Pricing Convertible Bonds Based On Black-Shcoles Formula
Advances in Social Science, Education and Humanities Research, volume 2044th International Conference on Economics, Social Science, Arts, Education and Management Engineering (ESSAEME 2018)Pricing Convertible Bonds Based on Black-Shcoles FormulaXin Du1, a, *, Lian Chen1, b1School of Economics and Management, Nanjing University of Science and Technology, Xuanwu District,Nanjing, Chinaadx 1992@foxmail.com, bllant116@yahoo.com*Corresponding authorKeywords: Convertible Bonds, Black-Scholes Formula, Risk NeutralAbstract: The article first introduces that convertible bonds is also a kind of financial derivativeswhich can be priced through using the Black-Shcoles formula, and derives PDE satisfied byconvertible bond prices, then calculates the pricing formula of convertible bonds, finally, uses theformula to calculate the theoretical price of a convertible bond in the market and compare the pricewith the market price. The average error rate between them is 6.61%, it shows that the pricingformula can accurately price convertible bonds. In the conclusion, it is taken into consideration thatintroducing stochastic interest rate model into the pricing equation in the future.1. IntroductionConvertible bonds is a type of bond that can be converted into the stock of the company thatissues the bond, when pre-agreed conditions are satisfied, and it usually has a lower coupon rate. Inessence, convertible bonds are based on the common corporate bonds, and with stock option whichallow purchasers to convert their purchased bonds into shares of companies that issues theconvertible bonds.Therefore, convertible bonds can be seen as a kind of financial derivative, and its target asset isthe company stock that it can convert into. As financial derivative, it is very important to accuratelyprice convertible bonds.Because convertible bonds contain stock options, the pricing of convertible bonds can refer tothe pricing of options. Among the pricing methods of options, the most famous is the Black-Scholesequation proposed by Black and Scholes (1973)[1]. This equation is a milestone in the pricing offinancial derivatives, because its pricing method can be used not only for option pricing but also forthe pricing of other financial derivatives, including the pricing of convertible bonds. Next, thearticle will use the method to price convertible bonds.2. Equation DerivationFirst, assume that the price ππ of the convertible bond is a function of the stock ππ and time π‘π‘,so that ππ ππ(ππ, π‘π‘). The stock price ππ satisfies the following stochastic process:ππππππππππ ππππππ ππππππ(1)Among them,is the stock return rate, ππ is the drift rate of return rate, ππ 2 is the variance ofππthe return rate, π§π§ is the Wiener process with a drift rate of 0 and a variance ratio of 1.According to ItΕ's lemma ππππ ππππ 1 2 ππ2 ππ 2ππ 2 ππ 2 ππππ ππππππππ(2)It can be seen that ππππ contains the risk termππππππππ. In order to eliminate the risk term, a combination ππ of convertible bonds and stocks can be constructed.Copyright 2018, the Authors. Published by Atlantis Press.This is an open access article under the CC BY-NC license 47
Advances in Social Science, Education and Humanities Research, volume 204Letππ ππ and then get ππ At this time, the asset portfolio is risk-free within the time π‘π‘, so So 1 2 ππ2 ππ 2ππ(3)ππ 2 ππ 2 π‘π‘(4) ππ ππππ π‘π‘(5)1 2 ππ2 ππ 2Soππ 2 ππ 2 π‘π‘ ππ( ππ 1 2 ππ ππ) π‘π‘(6)ππ 2 ππ 2 ππππ(7)ππ max(ππππππ , πΉπΉ), when π‘π‘ ππ(8) ππππ 2 ππ 2(7) is the PDE satisfied by the price ππ of the convertible bond.The options contained in the convertible bond are American call options, according toHongzhong Liu(2003)[3], American call option will not be executed before the maturity date. So theholder of the convertible bond will not be executed ahead of time. The boundary condition of theconvertible bond isAmong them, ππ is the expiration time, ππ is the conversion ratio, and πΉπΉ is the face value of thebond, usually taken as 100.3. Equation SolvingWith an appropriate transformation, the explicit solution of the equation can be found, which isthe expression of the convertible bond price.We can see that the above PDE does not contain any variables that are affected by investor riskpreference, so we can assume that all investors are risk neutral. This means that the required returnon investment by investors is the risk-free interest rate ππ.Therefore, for the price of the initial convertible bond v, the following formula is right:v ππ ππππ E[max(ππππππ , πΉπΉ)]Through appropriate deformation, we can get:Let πΊπΊ ππππππSubstituting (11) into (2), getv ππ ππππ πΉπΉ ππππ ππππ E[max(ππππ πΉπΉ/ππ, 0)] 1ππ, 2 πΊπΊ ππ 2 Soππππ ππ Soππππππππ ππ 12ππ 22(10) 0(11) ππππ ππππππ(12) ππππ ππππππ(13)ππ 2ππ 2 (9), 148
