The Implementation of Models with Jumps and
5.1 Covariance Estimation and Testing on the Ex- Ex-istence of Jumps
We start explaining the concept of realized volatility independently intro-duced by Andersen et al. [2]; Barndorff-Nielsen and Shephard [9]; Comte and Renault [34]. Multivariate generalizations (realized co-variation) are given e.g. inBarndorff-Nielsen and Shephard [10]; An-dersen et al. [4] containing asymptotic theory on correlations, regres-sions, and an analysis how covariances may change over time. A good sur-vey paper is Andersen et al. [3]. A partial generalization of the one-dimensional case, the so-called bipower variation, is given in Barndorff-Nielsen and Shephard [11]. Roughly speaking, realized volatility con-tains jumps and the remaining continuous elements whereas bipower only contains the continuous part, so that the difference describes the jump part.
151
Hence, one can test upon jumps, seeBarndorff-Nielsen and Shephard [12]. The whole concept is in fact quite difficult as time series are observed on a discrete basis. A finer and finer time grid is necessary. We therefore assume that the underlying stochastic processS is an exponential L´evy pro-cess of finite variation, i.e. logS is a Brownian motion plus a compound Poisson process.
This assumption further implies that the sample moments of the returns satisfy a certain scaling property (see Cont and Tankov [32, Equation 7.52]), a singularity spectrum can be estimated to find out more about the actual continuous/jump relation of return time series, see Cont and Tankov [32, Chapter 7]. Actually this study shows that we are not at the end of the story by either using a geometric Brownian motion or jump-diffusions. Both singularity spectra do not exactly meet the form of the empirical estimate, seeCont and Tankov [32, Section 7.6.2].
Nevertheless, jump-diffusions seem to be the better alternative. This is also confirmed in several of the above articles. Jumps are e.g. often caused by macroeconomic announcements. We therefore continue with a definition of realized covariance.
Remark 5.1.1. In this section, we describe a method to separate variation caused by the continuous and the jump parts of a time series. Moreover, we allow for a certain time dependence that is supposed to be variable on a short time basis which might help to be quick off the mark in daily trading. Clearly, both requests are possible only, if data are observed in very short time intervals to approximate time-continuous observations. The right choice are high-frequency data, i.e. the stock price is observed every time it is traded, optimally we receive tick by tick data. However, on a long term basis, it might be wise to sit out short time volatility in the market. Then it is not important to observe changes based on a daily basis. A reliable and robust estimation on a longer time horizon is preferred. In particular, this is crucial when optimizing the risk-return relation in a credit portfolio. We then rely on our estimation on discrete time series techniques. A famous model is the so-called multivariate GARCH (MGARCH). A high developed literature can be found for all kinds of equity data, see [18; 19; 39; 43; 44; 45; 55; 70; 107].
A good survey is presented inKroner and Ng[83]. A stochastic extension is given by (stochastic) factor models, see e.g.Chib et al. [29]. Moreover, we think for an analysis of credit portfolio management, actual default events on a yearly basis are more natural to be looked at. An extensive treatment is described inBluhm and Overbeck[16] and Niethammer[104, 105].
Realized Covariation
Let ˘X := logS(1), ˘Y := logS(2) be semimartingales and pl be any sequence of random partitions tending to identity (0 = τ0l ≤ τ1l ≤ ... ≤ τkl
l and τil stopping times), then, see e.g.Jacod and Shiryaev[68];Protter [110]:
X˘0Y˘0+X where ˘Xc denotes the continuous and ˘Xd the discontinuous part of ˘X. Sup-pose that ˘X and ˘Y are discretely measured at the same time points based on intervals of time of lengthδ >0.We obtain a discretized version ˘Xδ/Y˘δ, of ˘X and ˘Y , defined as ˘Xδ⌊t
δ⌋/Y˘δ⌊t
δ⌋. [⌊x⌋ denotes the Gauß-functional, i.e.
the integer part ofx.] The realized covariance attis then defined as RCtδ( ˘X,Y˘) = As a special case in the Brownian model,dlogS=σdW+µdt,we obtain
RCtδ→P Z t
t−1
σsσs′ds.
So from the dataSδj,j= 1, ...,⌊Tδ⌋, we get a new data setRCtδ,t= 1, ..., T. Moreover, if we set ˘X= ˘Y, we obtain an estimator for the realized volatility of ˘X. So in the one-dimensional case, we can perform a regression, e.g. an ARFIMA(p, d, q), see Andersen et al. [4]:
φ(L)(1−L)d( q
RCtδ−µ) =ψ(L)ut, ut∼ N(0, σ2)
The method becomes quite difficult in a multivariate setting or if jumps are taking into account. Jumps are discussed in the next subsection by filtering out jumps by the bipower variation. In the multivariate setting we face an additional problem; different price processes are non-synchronously traded, see e.g. Hayashi and Yoshida [64]. Moreover, apart from the fact that high frequency data are still discrete, an adjustment with respect to the given market micro structure has to be made, i.e. a bid-and-ask spread has to be smoothed out, see e.g. Hansen et al. [63]. A forecasting model also relying on fractionally integrated ARMA processes is given byChiriac and Voev [30].
