Greg Orosi

Classroom Capsules are short (1–3 pages) notes that contain new mathematical insights on a topic from undergraduate mathematics, preferably something that can be directly introduced into a college classroom as an effective teaching strategy or tool. Classroom Capsules should be prepared according to the guidelines on the inside front cover and submitted through Editorial Manager.

We present a novel method for extracting the risk‐neutral probability of default (PD) of a firm from American put option prices. Building on the idea of a default corridor proposed by Carr and Wu, we derive a parsimonious closed‐form formula for American put option prices from which the PD can be inferred. The method is easy to implement. Our empirical results based on seven large US firms for the period 2002–2010 show that, in some cases, our option‐implied PD can provide a more accurate estimate of default probability than the estimates implied from credit default swaps.

In this paper, we provide an alternative framework for constructing an arbitrage-free European-style option surface. The main motivation for our work is that such a construction has rarely been achieved in the literature so far. The novelty of our approach is that we perform the calibration and interpolation in the put option space. To demonstrate the applicability of our technique, we extract the model-free implied volatility from S&P 500 index options. Subsequently, we compare its information content to that of the CBOE VIX index. Our empirical tests indicate that information content of the option-implied volatility values based on our method are superior to the VIX index.

In this paper, we present a novel method to extract the risk-neutral probability of default from American put option prices. Under the assumptions of Carr and Wu (2011), we derive a closed form expression for American put options from which the probability of default can be inferred. Our empirical results based on four large institutions in the US during the 2007-2009 crisis show that directly applying the methodology of Carr and Wu (2011) can overestimate the probability of default. Our model also allows us to test the effect of positive equity recovery on estimated default probabilities.

In this paper, we provide an alternative framework for constructing an arbitrage-free European-style option surface. The main motivation for our work is that such a construction has rarely been achieved in the literature so far. The novelty of our approach is that we perform the calibration and interpolation in the put option space. To demonstrate the applicability of our technique, we extract the model-free implied volatility from S&P 500 index options. Subsequently, we compare its information content to that of the CBOE VIX index. Our empirical tests indicate that information content of the option-implied volatility values based on our method are superior to the VIX index.

Using a symmetry-based argument, we prove the well-known formula for the sum of the first n integers.

Using geometric arguments, we prove the well-known formula for the sum of the first n integers squared.