Advances in Social Science, Education and Humanities Research, volume 204According to risk neutral, take ππ ππIntegrate (15)ππππππππ ππππππ0 ππ( ππ ππππππππ ππ(ππππππ0 ππ ππ 22ππ 22 ππ, ππ 2 ππ)(14) ππ, ππ 2 ππ)(15)πΉπΉπΉπΉππ ππππ E max ππππ , 0 ππ0 ππ(ππ1 ) ππ ππππ ππ(ππ2 )ππππv ππ ππππ πΉπΉ ππππ0 ππ(ππ1 ) ππ ππππ πΉπΉπΉπΉ(ππ2 )Andππ1 ππ2 ππππ(ππππ0 πΉπΉ) ππ ππ 2 2 ππππ ππππππ(ππππ0 πΉπΉ ) ππ ππ 2 2 ππππ ππ(16)(17)(18)(19)ππ(π₯π₯) is the cumulative probability distribution function of the standard normal distribution. Inthis way, the expression of the convertible price is obtained.It can be seen that ππ ππππ πΉπΉ in the formula is the value of the bond part, and ππππ0 ππ(ππ1 ) ππ ππππ πΉπΉππ(ππ2 ) is the value of the call option contained in the convertible bond. So the formula isessentially decomposing convertible bonds into bonds and options and pricing them separately. Ifthe convertible bonds pay interest, the present value of the unpaid interest rate should also be added,and it is recorded as ππ. So there is4. Empirical Researchingv ππ ππππ πΉπΉ ππππ0 ππ(ππ1 ) ππ ππππ πΉπΉπΉπΉ(ππ2 ) ππ(20)This paper uses β110030.SHβ as an example to calculate the theoretical price of the convertiblebonds from 2018/1/2 to 2018/1/31 by using the above formula, and compares them with marketprices.Table 1 Results of empirical researchingDateStock Price Theoretical Price Market Price Error 96.57103.3210109.67216.15%149
Advances in Social Science, Education and Humanities Research, volume 579109.74385.37%6.30%The stock price is the closing price of the corresponding date. According to the terms of theconvertible bond, the conversion price is 7.24. The risk-free interest rate is 3.82%, the stock annualvolatility is calculated to be 0.2922, face value of 100, maturity date of 2 years, one-year interest of1.5 and biennial interest of 2. The results are shown in the Table 1 and Figure 8/1/272018/1/282018/1/292018/1/302018/1/3198Market priceTheoretical priceFigure 1 Market price and theoretical price5. ConclusionIt can be seen that there is a certain gap between the market price and the theoretical price,resulting in a variety of reasons for the error, such as the error caused by the volatility estimation,the model setting error and the error caused by the risk-free interest rate fluctuation. The model canbe further modified, for example taking into account the effects of interest rate fluctuations, andintroducing the stochastic interest rate model into the pricing formula of the convertible bond.The average error rate calculated is 6.61%, and the error rate is small, indicating that this formulacan accurately price convertible bonds.References[1] Black F, Scholes M. (1973) The Pricing of Options and Corporate Liabilities. Journal ofPolitical Economy, 81, 637-659.[2] Brennan M.J., Schwartz E.S. (1980) Analyzing convertible bonds. Journal of Financial andQuantitative Analysis, 15(4), 907-929.[3] Hongzhong Liu. (2003) Investments. Higher Education Press, 288-289.[4] Feng Ye. (2004) The Study of Pricing Chinaβs Convertible Bonds. Fudan University.[5] Qiβnan Lai, Changhui Yao and Zhicheng Wang. (2005) Empirical Study on the Pricing ofConvertible Bonds in China. Journal of Financial Research, 9, 105-121.[6] Qiyuan Zhou. (2007) Research on Pricing and Analyzing Convertible Bonds. Shanghai JiaoTong University.150
Advances in Social Science, Education and Humanities Research, volume 204151
Convertible Bonds, Black-Scholes Formula, Risk Neutral . Abstract: The article first introduces that convertible bonds also a kind of financial derivatives is which can be priced through using the Black-Shcoles formula, and derives PDE satisfied by convertible bond prices, then calculates the pricing formula of convertible bonds, finally, uses the
The rst and only time so far that contingent convertible bonds were used as a regulatory instrument was in Switzerland, when the Credit Suisse Group AG issued 2 billion of these new bonds on February 14th, 2011. The coupon payments of the contingent convertible bonds were substantially higher than for normal debt: 7.875% vs. 4% on average.
Bonds written by an insurance company for construction projects are referred to as contract surety bonds. The main types of contract surety bonds are: bid bonds, performance bonds, payment bonds, and warranty bonds (sometimes called maintenance bonds).The two basic functions of these bonds are: Prequalificationβassurance that the bonded
public safety as are secured bonds. Unsecured bonds are as effective at achieving court appearance as are secured bonds. Unsecured bonds free up more jail beds than do secured bonds because: (a) more defendants with unsecured bonds post their bonds; and (b) defendants with unsecured bonds have faster release-from-jail times.
Convertible Top 156 The convertible top and tonneau cover movement can cease in any of the positions 1, 2 or 3, as shown. If the movement ceases in position (1), release the tension in the convertible top cover and tension bow (rear of convertible top) by lifting the front of the
3. VALUATION OF Bonds AND Stock Objectives: After reading his chapter, you will 1. Understand the role of bonds in financial markets. 2. Distinguish between different types of bonds, such as zero-coupon, perpetual, discount, convertible, and junk bonds and apply the bond pricing formulas to evaluate these bonds. 3.
A convertible bond is a corporate bond that is (irrevocably) convertible at the holderβs option into a specified number of equity shares, whereas an exchangeable bond is convertible into shares of a different corporate entity. Hence, a convertible
In periods of elevated volatility, the asymmetrical profile of convertible bonds is therefore generally appealing to investors. Higher equity prices boost the value of the convertible bond. On the other hand, lower equity prices push the conversion option out of the money and make the convertible
Perfusionists certified by the American Board of Cardiovascular Perfusion through December 31, 2021. LAST FIRST CITY STATE COUNTRY Al-Marhoun Sarah New Orleans LA Alouidor Benjamin Los Angeles CA Alpert Bettina P. Marlborough MA Alpha Debra Reynolds Zionville IN Alshi Hanin Nooraldin H. Jeddah MA Saudi Arabia
