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Figure 8.1: VaR Term Structures using NGARCH and Monte Carlo Simulation

Figure 8.1: VaR Term Structures using NGARCH and Monte Carlo Simulation

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Monte Carlo Simulation



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• In Figure 8.1, the VaR is simulated using Monte Carlo on

an NGARCH model

• We use MCS to construct VaR per day as a function of

horizon K for two different values of σt+1

• In the left panel the initial volatility is one-half the

unconditional level and in the right panel σt+1 is three

times the unconditional level.

• The horizon goes from 1 to 500 trading days

• The VaR coverage level p is set to 1%

Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen



Monte Carlo Simulation



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Figure 8.1 shows that the term structure of VaR is initially

upward sloping both when volatility is low and when it is

high

• The VaR term structure is also driven by the term structure

of skewness and kurtosis and other moments

• Kurtosis is strongly increasing at short horizons and then

decreasing for longer horizons

• This hump-shape in the term structure of kurtosis creates

the hump in the VaR as seen in the right panel of Figure

8.1 when the initial volatility is high

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



Figure 8.2: ES Term Structures using NGARCH and

Monte Carlo Simulation



Notes to Figure: The left panel shows the S&P 500 ES per day

across horizons when the current volatility is one half its long run

value. The right panel assumes the current volatility is 3 times its

long run value.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Monte Carlo Simulation



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• In Figure 8.2, the ES is simulated using Monte Carlo on an

NGARCH model

• Here we plot the ESPt+1:t+K per day, against horizon K

• The coverage level p is again set to 1% and the horizon

goes from 1 to 500 trading days

• Note that the slope of the ES term structure in the left panel

of Figure 8.2 is steeper than the corresponding VaR term

structure in the left panel of Figure 8.1

• The hump in the ES term structure in the right panel of

Figure 8.2 is more pronounced than the hump in the VaR

term structure in the right panel of Figure 8.1

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen



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Figure 8.1: VaR Term Structures using NGARCH and Monte Carlo Simulation

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