Mathematical Finance — Bachelier Congress 2000

Louis Bachelier defended his thesis “Theory of Speculation” in 1900. He used Brownian motion as a model for stock exchange performance. This conversation with Bernard Bru illustrates the scientific climate of his times and the conditions under which Bachelier made his discoveries. It indicates that Bachelier was indeed the right person at the right time. He was involved with the Paris stock exchange, was self-taught but also took courses in probability and on the theory of heat. Not being a part of the “scientific establishment,” he had the opportunity to develop an area that was not of interest to the mathematicians of the period. He was the first, to apply the trajectories of Brownian motion, and his theories prefigure modern mathematical finance. What, follows is an edited and expanded version of the original conversation with Bernard Bru.

Bernard Bru is the author, most, recently, of Borel, Lévy, Neyman, Pearson et les autres [38]. He is a professor at, the University of Paris V where he teaches mathematics and statistics. With Marc Barbut, and Ernest, Coumet, he founded the seminars on the history of Probability at, the EHESS (École des Hautes Études en Sciences Sociales), which bring together researchers in mathematics, philosophy and the humanities.

The origins of much of the mathematics in modern finance can be traced to Louis Bachelier’s 1900 dissertation on the theory of speculation, framed as an option-pricing problem. This work marks the twin births of both the continuous-time mathematics of stochastic processes and the continuous-time economics of derivative-security pricing. In solving his option-pricing problem, Bachelier provides two different derivations of the classic partial differential equation for the probability density of what later was called a Wiener process or Brownian motion process. In one derivation, he writes down a version of what is now commonly called the Chapman—Kolmogorov convolution probability integral in one of the earliest examples of that integral in print. In the other, he uses a limit argument applied to a discrete-time binomial process to derive the continuous-time transition probabilities. Bachelier also develops the method of images (or reflection) to solve for a probability function of a diffusion process with an absorbing barrier. All this in his thesis five years before the publication of Einstein’s mathematical theory of Brownian motion.

People in financial mathematics make their living (in part) by Itô’s lemma: df(b) = f′(b) db + 1/2 f″(b)dt in which f is a smooth function and b(t): t ≥ 0 is the standard Brownian motion BM(1) starting at b(0) = 0. It lies at the back of Itô’s local representation dx = σ(x)db + m(x)dt of the general 1-dimensional diffusion x with paths x(t) : t ≥ 0 and infinitesimal operator G = 1/2σ ²(x)d ²/dx ² + m(x)d/dx, as suggested by the simplest example x(t) = x(0) + σb(t) + mt with constant σ and m. I will speak about a less familiar reduction of x to standard Brownian motion, by change of scale and clock, and other aspects of this main idea. It is appropriate to do so on the centenary of Bachelier’s thesis in which is to be found the very first use of Brownian paths per se. P. Lévy, K. Itô, and P. Malliavin, to name only the most illustrious, have gone much more deeply into the path function picture, but Bachelier had the idea first, way before its general recognition as the best way of thinking about motion subject to chance. I do not give financial applications, but see for instance Geman & Yor [1993] and Donati-Martin, Matsumoto, & Yor [2000].

There are many situations where we want to estimate the probability of a rare event with some precision. For example if we toss a fair coin n times the probability of getting n heads in a row is clearly small, and for large n is very small. We know its exact value as 2⁻ⁿ and in logarithmic scale we can write it as e ^{−n log 2}. While in this case the exact probability is easy to evaluate, there are many situations in which a direct exact calculation is impossible and we need to develop indirect methods that will provide us with estimates.

We investigate the term structure of forward and futures prices for models where the price processes are allowed to be driven by a general marked point process as well as by a multidimensional Wiener process. Within an infinite dimensional HJM-type model for futures and forwards we study the properties of futures and forward convenience yield rates. For finite dimensional factor models, we develop a theory of affine term structures, which is shown to include almost all previously known models. We also derive two general pricing formulas for futures options. Finally we present an easily applicable sufficient condition for the possibility of fitting a finite dimensional futures price model to an arbitrary initial futures price curve, by introducing a time dependent function in the drift term.

We propose two different, classes of analytical models for the dynamics of an asset, price that, respectively lead to skews and smiles in the term structure of implied volatilities. Both classes are based on explicit, SDEs that, admit, unique strong solutions whose marginal densities are also provided. We then consider some particular examples in each class and explicitly calculate the European option prices implied by these models.

