# Research on Pricing Methods of Convertible Bonds Based on Deep Learning GAN Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Pricing Method of Convertible Bonds Based on B–S Option Pricing Model

#### 2.2. Pricing Method of Convertible Bonds Based on Tree Graph Method

#### 2.3. Pricing Method of Convertible Bonds Based on Finite Difference Method

#### 2.4. Pricing Method of Convertible Bonds Based on Least Square Monte Carlo Simulation Method

#### 2.5. Debentable Pricing Method Based on Machine Learning Model

#### 2.6. Review of GAN Model Applications in Various Fields

#### 2.7. Overview

## 3. Research Design

#### 3.1. Pricing Theory of Convertible Bonds

#### 3.1.1. Traditional Convertible Bond Pricing Theory

- (1)
- B–S option pricing theory

- (2)
- Binary tree pricing method

- (3)
- Finite difference method

#### 3.1.2. Least Square Monte Carlo (LSM) Model

#### 3.1.3. Comparison of Traditional Pricing Methods

#### 3.2. Theoretical Analysis of Neural Network Model

#### 3.2.1. LSTM Model

- (1)
- Forget the door

- (2)
- Input door

- (3)
- Cell state

- (4)
- Output gate

#### 3.2.2. Generative Adversarial Network (GAN)

- (1)
- The basic principle of generating adversarial network model.

#### 3.2.3. Wasseratein Generated Network Model (WGAN)

#### 3.2.4. LSM Improved Model

## 4. Data and Empirical Analysis

#### 4.1. LSM Pricing Model

#### 4.1.1. Model Description

#### 4.1.2. Sample Data and Descriptive Statistics

#### 4.1.3. Parameter Estimation

- (1)
- Risk-free interest rate

- (2)
- Stock price volatility

- Descriptive statistics of the logarithmic rate of return of stock price;
- Test the stationarity of logarithmic rate of return;
- Autocorrelation and ARCH effect test;
- GARCH model is used to estimate stock price volatility, and unconditional stock price volatility is obtained.

^{−9}, and H0 hypothesis is rejected, indicating that the return rate of logarithm of stock does not obey the normal distribution.

#### 4.1.4. Empirical Results

- (1)
- Pricing results on the first day of listing

- (2)
- Multi-node pricing results within the duration

#### 4.2. Long and Short-Term Memory Network (LSTM) Pricing Model

#### 4.2.1. Model Description

- (1)
- Step Length

- (2)
- Hidden layer

- (3)
- Activation function

#### 4.2.2. Sample Data and Descriptive Statistics

#### 4.2.3. Data Preprocessing

- (1)
- Quarterly financial data/monthly macro indicators

- (2)
- Debt rating and subject rating

- (3)
- Normalization

#### 4.2.4. Empirical Results

#### 4.3. WGAN Pricing Model

#### 4.3.1. Model Description

- (1)
- Generator structure

- (2)
- Discriminator structure

#### 4.3.2. Empirical Results

#### 4.4. Improved Model of LSM

#### 4.4.1. Model Description

#### 4.4.2. Empirical Results

- (1)
- Generate sequence authenticity evaluation index

- (2)
- Pricing results of multi-node pricing within the duration

#### 4.5. Comparative Analysis of Pricing Effect of Each Model

## 5. Conclusions

- (1)
- The traditional LSM pricing model has a large error in the first-day pricing, indicating that the pricing function of this model needs to be further improved;
- (2)
- Among the four pricing models, the LSTM pricing model and WGAN pricing model have the best pricing effect. From the perspective of the MAPE index, the pricing effect of the WGAN pricing model (0.14%) is better than that of the LSTM pricing model (0.21%), and the pricing effect of the LSM improved model (1.17%) is better than that of the traditional LSM model (2.26%);
- (3)
- Applying the generative deep learning model GAN to the pricing of convertible bonds can avoid strict assumptions and significantly improve the pricing effect of the traditional model.

