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Abstract:
The price of a CMS based derivative is largely affected by the value of swaption volatilities at extreme strikes. In this article, we propose a very simple procedure for stripping consistently implied volatilities and CMS adjustments from the market quotes of swaption smiles and CMS swap spreads.

swaption, CMS, volatility smile, volatility skew, convexity adjustment, SABR, stochastic volatility, calibration

Abstract:
The SABR closed-form formula (Hagan et. al 2002) is the standard for smile-consistent pricing in the swaption market. Here we address the issue of turning SABR assumptions into a consistent and arbitrage-free term structure model in the BGM/Libor Market Model framework. We compute the joint dynamics followed by Libor rates and stochastic volatility of SABR kind under the general pricing measures used for interest rate derivatives, and we observe that the volatility dynamics is non-standard. Based on the analysis of the equation found, we develop and justify theoretically a few approximations aimed at making these no-arbitrage dynamics compatible with the use of the SABR closed-form formula. Then the formulas developed above are confronted both with alternative numerical implementations and with market data. We verify that the formulas for no-arbitrage corrections are acceptably precise, maintain good fitting, and produce regular Libor parameters. Finally we verify that the no-arbitrage corrections to the volatility dynamics make the out-of-calibration-sample prices implied by the model closer to market quotations, compared to prices implied by a trivial multivariate SABR neglecting such corrections.

stochastic volatility, SABR, no-arbitrage,libor market model, BGM

Abstract:
In the current markets, options with different strikes or maturities are usually priced with different implied volatilities. This stylized fact, which is commonly referred to asfsmile effect, can be accommodated by resorting to specific models, either for pricing exotic derivatives or for inferring implied volatilities for non quoted strikes or maturities. The former task is typically achieved by introducing alternative dynamics for the underlying asset price, whereas the latter is often tackled by means of statical adjustments or interpolations.

In this article, we deal with this latter issue 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 (the 0Delta straddle, the risk reversal and the vega-weighted butterfly), 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 5Delta put to the 5Delta call.

In the following, we will review this market procedure for a given currency. In particular, we will derive closed-form formulas so as to render its construction more explicit. We will then test the robustness (in a static sense) of the resulting smile, in that changing consistently the three initial pairs of strike and volatility produces eventually the same implied volatility curve. We will also show that the same procedure applied to Europeanstyle claims is consistent with static-replication results and consider, as an example, the practical case of a quanto European option. We will finally prove that the market procedure can also be justified in dynamical terms, by defining a hedging strategy that is locally replicating and self-financing.

FX option, smile, consisten pricing, stochastic volatility

Abstract:
In this paper we start by introducing the standard moment-matching procedure that one can apply to simulate the average price of a basket of basic assets. The basic idea is that of approximating the actual process of the basket value by a sufficiently simple stochastic process. The expression "sufficiently simple" should be interpreted as "simple enough to allow for analytic solutions to the pricing problem at hand".

The approximation happens on the basis of a moment matching principle, which can be stated as follows: set the parameters of the approximating process so that as many moments of the actual basket-price process as possible are exactly reproduced. With the usual lack of fantasy, the market choice of an approximating process seems to have fallen onto the lognormal one. The distinctive parameters of such a process being only two (the average return and the return's standard deviation over the time horizon set by the option to price) the moment matching procedure can only match the first two moments of the original distribution. The lengthy calculations can be performed so as to take into account the effect of dividends, either continuous or discrete (but in any case deterministic, both in payment dates and in amounts). A more compact formulation of this method is obtained by resorting to forward prices, which incorporate interest rates and dividends. We describe this basic framework in detail and then move to the three moments matching procedure, obtained by shifting the approximating Lognormal basket by a deterministic constant parameter. This new parameter allows to fit the first three moments without losing analytical tractability, in that we can immediately characterize the distributional properties of the resulting process trivially.

We then move to an empirical analysis of the two and three moments matching approximations, where we study the case of a basket of two equities in the Italian stock exchange and compare results by resorting to a Monte Carlo simulation to obtain the "true" distribution and statistics of the basket.

We subsequently move to analyze specifically the implications of the three moments method as far as a call option pricing is concerned.

The second part of the paper address the problem of computing a synthetic but at the same time rigorous measure of the deviation of the approximated baskets distributions from the real basket distribution. To characterize rigorously this distributional discrepancy, we introduce both the Kullback-Leibler information and the Hellinger distances in suitable spaces of probability densities, and explain how this can help us in our investigation. We compute the distances of the real basket from the parametric families of densities being used in the two and three moments approximations through Monte Carlo simulation. The two families are respectively the lognormal and shifted lognormal families. Finally, we try and isolate the variables and the situations causing this distance to increase drastically via a case study, thus characterizing the case where the two and three moments approximations can fail.

