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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Time-Consistent Portfolio Management SIAM J. Finan. Math. 3, pp. 1-32 (32 pages) Online Publication Date: January 03, 2012
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This paper considers the portfolio management problem for an investor with finite time horizon who is allowed to consume and take out life insurance. Natural assumptions, such as different discount rates for consumption and life insurance, lead to time inconsistency. This situation can also arise when the investor is in fact a group, the members of which have different utilities and/or different discount rates. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioral finance. Following [I. Ekeland and T. A. Pirvu, Math. Financ. Econ., 2 (2008), pp. 57–86], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution in the case of constant relative risk aversion utilities. Our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting and is therefore known in the literature as the “consumption puzzle.” Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium. |
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Optimal Trading with Stochastic Liquidity and Volatility SIAM J. Finan. Math. 3, pp. 163-181 (19 pages) Online Publication Date: January 31, 2012
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We consider the problem of mean-variance optimal agency execution strategies, when the market liquidity and volatility vary randomly in time. Under specific assumptions for the stochastic processes satisfied by these parameters, we construct a Hamilton–Jacobi–Bellman equation for the optimal cost and strategy. We solve this equation numerically and illustrate optimal strategies for varying risk aversion. These strategies adapt optimally to the instantaneous variations of market quality. |
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Pricing and Hedging in Affine Models with Possibility of Default SIAM J. Finan. Math. 3, pp. 328-350 (23 pages) Online Publication Date: April 10, 2012
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We propose a general framework for the simultaneous modeling of equity, government bonds, corporate bonds, and derivatives. Uncertainty is generated by a general affine Markov process. The setting allows for stochastic volatility, jumps, the possibility of default, and correlation between different assets. We show how to calculate discounted complex moments by solving a coupled system of generalized Riccati equations. This yields an efficient method to compute prices of power payoffs. European calls and puts as well as binaries and asset-or-nothing options can be priced with the fast Fourier transform methods of Carr and Madan [J. Comput. Finance, 2 (1999), pp. 61–73] and Lee [J. Comput. Finance, 7 (2005), pp. 51–86]. Other European payoffs can be approximated with a linear combination of government bonds, power payoffs, and vanilla options. We show the results to be superior to using only government bonds and power payoffs or government bonds and vanilla options. We also give conditions for European continent claims in our framework to be replicable if enough financial instruments are liquidly tradable and study dynamic hedging strategies. As an example we discuss a Heston-type stochastic volatility model with possibility of default and stochastic interest rates. |
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Option Pricing in Multivariate Stochastic Volatility Models of OU Type SIAM J. Finan. Math. 3, pp. 66-94 (29 pages) Online Publication Date: January 17, 2012
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We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets, while still being highly analytically tractable. First, we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multiasset options efficiently. To show the applicability of our results, we propose a concrete specification, the Ornstein–Uhlenbeck (OU)–Wishart model, where the dynamics of each individual asset coincide with the popular $\Gamma$-OU Barndorff-Nielsen–Shepard model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form. |
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The Relaxed Investor with Partial Information SIAM J. Finan. Math. 3, pp. 304-327 (24 pages) Online Publication Date: April 10, 2012
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We consider an investor in a financial market consisting of a riskless bond and several risky assets. The price processes of the risky assets are geometric Brownian motions where either the drifts are modeled as random variables assuming a constant volatility matrix or the volatility matrix is considered random and drifts are assumed to be constant. The investor is only able to observe the asset prices but not all the model parameters and hence information is only partial. A Bayesian approach is used with known prior distributions for the random model parameters. We assume that the investor can only trade at discrete-time points which are multiples of $h>0$ and investigate the loss in expected utility of terminal wealth which is due to the fact that the investor cannot trade and observe continuously. It turns out that in general a discretization gap appears, i.e., for $h\to 0$ the expected utility of the $h$-investor does not converge to the expected utility of the continuous investor. This is in contrast to results under full information in [L.C.G. Rogers, Finance Stoch., 5(2001), pp. 131–154]. We also present simple asymptotically optimal portfolio strategies for the discrete-time problem. Our results are illustrated by some numerical examples. |
