Downloadable papers and reports

"Le nostre misture riproducono grazie all'arte l'essenza che si vuole invocare, moltiplicano il potere di ciascun elemento."
- Ipazia in Baudolino
(2000) - Umberto Eco

"Interest Rates and The Credit Crunch: New Formulas and Market Models"
Abstract   .pdf [629] 
"Smile-consistent CMS adjustments in closed form: introducing the Vanna-Volga approach"joint with A. Castagna and M. Tarenghi (2009).
Abstract   .pdf [211k] 
"No-arbitrage conditions for cash-settled swaptions" (2007)
Abstract   .pdf [151k]
A version of this paper appeared in Risk (February 2008).
"Consistent Pricing of FX Options", joint with A. Castagna (2006).
Abstract   .pdf [189k]
A version of this paper appeared in Risk (January 2007).
"Swaption skews and convexity adjustments", joint with A. Pallavicini (2005).
Abstract   .pdf [216k]
A version of this paper appeared in Risk (August 2006).
"Mixing Gaussian Models to Price CMS Derivatives", joint with A. Pallavicini (2005).
Abstract   .pdf [179k]
"Pricing Inflation-Indexed Options with Stochastic Volatility", joint with N. Moreni (2005).
Abstract   .pdf [283k]
A version of this paper appeared in Risk (March 2006).
"Consistent Pricing and Hedging of an FX Options Book", joint with L. Bisesti and A. Castagna (2005).
Abstract   .pdf [215k]
A version of this paper appeared in Kyoto Economic Review (2005).
"Pricing Inflation-Indexed Derivatives" (2004).
Abstract   .pdf [245k]
A version of this paper appeared in Quantitative Finance (2005).
"A Shifted Lognormal LIBOR Model with Uncertain Parameters", joint with E. Errais and G. Mauri (2004).
Abstract   .pdf [300k]  
"Smile at the Uncertainty", joint with D. Brigo and F. Rapisarda (2003).
.pdf [192k]
A version of this paper appeared in Risk (May 2004).
"A Multi-Stage Uncertain Volatility Model" (2002).
.pdf [136k]
"Parameterizing Correlations: A Geometric Interpretation", joint with F. Rapisarda and D. Brigo (2002).
Abstract  .pdf [330k]
"Pricing the Smile in a Forward LIBOR Market Model" (2002).
.pdf [330k]
A reduced version of this paper appeared in Quantitative Finance (2003).
"Lognormal-Mixture Dynamics Under Different Means", joint with D. Brigo and G. Sartorelli (2002).
.pdf [152k]
"Alternative Asset-Price Dynamics and Volatility Smile", joint with D. Brigo and G. Sartorelli (2002).
.pdf [255k]
A version of this paper appeared in Quantitative Finance (2002).
"Approximated Moment-Matching Dynamics for Basket-Options Simulation", joint with D. Brigo, F. Rapisarda and R. Scotti (2001).
.pdf [1,749k]
A version of this paper appeared in Quantitative Finance (2004).
"Lognormal-Mixture Dynamics and Calibration to Market Volatility Smiles", joint with D. Brigo (2001).
.pdf [678k]
A version of this paper appeared in the International Journal of Theoretical & Applied Finance (2002).
"On Deterministic-Shift Extensions of Short-Rate Models", joint with D. Brigo (2000).
.pdf [286k]
A reduced version of this paper appeared in Finance and Stochastics (July 2001).
"Discrete Time vs Continuous Time Stock-price Dynamics and Implications for Option Pricing", joint D. Brigo (2000).
.pdf [218k]
A short version of this paper appeared in Finance and Stochastics (April 2000).

"Pricing and Static Replication of FX Quanto Options"  .pdf [129k] 
"No-Arbitrage Conditions for a Finite Options System"  .pdf [146k] 
"A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model"  .pdf [138k] 
"Pricing of Options on two Currencies Libor Rates"  .pdf [151k] 

"Interest Rates and The Credit Crunch: New Formulas and Market Models"
We start by describing the major changes that occurred in the quotes of market rates after the 2007 subprime mortgage crisis. We comment on their lost analogies and consistencies, and hint on a possible, simple way to formally reconcile them. We then show how to price interest rate swaps under the new market practice of using different curves for generating future LIBOR rates and for discounting cash flows. Straightforward modifications of the market formulas for caps and swaptions will also be derived.
