Chapter 2 Literature review
2.2 Credit risk
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distribution, and then applied the method of Monte Carlo Simulation to price an option. Cariboni and Schoutens (2007) supported that the asset value follows a VG model to calculate the credit default swap. Hirsa and Madan (2004) applied the VG model to price the American option.
Albrecher and Predota (2004) considered a NIG model to price the Asian option compared with the GBM model. Kalemanova, Schmid and Werner (2007) applied the NIG distribution to construct the factor copula model and calculated the value of synthetic CDO. Stentoft (2008) proposed a GARCH framework and Normal inverse Gaussian distribution to price American options.
We evaluate a CB based on the Lévy process generating the distribution of underlying stock price as a non-normal distribution. When using the Lévy process, we not only need to take the mean and variance of asset return into consideration but also need to consider the skewness and kurtosis.
Thus, we believe that the value of CB using the Lévy model is not only more accurate compared with the traditional GBM model but also fits the financial data well. In the empirical investigation, we apply the maximum likelihood estimator (MLE) method to estimate the parameters of Lévy model, which is different from other studies using the calibration.
2.2 Credit risk
There are two ways such as structural-form and reduced-form approach to measure the credit risks. The structural-form approach is called as option-pricing model. Merton (1974) first applied the capital structure to calculate the default probability. However, this paper assumed if firms can’t pay principal, firms default. Hence, firms only default at maturity. For solving this problem, Black and Cox (1976) considered firms may default before the maturity. If the firm’s values trigger the
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boundary, firms will default.
The default time is given an exogenous variable in the reduced-form approach. Jarrow and Turnbull (1995) first applied the reduced-form approach to price the financial securities. The recovery is exogenously given and they used the martingale method to calculate the default probability. Afterward, Jarrow, Lando and Turnbull (1997) used the Markov chain to describe the movement of credit ratings. They applied the transition matrix to measure the default probability.
Lando (1998) generalized version of a Markovian model proposed by Jarrow, Lando and Turnbull (1997). This model allows the intensities depend on the sate variables which control the term structure or possibly other economic variables. In Duffie and Singleton (1997), the intensity model of default with recovery is proposed to analyze the term structure of swap spreads which are likely to depend on other factor such as liquidity effect.
The rest of this paper is organized as follows. Section 2 discusses the defaultable instruments using the reduced-form approach and describes the dynamics of intensity rate as the function of stock prices. Section 3 discusses the Lévy process and its estimation technique. Section 4 describes the valuation of CB and adopts the least squares Monte Carlo algorithm to price CBs. Section 5 shows the data description, empirical results. Finally, we make the conclusions.
‧ Chapter 3 Methodology
3.1 Defaultable zero-coupon bond
Unlike Treasury bonds, corporate bonds and common stocks are subject to the credit risk. So, we introduce the reduced-form approach of credit risk to value these securities. Also, we assume that a probability is given, which is the risk-neutral measure. Following Duffie and Singleton (1999), framework, the defaultable zero-coupon bond is expressed as
],
3.2 Defaultable stock valuation
Traditionally, in most literature, the normality assumptions are popularly applied in pricing derivatives, where the Brownian motion is adopted. The stock price subjected to the credit risk is assumed to follow a geometric Brownian motion (GBM). Let the stock returns can be described as follows:
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where t is the expected instantaneous stock returns, is the standard deviation of instantaneous returns and is the standard Brownian motion under the physical measure. Thus, we can describe the dynamics of stock price under the risk-neutral measure as follows:
P
Wt
(3) where is the standard Brownian motion under the risk-neutral measure,
), )
exp((
0
Q t GBM
t t
t S r t W
S
Q
Wt GBM is the
compensator term of GBM model which makes sure that the stock price is under the risk-neutral measure, i.e. 2.
2 1
GBM
3.3 The dynamics of intensity rate
We know that when the stock price of firm is low, it tends to default. Thus, the intensity rate has negative effect on the stock price. Following Takahashi, Kobayashi and Nakagawa (2001), the intensity rate is the decreasing function of stock price, which is easily observed and traded frequently in the financial market. Also, Ayache, Forsyth and Vetzal (2004) considered the intensity rate increases as the stock price decreases. Hence, the intensity function is set as
( , ) ( ) ( )( ) ,
0 0
S
S S S
t
St t t (4)
where )(S0 is the estimated intensity rate at St S0. In Muromachi (1999), the formal function like Equation (4) was found to fit the bonds rated BB+ and below in Japanese market. Moreover, the values of are reasonably suggested range from –1.2 to –2.0.
