Two-dimensional COS method

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1 Two-dimensional COS method Marjon Ruijter Winterschool Lunteren 22 January /29

2 Introduction PhD student since October 2010 Prof.dr.ir. C.W. Oosterlee). CWI national research center for mathematics and computer science. CPB Netherlands Bureau for Economic Policy Analysis. Impact of climate change on investments and policy decisions. Financial mathematics. 2/29

3 Stochastic optimization - 2D Climate-economics problem Two stochastic processes: temperature and capital. Goal: maximize expected utility [ T ] vt, x 1, x 2 )= max E e ρs t) UC s ))ds F t. 1) {a s,c s} t 3/29

4 Stochastic optimization - 2D Climate-economics problem Two stochastic processes: temperature and capital. Goal: maximize expected utility [ T ] vt, x 1, x 2 )= max E e ρs t) UC s ))ds F t. 1) {a s,c s} t Rainbow option pricing problem Two stochastic processes: asset price 1 and asset price 2. Goal: maximize expected profit [ ] vt, x 1, x 2 )= max E e rτ t) gxτ 1, Xτ 2 ) F t. 2) τ [t,t ] 3/29

5 Stochastic optimization - 2D Climate-economics problem Two stochastic processes: temperature and capital. Goal: maximize expected utility [ T ] vt, x 1, x 2 )= max E e ρs t) UC s ))ds F t. 1) {a s,c s} t Rainbow option pricing problem Two stochastic processes: asset price 1 and asset price 2. Goal: maximize expected profit [ ] vt, x 1, x 2 )= max E e rτ t) gxτ 1, Xτ 2 ) F t. 2) τ [t,t ] The math is almost) the same 3/29

6 Table of Contents 1 Rainbow options 2 2D-COS formula 3 European options 4 Bermudan options 5 Heston stochastic volatility model 6 Conclusion 4/29

7 Financial mathematics In financial markets, traders deal in assets and options. The payoff of an option depends on the value of the underlying asset prices). Asset price X t is stochastic. 5/29

8 Financial mathematics In financial markets, traders deal in assets and options. The payoff of an option depends on the value of the underlying asset prices). Asset price X t is stochastic. Payoff call: g call x) =maxx K, 0) 3) Payoff put: g put x) =maxk x, 0) 4) 30 call 30 put gx) gx) x x Figure 1: Payoff call and put option, strike price K = 100 1D). 5/29

9 2 correlated asset prices Two stochastic asset price processes, X 1 t and X 2 t. 6/29

10 2 correlated asset prices Two stochastic asset price processes, Xt 1 and Xt 2. For example, correlated geometric Brownian motions: dxt 1 = μ 1Xt 1 dt + σ 1Xt 1 dw t 1, 5) dxt 2 = μ 2Xt 2 dt + σ 2Xt 2 dw t 2, 6) with dwt 1 dwt 2 = ρdt. 130 Correlated Asset Paths Asset Price Time to Expiry 6/29

11 Payoff rainbow options Basket option: weighted sum or average of different assets, e.g., gx 1, x 2 )=max 1 2 x 1 + x 2 ) K, 0 ). 7) gx 1,x 2 ) x x 1 Figure 2: Basket option. 7/29

12 Payoff rainbow options Basket option: weighted sum or average of different assets, e.g., gx 1, x 2 )=max 1 2 x 1 + x 2 ) K, 0 ). 7) gx 1,x 2 ) 10 5 gx 1,x 2 ) x x x x Figure 2: Basket option. Call on maximum option: Figure 3: Call on max. option. gx 1, x 2 )=maxmaxx 1, x 2 ) K, 0). 8) 7/29

13 European, American, and Bermudan-style European-style: you buy the option now, wait until terminal time T, then the option may be exercised. American-style: may be exercised at any time before the terminal time T. Bermudan-style: fixed exercise dates t m m =1,...,M) atwhich you can exercise the option. Financial mathematics: efficient computation of option price. vt 0, x 0 )=e rδt E [vt, X T )]. 9) 8/29

14 COS method Based on Fourier cosine series expansions. 9/29

15 COS method Based on Fourier cosine series expansions. Pricing financial and real options: European options F. Fang, C.W. Oosterlee, 2008), Bermudan and American options F. Fang, C.W. Oosterlee, 2009), Swing options, which are frequently used in energy markets B. Zhang, C.W. Oosterlee, 2010), Asian-style options B. Zhang, C.W. Oosterlee, 2011), Optimal dike height, M.J. Ruijter, master thesis, 2010),... 9/29

