""" Diffusion 2: jump diffusion, stochastic volatility, stochastic time Created on Tue Dec 08 15:03:49 2009 Author: josef-pktd following Meucci License: BSD contains: CIRSubordinatedBrownian Heston IG JumpDiffusionKou JumpDiffusionMerton NIG VG References ---------- Attilio Meucci, Review of Discrete and Continuous Processes in Finance: Theory and Applications Bloomberg Portfolio Research Paper No. 2009-02-CLASSROOM July 1, 2009 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1373102 this is currently mostly a translation from matlab of http://www.mathworks.com/matlabcentral/fileexchange/23554-review-of-discrete-and-continuous-processes-in-finance license BSD: Copyright (c) 2008, Attilio Meucci All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. TODO: * vectorize where possible * which processes are exactly simulated by finite differences ? * include or exclude (now) the initial observation ? * convert to and merge with diffusion.py (part 1 of diffusions) * which processes can be easily estimated ? loglike or characteristic function ? * tests ? check for possible index errors (random indices), graphs look ok * adjust notation, variable names, more consistent, more pythonic * delete a few unused lines, cleanup * docstrings random bug (showed up only once, need fuzz-testing to replicate) File "../diffusion2.py", line 375, in x = jd.simulate(mu,sigma,lambd,a,D,ts,nrepl) File "../diffusion2.py", line 129, in simulate jumps_ts[n] = CumS[Events] IndexError: index out of bounds CumS is empty array, Events == -1 """ import numpy as np #from scipy import stats # currently only uses np.random import matplotlib.pyplot as plt class JumpDiffusionMerton: ''' Example ------- mu=.00 # deterministic drift sig=.20 # Gaussian component l=3.45 # Poisson process arrival rate a=0 # drift of log-jump D=.2 # st.dev of log-jump X = JumpDiffusionMerton().simulate(mu,sig,lambd,a,D,ts,nrepl) plt.figure() plt.plot(X.T) plt.title('Merton jump-diffusion') ''' def __init__(self): pass def simulate(self, m,s,lambd,a,D,ts,nrepl): T = ts[-1] # time points # simulate number of jumps n_jumps = np.random.poisson(lambd*T, size=(nrepl, 1)) jumps=[] nobs=len(ts) jumps=np.zeros((nrepl,nobs)) for j in range(nrepl): # simulate jump arrival time t = T*np.random.rand(n_jumps[j])#,1) #uniform t = np.sort(t,0) # simulate jump size S = a + D*np.random.randn(n_jumps[j],1) # put things together CumS = np.cumsum(S) jumps_ts = np.zeros(nobs) for n in range(nobs): Events = np.sum(t<=ts[n])-1 #print n, Events, CumS.shape, jumps_ts.shape jumps_ts[n]=0 if Events > 0: jumps_ts[n] = CumS[Events] #TODO: out of bounds see top #jumps = np.column_stack((jumps, jumps_ts)) #maybe wrong transl jumps[j,:] = jumps_ts D_Diff = np.zeros((nrepl,nobs)) for k in range(nobs): Dt=ts[k] if k>1: Dt=ts[k]-ts[k-1] D_Diff[:,k]=m*Dt + s*np.sqrt(Dt)*np.random.randn(nrepl) x = np.hstack((np.zeros((nrepl,1)),np.cumsum(D_Diff,1)+jumps)) return x class JumpDiffusionKou: def __init__(self): pass def simulate(self, m,s,lambd,p,e1,e2,ts,nrepl): T=ts[-1] # simulate number of jumps N = np.random.poisson(lambd*T,size =(nrepl,1)) jumps=[] nobs=len(ts) jumps=np.zeros((nrepl,nobs)) for j in range(nrepl): # simulate jump arrival time t=T*np.random.rand(N[j]) t=np.sort(t) # simulate