A Green′s function for a convertible bond using the Vasicek model

Journal of Applied Mathematics, Aug 2018

We consider a convertible security where the underlying stock price obeys a lognormal random walk and the risk-free rate is given by the Vasicek model. Using a Laplace transform in time and a Mellin transform in the stock price, we derive a Green′s function solution for the value of the convertible bond.

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A Green′s function for a convertible bond using the Vasicek model

Hindawi Publishing Corporation Journal of Applied Mathematics R. MALLIER A. S. DEAKIN We consider a convertible security where the underlying stock price obeys a lognormal random walk and the risk-free rate is given by the Vasicek model. Using a Laplace transform in time and a Mellin transform in the stock price, we derive a Green's function solution for the value of the convertible bond. 1. Introduction A convertible bond is defined to be (e.g., Jorion [ 10 ]) a bond issued by a corporation that can be converted into the equity of that corporation at certain times using a predetermined exchange ratio. This entails the creation of new shares issued by the corporation if and when conversion occurs, and the existing shares are diluted by the creation of the new shares. The option to convert is solely at the discretion of the bond holder who will do so only if it is beneficial. Typically, firms issue convertible bonds because they offer a lower interest cost and less restrictive covenants than a nonconvertible bond, but the drawback is that the issuer will be confronted with capital structure uncertainty. Convertible bonds are often subordinated debentures, and because of this, the bond rating agencies have usually rated convertibles one class below that of a straight debenture (Dialynas et al. [ 7 ]), and typically issuing convertibles will not affect a company?s rating. In return for a reduced yield, an investor will receive a security with considerable upside potential along with downside protection. Conceptually, the behavior of a convertible bond can be segmented into four regions (e.g., Dialynas et al. [ 7 ]). In the late 1990s, most new issue convertibles were balanced converts, with around a 25% conversion premium, where the conversion premium is the excess an investor would pay to acquire the stock by buying the convertible and immediately converting rather than buying the stock itself. Typically, the price of a balanced convert responds materially to changes in both the underlying stock price and the spot interest rate, with a correlation of about 55% to 80% with changes in the stock price. A second category is equity substitute converts, where the conversion premium is less than 15%, usually because of rises in the price of the underlying. Typically, the price of an equity substitute responds much more to changes in the stock price than to interest rate changes. A third category is busted converts, where the underlying stock price has declined so significantly that the conversion option is worth very little and the value of the convertible approaches that of an otherwise identical nonconvertible bond. A fourth category is distressed converts, which are busted converts where the stock price has fallen so much that there is a significant chance of bankruptcy. As of 2000, the market value of convertible securities outstanding globally was in excess of $400 billion US, with approximately $200 billion in the USA alone (Dialynas et al. [ 7 ]). Given the size of the market for these securities, the pricing of convertible bonds is obviously an important problem. Traditionally, convertibles were valued based on the premise that buying a convertible is equivalent to buying the stock at a premium and recouping that premium from the coupons on the convertible, and the payback period is the time taken to recover the premium. More recently, however, contingent claims analysis has been used to value convertibles, which is the approach taken in the present study, and this approach dates back to the work of Ingersoll [ 9 ] and Brennan and Schwartz [ 3 ]. Brennan and Schwartz originally used the firm value as the underlying variable, and later (Brennan and Schwartz [ 4 ]) extended their analysis to include stochastic interest rates and also to include the value of the stock rather than that of the firm (McConnell and Schwartz [ 11 ]). Almost all of this earlier work led to a numerical rather than an analytical solution of the underlying equations for the value of a convertible bond, typically using binomial trees; by contrast, the present work in entirely analytical. In our analysis, we consider a convertible bond, whose value depends on both the price of the underlying stock, which is assumed to obey a lognormal random walk with constant volatility, as in the Black-Scholes option pricing model, and on the interest rate, which is assumed to follow a random walk given by the Vasicek [ 13 ] model. We will say more about this interest rate model and its advantages and disadvantages in Section 3. By constructing a risk-free portfolio, it is possible to go from the stochastic differential equations for the stock price and the spot rate to a PDE for the value of the convertible (e.g., Wilmott [ 14 ]), and that PDE is the starting point for our analysis in the next section. In our analysis, we consider this PDE, and by using a double integral transform, specifically a Laplace transform in time an (...truncated)


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R. Mallier, A. S. Deakin. A Green′s function for a convertible bond using the Vasicek model, Journal of Applied Mathematics, 2, DOI: 10.1155/S1110757X02203058