# Jingsong trading options

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This is an open jingsong trading options article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. To price convertible bonds more precisely, least squares Monte Carlo LSM method is used in this paper for its advantage in handling the dependence of derivatives on the path, and dynamic credit risk jingsong trading options used to replace the fixed one to make the value of convertible bonds reflect the real credit risk.

In the empirical study, we price convertible bonds based on static credit risk jingsong trading options dynamic credit risk, respectively. Empirical results indicate that the ICBC convertible bond has been overpriced, resulting from the underestimation of credit risk.

In addition, when there is an issue of dividend, the conversion price will change in China's convertible bonds, while it does not change in the international convertible bonds. So we also empirically study the **jingsong trading options** between the convertible bond's prices jingsong trading options assuming whether the conversion price changes or not.

It is hard to be valued because of its characters of both equity and bond, in addition to its varieties of terms. Theoretical research on convertible bonds was initiated by Ingersoll [ 1 ], who applied the Black-Sholes-Merton model of pricing options.

Following his work, Brennan and Schwartz [ 2 ] firstly used corporate value as the basic variable to price convertible bonds. However, corporate value was soon replaced by jingsong trading options price for its simplicity of observation and measurement, which was first introduced by McConnell and Schwartz [ 3 ].

Brennan and Schwartz [ 4 ] considered stochastic interest rate firstly and constructed a two-factor pricing model for convertible bonds. Then, stochastic jingsong trading options risk was introduced by Davis and Lischka [ 5 ]. From the above tendency we can see that researchers gradually jingsong trading options factors to increase the accuracy of valuation. To solve the more and more complex models, Monte Carlo simulation became widely used.

Credit risk is an important factor in convertible bonds valuation and is paid more attention than before in China. Currently, there are two methods to measure the credit risk. The first one is credit spread, which is firstly used by McConnell and Schwartz [ 3 ] to value convertible bonds.

Following their work, Tsiveriotis and Fernandes [ 11 ] split convertible bond into two components: The second method is to use default density, which was used by Duffie and Singleton [ 12 ] to price corporate bonds.

Then, Ayache et al. Comparing the two methods, jingsong trading options first one is more **jingsong trading options** used for its simplicity and convenience. However,the estimation of the spread is essential. To get more accurate price, researchers began to employ dynamic credit risk. Davis and Lischka [ 5 ] supposed that hazard rate obeys a Brownian motion, with the Vasicek model of interest rate, and then established a two and a half model.

But they did not verify the effect of the model using real data. Currently, there is not any model applying dynamic credit risk to price convertible bonds. **Jingsong trading options** rest of this paper is organized as follows.

Section 2 gives the basic framework of convertible bond pricing by least squares Monte Carlo simulation. Section 3 derives the equation of dynamic credit spread. Section 4 is the empirical part, including the effect of dynamic credit spread and the comparison of price obtained by assuming whether the conversion price changes or not.

Finally, Section 5 concludes the paper. Besides the common debt, convertible bonds are embedded with many options, such as conversion option, call option, put option, and option to lower the conversion price.

So, we should compare the value of these options jingsong trading options when pricing the convertible bonds. At the expiration, the final boundary can be. Every moment before the maturity of the convertible bond, investors and issuers will gamble over the benefit. Investors will maximize the value of convertible bonds, while the issuer will minimize the value of convertible bonds from exercising the call option.

To make it clear, we use Table 1 to show the rules of option exercise. The basic framework of convertible bonds pricing by LSM when the credit risk is static is as follows.

Considering the stock volatility jingsong trading options one of the important parameters affecting stock path, we assume that the stock price follows the stochastic volatility model presented by Heston [ 14 ]: We split the duration of convertible bonds into sections on average and assume that the convertible bonds can only be exercised at these discrete times.

We can generate random numbers by Monte Carlo simulation and then get paths from the Heston model. Then, the stock price matrix is obtained. Applying the optimal stopping theory, we compare the value of continuation and immediate exercise. If the latter one is bigger, then we get the stopping time denoted by and stopping value denoted by. We assume to be jingsong trading options complete probability space and assume to be the finite time horizon.

The jingsong trading options variable is the simulated stock price, from which we calculate the cash flow on the path at time and the temporary optimal stopping value. Then we take the expectation of the cash flows discounted by the risky interest ratejingsong trading options get the value of continuation as follows: The least square regression process is as follows.