We next try to separate jumps from the remaining process:
Bipower Variation
We already know that the realized volatility converges to [ ˘X]. We next describe a measure - bipower variation - that converges to [ ˘Xc] so that the difference with RC describes the jumps. A test on the presence on jumps is then designed by considering the hypothesis [ ˘X] = [ ˘Xc]. Under this hypothesis it is implied thatP
(∆ ˘X)2= 0.The next considerations rely on Barndorff-Nielsen and Shephard [12].
The 1,1-order bipower variation process (BPV){X˘}[1,1]t is defined as the probability limit of
⌊δt⌋
X
j=2
|X˘jδ−X˘(j−1)δ||X˘(j−1)δ−X˘(j−2)δ|, δ↓0, (5.1)
provided the limit exists. Under our assumption thatdX˘t=dlogSt=µdt+ σsdW +Jt, where Jt is a compound Poisson process and σ is independent ofW, we have
{X˘}[1,1]t = (E|u|)2[ ˘Xc]t, where u ∼ N(0,1) such that E|u| = √√2
π. Hence, [ ˘Xd]t can be consistently estimated by
[ ˘Xδ]t−(E|u|)−2{X˘δ}[1,1]t , where{X˘δ}[1,1]t is defined as the realized BPV:
{X˘δ}[1,1]t =
⌊δt⌋
X
j=2
|X˘jδ−X˘(j−1)δ||X˘(j−1)δ−X˘(j−2)δ|.
Barndorff-Nielsen and Shephard [12] further derive the distribution of [ ˘Yδ]−(E|u|)−2{X˘δ}[1,1]t and an adjusted feasible statistic to test on [ ˘X] = [ ˘Xc],i.e. no jumps.
The last lines were supposed to give an idea what can be done to es-timate covariance, jumps, and to separate them from the diffusion compo-nent. We refer for further details toBarndorff-Nielsen and Shephard [12];Cont and Tankov [32]. We next assume that the necessary param-eters are already estimated and given:
5.2 Comparison: Brownian vs. L´ evy Model
In this section, we illustrate our results about the optimal exponential port-folio and the convergence of thep-th problem. We start considering a single stock, i.e. N = 1. In the jump component, we consider a Poisson process and vary the intensityγ and the jump sizea. The optimal portfolio (number
of stocks held) w.r.t. an exponential utility function with α = 1 has been proven to be−θeSt−1,where θe is the solution of
0 =θeσ2+ ((ea−1)eθe(ea−1)γ+ (µ−r), (5.2) see e.g. Section 4.5. If a is tending to minus infinity, we observe that θe = −(µ−σ2r−γ). We will only consider −θe. It stays constant over time and represents the optimal amount we invest into the stock. −θe will be therefore called optimal exponential portfolio throughout this section.
0 0.2
0.4 0.6
0.8
1 −7 −6 −5 −4 −3 −2 −1 0 1
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2 2.5
jump size Optimal Exponential Portfolio: Continuous vs. Jump Process,
µ=0.1, σ =0.3
γ
ϑ(t)*S(t−)=−θe vs. ϑ(t)*S(t)=µ/σ2= 1.1111
continuous jump
Figure 5.1: Influence of jump size and intensity on the optimal portfolio Figure 5.1 shows the influence of jump size and intensity upon the op-timal exponential portfolio (−θe) in comparison to the optimal exponential portfolio if no jumps are taken into account (µ−r)/σ2. In detail, we set r= 0, µ= 0.1, σ= 0.3. A numerical solver of Equation 5.2 then yields op-timal exponential portfolios in a model with jumps (−θe, optimal invested amount into the stock) for different jump sizes a and different intensities γ (the black/grey graph). The white graph, a hyperplane parallel to the hyperplane with θe = 0, sets γ = 0 and therefore represents the constant optimal invested amount µ/σ2,if we only consider the geometric Brownian
motion of the jump diffusion. The difference of both graphs thus describes the importance of jumps for the optimal portfolio. Clearly, if the jump height is negative, the optimal portfolio with jumps is less than the portfo-lio if jumps are not considered and vice versa. We next analyze the portfoportfo-lio when jumps are taken into account. We observe that a low constanta, say -2, can be already interpreted as a default. The portfolio with jumps is almost equal to−θe= (µ−r−γ)σ2 . Furthermore, if the intensity is larger than the drift adjusted by the constant interest rate, we try to sell the stock, i.e.
if the intensity is too large relative to the obtained return µ, shortselling is suggested. We obtain a negative optimal portfolio. If the jump size is smaller than a small negative constant, the intensityγ almost solely drives the portfolio. In particular, if the jump size becomes very negative, i.e. sim-ulating a sudden jump to default, the portfolio tends to be equal to µ−σr2−γ. Nevertheless, a stock is always positive, also after a large negative jump.