jump size ww = np.random.binomial(1, p, size=(N[j])) S = ww * np.random.exponential(e1, size=(N[j])) - \ (1-ww) * np.random.exponential(e2, N[j]) # put things together CumS = np.cumsum(S) jumps_ts = np.zeros(nobs) for n in range(nobs): Events = sum(t<=ts[n])-1 jumps_ts[n]=0 if Events: jumps_ts[n]=CumS[Events] jumps[j,:] = jumps_ts D_Diff = np.zeros((nrepl,nobs)) for k in range(nobs): Dt=ts[k] if k>1: Dt=ts[k]-ts[k-1] D_Diff[:,k]=m*Dt + s*np.sqrt(Dt)*np.random.normal(size=nrepl) x = np.hstack((np.zeros((nrepl,1)),np.cumsum(D_Diff,1)+jumps)) return x class VG: '''variance gamma process ''' def __init__(self): pass def simulate(self, m,s,kappa,ts,nrepl): T=len(ts) dXs = np.zeros((nrepl,T)) for t in range(T): dt=ts[1]-0 if t>1: dt = ts[t]-ts[t-1] #print dt/kappa #TODO: check parameterization of gamrnd, checked looks same as np d_tau = kappa * np.random.gamma(dt/kappa,1.,size=(nrepl)) #print s*np.sqrt(d_tau) # this raises exception: #dX = stats.norm.rvs(m*d_tau,(s*np.sqrt(d_tau))) # np.random.normal requires scale >0 dX = np.random.normal(loc=m*d_tau, scale=1e-6+s*np.sqrt(d_tau)) dXs[:,t] = dX x = np.cumsum(dXs,1) return x class IG: '''inverse-Gaussian ??? used by NIG ''' def __init__(self): pass def simulate(self, l,m,nrepl): N = np.random.randn(nrepl,1) Y = N**2 X = m + (.5*m*m/l)*Y - (.5*m/l)*np.sqrt(4*m*l*Y+m*m*(Y**2)) U = np.random.rand(nrepl,1) ind = U>m/(X+m) X[ind] = m*m/X[ind] return X.ravel() class NIG: '''normal-inverse-Gaussian ''' def __init__(self): pass def simulate(self, th,k,s,ts,nrepl): T = len(ts) DXs = np.zeros((nrepl,T)) for t in range(T): Dt=ts[1]-0 if t>1: Dt=ts[t]-ts[t-1] lfrac = 1/k*(Dt**2) m = Dt DS = IG().simulate(lfrac, m, nrepl) N = np.random.randn(nrepl) DX = s*N*np.sqrt(DS) + th*DS #print DS.shape, DX.shape, DXs.shape DXs[:,t] = DX x = np.cumsum(DXs,1) return x class Heston: '''Heston Stochastic Volatility ''' def __init__(self): pass def simulate(self, m, kappa, eta,lambd,r, ts, nrepl,tratio=1.): T = ts[-1] nobs = len(ts) dt = np.zeros(nobs) #/tratio dt[0] = ts[0]-0 dt[1:] = np.diff(ts) DXs = np.zeros((nrepl,nobs)) dB_1 = np.sqrt(dt) * np.random.randn(nrepl,nobs) dB_2u = np.sqrt(dt) * np.random.randn(nrepl,nobs) dB_2 = r*dB_1 + np.sqrt(1-r**2)*dB_2u vt = eta*np.ones(nrepl) v=[] dXs = np.zeros((nrepl,nobs)) vts = np.zeros((nrepl,nobs)) for t in range(nobs): dv = kappa*(eta-vt)*dt[t]+ lambd*np.sqrt(vt)*dB_2[:,t] dX = m*dt[t] + np.sqrt(vt*dt[t]) * dB_1[:,t] vt = vt + dv vts[:,t] = vt dXs[:,t] = dX x = np.cumsum(dXs,1) return x, vts class CIRSubordinatedBrownian: '''CIR subordinated Brownian Motion ''' def __init__(self): pass def simulate(self, m, kappa, T_dot,lambd,sigma, ts, nrepl): T = ts[-1] nobs = len(ts) dtarr = np.zeros(nobs) #/tratio dtarr[0] = ts[0]-0 dtarr[1:] = np.diff(ts) DXs = np.zeros((nrepl,nobs)) dB = np.sqrt(dtarr) * np.random.randn(nrepl,nobs) yt = 1. dXs = np.zeros((nrepl,nobs)) dtaus = np.zeros((nrepl,nobs)) y = np.zeros((nrepl,nobs)) for t in range(nobs): dt = dtarr[t] dy = kappa*(T_dot-yt)*dt + lambd*np.sqrt(yt)*dB[:,t] yt = np.maximum(yt+dy,1e-10) # keep away from zero ? dtau = np.maximum(yt*dt, 1e-6) dX = np.random.normal(loc=m*dtau, scale=sigma*np.sqrt(dtau)) y[:,t] = yt dtaus[:,t] = dtau dXs[:,t] = dX tau = np.cumsum(dtaus,1) x = np.cumsum(dXs,1) return x, tau, y def schout2contank(a,b,d): th = d*b/np.sqrt(a**2-b**2) k = 1/(d*np.sqrt(a**2-b**2)) s = np.sqrt(d/np.sqrt(a**2-b**2)) return th,k,s if __name__ == '__main__': #Merton Jump Diffusion #^^^^^^^^^^^^^^^^^^^^^ # grid of time values at which the process is evaluated #("0" will be added, too) nobs = 252.