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Abstract:
This paper contains a statistical description of the whole U.S. forward rate curve (FRC), based on data from the period 1990-1996. We find that the average deviation of the FRC from the spot rate grows as the square-root of the maturity, with a proportionality constant which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a `Value-at-Risk' type of pricing. The instantaneous FRC however departs from a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behavior of the spot itself. We show that this is consistent with the volatility `hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the FRC, suggesting an analogy with the motion of a vibrating string subject to random perturbations. This shape dependent term could lead, in principle, to arbitrage. However, this arbitrage cannot be implemented in practice because of transaction costs. We suggest that the presence of transaction costs (or other market `imperfections') is crucial for model building, for a much wider class of models becomes eligible to represent reality.

Abstract:
An investment horizon is in practice not frequently known with certainty at the initial investment date. This paper addresses the problem of pricing and hedging a random cash-flow received at a random date in a general stochastic environment. We first argue that specific timing risk is induced by the presence of an uncertain time-horizon if and only if the random time under consideration is not a stopping time of the filtration generated by prices of traded assets. In that context, we provide an explicit characterization of the set of equivalent martingale measures, as well as a necessary and sufficient condition for a convenient separation between adjustment for market risk and timing risk. These results allow us to clarify the definition of the market price for timing risk, and lead to general pricing formulae and explicit hedging strategies for random cash-flows in the presence of timing risk. Potential applications are the valuation of employee stock options, real options, catastrophe insurance contracts, credit derivatives, callable and convertible bonds, mortgage-backed securities, as well as any other asset featuring an embedded prepayment option.

Abstract:
While there are now a number of empirical studies on the subject, very little is known on the market price for default risk from a theoretical perspective. This paper is a first step in the direction of an equilibrium model for the pricing of defaultable securities in an incomplete market setup. We first provide an explicit characterization of the set of equivalent martingale measures consistent with no arbitrage in the presence of default risk, as well as a necessary and sufficient condition for a convenient separation between adjustments for market risk and default risk. That result allows us to spell out an unambiguous definition of the market price for default risk as the logarithm of the ratio of the risk-adjusted probability of default to the original probability of default. It also suggests the following question: how should the original probability of default be adjusted to account for agents' risk-aversion? We address this question in a dynamic continuous-time equilibrium setup, and obtain a defaultable version of a standard consumption-based capital asset pricing model. In particular, we confirm the intuition that the correlation between default risk and market risk is a key ingredient of the equilibrium price for default risk, and obtain a quantitative estimate of the magnitude of the effect. Our model is consistent with empirical findings in that it predicts that the term structure of credit spreads can be upward sloping with a non-zero intercept. The theory is illustrated by an application to the valuation of employee compensation packages, which may be regarded as peculiar, yet natural, examples of defaultable securities.

Abstract:
A new class of risk measures called cash sub-additive risk measures is introduced to assess the risk of future financial, nonfinancial and insurance positions. The debated cash additive axiom is relaxed into the cash sub-additive axiom to preserve the original difference between the numeraire of the current reserve amounts and future positions. Consequently, cash sub-additive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented and in such contexts cash additive risk measures cannot be used. Several representations of the cash sub-additive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sub-linear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash sub-additive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash sub-additive risk measure.

Risk measures, Fenchel-Legendre transform, model uncertainty, inf-convolution, backward stochastic differential equations

Abstract:
The present paper addresses the problem of computing implied volatilities of options written on a domestic asset based on implied volatilities of options on the same asset expressed in a foreign currency and the exchange rate. It proposes an original method together with explicit formulas to compute the at-the-money implied volatility, the smile's skew, convexity, and term structure for short maturities. The method is completely free of any model specification or Markov assumption; it only assumes that jumps are not present. We also investigate how the method performs on the particular example of the currency triplet dollar, euro, yen. We find a very satisfactory agreement between our formulas and the market at one week and one month maturities.

Implied volatility, foreign exchange options.

Abstract:
The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization.

This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic
approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs.

risk measure, risk transfer, optimal design, hedging strategy, indifference pricing, BSDE

Abstract:
We are concerned with a classic portfolio optimization problem where the admissible strategies must dominate a floor process on every intermediate date (American guarantee). We transform the problem into a martingale, whose aim is to dominate an obstacle, or equivalently its Snell envelope. The optimization is performed with respect to the concave stochastic ordering on the terminal value, so that we do not impose any explicit specification of the agent's utility function. A key tool is the representation of the supermartingale obstacle in terms of a running supremum process. This is illustrated within the paper by an explicit example based on the geometric Brownian motion.

Abstract:
A new class of risk measures called cash subadditive risk measures is introduced to assess the risk of future financial, nonfinancial, and insurance positions. The debated cash additive axiom is relaxed into the cash subadditive axiom to preserve the original difference between the numéraire of the current reserve amounts and future positions. Consequently, cash subadditive risk measures can model stochastic and/or ambiguous interest rates or defaultable contingent claims. Practical examples are presented, and in such contexts cash additive risk measures cannot be used. Several representations of the cash subadditive risk measures are provided. The new risk measures are characterized by penalty functions defined on a set of sublinear probability measures and can be represented using penalty functions associated with cash additive risk measures defined on some extended spaces. The issue of the optimal risk transfer is studied in the new framework using inf-convolution techniques. Examples of dynamic cash subadditive risk measures are provided via BSDEs where the generator can locally depend on the level of the cash subadditive risk measure.

Abstract:
This paper provides an exact formula for the pricing of floating-rate notes (FRNs) in the very general situation when the floater is not priced at par. In an earlier work, Ramaswamy and Sundaresan (1986) also addressed the pricing of FRNs in a stochastic interest rates environment: Their results relied on partial differential equations (PDEs) that did not result in analytical solutions. The model presented in this paper uses martingale theory in providing closed- formed solutions for the price of FRNs with discrete coupon payments and with an arbitrary lag between the reset dates and the ex-coupon dates.The two results are (1) a valuation formula that gives the price of the floater as the sum of discount factors multiplied by the expected future coupon payments, the expectation being taken under an appropriate probability measure; (2) the explicit expression of each expected coupon, which does not need any knowledge of the probability mentioned above, since it is exactly the forward rate plus a term only related to the volatility of the term structure. As a natural application, the paper provides an explicit solution to the pricing of interest rate swaps exchanging a sequence of fixed-rate cash flows in arrears for variable- rate cash flows.

Abstract:
We present the solution of a portfolio optimization problem for an economic agent endowed with a stochastic insurable stream, under a liquidity constraint over the time interval [0,T]. Generally, the existence of labor income complicates the agent's decisions. Moreover, in the real world the economic agents are restricted in their ability to borrow against their future labor income. We deal with this kind of liquidity constraint following the lines of American option valuation which allows us to give a precise characterization of the optimal consumption as well as the terminal wealth. In a Markovian case, with infinite horizon and HARA utility, we obtain a closed form solution.