Abstract:
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.

swaption, CMS, volatility smile, volatility skew, convexity adjustment, Gaussian model, Hull and White model, stochastic volatility, uncertain volatility, calibration

Abstract:
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.

credit crunch; credit, liquidity, market rates, forward curve, discount curve, bootstrapping, FRAs, swaps, caps, swaptions, LIBOR market models, measure changes, stochastic volatility, Heston volatility, closed form formulas

Abstract:
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, and for the first time we derive the related explicit dynamics and show that it leads to a stochastic differential equation admitting a unique strong solution. We also provide 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.

Abstract:
We introduce a simple extension of a shifted geometric Brownian motion for modelling forward LIBOR rates under their canonical measures. The extension is based on a parameter uncertainty modelled through a random variable whose value is drawn at an infinitesimal time after zero. The shift in the proposed model captures the skew commonly seen in the cap market, whereas the uncertain volatility component allows us to obtain more symmetric implied volatility structures.

We show how this model can be calibrated to cap prices. We also propose an analytical approximated formula to price swaptions from the cap calibrated model. Finally, we build the bridge between caps and swaptions market by calibrating the correlation structure to swaption prices, and analyzing some implications of the calibrated model parameters.

Libor models, caps, swaptions

Abstract:
In the present paper we show how to extend any time-homogeneous short-rate model and analytically tractable short-rate model (such as Vasicek (1977), Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model which 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.

Short-rate models, Analytical tractability, Yield-Curve fitting, Vasicek's model, Dothan's model, Cox-Ingersoll-Ross' model, Longstaff and Schwartz's model, Monte Carlo evaluation

Abstract:
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.

Stochastic Differential Equations, Fokker--Planck Equation, Exponential Families, Stock Price Models, Black and Scholes model, Option Pricing, Trading Time Grid, Delta-Markovianity, Market Incompleteness, Option replication error

Abstract:
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.

hedging, local volatility, stochastic volatility, sabr

Abstract:
We introduce a new forward CPI model that is based on a multi-factor volatility structure and leads to SABR-like dynamics for forward inflation rates. Our approach is the first in the financial literature to reconcile zero-coupon and year-on-year quotes, granting, at the same time, a both fast and accurate calibration to market data. Explicit formulas for year-on-year caps/floors as well as for zero-coupon options are then derived in terms of the SABR volatility form. An example of calibration to market data is finally provided.

Inflation, SABR dynamics, closed-form formulas, calibration

Abstract:
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.

correlation, stochastic volatility, libor market model

Abstract:
We extend the model presented in Bonollo et al. by introducing a multiscenario framework that allows for a richer and more realistic specification, including non-static (stochastic) probabilities of default and losses given default. Though more complex from a computational point of view, the model with scenarios is still tractable analytically, yielding results in closed form expressions. The approximated value at risk has been calculated by generalizing the procedure exposed in Bonollo et al. for the single scenario case, in the presence of granularity in the exposures, sector concentration and contagion. The outcome is not simply a weighted sum of the VaRs in the individual scenarios, but results in a more involved function of the single scenarios’ parameters. The theoretical model description is complemented with an in-depth numerical
analysis.

Basel II, second pillar, credit VaR, analytical formula, contagion risk, sectoral risk, stochastic default probability, stochastic recovery, scenario

Abstract:
This paper deals with the effects of concentration (single name and sectoral) and contagion risk on credit portfolios. Results are obtained for the value at risk of the portfolio loss distribution, in the analytical framework originally developed by Vasicek in 1991 [1]. VAR is expressed as a sum of terms: the first contribution represents the value at risk of a hypothetical single-factor homogeneous portfolio, the remaining terms are corrections due to contagion, imperfect granularity and multiple industry-geographic sectors. A detailed numerical analysis is also presented.

Basel II, second pillar, credit VaR, analytical formula, contagion, sectorial risk

Abstract:
Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Mercurio and Moraleda (1996) and Moraleda and Vorst (1996a) have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, both models are Gaussian, and hence interest rates may become negative. In this paper we propose a family of interest rate models where (i) humped volatility structures are possible; (ii) the interest rate volatility can depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. We propose, for instance, models of the type of the proportional Ritchken and Sankarasubramanian (1995) and the Black and Karasinski (1991) model. To gain the computational tractability (iii), we show how to embed all models in this paper in either the Ritchken and Sankarasubramanian (1995) or the Hull and White (1990, 1994) class of models.