The stock has still got the chance to recover to an arbitrary high value. A real default corresponds to the case a= −∞ which is not included in the model. In practice however, our Brownian-Poisson- model presents a good approximation. It is not unrealistic to assume that the company might fully recover with a certain small probability after a default. In particular, as in practice it is well-known that definitions of default can vary a lot (failure to pay, restructuring, bankruptcy).
Jumps and variation in the Brownian motion drive the variation of the whole process. Figure 5.2 therefore shows the influence of changes in the intensity and the variance of the Brownian part upon the optimal portfolio while the total variation stays constant. We setµ= 0.1,a=−4, r= 0,and vary γ and σ such that V = σ2+ (ea−1)2γ stays constant. We plot the optimal portfolio for three different values ofV and analyze the changes: if the intensity is lower thanµ, the portfolio is positive. Increasing the intensity causes a lower investment in the stock. A positive portfolio is approximately only obtained ifµ > γ. σ2 only steers the scaling but does not change the sign. That is not surprising as we face an approximate “default scenario”
becausea=−4.A shift of the total variation V, 20% up and down, results in an almost parallel shift.
Summarizing, the graph (Figure 5.2) shows that the portfolio varies a lot dependent on the driver of variation, i.e. caused by a high variation in the diffusion (volatility σ) or the jump part (intensityγ). The separation of jumps and diffusion discussed in Section 5.1 is therefore important.
Finally, we illustrate the convergence of the Girsanov parameters (θq→ θe), in a two and a three-dimensional setting in Figure 5.3 and 5.4. We numerically calculateθq for a reasonable grid ofq in the interval (1,1.5) for multidimensional exponential Poisson processes with a Brownian component (by solving (4.2)). θeis derived as above. We plotq against the components of θq and (1, θe) in the same figure. The used intensity and jump size is given in the figures. We impressively observe convergence. Moreover, the
0 0.1 0.2 0.3 0.4 0.5 0
0.1 0.2
0.3 0.4
0.5
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4
σ2
change in portfolio with fixed total variance V V=σ2+(ea−1)2 γ , a=jump size= −4
γ
−θ e = optimal portfolio
V=0.4 V=0.4*1.2=0.48 V=0.4*0.8
Figure 5.2: Decomposition of the variance of a stock
obtained optimal portfolios give an intuition how the portfolio changes in a three-dimensional case when jumps are taken into account.
5.3 Concluding Remarks
This chapter explains a method to estimate the parameters of the processes considered in this thesis, i.e. the covariance matrix and the jump measure.
Moreover, we illustrate the results of this thesis.
We just mention one estimation method to give an idea how our mod-els could be implemented. We explain realized covariation/volatility and bipower variation. In the one-dimensional case, the difference of both num-bers indicates if there are jumps in the data. There are many more methods, we refer to the existing literature, see e.g. [32] for a good overview.
In the second part of the chapter, we analyze the influence of the model parameters on the optimal exponential portfolio. We observe that the de-composition of the total variation into jumps and variance is very important for the optimal exponential portfolio. Moreover, our graphs show that also
a credit risk example can be included in our setting. We conclude, that we should invest only, if the drift minus the default intensity and the riskless return is positive.
Finally, we illustrate convergence of the Girsanov parametersθq to θe in a two and three-dimensional setting.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
−0.4677
−0.4676
−0.4676
−0.4676
−0.4676
−0.4676
θ1q
Multidimensional Case:
Probable small positive jump θ1q, unprobable high negative jump θq2
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
−0.0961
−0.0961
−0.0961
−0.096
−0.096
θ2q
jump size S1=0.25, jump size S2=−0.75 σ=(0.3 0.1 0.12, 0.1 0.4 0.15 ) µ=[0.05; 0.06]; γ=(0.05,0.0005)
q
q e
q e
Figure 5.3: Convergence of the portfolio in a model with two stocks
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
−0.32
−0.31
−0.3
−0.29
θ1 q
Return vs. Negative Shocks
q e
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
−0.83
−0.82
−0.81
−0.8
−0.79
θ2 q
jump size S
1=−0.01, jump size S
2=−1, jump size S
3=0.01 σ=(0.3 0.1 0.12 0.11, 0.1 0.5 0.15 0.12, 0.1 0.5 0.6 0.13 )
µ=[0.07; 0.12; 0.05]; γ1=1 γ2=0.01
Optimal portfolio continuous vs. Lévy processes
ϑcexp=( 0.5236 1.0203 −0.6596) vs. ϑdexp=(0.2938 0.8032 −0.4749)
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0.47 0.48 0.49 0.5
q θ3q
Figure 5.4: Convergence of the portfolio in a model with three stocks
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