--------------------------------------------------- # # # Chapter 4: The ARCH Time Series Model # # # --------------------------------------------------- In the last chapter we simulated the model S(t_k) = S(t_{k-1}) * [ 1 + vol(t_{k-1})*phi_k ] (1) with vol^2(t_{k-1}) = stddev_d^2(t_{k-1}) = 1/d * [ret^2(t_{k-1})+...+ret^2(t_{k-d})] (2) and we saw, that the volatility in that model converged to zero, the simulated paths stayed constant after a while. Apparently vol(t_{k-1}) = 0 and S(t_k) = const is a solution of the equations (1,2) for arbitrary random numbers phi_k. On the other hand, we saw that the model S(t_k) = S(t_{k-1}) * [ 1 + bsvol * phi_k ] (3) with a constant Black-Scholes volatility bsvol does not have this problem, the simulated paths looked quite reasonable. However, the data analysis of chapter 2 clearly revealed that volatility is not constant and a good model should take this into account. Therefore, as a compromise between the models (1,2) and (3), we make the Ansatz: vol^2(t_{k-1}) = w0*bsvol^2 + w1*stddev_d(t_{k-1})^2 (4) = w0 * bsvol^2 + w1 * 1/d * [ret^2(t_{k-1})+...+ret^2(t_{k-d})] with some positive weights w0 and w1 with w0 + w1 = 1 . The price dynamics (1) with vol-specification (4) is called the ARCH-model or more precisely the ARCH(d)- model, since the history of the past d days is taken into account. Before we simulate a couple of paths, let's make some remarks: Remark 1) The letters A, R, C and H stand for AR = AutoRegressive, that means, future returns depend on past returns, and: CH = Conditional Heteroskedastic, that means, that the current, instantaneous volatilities are time dependent. There are conditional or instantaneous variances or volatilities, vol^2(t_{k-1}) = E[ ret^2(t_k) | t_{k-1} ] -- the notation on the right meaning that until time t_{k-1} everything is known and the expectation is taken only over phi_k -- and unconditional variances or volatilities, volbar^2 = E[ ret^2(t_k) ] where the expectation is taken over all phi_k,...,phi_1. Heteroskedastic simply means: not constant, but time dependent. Remark 2) Instead of vol-specification (4) one could make the more general Ansatz vol^2(t_{k-1}) = alpha0 + alpha1 * ret^2(t_{k-1}) + ... + alphad * ret^2(t_{k-d}) (5) with some d+1 parameters alpha0,...,alphad. Typically the specification (5) can be found in the books as the definition of the ARCH(d)-model. However, if one really wants to fit the model to actual time series data, one would merely choose a parametrization (4), eventually not necessasarily with equally weighted returns, but with linearly or exponentially weighted returns. The GARCH-model discussed in chapter 7 is basically an ARCH(d)-model with exponentially weighted returns. In addition, the parameters w0 and bsvol in (4) have an immediate intuitive meaning: as we will see in later chapters when we estimate the parameters with the maximum likelihood method, bsvol is usually very close to the standard deviation of the realized returns calculated over the full length of the time series. Thus, we don't have to estimate this parameter actually, but can calculate it directly as sd(ret), the standard deviation of the return series. Remark 3) In 2003, Robert Engle from New York Uni- versity (together with Clive Granger) was awarded the Nobel Prize in Economics "for methods of analy- zing economic time series with time-varying vola- tility (ARCH)". Now let's start and simply take a look at some simulated paths. To this end, we modify our pathNaivArch-function from the last chapter: # # Start R-Session: # pathArchd = function( N , bsvol , w0 , d ) { S = rep(0,N) ret = rep(0,N) phi = rnorm(N) S0 = 100 vol0 = bsvol # we put startvol to bsvol vol = rep(vol0,N) # ARCH-vol sumretd = 0 # the first step, k=1, is special: S[1] = S0*(1+vol0*phi[1]) ret[1] = (S[1]-S0)/S0 vol[1] = sqrt( w0*bsvol^2 + (1-w0)*ret[1]^2 ) sumretd = sumretd + ret[1]^2 for(k in 2:N) { if(k<=d) { S[k] = S[k-1]*(1+vol[k-1]*phi[k]) ret[k] = (S[k]-S[k-1])/S[k-1] sumretd = sumretd + ret[k]^2 vol[k] = sqrt( w0*bsvol^2 + (1-w0)*sumretd/k ) } else { S[k] = S[k-1]*(1+vol[k-1]*phi[k]) ret[k] = (S[k]-S[k-1])/S[k-1] sumretd = sumretd + ret[k]^2 sumretd = sumretd - ret[k-d]^2 vol[k] = sqrt( w0*bsvol^2 + (1-w0)*abs(sumretd)/d ) } }#next k par(mfrow=c(2,1)) plot(S,type="l",ylim=c(0,250)) plot(ret,type="l",ylim=c(-0.08,0.08)) result = list(S,ret) names(result) = c("S","ret") return(result) } # # let's check: # # this should be Black-Scholes: res = pathArchd( N=2500 , bsvol=0.01 , w0=1 , d=15 ) # returns are uniformly distributed:

# this is with variable vol: res = pathArchd( N=2500 , bsvol=0.01 , w0=0.5 , d=15 ) # vol looks still quite uniform:

res = pathArchd( N=2500 , bsvol=0.01 , w0=0.1 , d=15 ) # this shows vol clustering:

res = pathArchd( N=2500 , bsvol=0.01 , w0=0.1 , d=10 ) # this shows vol clustering:

res = pathArchd( N=2500 , bsvol=0.01 , w0=0.1 , d=250 ) # for large d, returns become more uniform:

res = pathArchd( N=2500 , bsvol=0.01 , w0=0.1 , d=5 ) # this shows vol clustering:

res = pathArchd( N=2500 , bsvol=0.01 ,w0=0.1 , d=1 )

res = pathArchd( N=2500 , bsvol=0.01 ,w0=0.01 , d=1 )

res = pathArchd( N=2500 , bsvol=0.01 ,w0=0 , d=1 ) # w0 > 0 necessary to stabilizes the model:

res = pathArchd( N=2500 , bsvol=0.01 ,w0=0 , d=15 ) # w0 > 0 necessary to stabilizes the model:

# a realistic set of parameters: res = pathArchd( N=2500 , bsvol=0.015 ,w0=0.15, d=15 )

This looks not unreasonable. Now we can start to fit the model to actual time series data. This we will do in the next chapter with the maximum likelihood method.

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