Finally, we will introduce a new LIBOR market model, which will be based on modeling the joint evolution of FRA rates and forward rates belonging to the discount curve. We will start by analyzing the basic lognormal case and then add stochastic volatility. The dynamics of FRA rates under different measures will be obtained and closed form formulas for caplets and swaptions derived in the lognormal and Heston (1993) cases
“Smile-consistent CMS adjustments in closed form: introducing the Vanna-Volga approach"
In this article, we introduce the Vanna-Volga approach for an alternative valuation of CMS convexity adjustments. Our pricing procedure leads to closed-form formulas that are extremely simple to implement and that retrieve, within bid-ask spreads, market data of CMS swap spreads.
"No-arbitrage conditions for cash-settled swaptions"
In this note, we derive no-arbitrage conditions that must be satisfied by the pricing function of cash-settled swaptions. The specific examples of a flat implied volatility and of a smile generated by the SABR functional form will be analyzed
"Consistent Pricing of FX Options"
We deal with the problem of inferring implied volatilities for non quoted strikes and analyze a possible solution in a foreign exchange (FX) option market. In such a market, in fact, there are only three active quotes for each market maturity, thus presenting us with the problem of a consistent determination of the other implied volatilities. FX brokers and market makers typically address this issue by using an empirical procedure to construct the whole smile for a given maturity. Volatility quotes are then provided in terms of the option’s delta, for ranges from the 5 delta put to the 5 delta call. In this article, we will review this market procedure for a given currency and prove related robustness and consistency results.
"Swaption skews and convexity adjustments"
We test both the SABR model and the shifted-lognormal mixture model as far as the joint calibration to swaption smiles and CMS swap spreads is concerned. Such a joint calibration is essential to consistently recover implied volatilities for non-quoted strikes and CMS adjustments for any expiry-tenor pair.
"Mixing Gaussian Models to Price CMS Derivatives"
In this article, we propose a simple interest rate model, which can well accommodate swaption smiles, while recovering market prices of CMS swap spreads. The model is based on a (possibly multi-factor) Gaussian short rate model coupled with parameter uncertainty. Examples of calibration to real market data will be presented as well as the pricing of some typical CMS-based derivatives.
"Pricing Inflation-Indexed Options with Stochastic Volatility"
In order to recover smile-consistent prices for inflation-indexed caps and floors, we propose a stochastic volatility model for forward consumer price indices, with volatility dynamics as in Heston (1993). Closed-form formulas for inflation-indexed caplets and floorlets based on the Carr and Madan (1998) Fourier transform approach are then derived. An example of calibration to market data and numerical details concerning our pricing procedure are finally given.
"Consistent Pricing and Hedging of an FX Options Book"
In the foreign exchange (FX) options market away-from-the-money options are quite actively traded, and quotes for the same type of instruments are available everyday with very narrow spreads (at least for the main currencies). This makes it possible to devise a procedure for extrapolating the implied volatilities of non-quoted options, providing us with reliable data to which to calibrate our favorite model.
In this article, we test the goodness of the Brigo, Mercurio and Rapisarda (2004) model as far as some fundamental practical implications are concerned. This model, which is based on a geometric Brownian motion with time-dependent coefficients that are not known initially and whose value is randomly drawn at an infinitesimal future time, can accommodate very general volatility surfaces and, in case of the FX options market, can lead to a perfect fit to the main volatility quotes.
We first show the fitting capability of the model with an example from real market data. We then support the goodness of our calibration by providing a diagnostic on the forward volatilities implied by the model. We also compare the model prices of some exotic options with the corresponding ones given by a market practice. Finally, we show how to derive bucketed sensitivities to volatility and how to hedge accordingly a typical options book.
