International Conference on Computer Information Systems and Industrial Applications (CISIA 2015)

The Pricing for Warrant Bonds under Fractional Brownian Motion

F.Y. Chen, Y.Y. Tan Y.Q. Li Insurance Professional College, School of mathematics and Computational Science, Changsha, Hunan, PR China Changsha University of Science and Technology Changsha, Hunan, PR China

Abstract———Assuming that the underlying follows Fractional warrants by using martingale pricing theory under a complete Brownian motion and that stochastic interest rate meets the market. Z.H. Li(2008) [6] made a study of the pricing for Vasicek model of interest rates, this paper establishes pricing WBS of Bao steel issued in 2008 by using B-S model of the model of Warrant Bonds and deduces the pricing formula of dilution effect, and concluded that the pricing of the part Warrant Bonds by utilizing risk-neutral theory. Finally, of WBS of Bao steel performed well, but the price of the this paper analyzes influence of concerned parameters of pricing warrant is high. H. Luo, H.M. Shen [7] studied the practical model on the value of Warrant Bonds by using the numerical application of WBS in our country. D. Zhu (2011)[8] used simulations. martingale pricing methods to derive the pricing formula of WBS. But there is an inadequacy, that is assuming that the Keywords-warrant bonds; fractional brownian motion; ; underlying stock price follows Geometric Brownian motion, risk-neutral pricing theory then it means that the changes of stock price are independent I. INTRODUCTION random variables, and that rate on assets follows normal distribution, which does not accord with the actual fluctuation Warrant Bonds (referred WBS) is one kind of special of stock price. Because a large number of empirical studies convertible bonds which is traded separately in equity and behavioral finance researches have shown that the warrants and bonds, and which is usually issued by listed fluctuation of the stock price is not random walk, but exhibits company. WBS consists of two parts: bond and the stock varying degrees of long-range dependence in different time; warrant. WBS gives the holders the right to buy issuing and the distribution of asset returns is not normal, but is company's in a certain period of time prior to characterized by "high kurtosis and fat tail"; this is a typical for the price and exercise ratio provided by the feature of financial asset returns. issuing company under the premise of maintaining the validity of the bonds when it is issued. There is essential difference This paper will improve the pricing model of WBS between WBS and ordinary convertible bonds(CBS) which established by D. Zhu, and derive the pricing formula of WBS lies in separable transactions about bond and option. After the by using risk-neutral valuation theory. Since Peter (1994) has holders of the WBS exercise the warrants rights, their claims proved the assumption that Fractional Brownian motion(FBM) still exist, and they can still recover the principal by the does not depend on independence and normal distribution, its expiration date and earn the interest. However, the creditors’ characteristic index and scale parameters can describe well the rights of the investors of CBS do not exist after their rights of of the financial markets, stock prices process and the warrant are exercised. Thus, WBS can be understood as a kind characteristics of “high kurtosis and fat tail” of asset yields' of innovative financial product of "obtaining warrants while distribution and so on. In addition, its self-similarity and long- buying bonds". range dependence is consistent with people's intuition of the financial markets, that is to say, future stock price will depend When WBS first appeared in the United States in 1970s, on not only the current price of the stock, but also the price many foreign scholars believed that it would become an over a considerable period of time [9]. Therefore, in order to alternative of the ordinary convertible bonds. Ingersoll more objectively reflect the reality of financial markets, this (1977)[1]proposed that bonds and warrants be combined with paper utilizes FBM to capture the fluctuations of the a financing portfolio equivalent to convertible bonds. Finnerty underlying stock price. (1986)[2]conducted a questionnaire for institutional investors buying WBS. The results showed that the discount of initial II. PRELIMINARIES issuance price of WBS would bring about the tax burden to the The value of WBS consists of two parts: pure bond value investors. And Payne [3] made a comprehensively and value of of the underlying stock. There is comparative study of correlation between WBS and CBS. In formula of maturity cash flow as follows: recent years, Chinese scholars have also made some positive  PC⋅ exploration on the pricing of WBS. K. Xu and T. Li (2007) P , S ≤ b υ  b T M V =  [4]made a empirical analysis of the pricing for China's first T ⋅  Pb C υ PSC+α β ( −υ ) , S > WBS based on Ma an shan Iron convertible bonds. H.Y. Hua,  b T T M X.J. Cheng (2007) [5] considered that the warrant in WBS is a Where P= MeiT denotes pure bond value calculated simple Bermuda warrant. They gave a theoretical price of b through coupon rate i (constant), M denotes the face value