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Optimal Control of Trading Algorithms: A General Impulse Control Approach SIAM J. Finan. Math. 2, pp. 404-438 (35 pages) Online Publication Date: June 07, 2011
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We propose a general framework for intraday trading based on the control of trading algorithms. Given a set of generic parameterized algorithms (which have to be specified by the controller ex-ante), our aim is to optimize the dates $(\tau_i)_i$ at which they are launched, the length $(\delta_i)_i$ of the trading period, and the value of the parameters $({\cal E}_i)_i$ kept during the time interval $[\tau_i,\tau_i + \delta_i)$. This provides the financial agent a decision tool for selecting which algorithm (and for which set of parameters and time length) should be used in the different phases of the trading period. From the mathematical point of view, this gives rise to a nonclassical impulse control problem where not only the regime ${\cal E}_i$ but also the period $[\tau_i,\tau_i+ \delta_i)$ have to be determined by the controller at the impulse time $\tau_i$. We adapt the weak dynamic programming principle of Bouchard and Touzi [SIAM J. Control Optim., 49 (2011), pp. 948–962] to our context to provide a characterization of the associated value function as a discontinuous viscosity solution of a system of partial differential equations with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide a simple example of application to a problem of optimal stock trading with a nonlinear market impact function. This shows how parameters adapt to the market. |
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Processes of Class Sigma, Last Passage Times, and Drawdowns SIAM J. Finan. Math. 3, pp. 280-303 (24 pages) Online Publication Date: April 03, 2012
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We propose a general framework for studying last passage times, suprema, and drawdowns of a large class of continuous-time stochastic processes. Our approach is based on processes of class Sigma and the more general concept of two processes, one of which moves only when the other is at the origin. After investigating certain transformations of such processes and their convergence properties, we provide three general representation results. The first allows the recovery of a process of class Sigma from its final value and the last time it visited the origin. In many situations this gives access to the distribution of the last time a stochastic process attains a certain level or is equal to its running maximum. It also leads to recently discovered formulas expressing option prices in terms of last passage times. Our second representation result is a stochastic integral representation that will allow us to price and hedge options on the running maximum of an underlying that are triggered when the underlying drops to a given level or, alternatively, when the drawdown or relative drawdown of the underlying attains a given height. The third representation gives conditional expectations of certain functionals of processes of class Sigma. It can be used to deduce the distributions of a variety of interesting random variables such as running maxima, drawdowns, and maximum drawdowns of suitably stopped processes. |
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SIAM J. Finan. Math. 2, pp. 839-865 (27 pages) Online Publication Date: October 12, 2011
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After the 2007 credit crisis, financial bubbles have once again emerged as a topic of current concern. An open problem is to determine in real time whether or not a given asset's price process exhibits a bubble. Due to recent progress in the characterization of asset price bubbles using the arbitrage-free martingale pricing technology, we are able to propose a new methodology for answering this question based on the asset's price volatility. We limit ourselves to the special case of a risky asset's price being modeled by a Brownian driven stochastic differential equation. Such models are ubiquitous both in theory and in practice. Our methods use sophisticated volatility estimation techniques combined with the method of reproducing kernel Hilbert spaces. We illustrate these techniques using several stocks from the alleged Internet dot-com episode of 1998–2001, where price bubbles were widely thought to have existed. Our results confirm the suspicions of the presence of bubbles in many of the dot-com stocks of 1998–2001. |
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Valuation Equations for Stochastic Volatility Models SIAM J. Finan. Math. 3, pp. 351-373 (23 pages) Online Publication Date: April 17, 2012
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We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset price is a martingale. |
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The Small-Maturity Smile for Exponential Lévy Models SIAM J. Finan. Math. 3, pp. 33-65 (33 pages) Online Publication Date: January 17, 2012
