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Abstract:
This paper focuses on a theoretical aspect of relations between the Black-Scholes implied volatility and the default probability in a general framework that the stock price becomes zero after default occurs. It is shown that the default probability of the company under a risk-neutral measure significantly links to the implied volatility skew at extremely small strike. Moreover, it is proved that the divergence speed of the implied volatility must be determined uniquely in any defaultable economy under arbitrage-free condition.

Implied Volatility, Default Probability, Arbitrage-Free Condition

Abstract:
This paper proposes an extended CreditGrades model called the L\'{e}vy CreditGrades model, which is driven by a L\'{e}vy process. In this setting, quasi closed-form formulae for pricing equity options to a reference firm and for calculating its survival probabilities are derived. Moreover, using three tractable L\'{e}vy CreditGrades models, we compute implied volatilities on equity options and term structures of credit default swaps (CDSs) and we examine the jump risk effects of the firm's asset value on short term CDS spreads and equity volatility skew. As a result, with this extension, our model is found to have more significant abilities than the original model introduced by Finger et al. [2002] and Stamicar and Finger [2005], and it is more appropriate for pricing both equity and credit derivatives simultaneously.

CreditGrades Model, L\'{e}vy Process, Equity Option, Credit Default Swap, Wiener-Hopf Factorization

Abstract:
This paper proposes a new scheme for static hedging of defaultable contingent claims. It is a generalization of the technique developed by Carr and Chou (1997), Carr and Madan (1998), and Takahashi and Yamazaki (2009a) into unified credit-equity modelings. The novel hedging strategy across credit and equity markets is presented, where any defaultable contingent claim is accurately replicated by a feasible number of plain vanilla equity options. Another point is that shorter maturity options are available to hedge longer maturity defaultable contingent claims. Through numerical examples, it is shown that the scheme is applicable to both structural and intensity-based models.

Static hedging, Default risk, Equity options, Defaultable bonds, Structural model, Intensity-based model

Abstract:
This paper proposes a new hedging scheme of European derivatives under uncertain volatility environments, in which a weighted variance swap called the polynomial variance swap is added to the Black-Scholes delta hedging for managing exposure to volatility risk. In general, under these environments one cannot hedge the derivatives completely by using dynamic trading of only an underlying asset owing to volatility risk. Then, for hedging uncertain volatility risk, we design the polynomial variance, which can be dependent on the level of the underlying asset price. It is shown that the polynomial variance swap is not perfect, but more efficient as a hedging tool for the volatility exposure than the standard variance swap. In addition, our hedging scheme has a preferable property that any information on the volatility process of the underlying asset price is unnecessary. To demonstrate robustness of our scheme, we implement Monte Carlo simulation tests with three different settings, and compare the hedging performance of our scheme with that of standard dynamic hedging schemes such as the minimum-variance hedging. As a result, it is found that our scheme outperforms the others in all test cases. Moreover, it is noteworthy that the scheme proposed in this paper continues to be robust against model risks.

European Derivatives, Black-Scholes Delta Hedging, Uncertain Volatility Risk, Polynomial Variance Swap

Abstract:
This paper proposes a pricing formula for residential mortgage-backed securities (RMBS) with the proportional hazard model. First, we develop basic models of mortgage contracts with prepayment risk in the intensity-based framework. Next, assuming the proportional hazard model to describe prepayment risk, which is used as a typical prepayment model both academically and in practice; a general pricing formula for not only RMBS, but also IO and PO is derived by using the cumulant expansion method. Furthermore, it is also shown that the formula is applicable to various types of the proportional hazard models. Finally, numerical examples based on Japanese RMBS market data demonstrate that the formulas very accurate and useful in practice.

Residential Mortgage-Backed Security, Prepayment Risk, Proportional Hazard Model, Cumulant Expansion