16 Fourier-cosine series expansion of function hx) on[a, b]: hx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) 10 / 29

17 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) 10 / 29

18 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=2 60 hx) x 10 / 29

19 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=3 60 hx) x 10 / 29

20 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=4 60 hx) x 10 / 29

21 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=6 60 hx) x 10 / 29

22 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=10 60 hx) x 10 / 29

23 Fourier-cosine series expansion of function hx) on[a, b]: N 1 ĥx) = H k cos kπ x a ), x [a, b], 10) with coefficients k=0 H k = 2 b a hy)cos kπ y a ) dy. 11) N=50 60 hx) x 10 / 29

24 1D-COS formula We use the COS formula to approximate expectations. vt, x) =E [vt, X T )] = vt, y)f y x)dy R 11 / 29

25 1D-COS formula We use the COS formula to approximate expectations. vt, x) =E [vt, X T )] = vt, y)f y x)dy b a R vt, y)f y x)dy 11 / 29

26 1D-COS formula We use the COS formula to approximate expectations. vt, x) =E [vt, X T )] = vt, y)f y x)dy b a = 2 R vt, y)f y x)dy k=0 V k T )F k x), with coefficients V k T ):= 2 b vt, y)cos kπ y a a F k x) := 2 b f y x)cos kπ y a a Exponential convergence in N for smooth density f y x). ) dy, 12) ) dy. 13) 11 / 29

27 1D-COS formula We use the COS formula to approximate expectations. vt, x) =E [vt, X T )] = vt, y)f y x)dy b a 2 R vt, y)f y x)dy N 1 k=0 V k T )F k x), with coefficients V k T ):= 2 b vt, y)cos kπ y a a F k x) := 2 b f y x)cos kπ y a a Exponential convergence in N for smooth density f y x). ) dy, 12) ) dy. 13) 11 / 29

28 Fourier coefficients F k : F k x) = b 2 2 a R f y x)cos kπ y a ) dy f y x)cos kπ y a ) dy 14) 12 / 29

29 Fourier coefficients F k : F k x) = = b f y x)cos 2 a 2 f y x)cos R 2 Re f y x)exp R kπ y a ) dy kπ y a ) dy ikπ y a ) ) dy 14) 12 / 29

30 Fourier coefficients F k : 2 b F k x) = f y x)cos kπ y a ) dy a 2 f y x)cos kπ y a ) dy R 2 = Re f y x)exp ikπ y a ) ) dy R 2 kπ ) ) = Re ϕ x e ikπ a b a 14) Characteristic function of random variable Y : ϕu) =E [exp iuy )] = expiuy)f y)dy. 15) For many asset price processes the characteristic function is available. 12 / 29 R

31 Fourier coefficients F k : 2 b F k x) = f y x)cos kπ y a ) dy a 2 f y x)cos kπ y a ) dy R 2 = Re f y x)exp ikπ y a ) ) dy R 2 kπ ) ) = Re ϕ x e ikπ a b a ) ) 2 kπ = Re x a ikπ ϕ levy e b a. 14) Characteristic function of random variable Y : ϕu) =E [exp iuy )] = expiuy)f y)dy. 15) For many asset price processes the characteristic function is available. 12 / 29 R

32 2D-COS formula In 1D: In 2D: vt, x) =e rδt E [vt, X T )] N 1 2 e rδt V k T )F k x). 16) vt, x) =e rδt E [vt, X T )] k=0

33 2D-COS formula In 1D: In 2D: vt, x) =e rδt E [vt, X T )] N 1 2 e rδt V k T )F k x). 16) vt, x) =e rδt E [vt, X T )] k=0

34 2D-COS formula In 1D: In 2D: vt, x) =e rδt E [vt, X T )] N 1 2 e rδt V k T )F k x). 16) vt, x) =e rδt E [vt, X T )] = e rδt vt, y)f y x)dy b 1 a 1 2 R b 2 a 2 2 k=0 N 1 1 N 2 1 e rδt k 1 =0 k 2 =0 This can be extended to higher dimensions. V k1,k 2 T )F k1,k 2 x). 17) 13 / 29

