Showing papers on "Mathematical finance published in 1999"

Essentials of Stochastic Finance: Facts, Models, Theory

Abstract: This important book provides information necessary for those dealing with stochastic calculus and pricing in the models of financial markets operating under uncertainty; introduces the reader to the main concepts, notions and results of stochastic financial mathematics; and develops applications of these results to various kinds of calculations required in financial engineering. It also answers the requests of teachers of financial mathematics and engineering by making a bias towards probabilistic and statistical ideas and the methods of stochastic calculus in the analysis of market risks.

Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example

Abstract: The paper represents a model for financial valuation of a firm which has control of the dividend payment stream and its risk as well as potential profit by choosing different business activities among those available to it. This model extends the classical Miller–Modigliani theory of firm valuation to the situation of controllable business activities in a stochastic environment. We associate the value of the company with the expected present value of the net dividend distributions (under the optimal policy). The example we consider is a large corporation, such as an insurance company, whose liquid assets in the absence of control fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, and drift is understood as potential profit. At each moment of time there is an option to reduce risk exposure while simultaneously reducing the potential profit—for example, by using proportional reinsurance with another carrier for an insurance company. Management of a company controls the dividends paid out to the shareholders, and the objective is to find a policy that maximizes the expected total discounted dividends paid out until the time of bankruptcy. Two cases are considered: one in which the rate of dividend payout is bounded by some positive constant M, and one in which there is no restriction on the rate of dividend payout. We use recently developed techniques of mathematical finance to obtain an easy understandable closed form solution. We show that there are two levels u0 and u1 with u0≤u1. As a function of currently available reserve, the risk exposure monotonically increases on (0,u0) from 0 to the maximum possible. When the reserve exceeds u1 the dividends are paid at the maximal rate in the first case and in the second case every excess above u1 is distributed as dividend. We also show that for M small enough u0=u1 and the optimal risk exposure is always less than the maximal.

Connecting discrete and continuous path-dependent options

Abstract: This paper develops methods for relating the prices of discrete- and continuous-time versions of path-dependent options sensitive to extremal values of the underlying asset, including lookback, barrier, and hindsight options. The relationships take the form of correction terms that can be interpreted as shifting a barrier, a strike, or an extremal price. These correction terms enable us to use closed-form solutions for continuous option prices to approximate their discrete counterparts. We also develop discrete-time discrete-state lattice methods for de- termining accurate prices of discrete and continuous path-dependent options. In several cases, the lattice methods use correction terms based on the connection between discrete- and continuous-time prices which dramatically improve con- vergence to the accurate price.

Dynamic programming and mean-variance hedging

Abstract: We consider the mean-variance hedging problem when asset prices follow Ito processes in an incomplete market framework. The hedging numeraire and the variance-optimal martingale measure appear to be a key tool for characterizing the optimal hedging strategy (see Gourieroux et al. 1996; Rheinlander and Schweizer 1996). In this paper, we study the hedging numeraire $\tilde a$ and the variance-optimal martingale measure $\tilde P$ using dynamic programming methods. We obtain new explicit characterizations of $\tilde a$ and $\tilde P$ in terms of the value function of a suitable stochastic control problem. We provide several examples illustrating our results. In particular, for stochastic volatility models, we derive an explicit form of this value function and then of the hedging numeraire and the variance-optimal martingale measure. This provides then explicit computations of optimal hedging strategies for the mean-variance hedging problem in usual stochastic volatility models.

Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark

Abstract: We consider the portfolio problem in continuous-time where the objective of the investor or money manager is to exceed the performance of a given stochastic benchmark, as is often the case in institutional money management. The benchmark is driven by a stochastic process that need not be perfectly correlated with the investment opportunities, and so the market is in a sense incomplete. We first solve a variety of investment problems related to the achievement of goals: for example, we find the portfolio strategy that maximizes the probability that the return of the investor's portfolio beats the return of the benchmark by a given percentage without ever going below it by another predetermined percentage. We also consider objectives related to the minimization of the expected time until the investor beats the benchmark. We show that there are two cases to consider, depending upon the relative favorability of the benchmark to the investment opportunity the investor faces. The problem of maximizing the expected discounted reward of outperforming the benchmark, as well as minimizing the discounted penalty paid upon being outperformed by the benchmark is also discussed. We then solve a more standard expected utility maximization problem which allows new connections to be made between some specific utility functions and the nonstandard goal problems treated here.