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3.4 The properties of Lévy process
Under the GBM model, the stock return follows a normal distribution. However, as mentioned in previous section, the return distributions of financial securities are always not normally distributed but exhibit fat tail and excess kurtosis. Hence, we will use the Lévy process to characterize the dynamics of stock price. A Lévy process is under the probability space , where is the sample space, F is the information filtration, and P denotes the physical measure, it has the following properties:
Xt
According to Lévy-Khintchine formula, we have
determine the distribution type of . Hence, we will use the Lévy process to characterize the dynamics of stock price. In the late 1980s and the 1990s, such as the normal inverse Gaussian (NIG)
Xt
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distribution (Barndorff-Nielsen, 1995), the variance Gamma (VG) distribution (Mandan, Carr and Chang, 1998) were widely applied.
In the Lévy process, the stock price under the risk-neutral measure can be described as follows:
St S0exp((rt t)tXt t), (5) where Xt is the Lévy process, is the compensator term of Lévy model which ensures that the
dynamics of stock price are under the risk-neutral measure.
Next, we introduce some properties of the NIG and VG model. Barndoff-Nielsen (1995) proposed the normal inverse Gaussian (NIG) model, which is an inverse Gaussian time-changed Brownian motion. Also, the NIG(,,,) with parameters 0, , 0 and
R
has a characteristic function given by
NIG(x,,,,)exp(ix)exp(( 2 ( ix)2 2 2)), (6) where )( is the characteristic function.
If we have the inverse Gaussian distribution I(, 2 2), the NIG model can be obtained by replacing the time t in the Brownian motion with a inverse Gaussian distribution. Hence, we can simulate XtNIG as follows:
1. generate I(, 2 2).
2. generate ) W ~ N(0,1.
3. set XtNIG I(t) I(t)W.
Hence, the NIG model can be represented as follows:
XtNIG I(t) I(t)W, (7)
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where is a location parameter, is the parameter and follows the inverse Gaussian distribution.
) I(t
In the NIG model, there are three parameters, which are the shape parameter , the skewness parameter and the excess kurtosis parameter In this case, the compensator term of the NIG . model is
NIG 2 (1)2, where 2 2.
Madan, Carr and Chang (1998) suggested the variance Gamma (VG) model, which is obtained by Brownian motion at a random time change given by a Gamma distribution. This model has two additional parameters determining the skewness and kurtosis which are and respectively.
Also, the characteristic function of the
v )
, , ,
( v
VG distribution is given by
) .
2 1 1
( ) exp(
) , , , ,
(x v ix ixv 2x2v 1v
(8)
If we have the Gamma distribution G(tv,v), the VG model can be obtained by replacing the time t in the Brownian motion with a Gamma distribution. Hence, we can simulate as follows:
VG
Xt
1. generate G(tv,v).
2. generate ) W ~ N(0,1.
3. set XtVG G(t)W(G(t)).
Hence, the VG model can be represented as follows:
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Moreover, Madan, Carr and Chang (1998) stated that the VG model could be expressed as the different parameters of two independent Gamma distributions. In this case, the compensator term of the VG model can be shown as
In addition, if we have the characteristic functions of these distributions, the basic moments can be obtained, i.e., mean, variance, skewness and kurtosis.
3.5 The estimation method
We discuss the method of parameter estimation for the NIG and VG model. We use the maximum likelihood estimator (MLE) to estimate the parameters of NIG and VG distributions. If we have the probability density functions of NIG and VG distributions, we can obtain the maximum log likelihood functions of NIG and VG model. The probability density function of NIG distribution is described as follows (Barndorff-Nielson, 1995):
,
The probability density function of VG distribution is described as follows (Madan, Carr and Chang, 1998):
K1
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Then, we can obtain the log likelihood function of NIG and VG model. They can be described respectively as follows:
where and denote the likelihood functions of NIG and VG distributions
respectively. In this application, the initial values of the parameters are set from the moment ()
LNIG LVG()
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method.
3.6 The valuation of convertible bond
The value of CB can be affected by many factors, such as the stochastic process of underlying stock price (denoted as ), the convertible price (denoted as ) and the credit risk of issuing company. In addition, there are many CBs’ provisions that influence the value of CB such as buy-back, sell-back, adjustment or reset of the conversion price. Brennan and Schwartz (1980) found that the value of CB is not significantly different under the assumption of constant or stochastic interest rates. Hence, for simplicity, we assume that the interest rate is constant. In addition, since the credit events have occurred frequently and the complex provisions such as sell-back and buy-back are popular in the market, we consider these conditions in measuring the value of CB.
S C
rt
We assume that the CB is a zero-coupon bond, in which, the par value is denoted as M. We propose that the valuation is proceeded under the probability space (,F,Q), where is the state; the filtration is the information process; and is the risk neutral probability.