35 Coefficients - most difficult part. ) ) F k1,k 2 x) 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 cos k 2 π y2 a2 b 2 a 2 dy 1 dy )

36 Coefficients - most difficult part. ) ) F k1,k 2 x) 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 cos k 2 π y2 a2 b 2 a 2 dy 1 dy ) We use the following goniometric relation: 2cosα)cosβ) =cosα + β)+cosα β). 19)

37 Coefficients - most difficult part. ) ) F k1,k 2 x) 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 cos k 2 π y2 a2 b 2 a 2 dy 1 dy ) We use the following goniometric relation: 2cosα)cosβ) =cosα + β)+cosα β). 19) Then 2F k1,k 2 x) =F + k 1,k 2 x)+f k 1,k 2 x), 20) where ) F ± k 1,k 2 x) := 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 ± k 2 π y2 a2 b 2 a 2 dy 1 dy )

38 Coefficients - most difficult part. ) ) F k1,k 2 x) 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 cos k 2 π y2 a2 b 2 a 2 dy 1 dy ) We use the following goniometric relation: Then where F ± k 1,k 2 x) := 2 b 1 a 1 2 2cosα)cosβ) =cosα + β)+cosα β). 19) = 2 b 1 a 1 2 b 2 a 2 Re 2F k1,k 2 x) =F + k 1,k 2 x)+f k 1,k 2 x), 20) b 2 a 2 f y x)cos R 2 R 2 f y x)exp ) k 1 π y1 a1 b 1 a 1 ± k 2 π y2 a2 b 2 a 2 dy 1 dy 2 ) ) ik 1 π y1 a1 b 1 a 1 ± ik 2 π y2 ±a2 b 2 a 2 dy 1 dy 2 21)

39 Coefficients - most difficult part. ) ) F k1,k 2 x) 2 2 b 1 a 1 b 2 a 2 f y x)cos k 1 π R y1 a1 b 1 a 1 cos k 2 π y2 a2 b 2 a 2 dy 1 dy ) We use the following goniometric relation: Then where F ± k 1,k 2 x) := 2 b 1 a 1 2 2cosα)cosβ) =cosα + β)+cosα β). 19) = 2 2 b 1 a 1 b 2 a 2 Re = 2 b 1 a 1 2 b 2 a 2 Re 2F k1,k 2 x) =F + k 1,k 2 x)+f k 1,k 2 x), 20) b 2 a 2 f y x)cos R 2 ϕ ) k 1 π y1 a1 b 1 a 1 ± k 2 π y2 a2 b 2 a 2 dy 1 dy 2 ik 1 π y1 a1 b 1 a 1 f y x)exp R 2 ) k1π b 1 a 1, ± k2π x b 2 a 2 exp ) ) ± ik 2 π y2 ±a2 b 2 a 2 dy 1 dy 2 ik 1 π a1 b 1 a 1 ik 2 π a2 21) b 2 a 2 ))

40 Approximate the terminal coefficients V k1,k 2 T ) with DCTs. Take Q max[n 1, N 2 ] grid-points and y n i i := a i +n i )b i a i Q and Δy i := b i a i, i =1, 2. 22) Q 15 / 29

41 Approximate the terminal coefficients V k1,k 2 T ) with DCTs. Take Q max[n 1, N 2 ] grid-points and y n i i := a i +n i )b i a i Q and The midpoint-rule integration gives us V k1,k 2 T ) = 2 b 1 a 1 2 b 2 a 2 b2 Q 1 Q 1 n 1 =0 n 2 =0 a 2 b1 a 1 gy n 1 1, y n 2 2 )cos Δy i := b i a i, i =1, 2. 22) Q ) ) gy)cos k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy 1 dy 2 k 1 π 2n 1+1 2Q ) cos k 2 π 2n 2+1 2Q ) b1 a 1 Q b 2 a 2 Q. The above 2D-DCT can be calculated efficiently by, for example, MATLAB s function dct2. 15 / 29

42 Results - European options Results geometric basket call under correlated geometric Brownian motion, vt 0, x 0 )= Table 1: N 2 = N 1 ). N Error 7.60e e e e e-13 CPU ms) log 10 error N 1 16 / 29

43 Jump-diffusion process The log-jump-diffusion process dst i =r λe[ej i 1])St i dt + σ ist i dw t i + S t i ej i 1)dq t, 23) with q t a Poisson process with intenstity λ, andj =J 1, J 2 ) bivariate normally distributed jumps. Figure 4: Density recovery ˆf X T x 0 ). 17 / 29

