Umberto Cherubini

Bivariate option pricing with copulas

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Fréchet bounds. Applications provided include prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, no-arbitrage pricing bounds, as well as values consistent with the independence of the underlying assets are provided. As a final reference value, a copula function calibrated on historical data is used.

Multivariate Option Pricing With Copulas

In this paper we suggest the adoption of copula functions in order to price multivariate contingent claims. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. We prove that such kernel is a copula function, and that its super -replication strategy is represented by the Frechet bounds. As applications, we provide prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, we provide no-arbitrage pricing bounds, as well as the values consistent with independence of the underlying assets. As a final reference value, we use a copula function calibrated on historical data.

Intertemporal budget constraint and public debt sustainability: the case of Italy

The theory of intertemporal budget constraint is applied to test Italian public debt sustainability, with the finding that current fiscal policy has not been following a sustainable path in the 1980s. In particular, we find that (i) while primary surplus is stationary, public debt is not, (ii) permanent shocks explain about 90% of forecast error variance of public debt, while playing a minor role in primary surplus and (iii) debt is not sustainable even if stochastic discount rates are accounted for.

A copula-based model of speculative price dynamics in discrete time

This paper suggests a new technique to construct first order Markov processes using products of copula functions, in the spirit of Darsow et al. (1992) [10]. The approach requires the definition of (i) a sequence of distribution functions of the increments of the process, and (ii) a sequence of copula functions representing dependence between each increment of the process and the corresponding level of the process before the increment. The paper shows how to use the approach to build several kinds of processes (stable, elliptical, Farlie-Gumbel-Morgenstern, Archimedean and martingale processes), and how to extend the analysis to the multivariate setting. The technique turns out to be well suited to provide a discrete time representation of the dynamics of innovations to financial prices under the restrictions imposed by the Efficient Market Hypothesis.

Liquidity and credit risk

The paper uses fuzzy measure theory to represent liquidity risk, i.e. the case in which the probability measure used to price contingent claims is not known precisely. This theory enables one to account for different values of long and short positions. Liquidity risk is introduced by representing the upper and lower bound of the price of the contingent claim computed as the upper and lower Choquet integral with respect to a subadditive function. The use of a specific class of fuzzy measures, known as g λ measures enables one to easily extend the available asset pricing models to the case of illiquid markets. As the technique is particularly useful in corporate claims evaluation, a fuzzified version of Mertons model of credit risk is presented. Sensitivity analysis shows that both the level and the range (the difference between upper and lower bounds) of credit spreads are positively related to the ‘quasi debt to firm value ratio’ and to the volatility of the firm value. This finding may be read as correlation between credit risk and liquidity risk, a result which is particularly useful in concrete risk-management applications. The model is calibrated on investment grade credit spreads, and it is shown that this approach is able to reconcile the observed credit spreads with risk premia consistent with observed default rate. Default probability ranges, rather than point estimates, seem to play a major role in the determination of credit spreads.

The Dependence Structure of Running Maxima and Minima: Results and Option Pricing Applications

We provide general results for the dependence structure of running maxima (minima) of sets of variables in a model based on (i) Markov dynamics; (ii) no Granger causality; (iii) cross-section dependence. At the time series level, we derive recursive formulas for running minima and maxima. These formulas enable to use a “bootstrapping” technique to recursively recover the pricing kernels of barrier options from those of the corresponding European options. We also show that the dependence formulas for running maxima (minima) are completely defined from the copula function representing dependence among levels at the terminal date. The result is applied to multivariate discrete barrier digital products. Barrier Altiplanos are simply priced by (i) bootstrapping the price of univariate barrier products; (ii) evaluating a European Altiplano with these values.

Pricing Vulnerable Options with Copulas

C ounterparty risk is usually defined as the risk which stems from the fact that the counterparty of a derivative contract is not solvent before or at expiration. As most of the derivative trading activity has been moving from standardized products quoted on futures-style markets, towards customized products traded on over-the-counter markets, the issue of counterparty risk evaluation has increasingly gathered momentum and is now one of the hot topics in option pricing theory. The corresponding options are named vulnerable. Every practitioner is well aware that accounting for counterparty risk has a substantial impact both on the evaluation and the hedging policy of a derivative contract. Any hedging strategy has in fact to take into account that the counterparty could go bankrupt at any moment during the life of the contract and that such default is relevant inasmuch as the contract is in the money, i.e., in the case in which it gives a positive payoff. Modifying the hedging strategy and the price of an option contract in this direction raises two main technical problems. The first has to do with the existence of different ways to represent market risk on the one side, credit (counterparty) risk on the other. We have in fact a large amount of pricing models for the non-vulnerable options, from Black-Scholes onwards, and different approaches to credit risk evaluation, structural or intensity-based. We would like to work in a full general environment, consistent with any default-free pricing rule and with any credit risk evaluation method. The second technical problem has to do with the dependence or association between the underlying asset of the derivative contract and the event of default of the counterparty, i.e., between market and credit risk.1 Accounting for dependence leads to a delicate technical issue, linked to the possibility of using multivariate distributions more general than the Gaussian one and being able to estimate their dependence parameters. In order to overcome both these difficulties, we will use copula functions to price vulnerable option contracts. As for the first task, copulas can embed any model for market and credit risk, namely for representing the probability of exercise of the option and that of default of the counterparty. As for the second, they allow us, for any given couple of marginal probabilities (the exercise and default one, separately considered), to couple them with great freedom, including with a Gaussian dependence structure. Copulas also readily provide us with pricing—and hedging—bounds, which are very useful since vulnerable option markets are typically incomplete ones. If we apply complete market pricing techniques to the evaluation of derivative contracts with counterparty risk, we implicitly assume that such source of Pricing Vulnerable Options With Copulas

Computational methods in finance: option pricing

Many computational methods familiar to scientists and engineers are now heavily used in todays financial markets. This survey looks at the history and the state of the art for one branch of computational finance, and explains why neural networks show special promise in setting correct prices for options.

Computing the Volume of n-Dimensional Copulas

Abstract A problem that is very relevant in applications of copula functions to finance is the computation of the survival copula, which is applied to enforce multivariate put–call parity. This may be very complex for large dimensions. The problem is a special case of the more general problem of volume computation in high-dimensional copulas. We provide an algorithm for the exact computation of the volume of copula functions in cases where the copula function is computable in closed form. We apply the algorithm to the problem of computing the survival of a copula function in the pricing problem of a multivariate digital option, and we provide evidence that this is feasible for baskets of up to 20 underlying assets, with acceptable CPU time performance.

Pricing and Hedging Credit Derivatives with Copulas

In this paper, we apply a copula function pricing technique to the evaluation of credit derivatives, namely a vulnerable default put option and a credit switch. Also in this case, copulas enable one to separate the specification of marginal default probabilities from their dependence structure. Their use is based here on no-arbitrage arguments, which provide pricing bounds and easy-to-implement super-replication strategies. Copyright Banca Monte dei Paschi di Siena SpA, 2003