Showing papers in "Finance and Stochastics in 2009"

Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets

Abstract: We consider the infinite-horizon optimal portfolio liquidation problem for a von Neumann–Morgenstern investor in the liquidity model of Almgren (Appl. Math. Finance 10:1–18, 2003). Using a stochastic control approach, we characterize the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. We furthermore analyze the sensitivities of the value function and the optimal strategy with respect to the various model parameters. In particular, we find that the optimal strategy is aggressive or passive in-the-money, respectively, if and only if the utility function displays increasing or decreasing risk aversion. Surprisingly, only few further monotonicity relations exist with respect to the other parameters. We point out in particular that the speed by which the remaining asset position is sold can be decreasing in the size of the position but increasing in the liquidity price impact.

Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem

Abstract: In this paper, we study a class of quadratic backward stochastic differential equations (BSDEs), which arises naturally in the utility maximization problem with portfolio constraints. We first establish the existence and uniqueness of solutions for such BSDEs and then give applications to the utility maximization problem. Three cases of utility functions, the exponential, power, and logarithmic ones, are discussed.

Quasi-Monte Carlo methods with applications in finance

Abstract: We review the basic principles of quasi-Monte Carlo (QMC) methods, the randomizations that turn them into variance-reduction techniques, the integration error and variance bounds obtained in terms of QMC point set discrepancy and variation of the integrand, and the main classes of point set constructions: lattice rules, digital nets, and permutations in different bases. QMC methods are designed to estimate s-dimensional integrals, for moderate or large (perhaps infinite) values of s. In principle, any stochastic simulation whose purpose is to estimate an integral fits this framework, but the methods work better for certain types of integrals than others (e.g., if the integrand can be well approximated by a sum of low-dimensional smooth functions). Such QMC-friendly integrals are encountered frequently in computational finance and risk analysis. We summarize the theory, give examples, and provide computational results that illustrate the efficiency improvement achieved. This article is targeted mainly for those who already know Monte Carlo methods and their application in finance, and want an update of the state of the art on quasi-Monte Carlo methods.

On irregular functionals of SDEs and the Euler scheme

Abstract: We prove a sharp upper bound for the error \(\mathbb {E}|g(X)-g(\hat{X})|^{p}\) in terms of moments of \(X-\hat{X}\) , where X and \(\hat{X}\) are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution to a stochastic differential equation at time T by the Euler scheme, and show that the approximation of the payoff of the binary option has asymptotically sharp strong convergence rate 1/2. This has consequences for multilevel Monte Carlo methods.

Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff

Abstract: Giles (Oper. Res. 56:607-617, 2008) introduced a multi-level Monte Carlo method for approximating the expected value of a function of a stochastic differential equation solution. A key application is to compute the expected payoff of a finan- cial option. This new method improves on the computational complexity of standard Monte Carlo. Giles analysed globally Lipschitz payoffs, but also found good per- formance in practice for non-globally Lipschitz cases. In this work, we show that the multi-level Monte Carlo method can be rigorously justified for non-globally Lip- schitz payoffs. In particular, we consider digital, lookback and barrier options. This requires non-standard strong convergence analysis of the Euler-Maruyama method.

Local volatility dynamic models

Abstract: This paper is concerned with the characterization of arbitrage-free dynamic stochastic models for the equity markets when Ito stochastic differential equations are used to model the dynamics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The main thrust of the paper is to characterize absence of arbitrage by a drift condition and a spot consistency condition for the coefficients of the local volatility dynamics.

Interacting particle systems for the computation of rare credit portfolio losses

Abstract: In this paper, we introduce the use of interacting particle systems in the computation of probabilities of simultaneous defaults in large credit portfolios. The method can be applied to compute small historical as well as risk-neutral probabilities. It only requires that the model be based on a background Markov chain for which a simulation algorithm is available. We use the strategy developed by Del Moral and Garnier in (Ann. Appl. Probab. 15:2496–2534, 2005) for the estimation of random walk rare events probabilities. For the purpose of illustration, we consider a discrete-time version of a first passage model for default. We use a structural model with stochastic volatility, and we demonstrate the efficiency of our method in situations where importance sampling is not possible or numerically unstable.

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

Abstract: We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Levy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Levy process. It can be discretized with exponentially decaying errors of the form O(exp (−aM b )) for some a,b>0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix–vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NMlog (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Levy models.

Pricing options under stochastic volatility: a power series approach

Abstract: In this paper we present a new approach for solving the pricing equations (PDEs) of European call options for very general stochastic volatility models, including the Stein and Stein, the Hull and White, and the Heston models as particular cases. The main idea is to express the price in terms of a power series of the correlation parameter between the processes driving the dynamics of the price and of the volatility. The expansion is done around correlation zero and each term is identified via a probabilistic expression. It is shown that the power series converges with positive radius under some regularity conditions. Besides, we propose (as in Alos in Finance Stoch. 10:353–365, 2006) a further approximation to make the terms of the series easily computable and we estimate the error we commit. Finally we apply our methodology to some well-known financial models.

A new higher-order weak approximation scheme for stochastic differential equations and the Runge-Kutta method

Abstract: The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge–Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.

