On the range of options prices

In this paper we consider the valuation of an option with time to expiration and pay-off function which is a convex function (as is a European call option), and constant interest rate , in the case where the underlying model for stock prices is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval , this interval being the biggest interval in which the values must lie, whatever model is used.     

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.     

Fast accurate binomial pricing

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.     

A note on the forward measure

For a Markov process , the forward measure over the time interval is defined by the Radon-Nikodym derivative , where is a given non-negative function and is the normalizing constant. In this paper, the law of under the forward measure is identified when is a diffusion process or, more generally, a continuous-path Markov process.     

Volatility of the short rate in the rational lognormal model

A recent article of Flesaker and Hughston introduces a one factor interest rate model called the rational lognormal model. This model has a lot to recommend it including guaranteed finite positive interest rates and analytic tractability. Consequently, it has received a lot of attention among practioners and academics alike. However, it turns out to have the undesirable feature of predicting that the asymptotic value of the short rate volatility is zero. This theoretical result is proved rigorously in this article. The outcome of an empirical study complementing the theoretical result is discussed at the end of the article. European call options are valued with the rational lognormal model and a comparably calibrated mean reverting Gaussian model. unsurprisingly, rational lognormal option values are considerably lower than the analogous mean reverting Gaussian option values. In other words, the volatility in the rational lognormal model declines so quickly that options are severely undervalued.     

An application of hidden Markov models to asset allocation problems

Filtering and parameter estimation techniques from hidden Markov Models are applied to a discrete time asset allocation problem. For the commonly used mean-variance utility explicit optimal strategies are obtained.     

On Leland's strategy of option pricing with transactions costs

We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland (1985).     

Processes of normal inverse Gaussian type

With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.     

A note on the existence of unique equivalent martingale measures in a Markovian setting

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.     

On a general class of one-factor models for the term structure of interest rates

We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .