A Note about the Black-Scholes Option Pricing Model under Time-Varying Conditions Yi-rong YING and Meng-meng BAI

Transcription

1 2017 2nd International Conference on Advances in Management Engineering and Information Technology (AMEIT 2017) ISBN: A Note about the Black-Scholes Option Pricing Model under Time-Varying Conditions Yi-rong YING and Meng-meng BAI College of Economics, Shanghai University, Shanghai, , China Keywords: Convertible bonds (CBs), Black-Scholes (BS) option pricing model, Partial differential equation Abstract By solving the second-order parabolic partial differential equation, the BS option pricing formula can be obtained; The BS pricing model under time-varying conditions is hard to get an explicit solution; We found that there is a same mistake in Zhige Ren s paper and Xiaochen Han s paper, and trying to put forward an idea to solve a sort of second-order parabolic partial differential equations Introduction In 1843, the American New York railway company issued the world s first convertible corporate bonds But during the next one hundred years, CBs in the rapid development of the securities market didn t get attention it deserves, only a few investment funds and hedge funds scattered all over the world focus on CBs In recent years, however, global CBs market has become increasingly mature and prosperity Especially the nature of the CBs with bonds and options, provide investors with a steady new choice The bond code Name of the bond Table 1 CBs listed on the nearly three years Launch date Repayment period (month) Unit value Coupon rate (%) Circulation (millon yuan) Interest rate type Geli CB 2015/01/ A Hangxin CB 2015/06/ A Sanyi CB 2016/01/ A Guomao CB 2016/01/ A Jiuzhou CB 2016/01/ A Baiyun CB 2016/03/ A Dianqi CB 2015/02/ A Guangqi CB 2016/02/ A Jiangnan CB 2016/04/ A Lanbiao CB 2016/01/ A Ge er CB 2014/12/ A Shunchang CB 2016/02/ B Qimu CB 2016/03/ A Huifeng CB 2016/05/ B Hongtao CB 2016/08/ B Denote: A stands for progressive rate type; B stands for fixed interest rate with interest-bearing Looked from the development trend of overseas convertible bond market, we found that the traditional CBs based on individual stocks, while modern CBs use stock index as the underlying securities In addition, from the perspective of the company's financing, convertible bonds provide a new refinancing mode for the company Therefore, the total value of the convertible bond markets hit a new record all over the world Despite Chinese CBs market is not mature currently, the convertible corporate market has become quickly and relatively important It is worth pointing out that even though the number of 372

2 issues is rather small, the issue sizes are typically very large Besides, convertible bonds are highly-hybrid financial derivatives They are important refinancing tools for listed companies, and essential investment targets among hedge funds and other institutional investors Most literature in pricing convertible bonds utilizes numerical methods including finite difference (Ayache et al, 2003), finite element, lattice-based and simulation methods (Ammann et al, 2008) Other studies on this topic choose the BS model, and typically substitute parameters with the coupon rate and historical volatility (Lai et al, 2005), rather than calibrating from data, and some use jump-diffusion models (Hua and Cheng, 2009) The BS partial differential equation may be the most important equation in the theory of finance because the dynamic process of all portfolios must satisfy it According to the specific problem and plus some essential boundary conditions, the BS pricing mode forms common theory to solve the pricing problem of general derivative financial products Therefore, this paper bases on the BS Partial differential equation and makes some supplements and validation And, we found that there is a same mistake in Ren s paper (Zhige Ren, 2015) and Han s paper (Xiaochen Han, 2015), and trying to put forward an idea to solve a sort of second-order parabolic partial differential equations Models Option is to give its owners within the prescribed time limit according to the specific price to buy or sell a certain quantity of a particular commodity, the option price is the option buyer paying to the seller for options contracts, option rights The rationality of the option price directly affects the investment risk, so the option pricing problem is an important aspect in the study of financial derivatives The BS option pricing model are widely used in the option pricing problem There are several BS option pricing assumptions: (i) stock volatility expectations during the option period is unchanged; (ii) the market there is no friction, namely, there is no tax and transaction costs; (iii) the validity period of financial assets in the options without bonuses and other income; (iv) the risk-free interest rate r is a constant; (v) the underlying asset price satisfy the geometric Brownian motion: = μ dt + σ dw (21) In the formula(21), signals underlying assets, [0,T], says the option maturity date, is expected to yield, signs for volatility, is a standard Brownian motion, E() = 0, Var() = Under the condition of above assumptions, BS option pricing model is as follows: + rs + δ s rc = 0 (22) Note: (, ) says the price of European call option in time In the long-term investment, risk-free interest rate is variable and the longer the time, the greater the risk, risk-free interest rate is smaller To start with, the risk-free interest rate volatility will be very big, but after reaching a certain value, the risk-free interest rate volatility will be increasing smaller till close to zero Therefore, the change characteristics of risk-free interest rate in accordance with exponential function According to the change characters of risk-free interest rate, this paper based on the BS model and assumed that risk-free interest rate is changing while other conditions keep still In addition, we assuming the risk-free interest rate and stock price, volatility is not relevant; the value of the risk-free rate in the time is ; the initial value for =0 is treasury rates, the assumption for expression as follows: r = r αe (23) Based on the principle of hedge and It o lemma theory knowledge, we gain the improved BS differential equation: 373