Let be the price of CB at time t with maturity date T,
T t
Ft
F ( )0 Q
) c
, ( Tt
CB , s and b be the converted,
sell-back and buy-back time respectively. Moreover, we assume min(c,s,b) is the terminated time of CB. The terminated time is decided according to the converted time c, sell-back time s and buy-back time b, these three values can be determined based on the comparison of the holding value and terminated value of CB.
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Based on the provision of conversion, the payoff of CB is if the CB is converted. Based on the provision of sell-back, the investors can sell the CB back to the issuers on the value of
M , where is a constant extra premium before a predetermined time. Also,
the issuers can buy the CB back from the investors on the value of M(1). Furthermore, if the CB is still remained until maturity, the value of CB is M at maturity date. Therefore, the payoff of CB at maturity date T can be shown as
The rational investors will terminate the CB if they expect the terminated value is larger than the holding value. The holding value and terminated value of CB can be calculated based on the generating stock price. In other words, after generating the stochastic process of underlying stock, we can determine the times of c, s and b, then evaluate the CB at maturity by Equation (11).
The first term in Equation (11) is the payoff when conversion happens, and the second term is the payoff that the investor sells back the CB to the issuer. The third term in Equation (11) is the payoff that the issuer buys the CB back from the investor. The last term is just the payoff that the investor holds the CB until maturity date. From the Equation (11), the value of CB at time under the risk-neutral measure can be described as
t
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3.7 Least squares Monte Carlo algorithm
The embedded call option of CB is an American-style, hence we must deal with the early-exercised problem. The least squares Monte Carlo Simulation (LSM) method proposed by the Longstaff and Schwarz (2001) is to evaluate the expected continued holding value compared with the conversion value and then to determine the optimal excise time at each simulated path, i.e., the investor may not convert the CB even if the stock price exceeds the convertible price. That means the investor expects the holding value is more than conversion value. In the LSM method, we assume the investor is allowed to convert the CB into common stock in points, , the continued holding vale of CB is defined as the expected discounted value of cash flow at time conditional on the time information. It can be estimated with a
least squares regression. For example, the conditional holding value is viewed as , where is the stock price, investor makes decision to hold the CB or convert the CB.
tk
The conditional excepted value is obtained by the backward method at each . When the CB is in the money at time , we assume the continued holding value of CB, denoted as , can be calculated by the regression, where is the path of stock price. Since the
continued holding value depends on the expectation of remaining discounted cash flows with respect to the risk-neutral measure , the holding value at time can be expressed as
1
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k
where is the risk-adjusted rate between time and , is the remaining discounted cash flows of each path between time and . Based on Equation (13), the holding
value is conditional on the known information at time ()
R tj tk C(w,tj,tk,T)
tj tk
tk
Ft .
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4.1 Data description
In this subsection, we describe five examples of CBs containing provisions or not and show why we adopt the Lévy model. Since the trading isn’t frequently in Taiwan CB market, we select the samples that contain relatively higher trading volume. There are five samples in this study, including Foxconn, Uni-President, the first issue of Synnex (denoted as Synnex 1), the second issue of Synnex (denoted as Synnex 2) and Yuanta respectively. The stock prices of these samples are obtained from the Taiwan Economic Journal (TEJ). The sample period of these samples orderly begins from November 10, 2006 to May 10, 2007, September 24, 2008 to March 23, 2009, June 25, 2008 to December 12, 2008, January 14, 2011 to July 15, 2011 and July 9, 2010 to December 24, 2010. All of the sample period is 120 trading days. The difference between five samples is the contract that Uni-President and Synnex 2 are three years without additional provisions and the others are five years with complex provisions such as sell-back and call-back. Table 1 briefly describes the properties of these CBs.
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Table 1: Convertible bond
Convertible bond Foxconn Synnex 1 Yuanta Uni-President Synnex 2
Date of issuing 2006/11/10 2008/6/24 2010/7/7 2007/10/25 2011/1/14 Duration 5 years 5 years 5 years 3 years 3 years
Issuing price ＄100 ＄97.7 ＄103 ＄103 ＄103.15
Coupon rate zero zero zero zero zero
Initial conversion price ＄316.55 ＄80 ＄20.8 ＄56 ＄86.8 Provision sell-back, buy-back sell-back, buy-back sell-back, buy-back none none
Premium of buy-back 0% 0% 0% 0% 0%
Premium of sell-back 0% 0% 0% 0% 0%
Interest rate 2.2% 2.6% 1.1% 2.8% 1.1%
Note: This table briefly describes the content of contracts for five samples.
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Many empirical studies in finance shows the return distributions of securities often have fat tail and excess kurtosis. These samples also have the properties. Moreover, the time period of return data begins from the whole year before the day that starts to evaluate plus the trading days. If the empirical returns exceed two standard deviations of returns, it may imply the returns of securities containing jump components. Figure 1 shows this phenomenon. Therefore, the CBs are suitable for applying the Lévy model, not the GBM model.