A closed-form solution to the problem of super-replication under transaction costs

Abstract: We study the problem of finding the minimal price needed to dominate European-type contingent claims under proportional transaction costs in a continuous-time diffusion model. The result we prove has already been known in special cases – the minimal super-replicating strategy is the least expensive buy-and-hold strategy. Our contribution consists in showing that this result remains valid for general path-independent claims, and in providing a shorter and more intuitive, financial mathematics-type proof. It is based on a previously known representation of the minimal price as a supremum of the prices in corresponding shadow markets, and on a PDE (viscosity) characterization of that representation.

Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences

Abstract: Analytic bounds on the reservation write price of European-style con- tingent claims are derived in the presence of proportional transaction costs in a model which allows for intermediate trading. The option prices are obtained via a utility maximization approach by comparing the maximized utilities with and without the contingent claim. The mathematical tools come mainly from the theories of singular stochastic control and viscosity solutions of nonlinear partial differential equations.

Some Applications of Impulse Control in Mathematical Finance

Abstract: We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.

On semi-linear degenerate backward stochastic partial differential equations

Abstract: In this paper we study the well-posedness and regularity of the adapted solutions to a class of linear, degenerate backward stochastic partial differential equations (BSPDE, for short). We establish new a priori estimates for the adapted solutions to BSPDEs in a general setting, based on which the existence, uniqueness, and regularity of adapted solutions are obtained. Also, we prove some comparison theorems and discuss their possible applications in mathematical finance.

Parameter estimation for discretely observed stochastic volatility models

Abstract: This paper deals with parameter estimation for stochastic volatility models. We consider a two-dimensional diffusion process (Y-t, V-t). Only (Y-t) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (V-t) rules the diffusion coefficient (volatility) of (Y-t) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (V-t), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from finance are treated, and numerical simulation results are given.

Minimal realizations of interest rate models

Abstract: We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the input-output map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and we give explicit formulas for bond prices in terms of the benchmark rates as well as for the computation of derivative prices.

Optimal stopping for a diffusion with jumps

Abstract: In this paper we give the closed form solution of some optimal stopping problems for processes derived from a diffusion with jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under diffusion-jump information.

Complete markets with discontinuous security price

Abstract: A parameterized family of financial market models is presented. These models have jumps intrinsic to the price processes yet have strict completeness, equivalent martingale measures, and no arbitrage. For each value of the parameter \(\beta (-2\leq\beta <0)\) the model is just as rich as the standard model using white noise (Brownian motion) and a drift; moreover as \(\beta\) increases to zero the model converges weakly to the standard model. A hedging result, analogous to the Karatzas-Ocone-Li theorem, is also presented.

An Introduction to Mathematical Finance: Options and Other Topics

Abstract: 1. Probability 2. Normal random variables 3. Geometric Brownian motion 4. Interest rates and present value analysis 5. Pricing contracts via arbitrage 6. The arbitrage theorem 7. The Black-Scholes formula 8. Valuing by expected utility 9. Exotic options 10. Beyond geometric Brownian motion models 11. Autoregressive models and mean reversion.

A generalization of the mutual fund theorem

Abstract: A generalization of the continuous time mutual fund theorem is given, with no assumptions made on the investors utility functions for consumption and final wealth, except that they are time-additive and non-decreasing. The extension is due to a new mathematical approach, using no more than simple properties of diffusion processes and standard linear algebra. The results are given for complete as well as certain incomplete markets. Moreover, optimal investment strategies that are known only for complete markets with a single risky asset, are automatically extended to complete and incomplete markets with multiple risky assets. An example is given.

Hedging contingent claims on semimartingales

Abstract: This paper extends the known results on the equivalence between market completeness and the uniqueness of martingale measures for finite asset economies, to the infinite asset case. Our arguments employ results from the theory of linear operators between locally convex topological vector spaces. This theory of linear operators provides an operational approach to the issue of completeness and uniqueness, that is also more closely connected with the mainstream of empirical asset pricing, than was hitherto available.

When are Options Overpriced? The Black-Scholes Model and Alternative Characterizations of the Pricing Kernel

Abstract: An important determinant of option prices is the elasticity of the pricing kernel used to price all claims in the economy. In this paper, we first show that for a given forward price of the underlying asset, option prices are higher when the elasticity of the pricing kernel is declining than when it is constant. We then investigate the implicaitons of the elasticity of the pricing kernel for the stochastic process followed by the underlying asset. Given that the underlying information process follows a geometric. Brownian motion, we demonstrate that constant elasticity of the pricing kernel is equivalent to a Brownian motion for the forward price of the underlying asset, so that the Black-Scholes formula correctly prices options on the asset. In contast, declining elasticiy implies that the forward price process is no longer a Brownian motion: It has higher volatility and exhibits autocorrelation. In this case, the Black-Scholes formula underprices all options.

Stochastic heat equation with random coefficients

Abstract: We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from Ito's formula for this anticipating stochastic integral.

In Honor of the Nobel Laureates Robert C. Merton and Myron S. Scholes: A Partial Differential Equation That Changed the World

Abstract: The Nobel Prize was given to Robert C. Merton and Myron S. Scholes for discovering a new method for determining the value of an option. This is known as the Black-Merton-Scholes option pricing formula. The purpose of this essay is to explain why the Black-Merton-Scholes option pricing formula is so important to the finance profession, the economics profession, the financial industry, and society at large. This is done by studying the history of the formula's development, the economic logic underlying the formula's derivation, and the formula's ramifications for the various professions.

Distributions of occupation times of Brownian motion with drift

Abstract: The purpose of this paper is to present a survey of recent developments concerning the distributions of occupation times of Brownian motion and their applications in mathematical nance. The main result is a closed form version for Akahori's generalized arc-sine law which can be exploited for pricing some innovative types of options in the Black & Scholes model. Moreover a straightforward proof for Dassios' representation of the -quantile of Brownian motion with drift shall be provided.

Consistency issues for numerical methods for variance control, with applications to optimization in finance

Abstract: The paper is concerned with numerical algorithms for the optimal control of diffusion-type processes when the noise variance also depends on the control. This problem is of increasing importance in applications, particularly in financial mathematics. We discuss the construction of numerical algorithms guaranteed to converge to the true minimum as the discretization level decreases and with acceptable numerical properties. The algorithms are based on the popular Markov chain approximation method. The basic criterion the algorithms must satisfy is a weak "local consistency" condition, which is essential for convergence to the true optimal cost function. This condition is often hard to satisfy by simple algorithms (with essentially only local transitions) when the variance is also controlled. Numerical "noise" can be introduced by the more convenient approximations. This question of "numerical noise" (also called "numerical viscosity") is dealt with in detail, and methods for eliminating or greatly reducing it are discussed.

Controlled Markov processes and mathematical finance

Abstract: These lectures are concerned with optimal control of Markov diffusion processes with complete state information, and with some applications in financial economics. Problems on a finite time horizon, and on an infinite horizon with discounted cost or ergodic (average cost per unit time) criterion are considered. We also consider risk sensitive stochastic control on an infinite horizon, with expected exponential-of-integral cost criteria. These problems are related, through a logarithmic transformation, to infinite horizon ergodic stochastic control and stochastic differential games. Risk sensitive control provides a link between stochastic and deterministic (robust control) approaches to disturbance attenuation. This is done by considering the robust control model as a small noise intensity limit of a corresponding risk sensitive model. In mathematical finance, we consider some models of optimal portfolio choice which are extensions of the classical Merton model. Problems of optimal long-term growth of expected utility of wealth are reformulated as infinite horizon risk sensitive control problems. Explicit solutions are given in absence of investment control constraints. In addition, a robust control approach to the Black-Scholes formula for pricing stock options is mentioned.