44 Jump-diffusion process The log-jump-diffusion process dst i =r λe[ej i 1])St i dt + σ ist i dw t i + S t i ej i 1)dq t, 23) with q t a Poisson process with intenstity λ, andj =J 1, J 2 ) bivariate normally distributed jumps. Table 2: Put-on-min option values ˆvt 0, x 0 )N 1 = N 2 = 125), they correspond to value in Clift, 2008). Figure 4: Density recovery ˆf X T x 0 ). S 2 0 S / 29

45 Spark spread option - 3D 3-dimensional GBM S t =[St power, St gas, S CO 2 t ]. Spark spread: net revenue from selling power. Payoff : gs T )=Ωmax ) S power T α g S gas T αco 2 S CO 2 T K, / 29

46 Spark spread option - 3D 3-dimensional GBM S t =[St power, St gas, S CO 2 t ]. Spark spread: net revenue from selling power. Payoff : gs T )=Ωmax ) S power T α g S gas T αco 2 S CO 2 T K, / 29

47 Spark spread option - 3D 3-dimensional GBM S t =[St power, St gas, S CO 2 t ]. Spark spread: net revenue from selling power. Payoff : gs T )=Ωmax ) S power T α g S gas T αco 2 S CO 2 T K, 0. vt 0, S 0 ) CPU time s) N N Q n/a n/a Calculation of the option s Greeks is straightforward. 18 / 29

48 Bermudan options Bermudan option: fixed exercise dates t m m =1,...,M) atwhich you can either exercise the option or continue. Option value in 1D: vt m, x) =max[gx), ct m, x)]. 24) 19 / 29

49 Bermudan options Bermudan option: fixed exercise dates t m m =1,...,M) atwhich you can either exercise the option or continue. Option value in 1D: vt m, x) =max[gx), ct m, x)]. 24) Coefficients V k at time t m : V k t m ):= 2 b a b a vt m, y)cos kπ y a b a ) dy. 25) 19 / 29

50 Bermudan options Bermudan option: fixed exercise dates t m m =1,...,M) atwhich you can either exercise the option or continue. Option value in 1D: vt m, x) =max[gx), ct m, x)]. 24) Coefficients V k at time t m : V k t m ):= 2 b a b a vt m, y)cos kπ y a b a ) dy. 25) Asset price X t Continue 85 Exercise Time t m 19 / 29

51 Bermudan option - 2D Coefficients V k1,k 2 at time t m : V k1,k 2 t m ):= with b2 b1 a 2 a 1 vt m, y)cos ) ) k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy 1 dy 2 vt m, x) =max[gx), ct m, x)]. 26) 20 / 29

52 Bermudan option - 2D Coefficients V k1,k 2 at time t m : V k1,k 2 t m ):= with b2 b1 a 2 a 1 vt m, y)cos ) ) k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy 1 dy 2 vt m, x) =max[gx), ct m, x)]. 26) 6 Left: Optimal exercise domains blue) and continuation domains green) at initial time t 0. J rectangular sub-domains. y y 1 20 / 29

53 Equidistant vs. non-equidistant grid x x 1 Figure 5: Equidistant grid. 21 / 29

54 Equidistant vs. non-equidistant grid x 2 x x x 1 Figure 5: Equidistant grid. Figure 6: Non-equidistant grid. 21 / 29

55 Recursive recovery V k1,k 2 t m )= b2 b1 a 2 = p + q a 1 vt m, y)cos k 1 π y 1 a 1 gy)cos G p C q ct m, y)cos b 1 a 1 ) cos k 1 π y 1 a 1 b 1 a 1 ) cos k 2 π y 2 a 2 b 2 a 2 ) dy 1 dy 2 k 2 π y 2 a 2 b 2 a 2 ) dy ) ) k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy 22 / 29

56 Recursive recovery V k1,k 2 t m )= b2 b1 a 2 = p + q a 1 vt m, y)cos k 1 π y 1 a 1 gy)cos G p C q ĉt m, y)cos b 1 a 1 ) cos k 1 π y 1 a 1 b 1 a 1 ) cos k 2 π y 2 a 2 b 2 a 2 ) dy 1 dy 2 k 2 π y 2 a 2 b 2 a 2 ) dy ) ) k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy 22 / 29

57 Recursive recovery V k1,k 2 t m )= b2 b1 a 2 = p + q a 1 vt m, y)cos k 1 π y 1 a 1 gy)cos G p C q ĉt m, y)cos b 1 a 1 ) cos k 1 π y 1 a 1 b 1 a 1 ) cos k 2 π y 2 a 2 b 2 a 2 ) dy 1 dy 2 k 2 π y 2 a 2 b 2 a 2 ) dy ) ) k 1 π y 1 a 1 b 1 a 1 cos k 2 π y 2 a 2 b 2 a 2 dy The resulting matrix-vector products Mu can be computed efficiently by a Fourier-based algorithm. The computation time achieved is ON log 2 N). 22 / 29

58 Algorithm 2D-COS method for pricing Bermudan rainbow options Initialisation: Calculate coefficients V k1,k 2 t M ). Main loop to recover ˆV t m ): For m = M 1to1: Determine the optimal continuation regions C q and early-exercise regions G p. Compute ˆV t m ) with the help of the FFT algorithm. Final step: Compute ˆvt 0, x 0 ) by inserting ˆV k1,k 2 t 1 ) into COS formula. 23 / 29

59 Results Bermudan option Error geometric basket option under GBM. N 1 is number of terms in series expansion N 2 = N 1 ). J is number of sub-domains. 24 / 29

60 Results Bermudan option Error geometric basket option under GBM. N 1 is number of terms in series expansion N 2 = N 1 ). J is number of sub-domains. log 10 error N J / 29

61 Results Bermudan option Error geometric basket option under GBM. N 1 is number of terms in series expansion N 2 = N 1 ). J is number of sub-domains. Call on maximum option log 10 error N J Table 3: Results call-on-max GBM). Andersen 2004) Shashi S 0 2D-COS Binomial SGM Direct ) ) ) 24 / 29

62 Heston model X t represents the asset price process and ν t is the variance process, with dwt 1 dwt 2 = ρdt) dx t =r 1 2 ν t)x t dt + ν t X t dwt 1, dν t = κ ν ν t )dt + η ν t dwt 2. 27) 25 / 29

63 Heston model X t represents the asset price process and ν t is the variance process, with dwt 1 dwt 2 = ρdt) dx t =r 1 2 ν t)x t dt + ν t X t dwt 1, dν t = κ ν ν t )dt + η ν t dwt 2. 27) 25 / 29

64 Heston model X t represents the asset price process and ν t is the variance process, with dwt 1 dwt 2 = ρdt) dx t =r 1 2 ν t)x t dt + ν t X t dwt 1, dν t = κ ν ν t )dt + η ν t dwt 2. 27) The variance process remains strictly positive if the Feller condition is satisfied, 2κū/η 2 1:=q Feller 0, otherwise it may reach zero. 25 / 29

65 European option, with Bermudan framework M = 12 time steps. 26 / 29

66 European option, with Bermudan framework M = 12 time steps. Feller satisfied q Feller =0.98), v = N e e e e e e e e-6 N e e e e-6 26 / 29

67 European option, with Bermudan framework M = 12 time steps. Feller satisfied q Feller =0.98), v = N e e e e e e e e-6 N e e e e-6 Feller not satisfied q Feller = 0.47), v = N e e e e e e e e-5 N e e e e-5 26 / 29

68 Feller not satisfied q Feller = 0.84), v = N e e e e e e e e-2 N e e e e-2 27 / 29

69 Bermudan put option N 1 Feller satisfied q Feller =0.98) N / 29

70 Bermudan put option N 1 Feller satisfied q Feller =0.98) N N 1 Feller not satisfied q Feller = 0.47) N / 29

71 Summary and conclusion COS method is based on Fourier-cosine series expansion. Exponential convergence for f C. Can be extended to higher dimensions for pricing rainbow options. Experiments with financial and spark options. Heston stochastic volatility model. Future research Higher dimensions, Gibbs phenomenon and filters, Asian options. Contact: m.j.ruijter@cwi.nl 29 / 29

Time-changed Brownian motion and option pricing

Time-changed Brownian motion and option pricing Peter Hieber Chair of Mathematical Finance, TU Munich 6th AMaMeF Warsaw, June 13th 2013 Partially joint with Marcos Escobar (RU Toronto), Matthias Scherer