Jan De Spiegeleer

Pricing of contingent convertibles under smile conform models

We look at the problem of pricing CoCo bonds where the underlying risky asset dynamics are given by a smile conform model, more precisely an exponential Levy process incorporating jumps and heavy tails. A core mathematical quantity that is needed in closed form in order to produce an exact analytical expression for the price of a CoCo is the law of the infimum of the underlying equity price process at a fixed time. With the exception of Brownian motion with drift, no such closed analytical form is available within the class of Levy process that are suitable for financial modeling. Very recently however there has been some remarkable progress made with the theory of a large family of Levy processes, known as β-processes, cf. Kuznetsov [12] and Kuznetsov et al. [14]. Indeed for this class of Levy processes, the law of the infimum at an independent and exponentially distributed random time can be written down in terms of the roots and poles of its characteristic exponent; all of which are easily found within regularly spaced intervals along one of the axes of the complex plane. Combining these results together with a recently suggested Monte-Carlo technique, due to Kuznetsov et al. [13], which capitalises on the randomised law of the infimum we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of β-processes, known as β-VG, which have similar characteristics to the classical Variance-Gamma model. The theory is put to work by performing two case studies. After calibrating our model to market data, we price and analyze one of the Lloyds CoCos as well as the first Rabo CoCo.

Efficient Pricing of Contingent Convertibles Under Smile Conform Models

We look at the problem of pricing CoCo bonds where the underlying risky asset dynamics are given by a smile conform model, more precisely an exponential Levy process incorporating jumps and heavy tails. A core mathematical quantity that is needed in closed form in order to produce an exact analytical expression for the price of a CoCo is the law of the infimum of the underlying equity price process at a fixed time. With the exception of Brownian motion with drift, no such closed analytical form is available within the class of Levy process that are suitable for financial modeling. Very recently however there has been some remarkable progress made with the theory of a large family of Levy processes, known as beta-processes, cf. Kuzentsov and Kuzentsov et al.. Indeed for this class of Levy processes, the law of the infimum at an independent and exponentially distributed random time can be written down in terms of the roots and poles of its characteristic exponent; all of which are easily found within regularly spaced intervals along one of the axes of the complex plane. Combining these results together with a recently suggested Monte-Carlo technique, due to Kuzentsov et al., which capitalises on the randomised law of the infimum we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of beta-processes, known as beta-VG, which have similar characteristics to the classical Variance-Gamma model. The theory is put to work by performing two case studies. After calibrating our model to market data, we price and analyze one of the Lloyds CoCos as well as the first Rabo CoCo.

Multiple Trigger CoCos: Contingent Debt Without Death Spiral Risk

This paper starts with the observation that the average issue size during 2012 of contingent convertible (CoCo) bonds was more than

Hunting for Black Swans in the European Banking Sector Using Extreme Value Analysis

1 bn. Typically a CoCo is converted into shares when a pre‐defined capital ratio such as the core tier 1 ratio (CT1) drops below a minimum level. In some other cases, the contingent convertibles investors will suffer from a pre‐defined haircut instead of a conversion into shares. Investors could dynamically hedge the equity exposure embedded within a contingent convertible by taking an offsetting short position in the underlying shares. This dynamic hedging can in some circumstances have a negative impact on the share price of the bank. It could indeed lead to a spiral of falling share prices. This so‐called death spiral effect can only be avoided if the size of the contingent convertible is moderate compared to the amount of outstanding public traded shares. In this contribution we advocate the use of contingent debt where there is more than one conversion trigger. Banks should move away from one large single CoCo issue towards issues with multiple accounting triggers spread across an extended range. This will alleviate the death spiral risk. The expected dynamic behavior of a CoCo bond has been modeled using a credit derivates approach. From these models we then quantify the equity sensitivity and the negative gamma resulting from the design of a contingent convertible and illustrate the possible pitfalls of a death spiral on the share price.

Machine learning for quantitative finance: fast derivative pricing, hedging and fitting

In financial risk management, a Black Swan refers to an event that is deemed improbable yet has massive consequences. In this communication we propose a way to investigate if the recent financial crisis was a Black Swan event for a given bank based on weekly closing prices and derived log-returns. More specifically, using techniques from extreme value methodology we estimate the tail behavior of the negative log-returns over two specific horizons: Pre-crisis: from January 1, 1994 until August 7, 2007 (often referred to as the official starting date of the credit crunch crisis); Post-crisis: from August 8, 2007 until September 23, 2014 (the cut-off date of our study).

CoCo Bonds and Implied CET1 Volatility

In this paper, we show how we can deploy machine learning techniques in the context of traditional quant problems. We illustrate that for many classical problems, we can arrive at speed-ups of several orders of magnitude by deploying machine learning techniques based on Gaussian process regression. The price we have to pay for this extra speed is some loss of accuracy. However, we show that this reduced accuracy is often well within reasonable limits and hence very acceptable from a practical point of view. The concrete examples concern fitting and estimation. In the fitting context, we fit sophisticated Greek profiles and summarize implied volatility surfaces. In the estimation context, we reduce computation times for the calculation of vanilla option values under advanced models, the pricing of American options and the pricing of exotic options under models beyond the Black–Scholes setting.

Data Mining of Contingent Convertible Bonds

In this work we introduce the notion of implied Core Equity Tier 1 volatility and the concept of a risk-adjusted distance to trigger. Using a derivatives-based valuation approach, we are able to derive the implied CET1 volatility from the market price of a CoCo bond in a Black-Scholes setting. The numerical results in this paper show how different contingent convertibles issued by the same bank and sharing a similar contractual CET1 trigger, have almost identical implied CET1 volatility levels. The same results confirm the difference in market risk between Tier 2 and Additional Tier 1 CoCo bonds. The ability to obtain an implied level for the CET1 volatility offers other interesting results. We are able to determine the implied CET1 level corresponding to a coupon cancellation. It further allowed us for example to look at the severity of one of the stress tests imposed by the ECB on European banks in November 2014. In that perspective we were also able to derive implied PONV CET1 levels.

Pricing Contingent Convertibles: A Derivatives Approach

This paper develops and applies sophisticated data mining techniques to detect in an early stage potential risks regarding the stability of institutions by making use of the market information of their issued contingent capital instruments. Data mining is an important new topic within the financial world. The detection of observations which are different from the majority, called outliers, can be of interest for market analysts, risk managers, regulators and traders. These exceptions might be caused by extraordinary circumstances that may potentially require extra hedging or can be seen as trading opportunities. They could also give regulators an early warning and signal for potential trouble ahead. In this paper we first explain and apply the new risk measure, called the Value-at-Risk Equivalent Volatility (VEV). The concept was introduced by the European authorities in the new PRIIPs1 regulation and needs to be implemented for all structured products by January 1st 2018. This risk-measure is an extension of the classical volatility measure by taking into account skewness and kurtosis. This measure however works in a one-dimensional setting. In this paper we apply outlier detection and the VEV concept to CoCo bonds. CoCos are hybrid high yield securities that convert into equity or write down if the issuing financial institution is in a distressed situation. Further we want to detect outliers in the CoCo market taking into account multiple variables such as the CoCo market returns and the underlying equity return. Based on a multiple-dimension distance we can detect CoCos that are outlying compared to previous time periods but also taking into account extreme moves of the market situation. To some extent, CoCos can be seen as derivative instruments with some capital ratio (CET1) as underlying driver. In this perspective, a CoCo market price is just the price of a derivative and hence contains forward looking information or at least the markets anticipated view on the financial health of the institution and the level of the relevant trigger. This paper develops data mining techniques that incorporate such forward looking view by comparing historical data with current CoCo market prices.