Interest-rate simulation under the real-world measure within a Gaussian HJM framework

Figures & data

Table 1. First market price of risk ϕ₁ with respect to the observable trend and average shape of the initial forward rate curve.

		Observable trend
Market price of risk, ϕ1		Falling	Stable	Rising
Shape of forward rate curve	Upward sloping	Negative	Negative	Near zero
	Flat	Negative	Near zero	Positive
	Downward sloping	Near zero	Positive	Positive

Note: It is assumed that volatility is lower than observable trend so that .

Table 2. Market prices of risks ϕ₁ and ϕ₂ with respect to the observable trend.

		Observable trend
		Bull steep	Steep	Bear steep	Bull flat	Flat	Bear flat
Market price of risk	ϕ1	Negative	Negative	Near zero	Negative	Negative	Near zero
	ϕ2	Negative	Negative	Negative	Positive	Positive	Positive

Note: It is assumed that forward rate curves are upward sloping in the sample period and that volatility is lower than observable trend so that .

Figure 1. Forward LIBOR in Japanese LIBOR/swap market from 2 April 2007 to 31 August 2009, where x_i = 0, 2, 5, 10 years. Period A is from 2 April 2007 to 16 June 2008, period B is from 16 June 2008 to 31 August 2009. The swap data are provided by Mizuho Information and Research Institute.

Figure 2. Implied forward LIBOR curves at three days (2 April 2007, 16 June 2008, and 31 August 2009). The observable trend is bear flat in period A and bull steep in period B.

Table 3. Eigenvalues and the market prices of risk for periods A and B.

	Eigenvalue, ρ^2	Contribution rate	Accumulated contribution rate, C	Market price of risk, ϕ	Market price of risk (LMM), ϕ
Period A: 2 April 2007–16 June 2008.
1st	7.61E−04	0.9305	0.9305	−0.256	0.038
2nd	4.53E−05	0.0554	0.9859	0.532	0.758
3rd	5.44E−06	0.0067	0.9926	0.598	0.746
4th	3.29E−06	0.0040	0.9966	0.495	0.114
5th	1.22E−06	0.0015	0.9981	0.896	0.205
6th	5.86E−07	0.0007	0.9988	−1.049	1.226
Period B: 16 June 2008–31 August 2009
1st	4.77E−04	0.8082	0.8082	−0.884	−1.094
2nd	8.35E−05	0.1413	0.9495	−1.886	−1.718
3rd	2.38E−05	0.0402	0.9897	−0.816	0.213
4th	3.14E−06	0.0053	0.9950	1.113	2.570
5th	1.31E−06	0.0022	0.9972	2.445	2.089
6th	5.57E−07	0.0009	0.9982	−0.557	0.550

Notes: Japanese LIBOR swap rates for April 2007 through August 2009 are used for covariance analysis, where (20 days). The parameters in this table are calculated by using the eight-factor model. The rightmost column presents the market prices of risk implied by the LMM for the same sample data (table 7.1 in [7]).