Smart expansion and fast calibration for jump diffusions

Abstract: Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and jump Poisson process We show that the accuracy of the formula depends on the smoothness of the payoff Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency As a consequence, the calibration of such model becomes very fast

Basket CDS pricing with interacting intensities

Abstract: We propose a factor contagion model for correlated defaults. The model covers the heterogeneous conditionally independent portfolio and the infectious default portfolio as special cases. The model assumes that the hazard rate processes are driven by external common factors as well as defaults of other names in the portfolio. The total hazard construction method is used to derive the joint distribution of default times. The basket CDS rates can be computed analytically for homogeneous contagion portfolios and recursively for general factor contagion portfolios. We extend the results to include the interacting counterparty risk and the stochastic intensity process.

Bias-correcting the realized range-based variance in the presence of market microstructure noise

Abstract: Market microstructure noise is a challenge to high-frequency based estimation of the integrated variance, because the noise accumulates with the sampling frequency. This has led to widespread use of constructing the realized variance, a sum of squared intraday returns, from sparsely sampled data, for example 5- or 15-minute returns. In this paper, we analyze the impact of microstructure noise on the realized range-based variance and propose a bias correction to the range-statistic. The new estimator is shown to be consistent for the integrated variance and asymptotically mixed Gaussian under simple forms of microstructure noise. We can select an optimal partition of the high-frequency data in order to minimize its asymptotic conditional variance. The finite sample properties of our estimator are studied with Monte Carlo simulations and we implement it using Microsoft high-frequency data from TAQ. We find that a bias-corrected range-statistic often leads to much smaller confidence intervals for the integrated variance, relative to the realized variance.

Numerical methods for Lévy processes

Abstract: We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Levy models.

MDP Algorithms for Portfolio Optimization Problems in pure Jump Markets

Abstract: We consider the problem of maximizing the expected utility of the terminal wealth of a portfolio in a continuous-time pure jump market with general utility function. This leads to an optimal control problem for piecewise deterministic Markov processes. Using an embedding procedure we solve the problem by looking at a discrete-time contracting Markov decision process. Our aim is to show that this point of view has a number of advantages, in particular as far as computational aspects are concerned. We characterize the value function as the unique fixed point of the dynamic programming operator and prove the existence of optimal portfolios. Moreover, we show that value iteration as well as Howard’s policy improvement algorithm works. Finally, we give error bounds when the utility function is approximated and when we discretize the state space. A numerical example is presented and our approach is compared to the approximating Markov chain method.

Adjoint-based Monte Carlo calibration of financial market models

Abstract: Adjoint methods have recently gained considerable importance in the finance sector, because they allow to quickly compute option sensitivities with respect to a large number of model parameters. In this paper we investigate how the efficiency of adjoint methods can be exploited to speed up the Monte Carlo-based calibration of financial market models. After analyzing the calibration problem both theoretically and numerically, we derive the associated adjoint equation and propose its application in combination with a multi-layer method, for which we prove convergence to a stationary point of the underlying optimization problem. Detailed numerical examples illustrate the performance of the method. In particular, the proposed algorithm reduces the calibration time for a typical equity market model with time-dependent model parameters from over three hours to less than ten minutes on a usual desktop PC.

Background filtrations and canonical loss processes for top-down models of portfolio credit risk

Abstract: In single-obligor default risk modeling, using a background filtration in conjunction with a suitable embedding hypothesis (generally known as ℍ-hypothesis or immersion property) has proven a very successful tool to separate the actual default event from the model for the default arrival intensity. In this paper we analyze the conditions under which this approach can be extended to the situation of a portfolio of several obligors, with a particular focus on the so-called top-down approach. We introduce the natural ℍ-hypothesis of this setup (the successive ℍ-hypothesis) and show that it is equivalent to a seemingly weaker one-step ℍ-hypothesis. Furthermore, we provide a canonical construction of a loss process in this setup and provide closed-form solutions for some generic pricing problems.

Stein’s method and zero bias transformation for CDO tranche pricing

Abstract: We propose an original approximation method, which is based on Stein’s method and the zero bias transformation, to calculate CDO tranches in a general factor framework. We establish first-order correction terms for the Gaussian and the Poisson approximations respectively and we estimate the approximation errors. The application to the CDO pricing consists of combining the two approximations.

Double-sided Parisian option pricing

Abstract: In this paper we derive Fourier transforms for double-sided Parisian option contracts. The double-sided Parisian option contract is triggered by the stock price process spending some time above an upper level or below some lower level. The double-sided Parisian knock-in call contract is the general type of Parisian contract from which also the single-sided contract types follow. The paper gives an overview of the different types of contracts that can be derived from the double-sided Parisian knock-in calls, and, after discussing the Fourier inversion, it concludes with various numerical examples, explaining the, sometimes peculiar, behavior of the Parisian option.

Hedging of American options under transaction costs

Abstract: We consider a continuous-time model of a financial market with proportional transaction costs. Our result is a dual description of the set of initial endowments of self-financing portfolios super-replicating an American-type contingent claim. The latter is a right-continuous adapted vector process describing the number of assets to be delivered at the exercise date. We introduce a specific class of price systems, called coherent, and show that the hedging endowments are those whose “values” are larger than the expected weighted “values” of the payoff process for every coherent price system used for the “evaluation” of the assets.

In which financial markets do mutual fund theorems hold true

Abstract: The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are: (i) Let N be the wealth process of the numeraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numeraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numeraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black–Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988). Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numeraire portfolio described above. (ii) If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.