In the capital structure models, the most famous are the Leland (1994) and Leland and Toft (1996) models. They extend the endogenous default approach by Black and Cox (1976) to include the bankruptcy costs and tax shield of debt, and analyze the static tradeoff theory of capital structure. Leland and Toft (1996) point out that equity in a capital structure model is not similar to a plain vanilla call option or an ordinary down-and-out barrier option.
In Leland model (1994), financial distress is happened when equity holders find that running company is unprofitable, given the debt is still serviced. Bankruptcy is determined endogenously rather than given a certain level of asset or other constraint exogenously.
There are two major assumptions as follows: (1) the firm’s activities such as financial structure and the capital structure decisions do not change when the decisions was made; (2) the face of debt remains static through time when it was issued. First, Leland (1994) provides the solution of the perpetual debt. Next, Leland (1994) derives the total value of the firm, which contains three terms: the firm’s asset value, plus the value of the tax shield of coupon payments, less the value of bankruptcy costs. Then Leland (1994) provides the bankruptcy asset level VB which is determined endogenously by maximizing the value of equity. Finally, VB is independent of time and it confirms the assumption of the constant bankruptcy-triggering asset level VB. However, optimal capital structure relates not only to leverage ratio but also the maturity of debt.
In Leland and Toft (1996) model, the assumption of financial distressed follows Leland (1994) model when shareholders find that running company is unprofitable. In this model, they can choose both maturity and amount of the debt. Their stationary debt structure is that they issue new finite maturity debt with continuous coupon and the same amount of principle will be retired at the same time. The result extends Leland’s (1994) closed-form solution to much more alternatives of possible debt structures and dividends payment and
develops the method to measure optimal maturity of debt as well as optimal amount of debt.
The Leland and Toft (1996) model has some different implications contrast to Leland (1994) model. First, they allow firm to choose its debt maturity, also explain why a firm issue short-term debt even that long-term debt provide more firm value. They find that short-term debt can reduces agency cost. Although long-term debt produces more tax benefit, short-term debt can balance bankruptcy and agency cost in determining the optimal maturity of the capital structure. Second, they find that Macaulay duration overstates the true duration of risky debt for bond portfolio management.
The Fan and Sundaresan (2000) model propose a game-theoretic setting that incorporates bargaining powers to the equity holders and the debt holders. If the firm is liquidated at the bankruptcy point, then debt holders receive liquidation value and equity holders receive nothing. However, if the firm is not liquidated, the firm value will be shared between equity holders and debt holders. The Fan and Sundaresan model shows that debt renegotiation encourages early default and increases credit spreads on corporate debt, given that shareholders can renegotiation in distress to avoid inefficient and costly liquidation. If shareholders do not have bargaining power, no strategic debt service occurs and the model converges to the Leland model.
The Francois and Morellec (2004) model extends the Fan and Sundaresan (2000) model to add the possibility of Chapter 11 proceeding. Shareholders hold a Parisian down-and-out option on the firm’s asset. Francois and Morellec solve the endogenous default barrier by maximizing equity value and providing closed-form solution for corporate debt and equity values. The sharing rule of cash flows during bankruptcy has a large impact on optimal leverage, while credit spread on corporate debt shows little sensitivity to the varying bargaining power.
2.2 The CEV Process and Option Pricing
Merton (1973) provided a closed-form solution for down-and-out options. Most of path-dependent options assume that the underlying asset follows geometric Brownian motion. This implies that price of the underlying asset is lognormally distributed. It has some drawbacks when the underlying assumption under Black-Scholes (1973) model.
Empirical studies have shown that stock prices are not lognormally distributed. If we use Black-Scholes model to price stock options, there exist well-known strike price bias (volatility smile). Therefore, we price path-dependent options when price of underlying asset follows the constant elasticity of variance (CEV) diffusion process. The CEV option pricing model was developed by Cox (1975). This process let the volatility of underlying asset linked to its price level, which is consist with the empirical evidence that the stock volatility tend to change as stock prices move up and down. The origin of the “volatility smile” is the negative correlation between stock price changes and volatility changes.
Nelson and Ramaswamy (1990) developed a simple binomial tree under the CEV process. They construct a binomial tree which its number of nodes grows linearly in the number of time intervals. It is shown how to construct computationally simple binomial process that converge weakly to commonly employed diffusion in financial models.
It is not always possible to build a very complex model with realistic features and solve the closed-form solution. As a result, we use a binomial lattice method to price corporate debt and model Chapter 11 proceedings developed by Broadie and Kaya (2007).
Their method takes the asset value of the firm as the primitive variable and prices equity and debt value as derivatives of this basic variable. The model generates results that are consistent with the limited liability of equity principle. It uses backward valuation, the continuation value of equity is known at each time step, so it can use to make bankruptcy decision. When the firm has a Chapter 11 bankruptcy alternative, their method can easily extend to the case by adding bankruptcy boundary and increasing the state space on the
lattice nodes if necessary. This numerical method (Broadie and Kaya (2007)) described the details of the implementation for three different models: Leland (1994), Francois and Morellec (2004), and Broadie, Chernov and Sundaresan (2005).
2.3 Parisian Options
A down-and-out Parisian option is options that know-out if the underlying asset price remains constantly beyond a given barrier over a pre-specified time interval, the so-called grace period. When a Parisian option is activated, its payoff at maturity is equal to standard European option while the payoff is zero if the option know-out before its maturity. The advantage of Parisian option with respect to standard barrier options is that it is more difficult to influence the option payoff. We use a variant of the lattice-based method, called the forward shooting grid (FSG), it has successfully applied to price path-dependent options, like Parisian option. The FSG approach was developed by Hull and White (1993) and Ritchken, Sankarasubramanian and Vijh (1993) for the pricing of American- and European-style Asian and lookback options. The FSG approach uses auxiliary state vector at each node on the lattice tree. The state vector is used to capture the path-dependent feature of the option contract, like grace period of the asset price.