Convertible and the convertible bonds we measure the efficiency

Convertible bonds are difficult to value, given their hybrid nature of containing
elements of debt and equity, and further complications arise due to the additional
American options and contractual complexities in which the possibility of the issuer to
exercise the option to call and the investor to exercise the option to put are dependent
upon the history of the underlying stock price. This paper uses nonparametric fixed
effect panel data model to empirically analyze the pricing problem of convertible
bonds in China. We obtain smooth function through nonparametric regression on the
historical trading data of the underlying and the derivative and get the smooth
function, then we construct simultaneous confidence band for smooth function and
regard it as rational pricing range, if the price of convertible bonds surpass the range,
it is mispriced. So the simultaneous confidence band can serve as arbitrage free interval
and denote arbitrage opportunities. The profitability and risk of arbitrage is an
important signal of market efficiency since arbitrage capitalize on market inefficiency
(Hogan et al., 2004; Jarrow et al., 2012), and the risk and profitability of arbitrage reflect
market efficiency (Bhattacharya and O’Brien, 2015). Therefore by measuring the
profitability and risk of arbitrage between the underlying and the convertible bonds
we measure the efficiency of convertible bonds market. The frequency of mispricing
among different convertible bonds and different time periods reflect the evolvement
and variability of pricing efficiency, arbitrage between the underlying and derivatives
corrects and capitalizes on mispricing as well as improves market efficiency.
The contribution of this paper lies in, first, we apply the nonparametric fixed effect
panel data model, a simple yet powerful methodology, to regress and forecast the
dynamic relations between the underlying stocks and the convertible bonds, without
the assumption of perfect market or any distribution of the prices, which is more
objective, simple and practical. Second, we construct simultaneous confidence band for
smooth function, the result of nonparametric regression, to find the pricing anomalies.
Simultaneous confidence band is the interval estimation of the smooth function,
without fundamental change of pricing mechanism, the prices outside simultaneous
confidence band denotes mispricing and the mispricing reflects the pricing efficiency
variation among individual convertible bond and across time. Third, mispricing
denotes arbitrage or trading opportunity, and arbitrage between the underlying and
the convertible bonds by relative value strategy capitalizes on inefficiency. Arbitrage
between convertible bonds and the underlying stocks kills two birds with one stone, as
arbitrageurs improve market efficiency while capitalize on market inefficiency. It is
noteworthy that the extreme abnormal prices means extreme profits as well as extreme
Research on
bond pricing
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risk and the arbitrageurs fail to correct such anomaly. If there is no fundamental
change in the pricing mechanism, there might be malicious manipulation, thus the
study provides some reference for supervision and regulation.
2. Literature review
Convertible bonds are issued by corporate issuers and are subject to the possibility of
default. According to the difference in treatment of default risk, there are two main
approaches for valuing convertible bonds. The first approach is structural approach,
which assumes default to be an endogenous event and bankruptcy happens when the
value of the firm’s assets falls below some low threshold level, and the basic underlying
state variable is the value of the issuing firm. This approach, pioneered by Merton
(1974), assumes that the firm value follows a stochastic diffusion process and default
happens when the firm value falls below the face value of the debt. Issuing convertible
bonds increases firm’s financial risk, Longstaff and Schwartz (1995) point out that
default happens as soon as the firm value reaches some predefined level common for all
issues of debt. Their values of spread are more comparable to the market observed
spreads than Merton’s. To obtain more realistic credit spreads, Zhou (2001) develops a
model that allow diffusion and jumps in the asset value process so the possibility of
instantaneous default caused by a sudden drop in firm value is considered. Ingersoll
(1977) introduces Black and Scholes (1973) option pricing model into the valuation of
convertible bonds. Under the assumption of perfect market the optimal policies for
call and conversion are considered and the closed form solution is obtained in the
framework of non-arbitrage equilibrium. Nyborg (1996) criticizes that convertible bond
value is assumed to be a function of the firm value, a variable not directly observed.
Some authors model the price of convertible bonds as a function of the stock price, a
directly obtained variable from exchange, to circumvent the problem. Ho and Pfeffer
(1996) divides the value of convertible bonds into three parts, the investment value,
i.e. the value of common corporate bond with identical principal, nominal rate and
maturity, the warrant value and value of redemption. The value of convertible bond is
the sum of investment value and warrant value minus the value of redemption, and the
binary tree model is applied to get the value. The structural approach allows the default
risk to be stochastic, and it takes into account the decision of corporate capital
structure, default, dividends, issuers and investors’ activities. It can describe the
optimal policies of issuers and investors therefore it is applicable for solving
comprehensive problems.
However, the capital structure become complicated after issuing hybrid financial
instruments, and because of asymmetric information, modeling firm value and capital
structure is difficult. The second approach is called reduced form approach or default
rate model. In contrast to the structural approach where default is an endogenous event
tied to the firm’s value and capital structure, in the reduced form models default is an
exogenous event. The default risk of a firm and its value are not explicitly related,
at any point in time, the probability of default is defined by a Poisson arrival process
and is described by a hazard function. The attractiveness of this approach is that the
convertible bond value can be modeled as a function of the stock price. Jarrow and
Turnbull (1995) introduce credit spread and term structure of interest rate and assume
default follows point process. Duffie and Singleton (1999) regard default as an
unpredictable event and follows hazard rate process, and it can be parameterized by
the percentage of decrease in firm’s market value when default happens. Tsiveriotis
and Fernandes (1998) develope the model of Goldman-Sachs (1994), ignoring the 


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