#1000 #252. ts = np.linspace(1./nobs, 1., nobs) nrepl=5 # number of simulations mu=.010 # deterministic drift sigma = .020 # Gaussian component lambd = 3.45 *10 # Poisson process arrival rate a=0 # drift of log-jump D=.2 # st.dev of log-jump jd = JumpDiffusionMerton() x = jd.simulate(mu,sigma,lambd,a,D,ts,nrepl) plt.figure() plt.plot(x.T) #Todo plt.title('Merton jump-diffusion') sigma = 0.2 lambd = 3.45 x = jd.simulate(mu,sigma,lambd,a,D,ts,nrepl) plt.figure() plt.plot(x.T) #Todo plt.title('Merton jump-diffusion') #Kou jump diffusion #^^^^^^^^^^^^^^^^^^ mu=.0 # deterministic drift lambd=4.25 # Poisson process arrival rate p=.5 # prob. of up-jump e1=.2 # parameter of up-jump e2=.3 # parameter of down-jump sig=.2 # Gaussian component x = JumpDiffusionKou().simulate(mu,sig,lambd,p,e1,e2,ts,nrepl) plt.figure() plt.plot(x.T) #Todo plt.title('double exponential (Kou jump diffusion)') #variance-gamma #^^^^^^^^^^^^^^ mu = .1 # deterministic drift in subordinated Brownian motion kappa = 1. #10. #1 # inverse for gamma shape parameter sig = 0.5 #.2 # s.dev in subordinated Brownian motion x = VG().simulate(mu,sig,kappa,ts,nrepl) plt.figure() plt.plot(x.T) #Todo plt.title('variance gamma') #normal-inverse-Gaussian #^^^^^^^^^^^^^^^^^^^^^^^ # (Schoutens notation) al = 2.1 be = 0 de = 1 # convert parameters to Cont-Tankov notation th,k,s = schout2contank(al,be,de) x = NIG().simulate(th,k,s,ts,nrepl) plt.figure() plt.plot(x.T) #Todo x-axis plt.title('normal-inverse-Gaussian') #Heston Stochastic Volatility #^^^^^^^^^^^^^^^^^^^^^^^^^^^^ m=.0 kappa = .6 # 2*Kappa*Eta>Lambda^2 eta = .3**2 lambd =.25 r = -.7 T = 20. nobs = 252.*T#1000 #252. tsh = np.linspace(T/nobs, T, nobs) x, vts = Heston().simulate(m,kappa, eta,lambd,r, tsh, nrepl, tratio=20.) plt.figure() plt.plot(x.T) plt.title('Heston Stochastic Volatility') plt.figure() plt.plot(np.sqrt(vts).T) plt.title('Heston Stochastic Volatility - CIR Vol.') plt.figure() plt.subplot(2,1,1) plt.plot(x[0]) plt.title('Heston Stochastic Volatility process') plt.subplot(2,1,2) plt.plot(np.sqrt(vts[0])) plt.title('CIR Volatility') #CIR subordinated Brownian #^^^^^^^^^^^^^^^^^^^^^^^^^ m=.1 sigma=.4 kappa=.6 # 2*Kappa*T_dot>Lambda^2 T_dot=1 lambd=1 #T=252*10 #dt=1/252 #nrepl=2 T = 10. nobs = 252.*T#1000 #252. tsh = np.linspace(T/nobs, T, nobs) x, tau, y = CIRSubordinatedBrownian().simulate(m, kappa, T_dot,lambd,sigma, tsh, nrepl) plt.figure() plt.plot(tsh, x.T) plt.title('CIRSubordinatedBrownian process') plt.figure() plt.plot(tsh, y.T) plt.title('CIRSubordinatedBrownian - CIR') plt.figure() plt.plot(tsh, tau.T) plt.title('CIRSubordinatedBrownian - stochastic time ') plt.figure() plt.subplot(2,1,1) plt.plot(tsh, x[0]) plt.title('CIRSubordinatedBrownian process') plt.subplot(2,1,2) plt.plot(tsh, y[0], label='CIR') plt.plot(tsh, tau[0], label='stoch. time') plt.legend(loc='upper left') plt.title('CIRSubordinatedBrownian') #plt.show()