S(0)e X(t) ) satisfies a PIDE { ( t + L(t,x) κx x ) v(t,x) = 0, v(t,x) = ϕ(s(0)e Bx ) where L acts on twice-differentiable functions g(x) as follows: L(t,x)g(x) = ( Θ(t) x x Σ ) x g(x) + (g(x+y) g(x))ν(dy) Ê n 13/33

18 Fourier Transform A function in the space domain g(x) can be transformed to a function in the frequency domain ĝ(ω), where ω is given in radians per second, and vice-versa using the continuous Fourier transform F [g](ω) F 1 [ĝ](x) 1 2π g(x)e iω x dx ĝ(ω)e iω x dω 14/33

19 Fourier Transform A function in the space domain g(x) can be transformed to a function in the frequency domain ĝ(ω), where ω is given in radians per second, and vice-versa using the continuous Fourier transform F [g](ω) F 1 [ĝ](x) 1 2π g(x)e iω x dx ĝ(ω)e iω x dω Continuous Fourier transform is a linear operator that maps spatial derivatives x into multiplications in the frequency domain F [ xg](ω) n = iω F [ x n 1 g ] (ω) = = (iω) n F [g](ω) 14/33

20 Pricing Framework in Fourier Space Applying the Fourier transform to the pricing PDE we obtain a PDE in frequency space { ( t + ˆL(t, ω) + κ + κω ω ) ˆv(t, ω) = 0, ˆv(T, ω) = ˆΦ(T, ω) 15/33

21 Pricing Framework in Fourier Space Applying the Fourier transform to the pricing PDE we obtain a PDE in frequency space { ( t + ˆL(t, ω) + κ + κω ω ) ˆv(t, ω) = 0, ˆv(T, ω) = ˆΦ(T, ω) The Fourier transform of the operator L(t,x) can be computed analytically ( ) ˆL(t, ω) = iωθ(t) 1 2 ω Σω + e iω z 1 ν(dz) 15/33

22 Pricing Framework in Fourier Space Applying the Fourier transform to the pricing PDE we obtain a PDE in frequency space { ( t + ˆL(t, ω) + κ + κω ω ) ˆv(t, ω) = 0, ˆv(T, ω) = ˆΦ(T, ω) The Fourier transform of the operator L(t,x) can be computed analytically ( ) ˆL(t, ω) = iωθ(t) 1 2 ω Σω + e iω z 1 ν(dz) Introduce a new coordinate system via frequency scaling ṽ(t, ω) = ˆv(t, e κ (t t ) ω) 15/33

23 Pricing Framework in Fourier Space Applying the Fourier transform to the pricing PDE we obtain a PDE in frequency space { ( t + ˆL(t, ω) + κ + κω ω ) ˆv(t, ω) = 0, ˆv(T, ω) = ˆΦ(T, ω) The Fourier transform of the operator L(t,x) can be computed analytically ( ) ˆL(t, ω) = iωθ(t) 1 2 ω Σω + e iω z 1 ν(dz) Introduce a new coordinate system via frequency scaling ṽ(t, ω) = ˆv(t, e κ (t t ) ω) The PDE reduces to an ODE in time parameterized by ω { ( t + L(t, ω) + κ ) ṽ(t, ω) = 0, ṽ(t, ω) = Φ(T, ω) 15/33

24 Pricing Framework in Fourier Space (cont.) Given the value of ṽ(t, ω) at time t 2 T, the constant coefficient ODE is easily solved to find the value at time t 1 < t 2 : ṽ(t 1, ω) = ṽ(t 2, ω) e Ψ κ(t 1,ω;t 2), where the frequency space propagator is Ψ κ (t 1, ω; t 2 ) = t2 t 1 L(s, ω)ds + Trκ(t 2 t 1 ) 16/33

25 Pricing Framework in Fourier Space (cont.) Given the value of ṽ(t, ω) at time t 2 T, the constant coefficient ODE is easily solved to find the value at time t 1 < t 2 : ṽ(t 1, ω) = ṽ(t 2, ω) e Ψ κ(t 1,ω;t 2), where the frequency space propagator is Ψ κ (t 1, ω; t 2 ) = t2 t 1 L(s, ω)ds + Trκ(t 2 t 1 ) The solution in terms of original coordinates (with t = t 1 ) is given by ˆv(t 1, ω) = ˆv(t 2, e κ (t 2 t 1) ω) e ˆΨ κ(t 1,ω;t 2) and the frequency space propagator is ˆΨ κ (t 1, ω; t 2 ) = t2 ˆL(s, e κ (s t 1) ω)ds + Tr κ(t 2 t 1 ) t 1 16/33

