Massimo Morini

Risky funding: a unified framework for counterparty and liquidity risk

We analyze the liquidity component in a derivative transaction where both counterparties can default, and the effect of a counterpartys default probability on his funding costs and benefits. The analysis shows that the value of a transaction is influenced not by the total cost of funding of a counterparty, but only by that component of the cost of funding corresponding to his bond-CDS basis spread, and this regulates which trades are possible in the market. Moreover, we find that the DVA can be represented as a funding benefit for the borrower, alternatively to the market standard that considers it a benefit coming from the borrowers own default risk.

Counterparty Risk Pricing: Impact Of Closeout And First-To-Default Times

In the absence of a universally accepted procedure for the credit valuation adjustment (CVA) calculation, we compare a number of different bilateral counterparty valuation adjustment (BVA) formulas. First we investigate the impact of the choice of the closeout convention used in the formulas. Important consequences on default contagion manifest themselves in a rather different way depending on which closeout formulation is used (risk-free or replacement), and on default dependence between the two entities in the deal. Second we compare the full bilateral formula with an approximation that is based on subtracting two unilateral credit valuation adjustment (UCVA) formulas. Although the latter might be attractive for its instantaneous implementation once one has a unilateral CVA system, it ignores the impact of the first-to-default time, when closeout procedures are ignited. We illustrate in a number of realistic cases both the contagion effect due to the closeout convention, and the CVA pricing error due to ignoring the first-to-default time.

The LIBOR model dynamics: Approximations, calibration and diagnostics

In this paper we consider several parametric assumptions for the instantaneous covariance structure of the LIBOR market model, whose role in the modern interest-rate derivatives theory is becoming more and more central. We examine the impact of each different parameterization on the evolution of the term structure of volatilities in time, on terminal correlations and on the joint calibration to the caps and swaptions markets. We present a number of cases of calibration in the Euro market. In particular, we consider calibration via a parameterization establishing a controllable one to one correspondence between instantaneous covariance parameters and swaptions volatilities, and assess the benefits of smoothing the input swaption matrix before calibrating.

Impact of the First to Default Time on Bilateral CVA

We compare two different bilateral counterparty valuation adjustment (BVA) formulas. The first formula is an approximation and is based on subtracting the two unilateral Credit Valuation Adjustment (CVA)s formulas as seen from the two different parties in the transaction. This formula is only a simplified representation of bilateral risk and ignores that upon the first default closeout proceedings are ignited. As such, it involves double counting. We compare this formula with the fully specified bilateral risk formula, where the first to default time is taken into account. The latter correct formula depends on default dependence between the two parties, whereas the simplified one does not. We also analyze a candidate simplified formula in case the replacement closeout is used upon default, following ISDAs recommendations, and we find the simplified formula to be the same as in the risk free closeout case. We analyze the error that is encountered when using the simplified formula in a couple of simple products: a zero coupon bond, where the exposure is unidirectional, and an equity forward contract where exposure can go both ways. For the latter case we adopt a bivariate exponential distribution due to Gumbel to model the joint default risk of the two parties in the deal. We present a number of realistic cases where the simplified formula differs considerably from the correct one.

No‐Armageddon Measure for Arbitrage‐Free Pricing of Index Options in a Credit Crisis

In this work, we consider three problems of the standard market approach to credit index options pricing: the definition of the index spread is not valid in general, the considered payoff leads to a pricing which is not always defined, and the candidate numeraire for defining a pricing measure is not strictly positive, which leads to a nonequivalent pricing measure. We give a solution to the three problems, based on modeling the flow of information through a suitable subfiltration. With this we consistently take into account the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show on market inputs that, while the pricing difference can be negligible in normal market conditions, it can become highly relevant in stressed market conditions, like the situation caused by the credit crunch.

