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

Abstract

This paper studies an interest-rate simulation for risk management under the real-world measure. First, this paper proposes a method to estimate the market price of risk from historical data in a Gaussian Heath, Jarrow, and Morton framework. Next, properties of the simulation are examined in connection with historical data. Finally, the market price of risk is roughly interpreted in regard to the historical change in interest rates. These results are explained through numerical examples.

Keywords:

• Interest-rate risk assessment
• Interest-rate simulation
• Real-world measure
• Market price of risk
• Gaussian HJM model

1. Introduction

The term structure model of interest rates was originally used for option pricing under a risk-neutral measure; recently, it has been applied in assessment of interest-rate risk, where the model should be constructed under the real-world measure. To simulate interest-rate dynamics under the real-world measure, it is necessary to estimate the market price of risk. In practice, the market price of risk has been assumed to be zero because how to estimate the market price of risk is not obvious. Moreover, the properties of the simulation in the real world have yet to be fully elucidated, and the numerical differences between a real-world simulation and a risk-neutral simulation are not known.^†

In econometrics, the market price of risk has been estimated by short-rate models. For example, Stanton (1997) used regression analysis on historical yield data, and Dempster et al. (2010) used the Kalman filter. These approaches use econometric software to numerically estimate the market price of risk, and for this reason, theoretical investigation of the properties of real-world simulation remains difficult.

Yasuoka (2013) introduced a LIBOR market model (LMM) under the real-world measure in the framework of Jamshidian (1997), and proposed a procedure to estimate the market price of risk under the assumption that this price is constant. Through this approach, the following results were obtained: (1) The simulation model in the LMM is similar to an empirical term structure model with historical drift and volatility. (2) It is shown how to determine the number of factors for simulation in connection with historical data. (3) When LIBOR volatility is low, the market price of risk is roughly explained by changes in the historical forward LIBOR curve, rather than by the state of the curve.

Although the LMM is useful to simulate the flexible dynamics of the yield curve, some practitioners prefer to use a short-rate model for risk management. For example, the Hull–White model (Hull and White, 1990), a special case of the Heath, Jarrow, and Morton (HJM) model (Heath et al., 1992), is popular. Therefore, it is reasonable to perform a study similar to Yasuoka (2013) on the HJM model. In this respect, the objective of this paper is to introduce a procedure for real-world simulation in a Gaussian HJM framework and to use this to study the market price of risk and properties of the simulation.

Section 2 introduces a procedure to estimate the market price of risk from historical data. Beginning with a full-factor model, it is shown that the real-world simulation model is an empirical term structure with historical drift and volatility. Section 3 studies how to decide a practical choice for the number of factors in the real-world simulation. Furthermore, the numerical meaning of the market price of risk is investigated in regard to historical trends in interest rates. Some studies have attempted to explain the market price of risk through state variables, for example the instantaneous spot rate by Stanton (1997). In contrast, our results explain the market price of risk through changes in the historical interest-rate curve, rather than through the curve state. Finally, Section 4 explains the obtained results through numerical examples.

2. Market price of risk and simulation

2.1. The HJM framework

This section briefly summarizes the framework of the HJM model. Let be a probability space, where T* is a time horizon, F_t is an augmented filtration, and P is the original measure. P is usually called the real-world measure (or alternatively, historical measure, etc.). f(t, T) denotes the instantaneous forward rate (hereinafter, the forward rate) with maturity T≤T* prevailing at time t≤T. We denote the inner product by · in . The dynamics of f(t, T) is assumed to be expressed by (1) where Z is a d-dimensional -Brownian motion, and α(t, T) and σ(t, T) are predictable processes satisfying some technical conditions (cf. Heath et al., 1992). Usually, σ(t, T) is referred to as volatility. The instantaneous spot rate (hereinafter, the spot rate) r is given by r(t)=f(t, t). The savings account B* is defined by , and denotes the price of the zero-coupon bond at time t with maturity T. Then, it follows that (2) where and Note that is referred to as price volatility. The discounted bond price is given by

Under the assumption of no arbitrage, Heath et al. (1992) shows that there exists ϕ(t) such that for arbitrary T. ϕ(t) is referred to as the market price of risk. Note that the use of plus/minus signs in this definition follows that in Jamshidian (1997) and Yasuoka (2013). Let be a measure equivalent to such that is a -Brownian motion. is known as a risk-neutral measure. Then, the discount price process is a -martingale for all T.

We then have (3) accordingly, (4) Although bond pricing is achieved under the interest-rate simulation for risk management is executed under .

Indeed, if we set , then it follows from equation (2) that We have (5) where the left-hand term is the excess return over r(t) of a T-maturity bond, and is a bond-price volatility. Thus, equation (5) coincides with a well-known definition of the market price of risk for the bonds market.

2.2. Market price of risk

The HJM model is Gaussian if σ(t, T) is a deterministic function in t and T. In what follows, an -valued volatility σ(t, T) is assumed to be always deterministic and continuous in t and T. Also, is deterministic and continuous.

Let be an arbitrary sequence of maturity dates with 0<T_i<T_i+1. Taking to be a d×d matrix, we assume that the rank of this matrix is equal to d. Assume that the volatility σ(t, T) is given. The market price of risk is assumed to be constant during the sample period. Setting , the forward rate process is expressed by (6)

For an integer n≥d, let be a sequence of lengths of time, where x_i<x_i+1. Let a time interval Δ t>0 be fixed, and be a sequence of observed times with t₁=0 and , where J+1 is the number of observed times. Naturally, we observe the instantaneous forward rate F(t_k, x_i) in the interest-rate market with a fixed length of time x_i from t_k to the maturity date t_k+x_i. Taking F(t_k, x_i) and , respectively, as f(0, x_i) and in equation (6) for each , we suppose that the following equation holds for all i. where x_i−s is identified with a maturity date at t=s.

To simplify the notation, we denote σ(0, x_i) and as σ_0i and υ_0i, respectively. Moreover, we denote the lth component of σ_0i as . Then, where the superscript T denotes transpose. The Euler integral implies (7) for all i and k, where . Let denote the sum of the squared difference between each side of equation (7) in the time series and cross sections, neglecting the random part such that (8) where we set and We determine ϕ as the solution that minimizes . Then, ϕ is estimated by the following proposition, where E^H[ ] denotes the sample mean over k=1, … , J.^†

Proposition 2.1

For a d-factor model with d≤n, let ϕ be a solution that minimizes . Then, ϕ is uniquely given as the solution of the linear equation (9) where (10)

Proof Taking as a d×n-matrix, we may assume that the rank of this matrix is d. Because it follows that Then, the partial derivative of in ϕ_l is (11) for l=1, … , d. We have and the Hessian of ε is given by (12) It holds that, for an arbitrary vector , (13) because is positive definite. Hence, the solution ϕ to the minimization problem is uniquely determined by solving . A constant vector is defined in equation (10). Upon substituting equation (10) into equation (11), the equation reduces to . This is equivalent to equation (9). From equation (13), is positive definite and thus rank d, so ϕ is obtained as a unique solution to equation (9). This completes the proof.

Note that for the setup, the Musiela parametrization (see Brace and Musiela, 1994) allows for an alternative approach from equation (6) to equation (7).

2.3. Property of simulation

Let σ(t, T) be Rⁿ-valued volatility. For a sequence of lengths of time we assume that is rank n. For , we obtain from the historical data. Let ϕ be the market price of risk obtained by Proposition 2.1, and let d=n.

We recall the forward rate process (6). Regarding x_i as a fixed maturity date at t=0, we set T_i=x_i for i=1, … , n. We consider an n-factor model as (14) for i=1, … , n. An n-factor model is called a full-factor model, and is said to be orthogonal if for all l, k with l≠k. The next proposition gives a single-period simulation under P by a full-factor model.

Proposition 2.2

If is rank n and orthogonal, then, for an arbitrary f(0, T_i) and Δ s>0 with Δ s<T₁, is simulated by (15)

Proof The Euler integral of equation (14) implies (16) for i=1, … , n. The inner product between and vanishes from equation (9) for all l. Since is rank n and orthogonal, the family of n-dimensional vectors spans . Hence, for all i. Substituting this and equation (10) into the second term of equation (16), we have equation (15).

For example, if the principal components of historical covariance are imposed on the volatility, then such volatility is orthogonal. Furthermore, represents a kind of historical change in the sample forward rate, Therefore, equation (15) is interpreted as a term structure model with historical drift and volatility.

