------------------------------------------------------ # # # Chapter 11: Simulated Likelihood Inference # # for Stochastic Volatility Models # # # ------------------------------------------------------ Consider the model S(t_k) = S(t_{k-1}) * [ 1 + vol(t_{k-1}) * phi_k ] (1) or ret(t_k) = vol(t_{k-1}) * phi_k (2) with a general stochastic volatility process vol_k = vol_k( vol_{k-1} , eps_k ) (3) where the (phi_k,eps_k) are mutually independent standard normal distributed random numbers. Let's abbreviate r_k := ret(t_k) v_k := vol(t_k) and assume that the time series under consideration has length N+1 (instead of N). Then the likelihood function for the model (1-3) is given by L = int_{R^N} prod_{k=1}^N 1/sqrt{2pi} 1/v_k * prod_{k=1}^N exp( -1/2*r_{k+1}^2/v_k^2 ) * prod_{k=1}^N 1/sqrt{2pi} exp( -1/2*eps_k^2 ) deps_k (4) where in the expression above v_k is a function of eps_k and v_{k-1}, v_k = v_k( eps_k , v_{k-1} ) (5) For example, for the GARCH-Diffusion model we had v_k^2 = v_{k-1}^2 + kappa * [ bsvol^2 - v_{k-1}^2 ] + beta * v_{k-1}^2 * eps_k (6) If we look at the expression (4), we see that it con- tains a factor prod_{k=1}^N 1/sqrt{2pi} 1/v_k (7) Thus, if we would choose v_k^2 = sigma^2 * eps_k^2 (8) this factor looks somehow similar to the quantities pi_N := phi_1^2 * ... * phi_N^2 (9) which we considered in Uebungsblatt 1-3 where we saw that we can't calculate these quantities with simple Monte Carlo. That is, for N = 2500, the following simple Monte Carlo sum does not converge for any reasonable M <= 10^6 or even M <= 10^9 simulations: mcsum = 1/M sum_{k=1}^M phi_{1,k}^2 * ... * phi_{N,k}^2 (10) Now, for the pi_N's we know that all factors phi_i^2 are independent and we can calculate E[pi_N] = E[phi_1^2] * ... * E[phi_N^2] (11) = 1 . The Monte Carlo analog of the calculation (11) would be [1/M sum_{k_1} phi_{1,k_1}^2 ] * ... * [1/M sum_{k_N} phi_{1,k_N}^2 ] = 1/M^N sum_{k_1,...,k_N} phi_{1,k_1}^2 * ... * phi_{1,k_N}^2 (12) Observe that in (10) and (12) the same set of random numbers can be used, namely N*M phi's. However, not too surprisingly, the calculation (12) has much better conver- gence properties. Let's make a quick check: #--------------------------------------------------------- # warmup excercise: improved MC-Calculation of the pi_N's: N = 2500 # length of time series M = 40000 # number of MC simulations phi = matrix(rnorm(N*M),nrow=M,ncol=N) # 100 Mio random numbers! # old method: piN = apply(phi*phi,1,cumprod) piN = t(piN) EpiN = colMeans(piN) par(mfrow=c(1,2)) plot(EpiN,ylim=c(0,3)) plot(log(EpiN)) # new method: EpiN = rep(0,N) EpiN[1] = mcsum = sum(phi[,1]^2)/M for(k in 2:N) { mcsum = sum(phi[,k]^2)/M EpiN[k] = EpiN[k-1]*mcsum } par(mfrow=c(1,2)) plot(EpiN,ylim=c(0,3)) plot(log(EpiN),ylim=c(-1,1)) # # end warmup excercise #--------------------------------------------------------- THUS: It is critical to perform an MC-sum at every factor, at every probability density factor for the r_k's, factor := 1/sqrt{2pi*v_k^2} * exp( -1/2*r_{k+1}^2/v_k^2 ) (13) and not just one sum at the very end. HOWEVER: For a stochvol model, it is not clear how to do this, since the factors (13) in the likelihood function (4) are not completely independent, the vol-function v_k depends, besides on the random number eps_k, also on the vol v_{k-1} of the previous factor, and this vol depends on the eps_{k-1}.. The following algorithm, which is taken from the very nice and very transparent publication of Pitt, Malik and Doucet: "Simu- lated Likelihood Inference for Stochastic Volatility Models" of 2014, is quite performant and pretty general, such that it can be applied to a large class of models. To be specific, we implement it for the GARCH-Diffusion model (6): logL_withComments = function( bsvol , w0 , d ) { N = length(ret) # length of time series M = length(eps[,1]) # number of MC-simus; the M x N matrix eps # will be precalculated, such that, when # varying model parameters, always the same # random numbers will be taken. w = 1 - 1/d kappa = (1-w)*w0 beta = (1-w)*(1-w0)*sqrt(2) vol0sq = bsvol^2 # we put start-vol = mean-vol, vol0 = volbar vol = rep(0,N) logL = 0 volsq_withinfo = rep( vol0sq , M ) # with return information volsq_withoutinfo = rep( vol0sq , M ) # without return information for( k in 1:(N-1) ) # same loop as in new method of warmup exercise above: { # for each factor_k we perform an MC-sum # GARCH-Diffusion vol-specification: v_k^2 =: volsq_k => # # volsq_k = volsq_{k-1} + kappa * [ vol0sq - volsq_{k-1} ] # + beta * volsq_{k-1} * eps_k # # this is a vector of length M = number MC-simus: # [k] [k-1] [k-1] volsq_withoutinfo = volsq_withinfo + kappa * ( vol0sq - volsq_withinfo ) + beta * volsq_withinfo * eps[,k] volsq_withoutinfo = abs( volsq_withoutinfo ) # the probability for the occurence of a given return ret_{k+1}, given # the value of the volatility v_k of the previous day, is given by the # probability density which we simply define as a quantity factor: # # factor := 1/sqrt{2pi*v_k^2} * exp( -1/2*r_{k+1}^2/v_k^2 ) # # this is also a vector of length M = number MC-simus: factor = dnorm( ret[k+1] , mean=0 , sd=sqrt(volsq_withoutinfo) ) # factor = prob( r_{k+1} | v_k ) # v_k without return information, but with eps_k and with v_{k-1} information # to make the algorithm more transparent, we now use two different names, # mcsum and normalization, for nearly the same quantity; the reason for this # will become more clear below where we define the quantity prob_for_vol: mcsum_factor = sum( factor ) / M # this is a sum over MC-simus and this # goes directly into the calculation of logL normalization = sum( factor ) # a normalization factor for the probabilities # prob( v_k | r_{k+1} ), defined below # the actual calculation of the log-Likelihood-function: logL = logL + log( mcsum_factor ) # now we have to calculate the v_k's, the volsq_withinfo, which we will # use for the calculation of the v_{k+1}'s in the GARCH_Diffusion equation # at the very top: # instead of considering the factor as a probability that a given return # ret_{k+1} realizes, that is, factor = prob( r_{k+1} | v_k ), we consider # it as a probability that, given this return ret_{k+1}, the squared vola- # tility vol(t_k)^2 will be equal to v_k^2 = volsq_withinfo : prob_for_vol = factor / normalization # prob_for_vol = prob( v_k | r_{k+1} ) # now, to take these probabilities into account, the v_k's which are used # for v_{k+1} = v_{k+1}( v_k , eps_{k+1} ) are simply drawn from the v_k's # without return information, the quantities volsq_withoutinfo, where each # volatility volsq_withoutinfo[j] in that vector comes with a probability # prob_for_vol[j]. Fortunately, R has the built-in function sample() which # does exactly this: volsq_withinfo = sample( volsq_withoutinfo, M , replace=TRUE , prob_for_vol ) # [k] # actually the algorithm does not only estimate the log-Likelihood function, # but it can also estimate the whole vol-process itself: we do this purely # for information and checking purposes, the quantity is not needed # in the algorithm: mcsum_volsq = sum( volsq_withinfo ) / M vol[k] = sqrt( mcsum_volsq ) }#next k result = list( logL , vol ) names(result) = c( "logL" , "vol" ) return(result) } # This completes the algorithm. Let's copy and paste the code without all the # comments to see that it is actually quite short: logL_withVol = function( bsvol , w0 , d ) { N = length(ret) M = length(eps[,1]) w = 1 - 1/d kappa = (1-w)*w0 beta = (1-w)*(1-w0)*sqrt(2) vol0sq = bsvol^2 vol = rep(0,N) logL = 0 volsq_withinfo = rep( vol0sq , M ) volsq_withoutinfo = rep( vol0sq , M ) for( k in 1:(N-1) ) { # GARCH-Diffusion: volsq_withoutinfo = abs( volsq_withinfo + kappa * ( vol0sq - volsq_withinfo ) + beta * volsq_withinfo * eps[,k] ) factor = dnorm( ret[k+1] , mean=0 , sd=sqrt(volsq_withoutinfo) ) normalization = sum( factor ) mcsum_factor = normalization / M logL = logL + log( mcsum_factor ) prob_for_vol = factor / normalization volsq_withinfo = sample( volsq_withoutinfo, M , replace=TRUE , prob_for_vol ) mcsum_volsq = sum( volsq_withinfo ) / M vol[k] = sqrt( mcsum_volsq ) }#next k result = list( logL , vol ) names(result) = c( "logL" , "vol" ) return(result) } # we check the algorithm on an artificially generated return set with # GARCH-Diffusion dynamics. We modify the GenerateStochVolPaths()-function # of Loesung9.R: GenerateGARCHDiffusionPath = function( N , bsvol , w0 , d ) { S0 = 100 S = rep(0,N) vol = rep(0,N) ret = rep(0,N) nu = rep(0,N) # nu := vol^2 phi = rnorm(N) eps = rnorm(N) w = 1 - 1/d kappa = (1-w)*w0 beta = (1-w)*(1-w0)*sqrt(2) vol0sq = bsvol^2 # the first step, as usual, is special: S[1] = S0 * ( 1 + bsvol * phi[1] ) ret[1] = (S[1] - S0)/S0 nu[1] = bsvol^2 + kappa*0 + beta*bsvol^2*eps[1] vol[1] = sqrt(abs(nu[1])) # now the actual simulation: for(k in 2:N) { S[k] = S[k-1] * ( 1 + vol[k-1] * phi[k] ) ret[k] = (S[k]-S[k-1])/S[k-1] # GARCH-Diffusion: nu[k] = nu[k-1] + kappa*(bsvol^2-nu[k-1]) + beta*nu[k-1]*eps[k] vol[k] = sqrt(abs(nu[k])) } par(mfrow=c(1,2)) plot(S,type="l",ylim=c(0,300),main="GARCH-Diffusion Preis") plot(vol,type="l",ylim=c(-0.02,0.06),main="GARCH-Diffusion Vol") abline(a=bsvol,b=0,col=2) result = list( S , vol , ret ) names(result) = c("S","vol","ret") return(result) } res = GenerateGARCHDiffusionPath( N=2500 , bsvol=1.5/100 , w0=0.15 , d=10 ) ret = res$ret plot(ret,type="l") # Now, let's try to recover the model parameters # bsvol=1.5/100 , w0=0.15 , d=10 # from the return data set: M = 500 N = length(ret) eps = matrix( rnorm(N*M) , ncol=N , nrow=M) bsvol = sd(ret) bsvol # first let's see whether we get a number at all: resL = logL_withVol( bsvol , w0=0.15 , d=10 ) resL$logL # ok, looks good # let's take a look at the estimated vol and compare to the exact vol # (of course, an exact vol we only have for artificially generated data; # for DAX or SPX we only have the return data ret): plot(res$vol,type="l",ylim=c(0,0.06)) lines(resL$vol,col="red") # quite okay! # to maximize logL, we don't need the estimated vols, so we just skip them # from the code. Since the sample()-function uses random numbers, we set a # seed at the beginning to make sure that when maximizing logL always the # same random numbers are used: logL = function( bsvol , w0 , d , seed=123 ) { set.seed(seed) N = length(ret) M = length(eps[,1]) w = 1 - 1/d kappa = (1-w)*w0 beta = (1-w)*(1-w0)*sqrt(2) vol0sq = bsvol^2 vol = rep(0,N) logL = 0 volsq_withinfo = rep( vol0sq , M ) volsq_withoutinfo = rep( vol0sq , M ) for( k in 1:(N-1) ) { volsq_withoutinfo = abs( volsq_withinfo + kappa * ( vol0sq - volsq_withinfo ) + beta * volsq_withinfo * eps[,k] ) factor = dnorm( ret[k+1] , mean=0 , sd=sqrt(volsq_withoutinfo) ) normalization = sum( factor ) mcsum_factor = normalization / M logL = logL + log( mcsum_factor ) prob_for_vol = factor / normalization volsq_withinfo = sample( volsq_withoutinfo, M , replace=TRUE , prob_for_vol ) } return(logL) } logL( bsvol=sd(ret), w0=0.15 , d=10 ) # we make a contour-plot in the (w0,d)-plane: # takes about 2:40 min for M=500, N=2500: w0 = seq(from=0.05,to=0.5,by=0.05) dd = 1:40 nw = length(w0) nd = length(dd) z = matrix( 0 , nrow=nd , ncol=nw ) for(i in 1:nd) { for(j in 1:nw ) { z[i,j] = logL( bsvol=sd(ret) , w0[j] , dd[i] ) } } # let's look at the result: contour( dd , w0 , z ) contour( dd , w0 , z , nlevels = 50 ) contour( dd , w0 , z , nlevels = 100 ,zlim=c(7300,7500)) # maximum close to w0 = 0.15, d=10 !! # we are able to recover model parameters! # more refined: # takes about 2:40 min for M=1000, N=2500: (half the number of points, # double number of MC-simus) w0 = seq(from=0.025,to=0.25,by=0.025) dd = 1:20 nw = length(w0) nd = length(dd) M = 1000 N = length(ret) eps = matrix( rnorm(N*M) , ncol=N , nrow=M) z = matrix( 0 , nrow=nd , ncol=nw ) for(i in 1:nd) { for(j in 1:nw ) { z[i,j] = logL( bsvol=sd(ret) , w0[j] , dd[i] ) } } contour( dd , w0 , z , nlevels = 100 ,zlim=c(7300,7500)) contour( dd , w0 , z , nlevels = 100 ,zlim=c(7400,7500)) max(z) # 7441.77 logL( sd(ret) , w0=0.15 , d=10 ) # 7438.83 logL( sd(ret) , w0=0.2 , d=10 ) # 7441.77 # okay, w0 a bit larger..

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