Arbitrage-Free Pricing of Credit Index Options: The No-Armageddon Pricing Measure and the Role of Correlation after the Subprime Crisis

In this work we consider three problems of the standard market approach to pricing of credit index options: the definition of the index spread is not valid in general, the usually considered payoff leads to a pricing which is not always defined, and the candidate numeraire one would use to define a pricing measure is not strictly positive, which would lead to a non-equivalent pricing measure. We give a general mathematical solution to the three problems, based on a novel way of modeling the flow of information through the definition of a new subfiltration. Using this subfiltration, we take into account consistently the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show that, while the related mispricing can be negligible for standard options in normal market conditions, it can become highly relevant for different options or in stressed market conditions. In particular, we show on 2007 market data that after the subprime credit crisis the mispricing of the market formula compared to the no arbitrage formula we propose has become financially relevant even for the liquid Crossover Index Options.

Credit Calibration with Structural Models: The Lehman Case and Equity Swaps Under Counterparty Risk

In this paper we develop structural first passage models (AT1P and SBTV) with time-varying volatility and characterized by high tractability, moving from the original work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini (2006)[15]. The models can be calibrated exactly to credit spreads using efficient closed-form formulas for default probabilities. Default events are caused by the value of the firm assets hitting a safety threshold, which depends on the financial situation of the company and on market conditions. In AT1P this default barrier is deterministic. Instead SBTV assumes two possible scenarios for the initial level of the default barrier, for taking into account uncertainty on balance sheet information. While in [19] and [15] the models are analyzed across Parmalats history, here we apply the models to exact calibration of Lehman Credit Default Swap (CDS) data during the months preceding default, as the crisis unfolds. The results we obtain with AT1P and SBTV have reasonable economic interpretation, and are particularly realistic when SBTV is considered. The pricing of counterparty risk in an Equity Return Swap is a convenient application we consider, also to illustrate the interaction of our credit models with equity models in hybrid products context.

A Note on Hedging with Local and Stochastic Volatility Models

The behaviour of a smile model when applied to hedging should be consistent with market evidence that asset prices and market smiles move in the same direction (Hagan et al. 2002). Local volatility models are criticized because not consistent with this desired behaviour, and this has been an important driver towards the use of stochastic volatility models. In this work we perform a simple analysis showing that, if we take into account explicitly the correlation between stochastic volatility and underlying asset which is typical of the most common stochastic volatility models, the hedging behaviour of stochastic volatility models does not always conform with the desired behaviour of a smile model in hedging. With further simple tests we show that the behaviour of local volatility and stochastic volatility models calibrated to market skew is less different than assumed in current market wisdom. Both approaches, when used consistently with model assumptions, do not show the desired behaviour in hedging, while for both models the desired behaviour is obtained in market practice by hedging techniques which are not fully consistent with rigorous model assumptions.

A Note on Correlation in Stochastic Volatility Term Structure Models

We present a simple methodology to guarantee that the total correlation structure in a Term Structure Model with one stochastic volatility factor remains positive semidefinite. We design the parameterization with the purpose of keeping as much freedom as possible for the correlation of interest rates and stochastic volatility, while letting the correlation among forward rates reproduce approximately the tendencies usually considered desirable in the market.

Efficient Analytical Cascade Calibration of the LIBOR Market Model with Endogenous Interpolation

The LIBOR Market Model (LMM) is rapidly becoming the industry standard approach for pricing interest rate derivatives like caps and swaptions. But while fast and exact calibration of model parameters to market cap quotes is possible, the problem is much more complicated for swaptions. The Cascade Calibration Algorithm (CCA) offers a solution, but implementing it can present serious numerical difficulties. Brigo and Morini explore how these can arise, and how they can be corrected by reducing the rank of the correlation matrix, for specific correlation structures. Numerical problems also arise when the data are not complete across maturities and swap lengths, which requires missing values to be introduced by interpolation. A new algorithm, the Endogenous Interpolation Cascade Calibration Algorithm, is presented to deal with this situation. The article shows that it is fast and accurate, and that it eliminates the numerical problems suffered by the CCA.