The following corollary shows the averages of interest rates after the single-period simulation. Comparing equation (17) with equation (18), we see the difference between a real-world simulation and a risk-neutral simulation.

Corollary 2.3

Under the assumption of Proposition 2.2, is normally distributed under both and with the following expectation. (17) (18) The standard deviations of in both simulations are equal to .

2.4. Volatility associated with principal component analysis

Returning to the setup in Section 2.2, we estimate the market price of risk more explicitly. We denote a sample covariance matrix by V such that (19) for i, j. We assume that V has rank d≤n. By the usual arguments pertaining to principal component analysis, the covariance matrix is decomposed into where and are the lth eigenvalue and the lth eigenvector, respectively. We may assume that all eigenvectors are chosen such that and ρ_l>0. e^l is also known as the lth principal component of the covariance, so it holds that For a positive integer k≤d, the accumulated contribution rate C_k is defined as .

Let σ(t, T) be a d-dimensional volatility satisfying then the market price of risk is obtained as follows.

Proposition 2.4

For a d-factor model with d≤n, if σ_0i is given as the principal components, then it follows that where .

Proof The left-hand side of equation (9) becomes . Similarly, the right-hand side of equation (9) becomes We have . From , we have . Since ρ_l≠0, we have for all l. This completes the proof.

We call ζ_l and the lth market price of risk (MPR) score and the lth volatility, respectively. The norm of is Thus, ρ_l represents the magnitude of the lth volatility, we call ρ_l the lth volatility risk. Consequently, the market price of risk is explained as the MPR score per unit risk of the volatility, this is analogous to Yasuoka (2013) on the LMM.

Note that positive and negative values of ϕ coincide with positive and negative MPR scores. Hence, MPR score is affected not only by the definition of ϕ, but also by the definition of e^l. To investigate positive and negative values of ϕ, we must fix their definitions. This is the reason that we assume for all l.

3. Numerical analysis of real-world simulation

To reduce the dimensions of the simulation to practical bounds, there are at least two approaches. One approach is to approximate the dynamics of the forward rate while retaining the arbitrage-free framework. This was already studied in Yasuoka (2013). Another approach is to approximate the expression (15) in Proposition 2.2. The point of the latter approach is that a historical and simple structure is retained. For the sake of convenience, we adopt the latter approach. In the following, we denote the d-factor simulated forward rate as f˜ to distinguish it from the full-factor model.

Proposition 3.1

For a full-factor model, we assume that σ_0i are given as the principal components, and that is rank n. If C_d is sufficiently close to 1 for some d<n, then the instantaneous forward rate at time Δ s is approximated by a d-factor model. (20)

Proof From the assumption, equation (15) holds. The second term in equation (15) does not depend on the number of factors. If C_d is sufficiently close to 1, it holds in the sense of probability, that Therefore, equation (20) is obtained.

Recall that Proposition 2.2 is introduced as a discrete-time version of an arbitrage-free model. In contrast, the above model is no more than an approximation of an arbitrage-free model. Indeed, in a d-factor model (d<n), does not vanish as it does in equation (20); hence, it is not trivial to obtain equation (20). However, the above approximation is mostly accurate if the number of factors is chosen such that C_d≈ 1.

Next we consider the numerical meaning of the market price of risk. The ith observable trend of the historical forward rate is defined by . Intuitively, this means the change of the forward rate curve during the period. The ith rolled trend of the historical forward rate is defined as . It follows that (21)

The first term is the ith observable trend, and the second term is almost equal to the sample mean of the forward rate slope.^†

For example, if the forward rate curve is upward sloping and remains unchanged, then equation (21) implies . This is known as the roll-down of the forward rate curve. Furthermore, when the upward sloping forward rate curve is rising fast, the rolled trend, that is, the roll-up of the forward rate curve, might be positive. Accordingly, the rolled trend corresponds to the roll-down or roll-up of the forward rate curve, which we are familiar with from financial experience. Proposition 2.2 shows that the real-world simulation reproduces the rolled trend of the entire forward rate curve. We next explain the meaning of the market price of risk in connection with the above trends.

Proposition 3.2

For a d-dimensional model with d≤n, we assume that the volatility σ_0i is given by principal components. If it holds that for all i, then we have the following approximations. (22)

Proof From Proposition 2.4, the MPR score is given by The assumption implies equation (22). Since , we have the second approximation. This completes the proof.

