Strategy Equilibrium Pricing Model

As the hybrid securities with the properties of the bonds and the stocks, convertible bonds give the investors the option to exchange for the stocks with preset ratio. Under a rapid development, convertible bonds have been the important financial instruments today, and pricing models of convertible bonds have also attracted a lot of attention of those people working in financial institution.Convertible bonds pricing models can be classified as two branches according to the underlying asset. Structural Approach assumes the underlying asset is the firm's value. And Reduced-Form Approach assumes the underlying asset is the stocks value of the firm. Since their different underlying asset, the two types of approach have their respective merits and undeniable value, but they also have some limitations.1. Reduced-Form Approach has a limitation on capturing default risk. As the underlying asset of Reduced-Form Approach, firm's stock price is always larger than zero, which excludes the possibility of the bankrupt of the firm. So this type of approach fails to capture default risk in nature. Although many methods have been involved to attempt to conquer the above limitation, these remedies can not reach a consist conclusion.2. Structural Approach has a limitation on manipulation. Through assuming the underlying asset as the firm's value, Structural Approach can capture default risk successfully. Because when the firm's value falls below the debt value, the bankrupt and default happen. But as long as the underlying asset assumption brings models the merit to capture default risk, it also brings a trouble. Since the firm's value is composed of the total market value of stocks, bonds and convertibles, which are issued by the firm, before getting convertibles'value, it is impossible to get the firm's value. So the firm's value is unobservable. For this reason, Structural Approach, including Merton (1974) model, Ingersoll (1977) model, Brennan & Schwartz (1977) model, Brennan &Schwartz (1980) model, Carayannopoulos (1996) model and Nyborg (1996) model, has the same limitation on manipulation, which restricted the development of Structural Approach.3. The existed models have a limitation on explaining the strategic interactions among different types of investors. In current market, we observe a lot of phenomena of the strategic interactions amongy different types of investors frequently. Usually 4 types of rational players are involved: unliquid shares holders, liquid shares holders, firm's managers (convertibles issuer) and convertibles holders. The strategic interactions among the 4 types of players are so complicated and can be classified as 3 relations: the competition between unliquid shares holders and liquid shares holders, the competition between firm's managers and total shares holders (Principal-agent relationship), and the competition between total shares holders and convertibles holders. Reduced-Form Approach has no capital structure parameter to describe the composition of stocks and debts, so this type of models have no idea about different types of investors and can not capture the strategic interactions. To Structural Approach, although it has the idea about different types of investors, it does not construct models by merging the strategic interactions. In sum, the existed models, whatever they belong to Reduced-Form Approach or Structural Approach, can not offer any rational explanation to the observed strategic interactions.To conquer the above limitations, this paper proposes a new pricing model, Strategy Equilibrium Pricing Model. The model's key viewpoint is that the convertible price results from the equilibrium allocation of the firm's value by competition among investors. So the pricing process can be regarded as a discounting process of strategic alloction of the firm's future stochastic cash flow under the competition among investors.Since the foundation of Strategy Equilibrium Pricing Model is the strategic interactions between shares holders and convertibles holders, we should solve 4 problems, which involve the existant of strategic interactions, the theoretic analysis of strategic interactions, the discounting of the allocation from strategic interactions and the acquirement of the strategic equilibrium price of convertibles under the condition of the unobservable firm's value. The relations of above 4 problems are: first whether the model can be constructed upon the investors' strategic interactions would be proved by the analysis of the existant of strategic interactions; secondly the strategic interactions can be abstracted into the standard mode of Game Theory through theoretic analysis. Thus the strategic part of the model is constructed; thirdly when stock investors and convertible investors choose none of their rights, the convertible investors'utility should be calculated by discounting the convertibles'future value, which introduces the third problem, the discounting of the allocation from strategic interactions; finally since the strategic part of the model only provides the mothod to calculate convertible price under the known firm's value. But the firm's value is unobservable. So the fourth problem, the