2.1 Option Pricing Model Reviews
Black and Scholes (1973) offer an explicit model for option pricing. They derive a close-form expression from Brownian motion for pricing the European option. They provide a new vision of pricing option.
Following the Black and Scholes model, Merton (1976) extends the Black and Scholes diffusion process model to the jump-diffusion process model. He is the first to derive the close-form expression of jump diffusion model. This model has an advantage to match the real world in that asset return sometimes has a discontinuous jump due to incomplete information. However, the assumption of this model is that rate of return follows log-normal distribution. It is not consistent with the empirical research that return distribution has left skewed and heavier tail than normal distribution.
Kou (2002) provides an option pricing approach under double the exponential jump diffusion process. This process has many good features, including the probability and tendency that up-side and down-side jumps could be given separately. Because of the nice features of the double exponential jump diffusion, the log-normal return assumption of Merton model could be corrected. In addition, double exponential jump diffusion process is easy to use for option pricing.
2.2 Structural Form Model
Early theorization of structural form model can be traced back to Merton (1974).
Merton provides an approach that can use corporate capital structure to price corporate debt and default risk. He points out that equity value could be considered as a call
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option which is priced by the Black-Scholes model (1973). There are some disadvantages of using Black-Sholes model when pricing equity value, such as ignoring that low liquidity makes corporate bond default and bond default happen only at maturity, hence the following literatures modified the original structural model of Merton.
Black and Cox (1976) extend the Black-Sholes model and solve the problem that bond default only occurs at maturity. It allows for corporate bond default anytime before maturity only if the bond value hits a pre-specific level. Once the bond value reaches the pre-specific level, the corporation goes into default or is liquidated immediately.
Although Black and Cox relax the assumption of default time of Black-Scholes and Merton framework, this model still shares some assumptions with the Merton model.
One of the drawbacks of this approach is that interest rate is assumed to be constant.
After Black and Cox, Longstaff and Schwartz (1995) develop a new approach to pricing risky bonds. This model incorporates the Black and Cox model with interest rate risk. This approach has an important advantage in that close-form expression for both risky fixed-rate and floating-rate bonds could be derived. It relaxes the assumption of a constant interest rate.
Another assumption of the Black and Cox model is that the remaining value of the firm at default has to be equal to the default boundary. Zhou (2001) provided a new model for solving this assumption. He combines Merton jump-diffusion process with the Black and Cox structural model; hence this model is able to endogenously produce random variation in recovery rate. Besides this, the jump-diffusion model solves another problem that the default rate reaches to zero when time maturity is in a very short-term.
Because of the features of a down-and-out Parisian option that expires if the underlying asset price goes down, hits a specific barrier level and stays below this level for a period window, Fujita and Ishizaka (2002) propose a new concept, “caution time,”
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for relaxing safety covenant. Their model states if firm value drops below the barrier, the bondholders will have observations on operation of the firm; this is what is meant by
“caution time”. If the time in which firm value stays below the barrier exceeds “caution time”, bondholders think the firm defaults. Also, if the firm value is below the barrier at maturity, bondholders believe the firm to be in default.
Francois and Morellec (2004) use the down-and-out Parisian option for modeling risky bonds under Chapter 11 of the U.S. Bankruptcy Code. They point out that Parisian option’s special feature of period window could fundamentally represent that a corporation renegotiate in financial distress under Chapter 11 of the U.S. Bankruptcy Code. This model lets bondholders and shareholders have an unambiguous effect on default incentives and credit spread.
Chen and Kou (2009) extend the model under the double exponential jump diffusion model of the barrier option framework for credit risk. This model presents that jump risk and endogenous default can have significant effect on credit spread. This model has more flexible shapes of jump to explain the empirical data than jump-diffusion model.
2.3 Parisian Option Reviews
Chesney, Jeanblanc and Yor (1997) define a new option called Parisian option which is extended from the barrier option framework. A down-and-out (up-and-out) Parisian option is an option that expires if the underlying asset price goes down (up), hits a specific barrier level and stays below (above) for a period window. Conversely, A down-and-in (up-and-in) Parisian option is an option that comes into existence if the underlying asset price goes down (up), hits a specific barrier level and stays below (above) the period window. They derive a formula based on the Brownian motion theory for pricing Parisian option.
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According to the definition of Parisian option, Avellaneda and Wu (1999) formulate a partial differential equation (PDE) for Parisian option. The PDE solves Parisian option pricing numerically on a trinomial lattice. They also characterize the value function of Parisian option in the continuous limit.
Bernard, Le Courtois and Quittard (2005) develop a new inverse Laplace which transforms the method used to price Parisian option. They provide a quick and simple numerical method to compute the price and Greeks of Parisian option.