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We derive a small-time expansion for out-of-the-money call options under an exponential Lévy model, using the small-time expansion for the distribution function given in [J. Figueroa-López and C. Houdré, Stochastic Process. Appl., 119 (2009), pp. 3862–3889], combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility $\sigma$ of the Gaussian component of the driving Lévy process is to increase the call price by $\frac{1}{2}\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility $\hat{\sigma}_{t}^{2}(k)$ at log-moneyness $k$, which sharpens the first order estimate $\hat{\sigma}_{t}^{2}(k)\sim \frac{\frac{1}{2}k^2}{t\log (1/t)}$ given in [P. Tankov, Pricing and hedging in exponential Lévy models: Review of recent results, in Paris-Princeton Lectures on Mathematical Finance, Springer, Berlin, 2011, pp. 319–359]. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model and apply this approach to the small-time pricing of at-the-money call options; we show that for $Y\in(1,2)$, $\lim_{t\to{}0}t^{-1/Y}\mathbb{E}(S_t-S_0)_{+}=S_{0}\mathbb{E}^{*}(Z_{+})$ and the corresponding at-the-money implied volatility $\hat{\sigma}_t(0)$ satisfies $\lim_{t \to 0}\hat{\sigma}_t(0)/t^{1/Y-1/2}=\sqrt{2\pi}\,\mathbb{E}^{*}(Z_{+})$, where $Z$ is a symmetric $Y$-stable random variable under $\mathbb{P}^*$ and $Y$ is the usual parameter for the CGMY model appearing in the Lévy density $\nu(x)=C x^{-1-Y}e^{-M x}{\bf 1}_{\{x>0\}}+C |x|^{-1-Y}e^{-G|x|}{\bf 1}_{\{x<0\}}$ of the process. |
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Primal and Dual Pricing of Multiple Exercise Options in Continuous Time SIAM J. Finan. Math. 2, pp. 562-586 (25 pages) Online Publication Date: August 18, 2011
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In this paper we study the pricing problem of multiple exercise options in continuous time on a finite time horizon. For the corresponding multiple stopping problem, we prove, under quite general assumptions, the existence of the Snell envelope, a reduction principle as nested single stopping problems, and a Doob–Meyer-type decomposition for the Snell envelope. The main technical difficulty arises from the fact that the price process of a multiple exercise option typically exhibits discontinuities from the right-hand side, even if the payoff process of the option is right-continuous. We also derive a dual minimization problem for the price of the multiple exercise option in terms of martingales and processes of bounded variation. Moreover, we explain how the primal and dual pricing formulas can be applied to compute confidence intervals on the option price via Monte Carlo methods, and we present a numerical example. |
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Portfolio Selection Using Tikhonov Filtering to Estimate the Covariance Matrix SIAM J. Finan. Math. 1, pp. 932-961 (30 pages) Online Publication Date: December 14, 2010
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Markowitz's portfolio selection problem chooses weights for stocks in a portfolio based on an estimated covariance matrix of stock returns. Our study proposes reducing noise in the estimation by using a Tikhonov filter function. In addition, we prevent rank deficiency of the estimated covariance matrix and propose a method for effectively choosing the Tikhonov parameter, which determines the filtering intensity. We put previous estimators into a common framework and compare their filtering functions for eigenvalues of the correlation matrix. We demonstrate the effectiveness of our estimator using stock return data from 1958 through 2007. |
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Asymptotic Approximations to Deterministic and Stochastic Volatility Models SIAM J. Finan. Math. 2, pp. 935-964 (30 pages) Online Publication Date: October 27, 2011
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The problem of pricing, hedging, and calibrating equity derivatives in a fast and consistent fashion is considered when the underlying asset does not follow the standard Black–Scholes model but instead the stochastic CEV (constant elasticity of variance) or SABR (stochastic alpha beta rho) model. The underlying process in the SABR model has the volatility as a stochastic function of the asset price. In such situations, trading desks often resort to numerical methods to solve the pricing and hedging problem. This can be problematic for complex models if real-time valuations, hedging, and calibration are required. A more efficient and practical alternative is to use a formula even if it is only an approximation. A systematic approach is presented, based on the WKB or ray method, to derive asymptotic approximations yielding simple formulas for the pricing problem. For the SABR model, default may be possible, and the original ray approximation is not valid near the default boundary, so a modified asymptotic approximation or boundary layer correction is derived. New results are also derived for the standard CEV model, which has deterministic volatility, as a special case of the SABR results. The applicability of the results is illustrated by deriving new analytical approximations for vanilla options based on the CEV and SABR models. The accuracy of the results is demonstrated numerically. |
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On the Convergence of Higher Order Hedging Schemes: The Delta-Gamma Case SIAM J. Finan. Math. 2, pp. 55-78 (24 pages) Online Publication Date: January 20, 2011
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Hedging errors induced by discrete rebalancing of the hedge portfolio of a delta-gamma hedging strategy are investigated. The rate of convergence of the expected squared hedging error as the number of adjustments of the hedge portfolio goes to infinity is analyzed. It is found that the delta-gamma strategy produces higher convergence rates than the usual delta strategy. |