"Pricing Inflation-Indexed Derivatives"
We start by briefly reviewing the approach proposed by Jarrow and Yildirim (2003) for modelling inflation and nominal rates in a consistent way. Their methodology is applied to the pricing of general inflation-indexed swaps and options. We then introduce two different market model approaches to price inflation swaps, caps and floors. Analytical formulas are explicitly derived. Finally, an example of calibration to swap market data is considered.
"A Shifted Lognormal LIBOR Model with Uncertain Parameters"
The lognormal LIBOR market model leads to flat implied volatility surfaces. To accommodate the typical smile effect in the caps and swaptions markets, one then usually resort to local- or stochastic-volatility models, possibly with jumps. This models are generally difficult to implement and rather time consuming when used to price and hedge exotic derivatives. However, it is possible to introduce stochasticity in the volatility in a very simple and intuitive manner, so as to calibrate the market implied volatility surfaces while preserving a great deal of analytical tractability. This is achieved by assuming that the forward rate dynamics are given by displaced geometric Brownian motions where the model parameters are not known at the initial time, but are discrete random variables whose values are drawn at an infinitesimal time. We refer to our model as to a Shifted Lognormal LIBOR Model with Uncertain Parameters (SLLMUP).
In this article, we analyze the SLLMUP analytical tractability by deriving caps and swaptions prices in closed form. We then illustrate how the model can accommodate market caps data and how the  instantaneous correlation parameters can be used for a calibration to swaptions prices. We finally analyze some important model's implications: i) we infer the swaptions smile implied by our joint calibration, and ii) we plot the evolution of some forward volatilities implied by the model.
"Smile at the Uncertainty"
The success of the Black-Scholes formula is mainly due to the possibility of synthesizing option prices through a unique parameter, the implied volatility. The Black-Scholes model, however, can not be used to price simultaneously all options in a given market because of the smile/skew effect commonly observed in practice. Moreover, historical analysis shows that volatilities are indeed stochastic.
Stochastic volatility models, therefore, seem to be a more realistic choice when modelling asset price dynamics for valuing derivative securities. However, only few examples retain enough analytical tractability.
The purpose of this paper is to propose a stochastic-volatility model that is analytically tractable as much as Black and Scholes’s and for which Vega hedging can be defined in a natural way. The model is based on an uncertain volatility whose random value is drawn, on an infinitesimal future time, from a finite distribution.
Our uncertain volatility model is equivalent to assuming a number of different possible scenarios for the asset forward volatility, which can therefore be hedged accordingly. An application to the FX market is explicitly considered.
"A Multi-Stage Uncertain Volatility Model".
We consider a simple uncertain-volatility model for the asset price underlying a given option market. The asset price volatility is assumed to follow a discrete (actually finite) Markov chain, which changes value on some fixed future times. The volatility chain is independent of the Brownian motion governing the future evolution of the asset. 
Modeling the volatility evolution in this way is equivalent to assuming different possible scenarios for the asset forward volatility. Closed form formulas for the claims that are explicitly priced under the Black-Scholes paradigm, are derived accordingly.
"Parameterizing Correlations: A Geometric Interpretation".
Finance as a whole is linked to correlations among the prices of assets (stocks, bonds, . . . ) or more generally of basic financial quantities (including interest rates, credit spreads, . . . ). Correlations are usually expressed in terms of matrices, which must satisfy four basic properties: A1) all their entries must lie in the interval [-1;1]; A2) diagonal entries must be equal to one; A3) the matrix must be symmetric; A4) the matrix must be positive semidefinite. 
In many applications, it is fundamental to be able to safely build and manipulate correlation structures, with which to model the joint behaviour of financial variables, both for pricing purposes and for risk management. This implies the need for a parameterization of a generic correlation matrix – a parameterization guaranteeing that the matrix will fulfill all four properties above. In this paper we will revise the “standard angles parameterization” (SAP) procedure described for example in Pinheiro and Bates [5], recently applied to financial modeling by Rebonato and Jaeckel [8], and discuss a geometric interpretation to it. 
Moreover, thanks to the new geometrical view, we will recast the SAP in a more parsimonious way, leading to the “triangular angles parameterization” (TAP), which is actually the form of SAP described in [5]. Finally, we consider an application to a demanding problem, that of sensibly parameterizing instantaneous correlations in the LMM.
