TRIVARIATE SUPPORT OF FLAT‐VOLATILITY FORWARD LIBOR RATES

This paper investigates the multivariate support of forward Libor rates in the one‐factor, constant volatilities Libor market model. The comparatively simple bivariate case was solved in Jamshidian (2008) in connection to the recent finding by Davis and Mataix‐Pastor (2007) of positive probability of negative Libor rates in the swap market model. The approach here builds on Jamshidian (2008) but becomes really effective only in the trivariate case, and there particularly for a special “flat‐volatility” case, leading to an analytic solution. The main idea is a certain recursion in the Libor market model by means of which the calculation of the support is reduced to a calculus of variation problem (with bounds on the slope).     

ASYMPTOTICALLY OPTIMAL PORTFOLIOS

This paper extends to continuous time the concept of universal portfolio introduced by Cover (1991). Being a performance weighted average of constant rebalanced portfolios, the universal portfolio outperforms constant rebalanced and buy-and-hold portfolios exponentially over the long run. an asymptotic formula summarizing its long-term performance is reported that supplements the one given by Cover. A criterion in terms of long-term averages of instantaneous stock drifts and covariances is found which determines the particular form of the asymptotic growth. A formula for the expected universal wealth is given.     

Scenario Simulation: Theory and methodology

This paper presents a new simulation methodology for quantitative risk analysis of large multi-currency portfolios. The model discretizes the multivariate distribution of market variables into a limited number of scenarios. This results in a high degree of computational efficiency when there are many sources of risk and numerical accuracy dictates a large Monte Carlo sample. Both market and credit risk are incorporated. The model has broad applications in financial risk management, including value at risk. Numerical examples are provided to illustrate some of its practical applications.     

BIVARIATE SUPPORT OF FORWARD LIBOR AND SWAP RATES

Based on a certain notion of "prolific process," we find an explicit expression for the bivariate (topological) support of the solution to a particular class of 2 × 2 stochastic differential equations that includes those of the three-period "lognormal" Libor and swap market models. This yields that in the lognormal swap market model (SMM), the support of the 1 × 1 forward Libor   L * _t   equals  [ l * _t , ∞)  for some semi-explicit  −1 ≤ l * _t ≤ 0  , sharpening a result of Davis and Mataix-Pastor (2007) that forward Libor rates (eventually) become negative with positive probability in the lognormal SMM. We classify the instances   l * _t < 0  , and explicitly calculate the threshold time at or before which   L * _t   remains positive a.s.     

LIBOR and swap market models and measures

A self-contained theory is presented for pricing and hedging LIBOR and swap derivatives by arbitrage. Appropriate payoff homogeneity and measurability conditions are identified which guarantee that a given payoff can be attained by a self-financing trading strategy. LIBOR and swap derivatives satisfy this condition, implying they can be priced and hedged with a finite number of zero-coupon bonds, even when there is no instantaneous saving bond. Notion of locally arbitrage-free price system is introduced and equivalent criteria established. Stochastic differential equations are derived for term structures of forward libor and swap rates, and shown to have a unique positive solution when the percentage volatility function is bounded, implying existence of an arbitrage-free model with such volatility specification. The construction is explicit for the lognormal LIBOR and swap “market models”, the former following Musiela and Rutkowski (1995). Primary examples of LIBOR and swap derivatives are discussed and appropriate practical models suggested for each.     

Option and Futures Evaluation With Deterministic Volatilities1

Several risk-neutral expectation formulae are derived in a general multifactor setting. Specializing to deterministic covariances of returns, they lead to formulae for forward and future prices as well as formulae for options on forward and futures contracts. the results are applicable to currencies, bonds, commodities with stochastic convenience yield, and stock indices. For currencies, a noarbitrage relation between domestic and foreign economies is formulated and applied to evaluate quanto futures and options.