26 Pricing Framework in Fourier Space (cont.) The scaled option prices in frequency space can be obtained from the scaled option prices in real space F [g](t, e κ (t 2 t 1) ω) = F [ğ](t, ω) e Tr κ (t2 t1), where ğ(t,x) g(t,xe κ (t 2 t 1) ) 17/33

27 Pricing Framework in Fourier Space (cont.) The scaled option prices in frequency space can be obtained from the scaled option prices in real space F [g](t, e κ (t 2 t 1) ω) = F [ğ](t, ω) e Tr κ (t2 t1), where ğ(t,x) g(t,xe κ (t 2 t 1) ) The final solution becomes v(t 1,x) = F 1 [ F [ v](t 2, ω) e ˆΨ(t 1,ω;t 2) ] (x) 17/33

28 Pricing Framework in Fourier Space (cont.) The scaled option prices in frequency space can be obtained from the scaled option prices in real space F [g](t, e κ (t 2 t 1) ω) = F [ğ](t, ω) e Tr κ (t2 t1), where ğ(t,x) g(t,xe κ (t 2 t 1) ) The final solution becomes v(t 1,x) = F 1 [ F [ v](t 2, ω) e ˆΨ(t 1,ω;t 2) ] (x) FST Method for Propagating Option Prices v m 1 = FFT 1 [ FFT [ v m ] e ˆΨ(t m 1,ω;t m) ] 17/33

29 Fourier Space Time-stepping Numerical Method European options v 0 = FFT 1 [ FFT [ v 1 ] e ˆΨ(t,ω;T) ] 18/33

30 Fourier Space Time-stepping Numerical Method European options v 0 = FFT 1 [ FFT [ v 1 ] e ˆΨ(t,ω;T) ] Bermudan/American options [ ] vm 1 = FFT 1 FFT [ v m ] e ˆΨ(t m 1,ω;t m), v m 1 = max { vm 1,v M}, where vm 1 represents the holding value of the option 18/33

31 Fourier Space Time-stepping Numerical Method European options v 0 = FFT 1 [ FFT [ v 1 ] e ˆΨ(t,ω;T) ] Bermudan/American options [ ] vm 1 = FFT 1 FFT [ v m ] e ˆΨ(t m 1,ω;t m), v m 1 = max { vm 1,v M}, where vm 1 represents the holding value of the option Barrier options v m 1 = FFT 1 [ FFT [ v m ] e ˆΨ(t m 1,ω;t m) ] ½ {x<b} + R ½ {x B} 18/33

32 Fourier Space Time-stepping Numerical Method European options v 0 = FFT 1 [ FFT [ v 1 ] e ˆΨ(t,ω;T) ] Bermudan/American options [ ] vm 1 = FFT 1 FFT [ v m ] e ˆΨ(t m 1,ω;t m), v m 1 = max { vm 1,v M}, where vm 1 represents the holding value of the option Barrier options v m 1 = FFT 1 [ FFT [ v m ] e ˆΨ(t m 1,ω;t m) ] ½ {x<b} + R ½ {x B} Exotic options, such as swings, can also be handled 18/33

33 Discrete Barrier Option Results N M Value Change Convergence Time (sec.) Option: Down-and-out barrier put S = 100, K = 105, T = 1, B = 90, R = 3 with daily monitoring Model: Merton jump-diffusion with mean reversion σ = 0.2, λ = 1.0, µ = 0.1, σ = 0.25, θ = 90.0, κ = 0.75, r = 0.05 Monte Carlo: % CI width of 114 sec. 19/33

34 1 Generalized Model for Commodity Spot Prices 2 Fourier Space Time-stepping framework 3 Computing Option Greeks 4 Dynamic and Static Hedging 20/33

35 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) 21/33

36 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). 21/33

37 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). The discrete method for computing Deltas is then given by k,m 1 = FFT 1 [iω k ˆv m 1 ] 21/33

38 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). The discrete method for computing Deltas is then given by k,m 1 = FFT 1 [iω k ˆv m 1 ] Higher order derivatives computed in similar manner 21/33

39 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). The discrete method for computing Deltas is then given by k,m 1 = FFT 1 [iω k ˆv m 1 ] Higher order derivatives computed in similar manner Theta t v(t,x) 21/33

40 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). The discrete method for computing Deltas is then given by k,m 1 = FFT 1 [iω k ˆv m 1 ] Higher order derivatives computed in similar manner Theta t v(t,x) Obtained directly from the pricing ODE t ṽ(t, ω) = ( L(t, ω) + κ ) ṽ(t, ω) 21/33