- (1)
- The factors affecting the pricing of convertible bonds can be selected more accurately and the prediction accuracy of the model can be improved;
- (2)
- Improve the Quant GANs model in the future, so that the generated samples can better reproduce the statistical characteristics of real sequences;
- (3)
- In order to avoid model failure on new data, we need to update the model in time and adjust and improve it according to new market conditions.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Name | Meaning |
---|---|

${B}_{t,T}$ | The theoretical price of the ordinary bond tranche |

I | The coupon rate for each interest payment date |

FV | Principal due for conversion |

R | Convertible bond yield to maturity (discount rate) |

T | The duration of a convertible bond |

Pricing Method | Advantage | Shortcoming |
---|---|---|

B–S model | (1) The thinking is clear and the analytical solution can be obtained (2) Easy to calculate | (1) Use only European options (2) Unable to reflect the value of the additional terms of the convertible bond |

Binary tree | (1) Ability to handle American options | (1) Large amount of computation (2) Additional clauses dealing with path dependence are relatively weak |

Finite difference method | (1) Ability to handle American options (2) By changing the terminal, more factors affecting the value of convertible bonds can be considered | (1) Large amount of computation (2) The method is limited by the partial differential equation of convertible bonds |

LSM simulation | (1) It can solve the problem of path dependence (2) Ability to handle American options | (1) The amount of calculation is large, and it needs to consume a large amount of memory, so it is difficult to achieve tens of thousands of simulations |

Security Code | Security Abbreviation | Term of Issue | Listing | Maturity | Transfer Price | Debt Rating | Subject Rating | Issue Size (100 Million) |
---|---|---|---|---|---|---|---|---|

128034.SZ | Jiangyin convertible bonds | 6 | 2018/2/14 | 2024/1/26 | 9.16 | AA+ | AA+ | 20 |

110059.SH | Shanghai Pudong Development convertible Bond | 6 | 2019/11/15 | 2025/10/27 | 15.05 | AAA | AAA | 500 |

110043.SH | Wuxi convertible bond | 6 | 2018/3/14 | 2024/1/30 | 8.9 | AA+ | AA+ | 30 |

110053.SH | Su Yin convertible bond | 6 | 2019/4/3 | 2025/3/13 | 7.9 | AAA | AAA | 200 |

110079.SH | Hangyin convertible bond | 6 | 2021/4/23 | 2027/3/28 | 17.06 | AAA | AAA | 150 |

113011.SH | Everbright bonds | 6 | 2017/4/5 | 2023/3/16 | 4.36 | AAA | AAA | 300 |

113021.SH | Citic bonds | 6 | 2019/3/19 | 2025/3/3 | 7.45 | AAA | AAA | 400 |

113042.SH | On the silver bond | 6 | 2021/2/10 | 2027/1/24 | 11.03 | AAA | AAA | 200 |

113050.SH | South Bank convertible bonds | 6 | 2021/7/1 | 2027/6/14 | 10.1 | AAA | AAA | 200 |

113052.SH | CCB convertible bond | 6 | 2022/1/14 | 2027/12/26 | 25.51 | AAA | AAA | 500 |

113055.SH | Convert bonds into silver | 6 | 2022/4/06 | 2028/3/2 | 14.53 | AAA | AAA | 80 |

113056.SH | Heavy silver bond | 6 | 2022/4/14 | 2028/3/22 | 11.28 | AAA | AAA | 130 |

113062.SH | Changyin convertible bond | 6 | 2022/10/17 | 2028/9/14 | 8.08 | AA+ | AA+ | 60 |

113065.SH | Qilu convertible bond | 6 | 2022/12/19 | 2028/11/28 | 5.87 | AAA | AAA | 80 |

113516.SH | Sunong convertible bonds | 6 | 2018/8/20 | 2024/8/2 | 6.34 | AA+ | AA+ | 25 |

128048.SZ | Zhang Bank convertible bonds | 6 | 2018/11/29 | 2024/11/12 | 6.06 | AA+ | AA+ | 25 |

128129.SZ | Green farmers swap bonds | 6 | 2020/9/18 | 2026/8/24 | 5.74 | AAA | AAA | 50 |

Security Abbreviation | Debt Rating | Listing | Risk-Free Rate (%) |
---|---|---|---|