© 2015. The authors - Published by Atlantis Press 738 T of convertible bonds, C denotes the agreed exercise price r() t −θ −k() T − t =δ − −k() T − s υ A( t , T )= (1 − e ) +θ ( T − t ) Z( t , T )∫ k (1 e ) dB ( s ) k , t (the conversion price), T denotes time to expiration of α Warrant Bonds, S denotes the stock price at time T , T T ∫ r()(,)(,) u du= A t T + Z t T denotes the ratio of the warrants attached to bonds(the Hence, t (5) amount of warrant which bonds subscribers can be free to obtain while buying one bond), and β denotes the right From the literature [10], proportion(the number of underlying stock which one T − − warrant assures the holders can purchase). var[(,)]Z t T= E [ Z2 (,)] t T = [(1δ − ek( T s ) )] 2 ds ∫t k =δ 2 − −−2k ( T − t ) + − k ( T − t ) + − σ 2 III. MODEL 3 [ 3e 4 e 2 k ( T t )]A ( t , T ) 2k and = , Namely, σ 2 A. The Pricing Model for Warrant Bonds E[ Z ( t , T )] 0 Z( t , T ): N (0, ( t , T )) (1) Let risk-free interest rate r() t be random, and meet the IV. THE PRICING FORMULA OF WARRANT BONDS UNDER Vasicek model commonly used in finance, FRACTIONAL BROWNIAN MOTION

=θ − + δ (1) A. Lemma 2 drt() k ( rtdt ()) dBt () [11] Provided ZN1 : (0,1) , ZN2 : (0,1) , Where k means reversion rate, θ denotes long-term CZZov( , ) = ρ , for any real number a,,,, b c d s , then the θ δ 1 2 interest mean, k,, are positive constants, B() t is a standard following equality is obtained: Brownian motion.

+ 1 2+ 2 + ρ + +ρ + − (2) The basic stock price corresponding to Warrant Bonds cZ1 dZ 2 2(c d 2 cd ) ac bd( ad bc) s E( e I+ ≥ ) = e Φ( ) follows Fractional Brownian motion process, and satisfies the {}aZ1 bZ 2 s a2+ b 2 +2ρ ab following differential equation

x − 1 s2 Where Φ = 1 2 denotes the function of dSt()= µ ()() tStdt+ σ StdBt () (), 0≤ tT ≤ ()xπ e ds H (2) 2 ∫−∞ standard normal distribution, proof refers to literature [11]. Where µ()t denotes the expected rate of return on stock B. Theorem σ price at time t , denotes the stock price volatility, µ()t is a Assuming that random interest rate meets Vasicek rate =ω > model and the underlying stock price follows the Fractional function of t ; BHH{ B ( t , ), t 0} is a FBM whose Hurst parameter is H , T denotes time to expiration, assuming Brownian motion process, the value of Warrant Bonds at any B() t and B() t are mutually independent. In the risk-neutral time t before expiration is: H

µ = 2 2H 2 H PC⋅ υ world, we obtain ()()t r t , then equation (2) can now be 1 2 1σ − −b + − A(,)(,)t T+ σ t T  2 (T t ) lnMS⋅ A ( t , T )  V( t , T , r ( t ), S ( t )) = P e 2 +αβ S() t Φ t  written as follows: b σ2(,)()t T+ σ 2 T 2HH − t 2   PC⋅ 2 b υ − +σ2 +1 σ 2 2H − 2 H − + 1σ ln⋅ A ( t , T ) ( t , T ) ( T t ) A(,)(,)t T2 t T  MSt 2  −αβCυ e Φ −  dSt()= rtStdt ()()+ σ StdB () (), t 0≤ t ≤ T σ2(,)()t T+ σ 2 T 2HH − t 2 H (3)   (6)

The following differential equation can be obtained by Proof. From the risk-neutral pricing theory using the Itoˆ formula: T −∫ r() u du V( t , T , r ( t ), S ( t )) =E[ et ⋅ V | F ] 2 2H − 1 T t dStln ( )= ( rt ( ) −σ Ht) dt + σ dBt( ) T T H −∫ r() s ds −∫ r() u du = t + t +αβ − E[ e P IPC⋅ υ | F ] E[ e ( P ( S Cυ )) IPC⋅ υ | F ] b {}S ≤ b t b T {}S > b t Accordingly, for all 0 ≤t ≤ T , T M T M

T Let V1 andV2 respectively denote the first part and second = −1 σ 2 2HH − 2 +σ − STSt() ()exp{∫ rsds ()2 ( T t ) ( BTBtHH () ())} t (4) part above formula.

B. Lemma 1 First, calculateV1 ,

[8] Solution of stochastic differential equation (1) is T −∫ r() s ds − − = t = A(,)(,)t T Z t T VPEe1 b [ I≤ PCb⋅ υ | FPet ] b Ee [ I≤ PCb⋅ υ | Ft ] u {}ST {}ST = −θ−k() u − t + θ + δ − k() u − s , for t≤ u ≤ T M M r()(()) u r t e ∫ e dB() s −A(,)(,) t T − Z t T t = P e E[ e I ⋅ |F ] b {exp[(,)AtT−1σ 2 ( T 2HH − t 2 ) + ZtT (,) +σ ( BTBt () − ())] ≤ PCb υ } t 2 HH MS⋅ t T T and =r() t −θ −−k() T − t +θ − +δ − −k() T − s ∫ rudu()k (1 e )() Tt∫ k (1 e )() dBs 2 t t Notice that: Z( t , T ): N (0,σ ( t , T )) , Proof refers to literature [8] − 2HH− 2 BH ( T ) BH ( t ): N (0, T t ) [12] Let

739 − Z(,) t T BHH()() T B t Geometric Brownian motion (hereafter GBM) and FBM. And then σ (,)t T : N(0,1) : N (0,1) , T2HH− t 2 These tests will consist of some simulation of the above two pricing models with some chosen parameters. In the second set, And since and are mutually independent, B() t BH () t we analyze the influences of different parameters in FBM − Z(t,) T BHH()() T B t model on the value of WBS. The following results and thenCov( σ ,) = 0. (,)t T T2HH− t 2 concerned figures are obtained by using Matlab. Thus, from Lemma 2, get: A. Comparison of WBS Prices Calculated by GBM Model and FBM Model

− −σ (,)t T Z(,) t T = A(t,) T σ (,)t T Now, for an illustration of the differences between the two V Pe E[ e I − ⋅ |F ] 1 b σ⋅Z(,) t T + σ 2HH − 2 ⋅BHH()() T B t ≤PCb υ − +1σ 2 2HH − 2 t {(,)t Tσ (,)t T T t lnMS⋅ A ( t , T )2 ( T t )} T2HH− t 2 t models: GBM and FBM, first, we compute the theoretical PC⋅ υ 2 2HH 2 2 1 2 b − +1σ − +σ −A(,)(,) t T + σ t T  lnMS⋅ A ( t , T )2 ( T t ) ( t , T )  prices of WBS under the two models. Table 1 presents the = P e 2 Φ t  b σ2(,)()t T+ σ 2 T 2HH − t 2 parameters for computing the theoretical prices of WBS. Apart   (7) from H, the other corresponding parameters about GBM model and FBM model are the same. The prices computed by Then calculateV2 , the above two pricing models are also presented in Table 2,

where St denotes the underlying stock price at time t; PGBM T T − r() u du − r() u du ∫t ∫t denotes the prices of WBS computed by the GBM model; VP= ()(−αβ CEeυ I≥ | F )+αβ Ee ( SI≥ | F ) 2 b {}ST Cυ t T{} ST Cυ t (8) PFBM denotes the prices of WBS calculated according to FBM model. Let V and V respectively denote the first part and 21 22 By comparing columns P and P in Table 2, the second part of above formula (8), V is calculated by the same GBM FBM 21 conclusions that we come to are (1) no matter it is about the methods of calculation of V1 , get: low time or high time to maturity, the prices given by FBM model are larger than the prices given by GBM model. This is

PC⋅ 2 b υ − +1 σ 2 2H − 2H +σ 2 − + 1 σ ln⋅ A ( t , T ) ( T t ) ( t , T ) mainly because the value of stock warrant in WBS will A(,)(,) t T2 t T  MSt 2  V=() P − αβ Cυ e Φ   21 b σ2 (,)()t T+ σ 2 T 2HH − t 2 increase when Hurst parameter H is greater than 0.5; (2) in the   (9) low time to maturity case, the prices calculated by FBM model and GBM model are close to each other, the differences are From formula (5), get: small. However, in high time to maturity case, the prices

T computed by FBM model are larger than the prices computed − ∫ r() u du − − = αβ t = αβ A(,)(,)t T Z t T V Ee[ SIPC⋅ | F ] eEeSI[ PC⋅ | F ] by GBM model. This is mainly because the value of stock 22 T {}S > b v t T {}S > b v t T M T M warrant in WBS is the increasing function of the time to − A(,) t T−1σ2 ( T 2HH − t 2 )(() +σ B T − B ()) t =αβS()[ t eA(,) t T E e 2 HH maturity, so the prices of WBS will increase; (3) apparently, ⋅I |F ] with the increase of S , the prices obtained by GBM and FBM AtT(,)−1σ2 ( T 2HH − t 2 )(,)( + ZtT +σ BTBt () − ()) ⋅ t t 2 HH >PCb υ {e ⋅ } MSt will both increase too, but the increasing amount is very small. 2HH 2 BHH()() T− B t 2 2HH 2 σ T− t ⋅ − 1 σ ()T −t 2HH 2 This is because the value of the bonds which is the main = αβ S( t) e2 E[ e T− t component of WBS is very large and the value of the stock ⋅ I − ⋅ |F ] σ Z(,) t T +σ 2HH − 2 BHH()() T B t >PCb υ − +1σ 2 2HH − 2 t {(,)t T σ (,)t T T t lnMS⋅ A ( t , T )2 ( T t )} warrant is very small when compared with other derivatives. T2HH− t 2 t ⋅ 1σ 2 2HH− 2 −PCb υ +  2 (T t ) lnMS⋅ A ( t , T )  =αβ S() t Φ t TABLE I . THE VALUATION OF THE CHOSEN PARAMETERS USED IN THESE  σ2+ σ 2 2HH − 2   (,)()t T T t  (10) MODELS. α Model θ r δ i β k H M Cv VV, and V are respectively substituted in type 1 21 22 GBM 0.8 0.05 0.06 0.2 0.05 0.4 0.5 0.5 100 25 , get V( t , T , r ( t ), S ( t )) FBM 0.8 0.05 0.06 0.2 0.05 0.4 0.5 0.7 100 25 V( t , T , r ( t ), S ( t )) TABLE II .PRICING RESULTS OF DIFFERENT PRICING MODELS. 2 2HH 2 PC⋅ υ 1 2 1σ − −b + −A(,)(,) t T + σ t T  2 (T t ) lnMS⋅ A ( t , T )  = P e2 +αβ S() t Φ t Underlying Low time to maturity: High time to maturity: b  σ2+ σ 2 2HH − 2   (,)()t T T t  = = = = stock price: St T2, t 1 T4, t 1 ⋅ 2 PCb υ − +1σ 2 2H − 2H +σ 2 − A(,)(,)t T+ 1σ t T ln⋅ A ( t , T ) ( T t ) ( t , T ) 2  MSt 2  −αβCυ e Φ −  PGBM PFBM PGBM PFBM σ2(,)()t T+ σ 2 T 2HH − t 2   15 105.0691 105.3500 109.4193 109.8588 20 105.7199 106.0474 110.1828 110.6666 V. SIMULATION STUDIES 25 106.4081 106.7623 110.9542 111.4762 The aim of this section is to show how to implement FBM 30 107.1004 107.4789 111.7266 112.2861 model for WBS and illustrates the effects of parameters of 35 107.7941 108.1962 112.4989 113.0959 40 108.4952 108.9168 113.2716 113.9059 pricing model. For these purposes, let’s make a report on two 45 109.2090 109.6437 114.0453 114.7161 sets of numerical experiments. In the first set, we compare the 50 109.9388 110.3788 114.8208 115.5267 theoretical prices calculated by the following models:

740 B. The Influence of Parameters in FBM Model theory). For young Chinese securities market, WBS is still a In what follows, we will illustrate the influence of different new financial product which has great development parameters of FBM model on the value of WBS. For the sake potential. It will play a positive role in broadening financing of simplicity, we will just consider the in-the-money case. channels for enterprises, enriching variety of securities and Indeed, using the same method, one can also discuss the booming the securities market. Therefore, how to price remaining cases: out-of-the-money and at-the-money. We take scientifically for WBS will be very important, which is worth- a first look at the influence of different Hurst parameter H on while for us to explore and study together. the values of WBS and then consider the influence of other ACKNOWLEDGEMENTS parameters σ,C and T on the values of WBS. The concerned v The authors wish to thank editors and referees for their δ= = parameters are θ =0.05, k =0.8, k =0.8, 0.2,T=2, t 1, σ =0.8, valuable comments and suggestions. All remaining errors are = = = = α = β = = the responsibility of the authors. This research was supported r r(1) 6%, H 0.7, i 5%, 0.4, 0.5, M 100, by the National Natural Science Foundation of China (No. = 和 = It only takes less than 10s to generate all Cv 25 St 26, 11171044). the pictures in Figure.1 and Fig 2 on a high-performance computer. Not surprisingly, Figure.1 indicates :(1) the value of REFERENCES WBS is an increasing function of H, σ and T; and (2)the [1] Ingersoll J, A valuation of convertible securities Financial Economics, 1977, 4(3):289-322. increasing parameter of Cυ comes along with a decrease of the [2] Finnerty J D, The case for issuing synthetic convertible bonds Midland. value of WBS. Journal, 1986, 84(3):73-82 [3] BC Payne, Convertible Bonds and Bond-warrant Packages: Contrasts in Figure.2 clearly displays the Value of WBS under different Issuer Profiles. Atlantic Economics, 1995,23(2):82-85. times to maturity and exercise prices, namely, a three [4] K. Xu, T. Li, An Empirical Analysis of pricing for "Ma an shan Iron and dimensional functional image of the value of WBS with regard Steel" Warrant Bonds Journal of Management, 2007,11 (3): 816-819. to time to maturity and exercise price. Figure.2 indicates that [5] H.Y. Hua, X.J. Cheng, The study of separately trading convertible bonds. the value of WBS will increase with the increase of time to Graduate Journal of Chinese Academy of Sciences, 2008, 25 (4): 439-444. maturity and decrease with the increase of the exercise price. [6] Z.H. Li, The Pricing Research of Bao steel WBS. Chengdu, University of Electronic Science and Technology, 2008.

107.5 108 107.4 [7] H. Luo, H.M. Shen, Practical application of Warrant Bonds in China, 107.3 107.5 107.2 Journal of Zhejiang University of Technology, 2009, 26 (5): 796-801. 107.1 107 107 [8] D. Zhu, The Martingale Pricing for WBS under stochastic interest 106.5 106.9 Value of WBS Value of WBS 106.8 Journal of Applied Mathematics, 2011,34 (2): 265-271. 106.7 106

106.6

105.5 106.5 [9] W.L. Huang, S.H. Li, Pricing of European option with proportional 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Hurst Parameter:H σ transaction costs in Fractional Brownian motion environment University 108 107.4 107 Journal of Applied Mathematics, 2011,26 (2): 201-208. 107.2 106 107 105 [10] Steven Shreve, Stochastic Calculus and Finance. New York: Springer- 106.8 104 106.6 Verlag, 1997. 103 Valueof WBS

Valueof WBS 106.4 102

101 106.2 [11] Dravid A, Richardson M, Sun T, Pricing Foreign Index Contingent 100 106 Claim: an Application to Nikkei Index Warrants. Derivative ,1993, 99 105.8 0 0.5 1 1.5 2 2.5 3 3.5 4 20 22 24 26 28 30 32 34 36 38 40 Time to maturity:T Exercise price 1(1):33-51. FIGURE I. . VALUE OF WBS. [12] Guasoni P, No arbitrage under transaction costs, with Fractional Brownian motion and beyond. .Mathematical Finance, 2006, 16(3):569- 582.

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105 Value of WBS of Value 100

95 8 6 40 4 35 30 2 25 0 20 Time to maturity Exercise price FIGURE II. VALUE OF WBS UNDER DIFFERENT TIMES TO MATURITY AND EXERCISE PRICES.

VI. CONCLUSION This paper has overcome the drawback of the pricing model for WBS proposed by some predecessors, used the experiences for reference that the financial assets price follows Fractional Brownian motion process which a large number of financial empirical studies have proved, and established the pricing model for WBS which draws financial markets closer to the actual conditions of financial markets. Assuming that the underlying stock price follows Fractional Brownian motion and stochastic interest rate meets Vasicek rate model, the pricing formula of Warrant Bonds is obtained by utilizing the risk-neutral pricing theory (namely, no arbitrage pricing

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