The term ‘no-good-deal pricing’ in this paper encompasses pricing techniques base on the absence of attractive investment opportunities — good deals — in equilibrium. We borrowed the term from [8] who pioneered the calculation of price bands conditional on the absence of high Sharpe Ratios. Alternative methodologies for calculating tighter-than-no-arbitrage price bounds have been suggested by [4], [6], [12]. The theory presented here shows that any of these techniques can be seen as a generalization of no-arbitrage pricing. The common structure is provided by the Extension and Pricing Theorems, already well known from no-arbitrage pricing, see [15]. We derive these theorems in no-good-deal framework and establish general properties of no-good-deal prices. These abstract results are then applied to no-goos-deal bounds determined by von Neumann-Morgenstern preferences in a finite state model¹. One important result is that no-good-deal bouns generated by an unbounded utility function are always strictly tighter than the no-arbitrage bounds. The same is not true for bounded utility functions. For smooth utility functions we show that one will obtain the no-arbitrage and the representative agent equilibrium as the two opposite ends of a spectrum of no-good-deal equilibrium restrictions indexed by the maximum attainable certainty equivalent gains.

We investigate a method for pricing the generic spread option beyond the classical two-factor Black-Scholes framework by extending the fast Fourier Transform technique introduced by Carr & Madan (1999) to a multi-factor setting. The method is applicable to models in which the joint characteristic function of the prices of the underlying assets forming the spread is known analytically. This enables us to incorporate stochasticity in the volatility and correlation structure — a focus of concern for energy option traders — by introducing additional factors within an affine jump-diffusion framework. Furthermore, computational time does not increase significantly as additional random factors are introduced, since the fast Fourier Transform remains two dimensional in terms of the two prices defining the spread. This yields considerable advantage over Monte Carlo and PDE methods and numerical results are presented to this effect.

Part A of this paper is a transcript of the third author’s lecture at the Bachelier Conference, July 1st, 2000; it is a summary of our joint works on this topic.

Statistical analysis of data from the financial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as subclasses hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients to model price processes. GH distributions generate in a canonical way Lévy processes, i.e. processes with stationary and independent increments. We introduce a model for price processes which is driven by generalized hyperbolic Lévy motions. This GH model is a generalization of the hyperbolic model developed by Eberlein and Keller (1995). It is incomplete. We derive an option pricing formula for GH driven models using the Esscher transform as martingale measure and compare the prices with classical Black-Scholes prices. The objective of this study is to examine the consistency of our model assumptions with the empirically observed price processes for underlyings and derivatives. Finally we present a simplified approach to the estimation of high-dimensional GH distributions and their application to measure risk in financial markets.

In order to manage interest rank risk exposure, bank treasurers allocate their portfolio of assets and liabilities to what they call standard “buckets”. In order words the portfolio of cash flows from assets and liabilities at various times are re-expressed as a portfolio of cash flows at certain specified standard times. These specified times usually correspond to important dates like expiry dates of hedging instruments (bank bills, bank bill futures, interest rate swaps, and so on). The exact choice of these dates will not concern us in this paper, but will be specified by the users of these results. Of course, this procedure can only be done in an approximate way. Traditionally, the real cash flows are assigned so that the portfolio of assets and liabilities of the bank portfolio match the portfolio of bucketed assets with respect to present value and (Macaulay) duration (or other risk measures like Reddington convexity), which could hold for a short time interval. We will not review the various methods used by practitioners, but present a method based on the use of Wiener Chaos expansions, (with respect to suitable forward probability measures), of assets and liabilities and to select bucketing in order to match various orders of the chaos expansion in the assets and liabilities portfolio and bucketed portfolio. In fact matching the first three orders of the chaos expansion is closely related to present value, duration, and convexity matching, (see Brace and Musiela [1]). In this framework it is possible to study and give results about optimal assignments, so that matching may hold over a working time horizon (a day, a week, and so on). Once assets and liabilities have been bucketed, the interest rate or exposure can be managed by the use of standard treasury products. This can involve a similar procedure to the one we have described already, by which we represent the bucketed cash flows in terms of the cash flows of a portfolio of hedging instruments.

The so-called intensity-based approach to the modelling and valuation of de-faultable securities has attracted a considerable attention of both practitioners and academics in recent years; to mention a few papers in this vein: Duffie [8], Duffie and Lando [9], Duffie et al. [10], Jarrow and Turnbull [13], Jarrow et al. [14], Jarrow and Yu [15], Lando [21], Madan and Unal [23]. In the context of financial modelling, there was also a renewed interest in the detailed analysis of the properties of random times; we refer to the recent papers by Elliott et al. [12] and Kusuoka [20] in this regard. In fact, the systematic study of stopping times and the associated enlargements of filtrations, motivated by a purely mathematical interest, was initiated in the 1970s by the French school, including: Brémaud and Yor [4], Dellacherie [5], Dellacherie and Meyer [7], Jeulin [16], and Jeulin and Yor [17]. On the other hand, the classic concept of the intensity (or the hazard rate) of a random time was also studied in some detail in the context of the theory of Cox processes, as well as in relation to the theory of martingales. The interested reader may consult, in particular, the monograph by Last and Brandt [22] for the former approach, and by Brémaud [3] for the latter. It seems to us that no single comprehensive source focused on the issues related to default risk modelling is available to financial researchers, though. Furthermore, it is worth noting that some challenging mathematical problems associated with the modelling of default risk remain still open. The aim of this text is thus to fill the gap by furnishing a relatively concise and self-contained exposition of the most relevant — from the viewpoint of financial modelling — results related to the analysis of random times and their filtrations. We also present some recent developments and we indicate the directions for a further research. Due to the limited space, the proofs of some results were omitted; a full version of the working paper [19] is available from the authors upon request.