Jiangyin convertible bonds | AA+ | 2018/2/14 | 5.6003 |

Shanghai Pudong Development convertible Bond | AAA | 2019/11/15 | 3.8117 |

Wuxi convertible bond | AA+ | 2018/3/14 | 5.4230 |

Su Yin convertible bond | AAA | 2019/4/3 | 4.0120 |

Hangyin convertible bond | AAA | 2021/4/23 | 3.6957 |

Everbright bonds | AAA | 2017/4/5 | 4.4287 |

Citic bonds | AAA | 2019/3/19 | 4.0818 |

On the silver bond | AAA | 2021/2/10 | 3.7587 |

South Bank convertible bonds | AAA | 2021/7/1 | 3.6836 |

CCB convertible bond | AAA | 2022/1/14 | 3.2402 |

Convert bonds into silver | AAA | 2022/4/06 | 3.4224 |

Heavy silver bond | AAA | 2022/4/14 | 3.3653 |

Changyin convertible bond | AA+ | 2022/10/17 | 3.0916 |

Qilu convertible bond | AAA | 2022/12/19 | 3.3693 |

Sunong convertible bonds | AA+ | 2018/8/20 | 4.8510 |

Zhang Bank convertible bonds | AA+ | 2018/11/29 | 4.5052 |

Green farmers swap bonds | AAA | 2020/9/18 | 3.6520 |

**Table 5.**Descriptive statistics of the logarithmic yield rate of Jiangbank convertible bonds and regular shares.

Name | Mean | Max | Min | STD | Jarque-Bera |
---|---|---|---|---|---|

Jiangyin convertible bonds | −0.001116 | 0.09561 | −0.10552 | 0.03721 | 38.85 (3.7 × 10^{−9}) |

Name | ADF | p-Value |
---|---|---|

Jiangyin convertible bonds | −13.279 | 0.00 |

Name | Lagrange Multiplier Statistics | p-Value |
---|---|---|

Jiangyin convertible bonds | 27.649 | 0.00 |

Security Code | Security Abbreviation | Historical Volatility | GARCH(1,1) |
---|---|---|---|

128034.SZ | Jiangyin convertible bonds | 0.5896 | 0.5372 |

110059.SH | Shanghai Pudong Development convertible Bond | 0.2140 | 0.2239 |

110043.SH | Wuxi convertible bond | 0.5056 | 0.4960 |

110053.SH | Su Yin convertible bond | 0.2098 | 0.3323 |

110079.SH | Hangyin convertible bond | 0.3931 | 0.4133 |

113011.SH | Everbright bonds | 0.1581 | 0.1602 |

113021.SH | Citic bonds | 0.2493 | 0.2652 |

113042.SH | On the silver bond | 0.1858 | 0.1871 |

113050.SH | South Bank convertible bonds | 0.3129 | 0.3123 |

113052.SH | CCB convertible bond | 0.3473 | 0.3467 |

113055.SH | Convert bonds into silver | 0.3495 | 0.3532 |

113056.SH | Heavy silver bond | 0.2510 | 0.2505 |

113062.SH | Changyin convertible bond | 0.2957 | 0.2954 |

113065.SH | Qilu convertible bond | 0.2859 | 0.2866 |

113516.SH | Sunong convertible bonds | 0.3746 | 0.3764 |

128048.SZ | Zhang Bank convertible bonds | 0.5175 | 0.5175 |

128129.SZ | Green farmers swap bonds | 0.4297 | 0.4361 |

Security Abbreviation | Actual Price | Estimated Price | Error Rate |
---|---|---|---|

Jiangyin convertible bonds | 97.7600 | 112.0035 | 14.56% |

Shanghai Pudong Development convertible Bond | 103.8901 | 107.4661 | 3.44% |

Wuxi convertible bond | 97.3947 | 115.2294 | 18.31% |

Su Yin convertible bond | 109.1290 | 117.3514 | 7.53% |

Hangyin convertible bond | 113.9763 | 120.0042 | 5.28% |

Everbright bonds | 102.9918 | 108.7224 | 5.56% |

Citic bonds | 107.9877 | 112.1466 | 3.85% |

On the silver bond | 100.1368 | 105.6658 | 5.52% |

South Bank convertible bonds | 120.2712 | 120.9079 | 0.52% |

CCB convertible bond | 111.0496 | 115.2653 | 3.79% |

Convert bonds into silver | 121.7108 | 124.3157 | 2.14% |

Heavy silver bond | 104.8973 | 109.5403 | 4.42% |

Changyin convertible bond | 118.7719 | 120.6737 | 1.60% |

Qilu convertible bond | 95.1164 | 111.7812 | 17.52% |

Sunong convertible bonds | 99.1781 | 118.5775 | 19.56% |

Zhang Bank convertible bonds | 102.3814 | 122.2723 | 19.42% |

Green farmers swap bonds | 109.0568 | 119.8994 | 9.94% |

Date | GARCH(1,1) | Actual Price | Estimated Price | Error Rate |
---|---|---|---|---|