acquirement of the strategic equilibrium price of convertibles under the condition of the unobservable firm's value, is introduced.1. The existant of strategic interactions. The existant of strategic interactions is the facticity and substantiality of the strategic interactions among investors. By MM theory's assumption of the independence between the firm's financing structure and investment structure, given the firm's investment projects, the probability distribution of the future stochastic cash flow generated by those projects is fixed. As the dividers of the stochastic cash flow, facing the fixed probability distribution, shares holder and convertibles holders will maximize their respective possible allocations through competition. Thus the future stochastic cash flow generated by the firm's investment projects can be devided into two parts. One part is allocated by shares holders, and can be considered as the cash flow generated by shares, the present value of which is the stocks present value. Another part is allocated by convertibles holders, and can be considered as the cash flow generated by convertibles, the present value of which is the convertibles present value. Through The Law of One Price, the sum of the stocks present value and the convertibles present value must consist with the firm's value calculated by discounting the future stochastic cash flow generated by the firm's investment projects. So, it proves the existant of strategic interactions. This existant settles the foundation of Strategy Equilibrium Pricing Model. The above content will be presented in Chapter 3.2. The theoretic analysis of strategic interactions. The theoretic analysis of strategic interactions is to analyse the investors'competition process through three aspects of strategies set, utility and Nash equilibrium based on Game Theory. From aspect of the strategies set, share holders'strategies set is {Hold, Call, Reset}, and convertible holders'strategies set is {Hold, eXecute, Put}. From aspect of the utilities, the share holders'utilities are represented by the stocks present value, and the convertible holders'utilities are represented by the convertibles present value. From the aspect of Nash equilibrium, since their strategic relation is classified into"Markovian Zero Sum Multistage Game With Perfect and Complete Information", the process to calculate Nash equilibrium of the game rolls back from the convertibles expiration to the valuation day. The above content will be presented in Chapter 3.3. The discounting of the allocation from strategic interactions. The discounting of the allocation from strategic interactions deals with how to calculate the convertible's present value by discounting its future value when the convertible continues existing. But when investors issue some provisions to end the convertible's life, its value can be decided by Convetible Issue Specification. The difficult point rests on how to calculate the holding value of convertibles when two types of investors all choose"Hold"strategy. Since then the holding value of convertibles should be calculated through method of discounting the future stochastic value. So the focus of the problem is fallen on how to get the discount rate. To get discount rate several techniques and their modifications have been invented. In the research note of Goldman Sachs (1994), the credit adjusted rate, which is the weighted average between riskless rate and risky rate by weight of conversion probability, is introduced to discount the convertible value of the two nodes one period in the future. And some researchers suggest using arbitrage ratio instead of conversion probability. Those techniques to get discount rate are naive and have the limitation of Reduced-Form Approach. So a new method, which rests on Risk Burden Ratio to calculate discount rate, is proposed. The above content will be presented in Chapter 4.4. The acquirement of the strategic equilibrium price of convertibles under the condition of the unobservable firm's value. The acquirement of the strategic equilibrium price of convertibles under the condition of the unobservable firm's value is the convertible value's deducing process by using the observed stock price under the limitation of the strategic equilibrium among investors. Since the proposed model assumes the underlying asset as the firm's value, it has the same limitation on manipulation as Structural Approach. To solve the problem, Theory of Corporation Market Value Allocation is proposed by viewing MM Theory from another angle. Then the convertible value can be calculated through the observed stock price. The above content will be presented in Chapter 4.After solving the above problems, the Strategy Equilibrium Pricing Model is actually constructed. The following work is to check the correction of the model. The paper uses Strategy Equilibrium model to price option and corporate debt, then compares the results with the result from Black-Scholes model and Merton model. The consistent results are reached. So it shows up the new model's compatibility with risk neutral pricing models, and its adaption to multiple financial instruments. The above content will be presented in Chapter 5 and Chapter 6.Since then, the paper starts the convertibles research of issueing provisions, issueing motivation and market pricing, to analyse some hotpot in Chinese convertible market, get some conclusions with academic value, and propose some constructive suggestions.In the part of the convertible issueing provisions, Put Provision and Reset Provision are focused. At the analyzing Put Provision of convertible by the theory of equilibrium point, the profound reason of HuaLing-convertible-put anecdote is revealed. It is caused by the discontinuity of convertible bonds price which results from the flaw of Put Provision. The flaw results in two equilibrium points. But none of the two points is Nash Equilibrium Point. At any point, investors have their Pareto Improvements. Stocks holders and convertibles holders compete against each other between those two points. The compete will not reach a equilibrium, which results in an unstable situation of convertible price and stock price. At analyzing Reset Provision, in efficient market if investors choose optimal strategies of Convert and Put, and issuer chooses optimal strategies of Call and Reset, Reset Provision totally counteracts the effect of Put Provision. The effort to protest investors through Put and Reset Provision, so as to get the advantage at the IPO of convertible bonds, is vanished. On the contrary it makes convertible too complicated to be accepted by ordinary investors. So we suggest changing the style of Put Provision to keep the continuity of convertible bonds price and getting rid of redundant Reset Provision to prevent their counteraction and protect the convertibles investors'benefit, or deleting the two provisions to simplify the complexity of convertible and increase its popularity among investors. The above content will be presented in Chapter 7.In the part of the convertible issueing motivation, the reason why the firm choose convertible as its financing instrument rests on the convertible price's neutral character with respect to the firm's business risk that solves the inverse choice before IPO and the ethical risk after IPO caused by the competition between investors and issuer with asymmetric information. Recent pricing model can not deal with the above analysis. To Reduced-Form Approach, the underlying asset is the stock price, but not the firm's value, so it has no parameter of the firm's business risk. To Structural Approach, although it has the parameter of the firm's business risk, it is not constructed upon the strategic interactions, so it also can not reveal the reason why firm issue convertibles. But to Strategy Equilibrium Pricing Model, it has the parameter of the firm's business risk as will as the strategic interactions, so it can reveal the convertible issueing motivation by researching its neutral character and how this character solves the inverse choice and ethical risk caused by the asymmetric information. To decrease the financing cost caused by asymmetric information, the convertible's parameters should be chosen to keep the neutral character as best as possible. So making the neutral character as one of the criteria of the convertible design is suggested. The above content will be presented in Chapter 8.In the part of the convertible market pricing, the practicability of pricing models is discussed. Since the Chinese convertible market is not an efficient market and existes a lot of arbitrages, many pricing models can not get consistent result with the market price for their common assumption of efficient market. Facing the long term inefficient market, I believe that it is caused by two limitations of market trading rule: none of short selling and low liquidity caused by double auction rule, and suggest developing the financing exchange trade to introduce short selling and progressing marketmaker rule to increase the convertibles'liquidity. The above content will be presented in Chapter 9.In sum, on account of the 3 main shortages of recent convertible pricing model, the paper proposes a modification of introducing strategic interaction into model. After the 4 problems, which involve the existant of strategic interactions, the theoretic analysis of strategic interactions, the discounting of the allocation from strategic interactions and the acquirement of the strategic equilibrium price of convertibles under the condition of the unobservable firm's value, are solved, Strategy Equilibrium Pricing Model is established. By the new model, the paper does 3 convertible researches on provisions, issueing motivation and market pricing and gets some conclusions. The above work gains its unique charaters:1. A new method to capture the risk of financial instrument is proposed. And Theory of Corporation Market Value Allocation is proposed to improve Structural Approach on its application.2. Facing the fact that Structural Approach has no binomial tree format, Strategy Equilibrium Pricing Model with the characters of easy operation and capturing default risk is constructed on binomial tree format in real world probability space.3. By Strategy Equilibrium Pricing Model, 3 convertible researches of issueing provisions, issueing motivation and market pricing are analysed and some uncovered conclusions are drawn.