Chapter 3 Methodology
3.7 Least squares Monte Carlo algorithm
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 GBM, NIG and VG model are -1.0671, -1.9530 and –2.3513 respectively. Other samples are also listed in Table 3. In Table 3, we find that the parameters of intensity rate are negative. Most of these values are between -1 and -2. Moreover, in the VG model, control the skewness. We find that
is almost zero, it means the distribution is symmetric.
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Table 3: Estimated parameters in different models
Foxconn Uni-President Synnex 1 Synnex 2 Yuanta
Notes: The parameters of GBM, NIG and VG model are estimated by MLE. The parameter of intensity rate is estimated by the calibration. (‧) means the standard error of the parameter.
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4.4 The performance of convertible bond
In this subsection, we display the performances of five samples with different contracts. After estimating the parameters, the LSM algorithm is applied to evaluate the theoretical value of CBs.
Moreover, if the model prices of CBs are obtained, the performance index, which is root mean square error (RMSE), can measure the performance of these prices. In this index, it gives the larger weight for the larger deviations, and can be calculated as follows:
1 ,
where is the number of trading days. Also, the pricing errors are defined as the deviation between market and model prices. It can be measured as follows:
n
The empirical results of five samples are displayed in Table 4. The RMSE values of Foxconn are 0.0536, 0.0496 and 0.0487 in the GBM, NIG and VG model respectively. In addition, the RMSE values of Uni-President are 0.1197, 0.1151 and 0.1072 in the GBM, NIG and VG model respectively. Also, the other samples are displayed in Table 4. In this table, we can find that both the performances of VG and NIG model are better than the GBM model, in which the VG model has the best performance except Synnex 1. It is shown that the NIG model is better than the other models.
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Table 4: Compariative performance of five samples
RMSE GBM Model NIG Model VG Model
Foxconn 0.0536 0.0496 0.0487 Uni-President 0.1197 0.1151 0.1072
Synnex 1 0.1478 0.1401 0.1467
Synnex 2 0.0482 0.0459 0.0367
Yuanta 0.1079 0.1015 0.0877
Note: In the performances of these five examples, the Lévy model is better than the GBM model.
Moreover, the pricing errors and SK ratios are shown in figure 3, where is the stock price and
S
K is the exercised price. In addition, the solid and dash lines represent the pricing errors and SK ratios respectively. We find that most of our samples show CBs are out-of-money, the pricing errors are positive like in Carayannopoulos (1996).
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Figure 3: The pricing errors of all samples
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In recent years, many firms gradually have increased their funds by issuing convertible bonds.
We find that the issuing volume of Taiwan’s CBs has grown up from 0.9 billion in 1999 to 290 billion in 2011. Therefore, convertible bonds as financing tools are more important for firms. The reason is that the coupon rate of convertible bonds is lower than the coupon rate of straight bonds, in this way, firms can reduce the financing cost by issuing convertible bonds instead of straight bonds. In addition, firms give investors the potential opportunity to obtain capital gains from debtors to shareholders since investors can switch convertible bonds into underlying common stocks. Hence, if the projects can’t bring better operating income and firms might not have enough cash to pay the principal of bonds, they issue CBs to avoid insolvency rather than straight bonds.
Moreover, some CBs are issued attached complex provisions such as buy-back and sell-back. Thus, this paper attempts to evaluate the prices of CB including two provisions.
Since the beginning of 2007, the financial crises, including the subprime mortgage and European sovereign debt crises, occurred constantly and the European sovereign debt crisis still continues taking place, the credit risk is more important in valuing securities. Therefore, we consider the credit risk in pricing CBs. In this paper, since the reduced-form approach is easy to use and flexible, we evaluate CBs by this method. In particular, the intensity rate is not only dynamic but also a decreasing function of stock price. In the previous pricing models such as Black and Scholes (1973), these models are mostly based on the assumption of the log asset return as the normal distribution. However, the financial empirical data show that the return distributions of asset have fat tail and excess kurtosis. Since the Lévy model such as NIG and VG model can capture the jump components, which are applied to fit the market situation well, we use the Lévy model to
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evaluate CBs. In the parameter estimation of Lévy model, unlike the calibration method, we adopt the maximum likelihood estimator to estimate the parameters. In the pricing method, the least squares Monte Carlo Simulation is used since it is more flexible than other methods like finite difference or binomial tree that can deal with early-exercise problems with complex provisions.
In the empirical studies, we take five samples as examples including Foxconn, Uni-President, Synnex 1, Synnex 2 and Yuanta respectively. Whatever the QQ plot or two-sample KS test, we show the Lévy model fits the data of five samples well compared with the traditional GBM model.
Moreover, the stock returns of four corporations display the jump components. Thus, the Lévy model is preferred to apply. For the performances of five samples, we also find the NIG and VG model are better than the GBM model. Furthermore, most of our samples show CBs are out-of-money, the pricing errors are positive.
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