2022/2/21 | 0.1756 | 105.9092 | 102.7697 | −2.96% |

2022/3/21 | 0.1684 | 104.7241 | 102.1407 | −2.46% |

2022/4/20 | 0.1563 | 104.8308 | 102.3051 | −2.40% |

2022/5/20 | 0.1513 | 105.3475 | 102.4839 | −2.71% |

2022/6/20 | 0.1559 | 105.0701 | 102.8459 | −2.11% |

2022/7/20 | 0.1559 | 105.4568 | 103.1392 | −2.19% |

2022/8/22 | 0.1568 | 106.3042 | 103.3065 | −2.81% |

2022/9/20 | 0.1588 | 106.3561 | 103.5676 | −2.62% |

2022/10/20 | 0.1578 | 106.3608 | 103.8714 | −2.34% |

2022/11/21 | 0.1618 | 105.3742 | 102.8606 | −2.38% |

2022/12/20 | 0.1667 | 104.0123 | 103.2316 | −0.75% |

2023/1/20 | 0.1590 | 104.916 | 103.5403 | −1.31% |

Date | Actual Price | LSM | Error Rate | LSTM | Error Rate | WGAN | Error Rate | LSM Improved | Error Rate |
---|---|---|---|---|---|---|---|---|---|

2022/2/21 | 105.90 | 102.76 | −2.96% | 106.36 | 0.43% | 106.21 | 0.29% | 106.89 | 0.93% |

2022/3/21 | 104.72 | 102.14 | −2.46% | 104.97 | 0.24% | 105.01 | 0.28% | 104.51 | −0.19% |

2022/4/20 | 104.83 | 102.30 | −2.40% | 105.10 | 0.26% | 105.02 | 0.18% | 105.07 | 0.22% |

2022/5/20 | 105.34 | 102.48 | −2.71% | 105.42 | 0.08% | 105.35 | 0.01% | 104.30 | −0.99% |

2022/6/20 | 105.07 | 102.84 | −2.11% | 105.20 | 0.13% | 105.26 | 0.19% | 104.38 | −0.65% |

2022/7/20 | 105.45 | 103.13 | −2.19% | 105.52 | 0.07% | 105.51 | 0.05% | 104.81 | −0.61% |

2022/8/22 | 106.30 | 103.30 | −2.81% | 106.61 | 0.30% | 106.64 | 0.32% | 103.80 | −2.35% |

2022/9/20 | 106.35 | 103.56 | −2.62% | 106.19 | −0.15% | 106.46 | 0.10% | 104.06 | −2.15% |

2022/10/20 | 106.36 | 103.87 | −2.34% | 106.13 | −0.21% | 106.29 | −0.06% | 104.12 | −2.10% |

2022/11/21 | 105.37 | 102.86 | −2.38% | 105.23 | −0.13% | 105.38 | 0.01% | 103.03 | −2.22% |

2022/12/20 | 104.01 | 103.23 | −0.75% | 104.27 | 0.25% | 104.00 | −0.01% | 103.55 | −0.44% |

2023/1/20 | 104.91 | 103.54 | −1.31% | 105.21 | 0.28% | 104.70 | −0.21% | 103.75 | −1.10% |

Mean | 105.38 | 104.35 | −2.26% | 105.52 | 0.13% | 105.49 | 0.10% | 104.35 | −0.98% |

Model Name | MAPE | RMSE |
---|---|---|

LSM | 2.26% | 2.4699 |

LSTM | 0.21% | 0.2463 |

WGAN | 0.14% | 0.1902 |

LSM improved | 1.17% | 1.4895 |

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**MDPI and ACS Style**

Ren, G.; Meng, T.
Research on Pricing Methods of Convertible Bonds Based on Deep Learning GAN Models. *Int. J. Financial Stud.* **2023**, *11*, 145.
https://doi.org/10.3390/ijfs11040145

**AMA Style**

Ren G, Meng T.
Research on Pricing Methods of Convertible Bonds Based on Deep Learning GAN Models. *International Journal of Financial Studies*. 2023; 11(4):145.
https://doi.org/10.3390/ijfs11040145

**Chicago/Turabian Style**

Ren, Gui, and Tao Meng.
2023. "Research on Pricing Methods of Convertible Bonds Based on Deep Learning GAN Models" *International Journal of Financial Studies* 11, no. 4: 145.
https://doi.org/10.3390/ijfs11040145