Figure 1: The returns of all samples
4.2 The goodness of fit test
In order to know how the fitted density adapts to the empirical data, in this subsection, we adopt two methods such as QQ plot and two-sample Kolmogorov–Smirnov (two-sample KS) to test
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the goodness of fitted distribution for the empirical data. In the QQ plot test, according to the Glivekno-Cantelli theorem, let a sample be , and its cumulative probability density function be
yn
y y1, 2,, )
, (yi
F , furthermore, the ordered sample Y(1),Y(2),,Y(n) satisfy the condition:
limmax|( 0.5) ( (); )| 0.
1
i
N i
N i N F Y
Also, the figure for the sequential pairs are based on the sequence of {((i0.5) N),F(Y(i);))}Ni1.
It is so called as the PP plot which is an approximately 450 straight line. Moreover, the figure depends on the following sequence called QQ plot that is also an approximately 450 straight line as {(Y(i),F1((i0.5)/N);))}iN1.
QQ plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quintiles against each other. If the plotted figure is deviated from the 450 straight line too far, which means that the empirical data do not fit the assumed distribution well.
The QQ plots of GBM, NIG and VG model are shown in figure 2. All of the data display the GBM model can’t capture fat tail well. In addition, the VG model of Synnex 1 also can’t capture this phenomenon. Even so, the empirical results show that the Lévy model is better than the GBM model in the fitness.
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Figure 2: The QQ plot of all samples
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In the description of two-sample KS test that is used to determine whether the empirical and fitted distribution differ significantly. Let be the cumulative probability density of empirical distribution, be the cumulative probability density of fitted distribution, the two-sample KS test can be calculated as
Fe
Ff
KS max|Fe(x) Ff(x)|.
R
x
This statistical test is based on the difference between the cumulative probability density of empirical distribution and the cumulative probability density of fitted distribution.
Table 2: KS two-sample test of five samples
KS two-sample test GBM Model NIG Model VG Model
Foxconn 0.0791 0.0627 0.0655 Uni-President 0.0741 0.0391 0.0511
Synnex 1 0.0765 0.0540 0.1092
Synnex 2 0.0647 0.0360 0.0575
Yuanta 0.0820 0.0632 0.0729
Note: The test is all at 10% significance level.
As shown in Table 2, the values of two-sample KS test for Foxconn are 0.0791, 0.0627 and 0.0655 in the GBM, NIG and VG model at 10% significance level respectively. Similarly, the values of two-sample KS test for Uni-President are 0.0741, 0.0391 and 0.0511 in the GBM, NIG and VG model at 10% significance level respectively. Also, the others are shown in Table 2. In GBM model, the cumulative probability density of empirical distribution doesn’t fit the cumulative probability density of fitted distribution among these samples. In the Lévy model, except the VG model of Synnex 1, the cumulative probability density of empirical distribution fits the cumulative
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probability density of fitted distribution well. In sum, whatever QQ plot or two-sample KS test, we find that the goodness of Lévy model is superior to the GBM model. Then, we estimate the parameters of these three models in the next subsection.
4.3 Parameter estimation of stock price and calibration of intensity rate
Before pricing CBs, we have to estimate the parameters of GBM, NIG and VG distribution by using the maximum likelihood method described in Equation (10). Moreover, if we have the parameters of Lévy and GBM model, then the parameter of intensity rate can be estimated by calibration. Thus, before pricing CBs, the parameter is calibrated by minimizing the difference between the market prices and the model prices given by
, ) (
min
arg stMarket stModel 2
where is the market CB price of each trading day, also, is the model CB price of each trading day.
Market
st stModel
The estimated parameters of Foxconn, Uni-President, Synnex 1, Synnex 2 and Yuanta are listed in Table 3. In Foxconn, the estimated values of GBM parameters and are 0.0024 and 0.0212 respectively. The estimated values of NIG parameters , , and are 0.0027, 47.7661, -2.1822 and 0.0235 respectively. The estimated values of VG parameters , , and are 0.0039, 0.0219, 0.5411 and -0.0023 respectively. Also, the parameters of intensity rate in the
The estimated parameters of Foxconn, Uni-President, Synnex 1, Synnex 2 and Yuanta are listed in Table 3. In Foxconn, the estimated values of GBM parameters and are 0.0024 and 0.0212 respectively. The estimated values of NIG parameters , , and are 0.0027, 47.7661, -2.1822 and 0.0235 respectively. The estimated values of VG parameters , , and are 0.0039, 0.0219, 0.5411 and -0.0023 respectively. Also, the parameters of intensity rate in the