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Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model SIAM J. Finan. Math. 2, pp. 22-54 (33 pages) Online Publication Date: January 13, 2011
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This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion factor process. The criterion, following earlier work by Bielecki, Pliska, Nagai, and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance). By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of this paper is to show that the risk-sensitive jump-diffusion problem can be fully characterized in terms of a parabolic Hamilton–Jacobi–Bellman PDE rather than a partial integro-differential equation, and that this PDE admits a classical $(C^{1,2})$ solution. |
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Good Deal Bounds Induced by Shortfall Risk SIAM J. Finan. Math. 2, pp. 1-21 (21 pages) Online Publication Date: January 11, 2011
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We consider, throughout this paper, an incomplete financial market which is governed by a possibly nonlocally bounded right-continuous with left-limits (RCLL) special semimartingale. We shall provide good deal bounds for contingent claims induced by shortfall risk in the framework of the Orlicz heart setting. We prove that the upper and lower bounds of such a good deal bound are expressed by a convex risk measure on an Orlicz heart. In addition, we obtain representation results for three types of model, which are an unconstrained portfolio model, a $W$-admissible model, and a predictably convex model. |
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Stochastic Switching Games and Duopolistic Competition in Emissions Markets SIAM J. Finan. Math. 2, pp. 488-511 (24 pages) Online Publication Date: August 09, 2011
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We study optimal behavior of energy producers under a CO$_2$ emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO$_2$ allowance price. Such duopolistic competition results in a new type of non–zero-sum stochastic switching game with finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible, and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed, and a numerical example is presented. |
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An Asymptotic Expansion with Push-Down of Malliavin Weights SIAM J. Finan. Math. 3, pp. 95-136 (42 pages) Online Publication Date: January 24, 2012
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This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multidimensional stochastic volatility models. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied. We provide an expansion formula for generalized Wiener functionals and closed-form approximation formulas in the stochastic volatility environment. In addition, we present applications of the general formula to expansions of option prices for the shifted log-normal model with stochastic volatility. Moreover, with some results of Malliavin calculus in jump-type models, we derive an approximation formula for the jump-diffusion model in the stochastic volatility environment. Some numerical examples are also shown. |
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Managing Risk with Short-Term Futures Contracts SIAM J. Finan. Math. 2, pp. 715-726 (12 pages) Online Publication Date: September 22, 2011
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Under the constraint of terminal risk, we search for an optimal deterministic strategy to reduce the running risk in hedging a long-term commitment with short-term futures contracts. An explicit solution is given if the underlying stock follows the simple stochastic differential equation $dS_t=\mu dt+\sigma dB_t$, where $B_t$ is the standard Brownian motion. Our result generalizes the result of Larcher and Leobacher in [Math. Finance, 13 (2003), pp. 331–344]. As an application, we provide a solution to the utility optimization problem posed in that paper. |
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Optimal Timing to Purchase Options SIAM J. Finan. Math. 2, pp. 768-793 (26 pages) Online Publication Date: October 11, 2011
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We study the optimal timing of derivative purchases in incomplete markets. In our model, an investor attempts to maximize the spread between her model price and the offered market price through optimally timing her purchase. Both the investor and the market value the options by risk-neutral expectations but under different equivalent martingale measures representing different market views. The structure of the resulting optimal stopping problem depends on the interaction between the respective market price of risk and the option payoff. In particular, a crucial role is played by the delayed purchase premium that is related to the stochastic bracket between the market price and the buyer's risk premia. Explicit characterization of the purchase timing is given for two representative classes of Markovian models: (i) defaultable equity models with local intensity; (ii) diffusion stochastic volatility models. Several numerical examples are presented to illustrate the results. Our model is also applicable to the optimal rolling of long-dated options and sequential buying and selling of options. |
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