41 Computation of Greeks - State Variables Delta Sk v(t,x) = xk v(t,x)/s k (t) Differentiation in real space computed via scaling in Fourier space xk v(t,x) = F 1 [iω k ˆv(t, ω)](x). The discrete method for computing Deltas is then given by k,m 1 = FFT 1 [iω k ˆv m 1 ] Higher order derivatives computed in similar manner Theta t v(t,x) Obtained directly from the pricing ODE t ṽ(t, ω) = ( L(t, ω) + κ ) ṽ(t, ω) The discrete method for computing Theta is then given by Θ m 1 = FFT 1 [ ( ˆL(t, ω) + κ ) ˆv m 1 ] 21/33

42 Computation of Greeks - Model Parameters In Fourier space, the sensitivity satisfies an ODE with source term { ( t + L κ )ṽ(t, ω) } = ( t + L κ ) ṽ(t, ω) + L κ ṽ(t, ω) = 0 22/33

43 Computation of Greeks - Model Parameters In Fourier space, the sensitivity satisfies an ODE with source term { ( t + L κ )ṽ(t, ω) } = ( t + L κ ) ṽ(t, ω) + L κ ṽ(t, ω) = 0 The ODE can be solved explicitly [ ] v(t,x) = F 1 ˆΨ κ (t, e κ (T t) ω; T) ˆv(t, ω) (x) 22/33

44 Computation of Greeks - Model Parameters In Fourier space, the sensitivity satisfies an ODE with source term { ( t + L κ )ṽ(t, ω) } = ( t + L κ ) ṽ(t, ω) + L κ ṽ(t, ω) = 0 The ODE can be solved explicitly [ ] v(t,x) = F 1 ˆΨ κ (t, e κ (T t) ω; T) ˆv(t, ω) (x) The discrete method for computing the sensitivity is then given by, m 1 = FFT 1 [ ˆΨκ (t m 1, e κ t m ω; t m ) ˆv m 1 ] 22/33

45 Computation of Greeks - Model Parameters In Fourier space, the sensitivity satisfies an ODE with source term { ( t + L κ )ṽ(t, ω) } = ( t + L κ ) ṽ(t, ω) + L κ ṽ(t, ω) = 0 The ODE can be solved explicitly [ ] v(t,x) = F 1 ˆΨ κ (t, e κ (T t) ω; T) ˆv(t, ω) (x) The discrete method for computing the sensitivity is then given by, m 1 = FFT 1 [ ˆΨκ (t m 1, e κ t m ω; t m ) ˆv m 1 ] Higher order derivatives computed in similar manner 22/33

46 Greeks Computation Errors Price Error Delta Error Absolute Error 10 5 N=4096 N=8192 N=16384 N=32768 Absolute Error 10 5 N=4096 N=8192 N=16384 N= Stock Price (S) Gamma Error Stock Price (S) Vega Error Absolute Error N=4096 N=8192 N=16384 N=32768 Absolute Error 10 5 N=4096 N=8192 N=16384 N= Stock Price (S) Theta Error Stock Price (S) Rho Error Absolute Error 10 5 N=4096 N=8192 N=16384 N=32768 Absolute Error 10 5 N=4096 N=8192 N=16384 N= Stock Price (S) Stock Price (S) 23/33

47 1 Generalized Model for Commodity Spot Prices 2 Fourier Space Time-stepping framework 3 Computing Option Greeks 4 Dynamic and Static Hedging 24/33

48 Dynamic Hedging Hedging portfolio for the option V consists of B units of cash, e units of the underlying asset S and N hedging instruments I with weights φ Π = φ I(t,S(t)) + es(t) + B V(t,S(t)) 25/33

49 Dynamic Hedging Hedging portfolio for the option V consists of B units of cash, e units of the underlying asset S and N hedging instruments I with weights φ Π = φ I(t,S(t)) + es(t) + B V(t,S(t)) The portfolio s value remains unchanged under small movements in price S Π = φ S I(t,S(t)) + e S V(t,S(t)) = 0 25/33

50 Dynamic Hedging Hedging portfolio for the option V consists of B units of cash, e units of the underlying asset S and N hedging instruments I with weights φ Π = φ I(t,S(t)) + es(t) + B V(t,S(t)) The portfolio s value remains unchanged under small movements in price S Π = φ S I(t,S(t)) + e S V(t,S(t)) = 0 Can also hedge against small movements in interest-rates, volatility, etc. Π = φ I(t,S(t)) V(t,S(t)) = 0 25/33