Proposition 3.2 shows that ϕ is largely determined by the rolled trend in the historical forward rate. Therefore, the numerical meaning of the market price of risk is roughly interpreted with respect to the observable trend. From equation (22), approximately represents the coordinates of the rolled trend in the principal component space. In other words, the lth MPR score ζ_l is approximated by the lth projection of the rolled trend to the space. The principal components e¹, e², and e³ are known to explain the movement of the forward rate curve in terms of level, slope, and curvature, respectively. Similarly, , and ζ₃ approximately represent the level, slope, and curvature factors of the rolled trend. Hence, ϕ_l roughly means the lth projection of the rolled trend per lth volatility risk.

Consider an investor who holds bonds for all maturities and expects the roll-down of the entire forward rate curve as an excess return. Then, the first market price of risk ϕ₁ is a risk-adjusted measure for the roll-down return of the entire forward rate curve. The values are interpreted in a similar way. Table 1 shows rough estimates of ϕ₁ in each case. Table 2 shows rough estimates of ϕ₁ and ϕ₂ in association with the two factors of the observable trend, where we assume that most of the forward rate curves in the sample data are upward sloping. These tables are exactly same as those under the LMM (tables 1 and 2 in Yasuoka, 2013).

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 .

4. Numerical examples

We have data for Japanese LIBOR swap rates from 2 April 2007 to 31 August 2009, which is the same data set used in Yasuoka (2013). Setting δ=0.5 (year) and x_i=δ i for i=1, 2, … , the 6-month forward LIBOR is obtained for this period. For convenience, we regard the forward LIBOR as the instantaneous forward rate in the following example. According to Yasuoka (2013), we divide the sample period into two periods. Period A is earlier and covers 2 April 2007 to 16 June 2008, period B is later and covers 16 June 2008 to 31 August 2009. Figure 1 presents the time series of the forward rates for x_i=0, 2, 5, 10 years and illustrates periods A and B. Figure 2 presents the forward rate curves at three days (2 April 2007, 16 June 2008, and 31 August 2009). These curves show that the observable trend is bear-flattening in period A, and bull-steepening in period B.

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.

In the data analysis, are directly implied from the LIBOR/swap rates, but is not since usually . So, we approximate by a linear interpolation such that Table 3 shows the eigenvalues, contribution rates, and the market prices of risk for each period. Since the observable trend is almost bear-flattening in period A, the ‘bear flat’ column of table 2 shows that ϕ₁ is near zero and ϕ₂ is positive. Table 3 shows that and in this period, which are roughly explained in table 2. The observable trend of period B is bull-steepening. Table 2 shows that ϕ₁ and ϕ₂ are both negative in the ‘bull steep’ column. The results are that and . These values are also consistent with table 2.

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]).

For comparison, the rightmost column presents the market price of risk that is implied by the LMM framework on the same sample (table 7.1 in [7]). As mentioned at the end of Section 3, the numerical interpretation of the market price of risk in this study is largely similar to that in the LMM. This commonality is reflected in that the first and the second marker prices of risk show similar tendency between two models.

5. Conclusion

The interest-rate simulation model has been introduced under the real-world measure in the Gaussian HJM framework. The procedure to estimate the market price of risk was presented under the assumption that market price of risk is constant. The market price of risk is interpreted by the change in the historical forward rate curve rather than the change in the state of the curve.

In particular, when the principal components are imposed on the volatilities, the simulation model is almost equivalent to a term structure model with historical drift and volatility. Accordingly, the mean of the forward rate obtained from the simulation is strongly affected by sample period selection.

Although the possibility of a negative interest rate is a drawback of the Gaussian model, this paper's approach is practical for interest-rate risk management in actual businesses. As an example of present use, the Hull–White model, which is a special case of the HJM model, is successfully used for risk management.

Additional information

Funding

This work was partially supported by JSPS KAKENHI [Grant Number 26380399].

Notes

^† In this paper, an interest-rate simulation under the real-world measure is referred to as a real-world simulation for short. Similarly, the simulation under a risk-neutral measure is referred to as a risk-neutral simulation.

^† For arbitrary historical data {A_k}_{k=1, … , J+1}, the sample mean is usually evaluated by using . In this sense, E^H[ ] is not the sample mean of A_k because .

^† For a more precise explanation, see note 2.