"Pricing the Smile in a Forward LIBOR Market Model".
We introduce two general classes of analytically-tractable diffusions for modeling forward LIBOR rates under their canonical measure.
The first class is based on the assumption of forward-rate densities given by the mixture of known basic densities. We consider two fundamental examples: i) a mixture of lognormal densities, and ii) a mixture of densities associated to “hyperbolic-sine” processes. We derive explicit dynamics, prove existence and uniqueness results for the solution to the related SDEs and obtain closed-form formulas for caps prices.
The second class is based on assuming a smooth functional dependence, at expiry, between a forward rate and an associated Brownian motion. This class is highly tractable: it implies explicit dynamics, known marginal  and transition densities and explicit caplet prices at any time. As an example, we analyze the dynamics given by a linear combination of geometric Brownian motions (GBM) with perfectly correlated  (decorrelated) returns. We finally construct a specific model in the second class that reproduces exactly the market caplet volatilities given in input.
Examples of the implied-volatility curves produced by the considered models are also shown.
"Lognormal-Mixture Dynamics under Different Means".
We prove existence and uniqueness of the strong solution to the SDE whose drift rate is a given constant and whose diffusion coefficient is defined so as to imply a marginal density that is given by a mixture of lognormal densities. Such densities, which can have different means, must fulfill the no-arbitrage conditions that typically arise in mathematical finance when using the given SDE for modeling the price dynamics of some financial asset.
"Alternative Asset-Price Dynamics and Volatility Smile".
We propose a general class of analytically tractable models for the dynamics of an asset price leading to smiles or skews in the implied volatility structure. The considered asset can be an exchange rate, a stock index, or even a forward Libor rate. The class is based on an explicit SDE under a given forward measure. The models we propose feature i) explicit asset-price dynamics, ii) virtually unlimited number of parameters, iii) analytical formulas for European options.
We then consider the fundamental case where the asset price density is given, at every time, by a mixture of lognormal densities. We also derive an explicit approximation of the implied volatility function in terms of the option moneyness. We finally introduce two other examples: the first is still based on lognormal densities, but it allows for different means in the distributions, the second is instead based on processes of hyperbolic-sine type.
"Approximated Moment-Matching Dynamics for Basket-Options Simulation".
The aim is this paper is to present two moment matching procedures for basket-options pricing and to test their distributional approximations via distances on the space of probability densities, the Kullback-Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the real simulated basket terminal distribution and the distributions used for the approximation, both in the lognormal and shifted-lognormal families. We isolate influences of instantaneous volatilities and instantaneous correlations, in order to assess which configurations of these variables have a major impact on the KLI and HD and therefore on the quality of the approximation. A number of numerical investigations is carried out.
"Lognormal-Mixture Dynamics and Calibration to Market Volatility Smiles".
We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, deriving explicit dynamics, closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data.
"On Deterministic-Shift extensions of Short-Rate Models".
In the present paper we show how to extend any time-homogeneous short-rate model (such as Vasicek (1977),  Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model that can reproduce any observed yield curve, through a procedure  that preserves the possible  analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model,  our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. Our approach can also be applied to the Dothan (1978) or Rendleman and Bartter (1980)  model, thus yielding a "quasi" lognormal short-rate model which fits any given yield  curve and for which there exist analytical formulae for prices of zero coupon bonds. We also consider the extension of time-homogeneous models without analytical formulae  but whose tree-construction procedures are particularly appealing, such as the Exponential Vasicek's. We explain why the CIR++ extended CIR model is the more interesting model obtained through our procedure. We also give explicit analytical formulae for bond options, hence swaptions, caps and floors, and we explain how  the model can be used for Monte Carlo evaluation of European path-dependent interest-rate derivatives. We finally hint at the same extension for multifactor models and explain its strong points for concrete applications.
"Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing".
In the present paper we construct stock price processes with the same marginal log-normal law as that of a geometric Brownian motion and also with the same transition density (and returns' distributions) between any two instants in a given  discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes', in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading. The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.