51 Dynamic Hedging Hedging portfolio for the option V consists of B units of cash, e units of the underlying asset S and N hedging instruments I with weights φ Π = φ I(t,S(t)) + es(t) + B V(t,S(t)) The portfolio s value remains unchanged under small movements in price S Π = φ S I(t,S(t)) + e S V(t,S(t)) = 0 Can also hedge against small movements in interest-rates, volatility, etc. Π = φ I(t,S(t)) V(t,S(t)) = 0 What about large movements? 25/33

52 Static Hedging - Minimize Portfolio Variance Minimize portfolio price variance under expected asset price movement, Kennedy, Forsyth, Vetzal (2009): arg min e n, φ n ξ E tn [ φn I n + e n S n V n ] 2 + (1 ξ)υn. where Υ n is the transaction cost to rebalance the portfolio: Υ n = N [ α k ( φ k,n φ k,n 1 ) ] 2 [ ] 2 I k,n + β(e n e n 1 )S n, k=1 26/33

53 Static Hedging - Minimize Portfolio Variance Minimize portfolio price variance under expected asset price movement, Kennedy, Forsyth, Vetzal (2009): arg min e n, φ n ξ E tn [ φn I n + e n S n V n ] 2 + (1 ξ)υn. where Υ n is the transaction cost to rebalance the portfolio: Υ n = N [ α k ( φ k,n φ k,n 1 ) ] 2 [ ] 2 I k,n + β(e n e n 1 )S n, k=1 Since the objective function is quadratic, the optimality requires F φ k,n = ξ E tn [ ( φ I + e S V )( 2 Ik ) ] + (1 ξ) φk,n Υ n = 0 F e n = ξ E tn [ ( φ I + e S V )( 2 S ) ] + (1 ξ) en Υ n = 0 26/33

54 Static Hedging - Minimize Portfolio Variance 27/33

55 Static Hedging - Minimize Price and Greeks Variance Minimize portfolio price and Greeks variance under expected asset price movement arg min e n, φ n ξ D w D E tn [ φn (D I n ) + e n (DS n ) (DV n )] 2 + (1 ξ)υn 28/33

56 Static Hedging - Minimize Price and Greeks Variance Minimize portfolio price and Greeks variance under expected asset price movement arg min e n, φ n ξ D w D E tn [ φn (D I n ) + e n (DS n ) (DV n )] 2 + (1 ξ)υn Since the objective function is quadratic, the optimality requires F = ξ [ ( φ )( w D E tn (D I) + e (DS) (DV) 2 (DIk ) )] φ k,n D +(1 ξ) φk,n Υ n = 0 F = ξ [ ( φ )( ) w D E tn (D ] I) + e (DS) (DV) 2 (DS) e n D +(1 ξ) en Υ n = 0 28/33

57 Static Hedging - Minimize Price and Greeks Variance 29/33

58 Loss Distribution and VaR - Constant Volatility 30/33

59 Loss Distribution and VaR - Dynamic Volatility 31/33

60 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces 32/33

61 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces Independent-increment, mean-reverting and interest-rate Lévy models are handled generically 32/33

62 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces Independent-increment, mean-reverting and interest-rate Lévy models are handled generically Option values are obtained for a range of spot prices - readily price path-dependent options 32/33

63 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces Independent-increment, mean-reverting and interest-rate Lévy models are handled generically Option values are obtained for a range of spot prices - readily price path-dependent options Two FFTs per time-step are required; no time-stepping for European options or between monitoring dates of discretely monitored options 32/33

64 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces Independent-increment, mean-reverting and interest-rate Lévy models are handled generically Option values are obtained for a range of spot prices - readily price path-dependent options Two FFTs per time-step are required; no time-stepping for European options or between monitoring dates of discretely monitored options One extra FFT required to compute each Greek 32/33

65 FST Framework Summary The Approach Consider the PIDE for the option price Transform the PIDE into ODE in Fourier space Solve the resulting ODE analytically Utilize FFT to efficiently switch between real and Fourier spaces Independent-increment, mean-reverting and interest-rate Lévy models are handled generically Option values are obtained for a range of spot prices - readily price path-dependent options Two FFTs per time-step are required; no time-stepping for European options or between monitoring dates of discretely monitored options One extra FFT required to compute each Greek Second order convergence in space and second order convergence in time for American options with penalty method 32/33

66 Thank You! Model FST Greeks Hedging Jackson, K. R., S. Jaimungal, and V. Surkov (2008). Fourier space time-stepping for option pricing with Lévy models. Journal of Computational Finance 12(2), Jaimungal, S. and V. Surkov (2008). A general Lévy-based framework for energy price modeling and derivative valuation via FFT. Working paper, available at Surkov, V. (2009). Efficient static hedging under generalized Lévy models. Working paper. More at 33/33