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I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.

We propose a non linear Langevin equation as a model for stock market fluctuations and crashes. This equation is based on an identification of the different processes influencing the demand and supply, and their mathematical transcription. We emphasize the importance of feedback e#ects of price variations onto themselves. Risk aversion, in particular, leads to an "up-down" symmetry breaking term which is responsible for crashes, where "panic" is self reinforcing. It is also responsible for the sudden collapse of speculative bubbles. Interestingly, these crashes appear as rare, "activated" events, and have an exponentially small probability of occurence. The model leads to a specific "shape" of the falldown of the price during a crash, which we compare with the October 1987 data. The normal regime, where the stock price exhibits behavior similar to that of a random walk, however reveals non trivial correlations on different time scales, in particular on the time scale over which operators perceive a change of trend.

In this paper we consider reserving and pricing methodologies for a pensions-type contract with a simple form of guaranteed annuity option. We consider only unit-linked contracts, but our methodologies and, to some extent, our numerical results would apply also to with profits contracts. The Report of the Annuity Guarantees Working Party, Bolton et al. (1997), presented the results of a very interesting survey as at the end of 1996 of life assurance companies offering guaranteed annuity options. There was no consensus at that time among the companies on how to reserve for such options. The Report discussed several approaches to reserving but concluded that it was unable to recommend a single approach. This paper is an attempt to fill that gap. We investigate two approaches to reserving and pricing. In the first sections of the paper we consider quantile, and conditional tail expectation, reserves. The methodology we adopt here is very close to that proposed by the Maturity Guarantees Working Party in its Report to the profession, Ford et al. (1980). We show how these policies could have been reserved for in 1985, and what would have been the outcome of using the proposed method.

: In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden, large fluctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. For discussion and classification, the notation of life time of numerical solutions is introduced. Thereby the task consists in construction of numerical solutions with lengthened life time up to eternal one. During the exposition we outline the role of implicitness for this `process of numerical regularization'. Boundedness (Nonnegativity) of some implicit numerical solutions can be proved at least for a class of linearly bounded models. Balanced implicit methods (BIMs) turn out to be very efficient for this purpose. Furthermore, the local property of conditional positivity of numerical solut...

Abstract. The objective of this paper is to describe a quantitative framework for o ering timber harvest bids on federal lands which includes special considerations of the volatility oftimber indices and the harvesting costs involved in harvesting timber. The advantage of such an approach is that it provides a precise framework in which various underlying considerations, such asvolatility and cost, may be systematically de ned, measured and evaluated. The valuations derived in this paper provide a market standard against which additional value that encompasses social or environmental welfare may beevaluated. We discuss a speci c USDA Forest Service timber sale as a case study to illustrate this approach. 1.

In recent years there has been a trend towards market consistent valuation in those institutions for which actuaries have responsibilities. The larger United Kingdom with-profits insurance companies are now preparing realistic balance sheets, both for internal purposes and also at the request of the Financial Services Authority. International accounting standards have been moving to a fair value approach. Pension fund accounting under FRS 17 has also moved in this direction. In this paper we examine the reasons for the adoption of market consistent valuation and discuss some of the commercial implications and corporate valuation. We consider the methods and assumptions which can be used to develop market consistent valuations of cash flows typically encountered in the liabilities of financial institutions, together with some of the problems inherent in the calibration of models used for the valuation of these cash flows. The volatility assumption is crucial to the valuation of options and guarantees, and we discuss the relationship between historical and implied volatility. While most insurance companies initially adopted formulae to value their with-profits

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 prefactor 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 form 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 behaviour 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

We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). Finally, we give a specific example to show how this idea can be sucessfully implemented for improving risk management. Empirical correlation matrices are of great importance for risk management and asset allocation. The probability of large losses for a certain portfolio or option book is dominated by correlated moves of its different constituents – for example, a position which is simultaneously long in stocks and short in bonds will be risky because stocks and bonds usually move in opposite directions in crisis periods. The study of correlation (or covariance) matrices thus has a long history in finance and is one of the cornerstone of Markowitz’s theory of optimal portfolios 1,2: given a