This paper introduces a class of two counters of jumps option pricing models. The stock price follows a jump-diffusion process with price jumps up and price jumps down, where each type of jumps can have different means and standard deviations. Price jumps can be negatively autocorrelated as it has been observed in practice. We investigate the volatility surfaces generated by this class of two counters of jumps option pricing models. Our formulae, like the jump-diffusion models with a single counter of jumps, are able to generate smiles, and skews with similar shapes to those observed in the options markets. More importantly, unlike the jump-diffusion models with a single counter of jumps, our formulae are able to generate term structures of implied volatilities of at-the-money options with ∩-shaped patterns similar to those observed in the marketplace.

We model deposit insurance as a European put option on the value of the bank in which bank assets follow a displaced lognormal diffusion process. We derive closed-form solutions for the value of the bank for bank equity holders, depositors, and the deposit insurer under three deposit insurance schemes that are representative of deposit insurance around the world. Doing so allows us to compute actuarially fair insurance premiums that are risk adjusted, include market information, and explicitly account for the diverging effects of safe versus risky assets on bank risk. We illustrate the use of the model on a sample of 212 U.S. bank holding companies and discuss practical considerations for implementing the model. Implications for our model as a market-based, early indicator of bank risk are considered.

This article presents a pure exchange economy that extends Rubinstein (1976) to show how the jump–diffusion option pricing model of Merton (1976) is altered when jumps are correlated with diffusive risks. A non-zero correlation between jumps and diffusive risks is necessary in order to resolve the positively sloped implied volatility term structure inherent in traditional jump diffusion models. Our evidence is consistent with a negative covariance, producing a non-monotonic term structure. For the proposed market structure, we present a closed form asset pricing model that depends on the factors of the traditional jump–diffusion models, and on both the covariance of the diffusive pricing kernel with price jumps and the covariance of the jumps of the pricing kernel with the diffusive price. We present statistical evidence that these covariances are positive. For our model the expected stock return, jump and diffusive risk premiums are non-linear functions of time.

This article presents a modification of Merton’s (1976) ruin option pricing model to estimate the implied probability of default from stock and option market prices. To test the model, we analyze all global financial firms with traded options in the US and focus on the subprime mortgage crisis period. We compare the performance of the implied probability of default from our model to the expected default frequencies based on the Moody’s KMV model and agency credit ratings by constructing cumulative accuracy profiles (CAP) and the receiver operating characteristic (ROC). We find that the probability of default estimates from our model are equal or superior to other credit risk measures studied based on CAP and ROC. In particular, during the subprime crisis our model surpassed credit ratings and matched or exceeded KMV in anticipating the magnitude of the crisis. We have also found some initial evidence that adding off-balance-sheet derivatives exposure improves the performance of the KMV model.