In recent years various suggestions concerning contingent claim valuation in incomplete markets have been made. We argue that some of them can be naturally interpreted in terms of neutral derivative prices which occur if derivative demand and supply are balanced. Secondly, we introduce the notion of consistent derivative pricing which is a way of constructing market models that are consistent with initially observed derivative quotations.

Many credit management systems, based on different underlying frameworks, are now available to measure and control default and credit risks¹. Homogeneous credit classes and associated transition matrix may thus be constructed within many different frameworks. For illustration, the KMV Corporation provides a transition matrix within a structural approach à la Black-Sholes-Merton (Crouhy-Galai-Mark [5]). Independentely from the underlying framework, a methodology based on credit classes may therefore be used to price any claim contingent on credit events among which the default.

On financial markets, option traders typically readjust their hedging portfolios when the underlying stock price has moved by a given percentage. This trading rule implies that rebalancing occurs at random times and that the Black and Scholes [4] model which relies on the assumption of continuous rebalancing is no longer appropriate. Although this continuity assumption is clearly unrealistic for transaction cost reasons, because prices are quoted in ticks, or because of the mere impossibility of continuous trading, it has received comparatively less academic attention than other assumptions such as constant volatility.

Within the mathematical finance literature, there have been several distinct classes of interest-rate model. The first historically was the family of spotrate models, where one proposes a model for the evolution of the spot rate of interest under the pricing measure, and then attempts to find expressions for the prices of derivatives; the models of Vasicek [16], Cox, Ingersoll & Ross [7], Black, Derman & Toy [3] and Black & Karasinski [4] are well-known examples of this type. Next came the whole-yield models, starting with Ho & Lee [10] in a discrete setting, and then in the continuous setting Babbs [1] and Heath, Jarrow & Morton [9]. Lately, there has been much interest in so-called market models, whose chief characteristic is the choice of some suitable numéraire process, relative to which the prices of various derivatives have some particularly tractable form; see Miltersen, Sandmann & Sondermann [12] and Brace, Gatarek & Musiela [5] for examples of such models. These three classes of models have been developed extensively; a thorough survey would be outside the aims of this paper, but we refer the reader to the excellent recent monograph of Musiela & Rutkowski [11] for more details and references.

We construct arbitrage free dynamics for the term structure of interest rates driven by infinitely many factors, each one representing a basic shape for the instantaneous forward rate curve in a given market. The consistency between a finitc-dimeusioual space of polynomials where the curve is day-to-day recovered and the proposed evolution equation is investigated. The main result is the developemeut of a historical-implicit hybrid calibration procedure for our infinite-dimensional shape factor model. In this context, we also derive a pricing formula for caplets.

We give a review of classical and recent results on maximization of expected utility for an investor who has the possibility of trading in a financial market. Emphasis will be given to the duality theory related to this convex optimization problem.

We then pass to the general case of an arbitrage-free financial market modeled by an ℝ ^d -valued semi-martingale. In this case some regularity conditions have to be imposed in order to obtain an existence result for the primal problem of finding the optimal investment, as well as for a proper duality theory. It, turns out, that, one may give a necessary and sufficient condition, namely a mild condition on the asymptotic behavior of the utility function, its so-called reasonable asymptotic elasticity. This property allows for an economic interpretation motivating the term “reasonable”. The remarkable fact, is that, this regularity condition only pertains to the behavior of the utility function, while we do not, have to impose any regularity conditions on the stochastic process modeling the financial market, (to be precise: of course, we have to require the arbitrage-freeness of this process in a proper sense; also we have to assume in one of the cases considered below that, this process is locally bounded; but, otherwise it, may be an arbitrary ℝ ^d -valued semi-martingale).

We state two general existence and duality results pertaining to the setting of optimizing expected utility of terminal consumption. We also survey some of the ramifications of these results allowing for intermediate consumption, state-dependent, utility, random endowment,, non-smooth utility functions and transaction costs.

Suppose that we are observing a random process X = (X _t ) on an interval [0,T]. The objects θ and τ introduced below are essential throughout the paper: