Convertible Bonds with Higher Loan Rate: Model, Valuation, and Optimal Strategy

We study the pricing problem for convertible bonds via backward stochastic differential equations (BSDEs). By virtue of reflected BSDEs and Malliavin derivatives, we establish the formulae for the fair price of convertible bonds and the hedging portfolio strategy explicitly. We also obtain the optimal conversion time when there is no dividends-paying for underlying common stocks. Furthermore, we consider the case that the loan rate is higher than riskless interest rate in a financial market, and conclude that it does not affect the price of convertible bonds actually. To illustrate our results, some numerical simulations are given and discussed at last

Simulation-Based Pricing of Convertible Bonds

We propose and empirically study a pricing model for convertible bonds based on Monte Carlo simulation. The method uses parametric representations of the early exercise decisions and consists of two stages. Pricing convertible bonds with the proposed Monte Carlo approach allows us to better capture both the dynamics of the underlying state variables and the rich set of real-world convertible bond specifications. Furthermore, using the simulation model proposed, we present an empirical pricing study of the US market, using 32 convertible bonds and 69 months of daily market prices. Our results do not confirm the evidence of previous studies that market prices of convertible bonds are on average lower than prices generated by a theoretical model. Similarly, our study is not supportive of a strong positive relationship between moneyness and mean pricing error, as argued in the literature.Convertible bonds, Pricing, American Options, Monte Carlo simulation

Dynkin Game of Convertible Bonds and Their Optimal Strategy

This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optimal stopping time problem with state constraint, and then in a Markovian setting, we investigate the optimal strategy by analyzing the properties of the corresponding free boundary, including its position, asymptotics, monotonicity and regularity. We identify situations when call precedes conversion, and vice versa. Moreover, we show that the irregular payoff results in the possibly non-monotonic conversion boundary. Surprisingly, the price of the convertible bond is not necessarily monotonic in time: it may even increase when time approaches maturity.Comment: 28 pages, 9 figures in Journal of Mathematical Analysis and Application, 201

Convertible Bonds: Risks and Optimal Strategies

Within the structural approach for credit risk models we discuss the optimal exercise of the callable and convertible bonds. The Vasi˘cekâmodel is applied to incorporate interest rate risk into the firmâs value process which follows a geometric Brownian motion. Finally, we derive pricing bounds for convertible bonds in an uncertain volatility model, i.e. when the volatility of the firm value process lies between two extreme values.Convertible bond, game option, uncertain volatility, interest rate risk

Valuation of Convertible Bonds

Convertible bonds are hybrid financial instruments with complex features. They have characteristics of both debts and equities, and usually several options are embedded in this kind of contracts. The optimality of the conversion decision depends on equity price, interest rate and default probability of the issuer. The decision making can be further complicated by the fact that most convertible bonds have call provisions allowing the bond issuer to call back the bond at a predetermined call price. In this thesis, we first adopt a structural approach where the Vasicek-model is applied to incorporate interest rate risk into the firm's value process which follows a geometric Brownian motion. Default is triggered when the firm's value hits a lower boundary. The complex nature of the firm's capital structure and information asymmetry may make it hard to model the firm's value and the capital structure. In this case, reduced-form models are applied for the study of convertible bonds. We adopt a parsimonious, intensity-based default model, in which the default intensity is modeled as a function of the pre-default stock price. We first analyze the contract features of the convertible bonds and show that callable and convertible bonds can be decomposed into a straight bond and a game option component. Then the no-arbitrage prices of the European- and American-style callable and convertible bonds are derived. In American-style contracts, the focus is on the analysis of the strategic optimal behavior. The bondholder and issuer choose their stopping times to maximize or minimize the expected payoff respectively. For the bondholder it is optimal to select the stopping time which maximizes the expected payoff given the minimizing strategy of the issuer, while the issuer will choose the stopping time that minimizes the expected payoff given the maximizing strategy of the bondholder. The no-arbitrage price can be approximated numerically by means of backward induction. In the structural model, the recursion is carried out alongside a recombining binomial tree. Whereas in the reduced-form approach, the optimization problem is formulated and solved with the help of the theory of doubly reflected backward stochastic differential equations. In practice, it is often a difficult problem to calibrate a given model to the available data. Determining the volatility of the firm's value process or stock price process is not a trivial problem. We therefore assume that the volatility of the firm's value process/stock price process lies between two extreme values, and combine it with the results on game option in incomplete market to derive certain pricing bounds for callable and convertible bonds. The maximizing strategy of the bondholder and the choice of the most pessimistic pricing measure from his perspective determine the lower bound of the no-arbitrage price. Whereas the minimizing strategy of the issuer and the most pessimistic expectation from his aspect construct the upper bound of the no-arbitrage price. Numerically, to make the computation tractable a constant interest rate is assumed. The pricing bounds can be calculated with recursion alongside a recombining trinomial tree or with the finite-difference method

Dynkin games with Poisson random intervention times

This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity

Convertible Bonds: Default Risk and Uncertain Volatility

Within a default intensity approach we discuss the optimal exercise of the callable and convertible bonds. Pricing bounds for convertible bonds are derived in an uncertain volatility model, i.e. when the volatility of the stock price process lies between two extreme values.Convertible bond, game option, uncertain volatility, interest rate risk

Pricing Hybrid Securities: The Case of Malaysian ICULS

This paper provides an indepth analysis of Irredeemable Convertible Unsecured Loan Stocks or ICULS. A Malaysian variant of the convertible bond, ICULS are a hybrid security. Despite their introduction and trading since the late 1980’s, not much work have been done on them. This paper presents the first empirical evidence on the pricing of ICULS. We propose a pricing model for ICULS, built on the replication technique of options. Using 30 months (2½ years) of daily price data, we test our model on a sample of 34 ICULS. Though on average ICULS are underpriced by 2.3%, we find an equal number of under and overpriced ICULS. Our findings show that not only does the market misprice ICULS, the mispricing is sustained over quite a while. Infact, even over a one year window period, marginal mispricing remains. We argue that issuers of ICULS benefit much more than investors do.Pricing of Hybrid Securities; Irredeemable Convertible Unsecured Loan Stocks or ICULS; Option Pricing, Pricing Efficiency

Adapted Downhill Simplex Method for Pricing Convertible Bonds

The paper is devoted to modeling optimal exercise strategies of the behavior of investors and issuers working with convertible bonds. This implies solution of the problems of stock price modeling, payoff computation and min-max optimization. Stock prices (underlying asset) were modeled under the assumption of the geometric Brownian motion of their values. The Monte Carlo method was used for calculating the real payoff which is the objective function. The min-max optimization problem was solved using the derivative-free Downhill Simplex method. The performed numerical experiments allowed to formulate recommendations for the choice of appropriate size of the initial simplex in the Downhill Simplex Method, the number of generated trajectories of underlying asset, the size of the problem and initial trajectories of the behavior of investors and issuers.Comment: 18 pages, 8 figure