On the relationship of the dynamic programming approach and the contingent claim approach to asset valuation

Abstract: We consider a general model for an investment producing a single commodity, and, assuming that there exists a traded asset spanning the corresponding market, we prove a “verification theorem” which relates the solution of an appropriate differential equation with the investment's contingent claim price. In this way, we show in a mathematically rigorous way that the contingent claim approach and the dynamic programming approach to the problem of asset valuation are equivalent, modulo parameter calibration. Our analysis can be used in a straightforward way to address a big number of investment models.

Existence of global stochastic flow and attractors for Navier–Stokes equations

Abstract: For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution.

Optimal Risk/Dividend Distribution Control Models: Applications to Insurance

Abstract: The current paper presents a short survey of stochastic models of risk control and dividend optimization techniques for a financial corporation. While being close to consumption/investment models of Mathematical Finance, dividend optimization models possess special features which do not allow them to be treated as a particular case of consumption/investment models. In a typical model of this sort, in the absence of control, the reserve (surplus) process, which represents the liquid assets of the company, is governed by a Brownian motion with constant drift and diffusion coefficient. This is a limiting case of the classical Cramer-Lundberg model in which the reserve is a compound Poisson process, amended by a linear term, representing a constant influx of the insurance premiums. Risk control action corresponds to reinsuring part of the claims the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company. This translates into controlling the drift and the diffusion coefficient of the approximating process. The dividend distribution policy consists of choosing the times and the amounts of dividends to be paid put to shareholders. Mathematically, the cumulative dividend process is described by an increasing functional which may or may not be continuous with respect to time. The objective in the models presented here is maximization of the dividend pay-outs. We will discuss models with different types of conditions imposed upon a company and different types of reinsurances available, such as proportional, noncheap, proportional in a presence of a constant debt liability, excess-of-loss. We will show that in most cases the optimal dividend distribution scheme is of a barrier type, while the risk control policy depends substantially on the nature of reinsurance available.

Financial markets : stochastic analysis and the pricing of derivative securities

Abstract: Basic concepts and objects of a financial market The elements of discrete stochastic analysis A stochastic model for a financial market. Arbitrage and completeness Pricing European options in complete markets. The binomial model and the Cox-Ross-Rubinstein formula Pricing and hedging American options in complete markets Financial computations on a complete market with the use of nonself-financing strategies Incomplete markets. Pricing of options and problems of minimizing risk The structure of prices of other instruments of a financial market. Forwards, futures, bonds The problem of optimal investment The concept of continuous models. Limiting transitions from a discrete market to a continuous one. The Black-Scholes formula Appendix 1 Appendix 2 Appendix 3 Hints for solving the problems Bibliography Subject index.

Some applications of occupation times of brownian motion with drift in mathematical finance

Abstract: In the last few years new types of path-dependent options called corridor options or range options have become well-known derivative instruments in European options markets. Since the payout proles of those options are based on occupation times of the underlying security the purpose of this paper is to provide closed form pricing formulae of Black & Scholes type for some signicant representatives. Alternatively we demonstrate in this paper a relatively simple derivation of the Black & Scholes price for a single corridor option { based on a static portfolio representation { which does not make use of the distribution of occupation times (of Brownian motion). However, knowledge of occupation times' distributions is a more powerful tool.

Exploding hedging errors for digital options

Abstract: In the complete market model of geometric Brownian motion, all kinds of exotic options can be priced and hedged perfectly using a delta hedging strategy which duplicates the option's payoff. If trading takes place in a frictionless market, this delta hedging strategy is said to eliminate the option writer's risk completely. It will be shown that for certain contingent claims, for example digital options, the hedge can fail completely if the underlying risky asset does not follow the assumed geometric Brownian motion. Indeed, the hedging error may diverge and delta hedging can actually increase the risk of the option writer.