Pricing Convertible Exchangeable Bonds

Convertible exchangeable bonds can be used to finance by some companiesin capital market, now it develops quickly in the oversea capital market. Between1998 and 2001 convertible exchangeable bonds worth of about 112 billion Euro wereissued in Europe alone. It is great significant to price convertible exchangeable bondsproperly for issuing companies designing issuance provisions, investors reasoninglyinvesting, and convertible exchangeable bonds market developing healthily.The convertible exchangeable bond is rather complicated finance derivative. Itoffers investors the right to convert a bond into a specified number of shares of stockof the designated firm. Except the common debt, it also includes many options,such as conversion option, call option, put option and so on. So, the convertibleexchangeable bond is a hybrid security that offers the market special investmentchallenges and opportunities.Convertible exchangeable bonds differ from convertible bonds. Convertible ex-changeable bonds that they are issued by a company (issuer) and can be exchangedfor the shares of another company. But convertible bonds can be exchanged forshares of the issuer. So those who invest in convertible exchangeable bonds beardefault risk of one company and the equity risk related to another company, whereasthose who invest in convertible bonds only bear the credit risk and the equity riskof the issuer alone.This paper studies the valuation of convertible exchangeable bond by modern fi-nancial engineering theory and partial derivative equation's theory. Some pricing for-mulas are obtained in some special cases. According to factors of impacting the valueof convertible exchangeable bonds, we respectively analyse characteristics of debtand equity of the convertible exchangeable bonds, and give the pricing of convertibleexchangeable bonds with existence of credit risk in both cases. Specially, When the price of put is restricted, we obtain a pricing formula of convertible exchangeablebond with credit risk. Furthermore, in this paper, with put and call provisions ,weuse a binomial tree to price convertible bond with no-default risk.Then we obtaina binomial tree pricing model of convertible exchangeable bonds with default risk.Anew way about pricing convertible bonds is proposed with proper discounted rateby binomial tree approach. The idia is that the convertible exchangeable bond isbroken down into equity value and debt value, because different parts of the con-vertible exchangeable bond have different default risk, we use different discountedrate to discount equity value and debt value of convertible exchangeable bonds. Itis worth pointing out that the model have following characteristics:simple,intuitiveand more importantly having the strong ability to combine other items in indentureof convertible exchangeable bonds. The following is our main results:(1) According to the equivalence between convertible exchangeable bond andconvertible bond without credit risk, we take model of pricing convertible bonds intopricing convertible exchangeable bonds.(2) Once there exists credit risk of the issuer of convertible exchangeable bond,from Tsiveriotis and Fernandes(1998), we obtain a pricing formula of zero-couponconvertible exchangeable bond with the put price P(tp) < F·exp[ff(r+rc)(T fftp)],where rc is credit spread. According to Proposition 1, we have thorem 1 as follow,Theorem 1 If there exists credit risk of the issuer,redemption price P(tp) F·exp[?(r +rc)(T ?tp)]+ ci·exp[?(r +rc)(ti ?tp)], call , then we drivea couple of free boundary of bi-obstacle problem on the cash-only-part U(S,t) andthe value V (S,t) of the convertible exchangeable bond by considering the conditonof convertible exchangeable bond, including the maturity condition, the conversioncondition, the redemption condition, the call condition.(5)For the convertible exchangeable bond, with put and call ,we use a binomialtree to price the convertible exchangeable bonds with no-default risk and default.Then we obtain a binomial tree pricing model of convertible exchangeable bonds withdefault risk. A new way about pricing convertible bonds is proposed with properdiscounted rate by the binomial tree approach. The idia is that the convertibleexchangeable bond is broken down into equity value and debt value, because equity-part and cash-only-part of the convertible exchangeable bond have di?erent defaultrisk, we use di?erent discounted rate to discount equity-part and cash-only-part ofconvertible exchangeable bond. Finally,we consider optimal conversion strategy,callstrategy and put strategy fully and price the convertible exchangeable bonds in thebinomial tree approach with default. It is worth pointing out that the model havefollowing characteristics: simple,intuitive and more importantly having the strongability to combine other items in indenture of convertible exchangeable bonds.