3 + (r α exp s + δ s r α exp c = 0 (24) According to the stochastic partial differential equations to solve this differential equation, Ren (2015) argued that the pricing formula of European call option is: c(s, t) = s N(d ) K exp r α exp (T t) N(d ) (25) = ( )(), = Han (2016) expanded general BS option pricing model into BS mode with paying dividends (the interest rate is not fixed); There are three assumptions: 1 The stock price changes on the basis of random equation: = () + () ; 2 The risk-free interest rate is = () ; 3 The dividend rate is = () ; Finally, we gain the continuous BS option pricing model: + () S + r(t) q(t)s r(t) C = 0 (26) The European call option with paying dividends is: C(S, t) = Se d = Analysis () ()()() () N(d ) Ke, d = () N(d ) (27) ()()() () = d σ (u)du We found that there is a mistake in Ren s paper A Black-Scholes option pricing mode based on the risk-free interest rate under varying conditions In fact, the BS option pricing equation is hard to get an explicit solution And it s not hard to see, the above (24) is the second order parabolic partial differential equation with variable coefficients, and its solution can see Kazemi (2017), the function (, ) = (); = ( ) Using infinite series method, however, a shortcoming is to convert the original equation for a set of equations, the solving process is quite tedious and complicated, not to mention to prove the convergence of the solution Denote R= r α exp, it is easy to calculate partial derivatives about formula (25): = ( ) () se Ke () Ke () R + 1 N(d ) + () e, (31) = N(d ) + () e (), (32) = ( ) 1 e + K e () d () ( ) + (33) () 374

4 ( ) () = (), ( ) = ( ) + () (), ( ) = ( ) = () () We simultaneous above formulas(31) (32) (33) and (25), then deduce the results: h, = () () () h, = ( ) ( ) + ( ) () ( ), ( ) Obviously we can see that if 0, then h, 0 is wrong That is to say, Ren (2015) derived the wrong formula in his paper In the same way, we prove the formulas in Han s paper as follows: () = () ( ) + ( ) () () ( ) + ( ), () = ( ) + () () (), ( ) = ( ) = () = () ( ) 1 () + () (), ( ) = ()()() () h(, ) = h (, ), h (, ) = () h (, ) = () h (, ) = () + () ()()() () () ()()() () () () () () (1 + ); +, ( ) () () = ( ) + () () () () Obviously, all h (, ),h (, ), h (, ) equal zero at the same time is almost impossible, therefore, we believe there is a same mistake in Han s paper Conclusions The initial BS partial differential equation is intuitive linear and easy to deduce result But obviously this paper analyzes the BS model under time-varying condition which is a nonlinear equation that can t use general method to compute The developed method is the infinite series 375 ;

5 method which turns the BS option pricing model into a set of equations To be honest, the solving process is tedious and complicated, let alone trying to prove its convergence The method to correct Ren s wrong formula should be establish a set of comparison principle, then use advantage function gradually approaching to gain second-order parabolic equation solution of time-varying coefficients Acknowledgement This work was supported by National Natural Science Foundation of China ( ) References [1] Ammann, M, Kind, A, Wilde, C, Simulation-based pricing of convertible bonds, Journal of Empirical Finance 15 (2008) [2] Ayache, E, Forsyth, PA, Vetzal, KR, Valuation of convertible bonds with credit risk, J Derivatives 11 (2003) 9 29 [3] Hua, H Y, Cheng, X J, Pricing Convertible Bond Based on the Jump-diffusion Process, Application of Statistics and Management (in Chinese) 28 (2009) [4] Kazemi, Mehdi Dehghan, Ali Foroush Bastani, Asymptotic expansion of solutions to the Black-Scholes equation arising from American option pricing near the expiry, Journal of Computational and Applied Mathematics 311 (2017) [5] Lai, Q N, Yao, C H, Wang, Z C, An Empirical Study of Convertible Bonds, Journal of Financial Research (in Chinese) 9 (2005) [6] Xiaochen Han, Jun Li, Yangyang Hou, Partial differential equations in the application of option pricing, Journal of Liaoning Technical University (Natural Science) 35 (2016) [7] Zhige Ren, Lang He, Zhangcan Huang, A Black-Scholes option pricing mode based on the risk-free interest rate under varing conditions, Mathematical journal (in Chinese) 35 (2015)