We put the value of continuation to be the dependent variable and the underlying stock price to be the independent variableand Laguerre polynomials are chosen to be the basis function to make the least square regression.

The procedure to get the estimated value of can be described by the following equations: Compare with the conversion value, call value, and the jingsong trading options value on the basis of the exercise rules of convertible bonds. If is bigger, then the optimal stopping value remains the same; if is smaller, then we get the new stopping time and jingsong trading options new stopping **jingsong trading options.** The convertible bond can be valued by discounting each back to time with the risky interest rate, and averaging over all paths: It is worth noticing that not all values of continuation are estimated by the Laguerre polynomials.

To increase the estimated accuracy, the following three conditions must be excluded. So, the value of continuation does not need to be estimated. So, there is no need to estimate the value of continuation, where is the discounted value of the call value until maturity.

Currently, there are two main methods to describe the dynamic credit risk. The first one is the two and a half model of Jingsong trading options and Lischka [ 5 ], who assumed that jingsong trading options hazard rate obeys a Brownian motion.

The second one is of Huang et al. In this paper, we describe the credit risk to be the dynamic credit spread linked with the stock price, because the credit risk is mostly affected by the stock price.

Moreover, the conversion price can be the benchmark of the stock price, and then we can derive the equation of dynamic credit spread as follows: The equation indicates that whenthe credit spread will decrease; whenthe credit spread will increase.

Finally, the risky interest jingsong trading options in the last section can be written to be. To test the performance of our model, we consider ICBC convertible bond, one of the largest convertible bonds issued in China, to do the empirical study. Before pricing we first need to estimate the related parameters. We get the term structure of risk-free interest rate based on cubic polynomial spline function using 15 treasury bonds traded in Shanghai Exchange.

We first assume that the credit risk is static and get paths of the stock price by LSM. Then we get the theoretical prices of ICBC convertible bond using our pricing framework. Figure 1 is the comparison of the theoretical prices and market prices. From Figure 1 we can see that the tendencies of the two lines fit well in the long run, so we can use this price framework to forecast market price of convertible bond and make investment decision. On the other hand, the market price is a little higher than jingsong trading options theoretical price, so ICBC convertible bond is a little overpriced.

We also define the variable AD to be the absolute deviation of the theoretical price from the jingsong trading options price as follows: Then we get the jingsong trading options of AD by Matlab and depict it in Figure 2. The following mean absolute deviation MAD is defined to describe the integral result of our model: In this section, we give the pricing results of ICBC convertible bond using the dynamic credit risk modeled by 6 and with.

We compare the market prices with the prices obtained from static credit risk and dynamic credit risk in Figure 3. We can see from Figure 3 that the market price of the ICBC convertible bond is higher than the prices obtained from both the static and dynamic credit risk models, which means that the ICBC convertible bond is overestimated, resulting from the underestimation of credit risk.

We can also see that the dynamic credit risk can reflect the real credit risk, since when the price of convertible bond goes downwards, the price obtained from dynamic credit risk is lower than those obtained from static credit risk.

This implies that we jingsong trading options use dynamic credit risk to price convertible bonds. In the above empirical study we price the ICBC convertible bond based on changing conversion price CP resulting from two issues of dividend.

So in this section we also give the prices of ICBC convertible bond assuming that the conversion price does not change. The pricing results together with those obtained from changing conversion price are shown in Figure 4.

Figure 4 indicates that, compared with the prices with changing CP obtained in Section 4. This demonstrates that when the company issues dividend the effect of unchanging CP is jingsong trading options.

Therefore, Chinese companies of convertible bonds are suggested to change CP when issuing dividend. That is, we cannot copy the international experience of unchanging CP. This paper studies the pricing of convertible bonds with dynamic credit risk using least squares Monte Carlo method.

We employ the dynamic credit spread changing with stock price. In empirical study, our model is proved to be effective and the comparison test demonstrates that the dynamic credit risk is important in convertible bond pricing.

The price obtained from dynamic credit risk can reflect the real credit risk. **Jingsong trading options,** the potential risk resulting from the overestimation of convertible bonds cannot be neglected by the investors. In addition, jingsong trading options also study the empirical effect of changing the conversion jingsong trading options when the issuer distributes dividend.

Consequently, the unchanged conversion price jingsong trading options lead to an unreasonable price. The authors declare that there is no conflict of interests regarding the publication of this paper. Home Journals About Us. Discrete Dynamics in Nature and Society.

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