Analytical approximations of local‐Heston volatility model and error analysis

This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.     

Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999) , as well as Ma and Zhang (2000) , using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.     

The asymptotic expansion of the regular discretization error of Itô integrals

We study an Edgeworth‐type refinement of the central limit theorem for the discretization error of Itô integrals. Toward this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. Two applications to finance are given; the asymptotics of discrete hedging error under the Black–Scholes model and the difference between continuously and discretely monitored variance swap payoffs under stochastic volatility models.     

A General Approach to Hedging Options: Applications to Barrier and Partial Barrier Options

In this paper we consider a Black and Scholes economy and show how the Malliavin calculus approach can be extended to cover hedging of any square integrable contingent claim. As an application we derive the replicating portfolios of some barrier and partial barrier options.     

Hedging Options: The Malliavin Calculus Approach versus the Δ-Hedging Approach

In this paper we consider a Black and Scholes economy and investigate two approaches to hedging contingent claims. We show that the general Malliavin calculus approach can generate the classical Δ-hedging formula under weaker conditions.     

PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest,   r > 0  , and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.     

Analysis of Error with Malliavin Calculus: Application to Hedging

The aim of this paper is to compute the quadratic error of a discrete time-hedging strategy in a complete multidimensional model. This result extends that of Gobet and Temam (2001) and Zhang (1999) . More precisely, our basic assumption is that the asset prices satisfy the d -dimensional stochastic differential equation   dXⁱ_t = Xⁱ_t ( bⁱ ( X_t ) dt +σ _{i , j} ( X_t ) dW^j_t )  . We precisely describe the risk of this strategy with respect to n , the number of rebalancing times. The rates of convergence obtained are     for any options with Lipschitz payoff and  1/ n ^1/4  for options with irregular payoff.     

Consumers’ privacy calculus: The PRICAL index development and validation

Although collecting personal information about consumers is crucial for firms and marketers, understanding of when and why consumers accept or reject information collection remains limited. The authors conceptualize a privacy calculus that represents a consumer’s trade–off of the valence and uncertainty of the consequences of the collection, storage, and use of personal information. For example, usage-based car insurance requires drivers to share data on their driving behavior in exchange for a discount (certain benefit) but at the risk of third parties intercepting location data for malicious use (uncertain disadvantage). Building on this conceptualization, the authors develop the privacy calculus (PRICAL) index. They empirically confirm the validity of the items (Study 1) and the index as a whole (Study 2). The PRICAL index is generally applicable and improves the explanation of behavioral intentions (Study 2) and actual behavior (Study 3), compared with currently used constructs (e.g., privacy concern, trust). Overall, the PRICAL index allows managers to understand consumers’ acceptance of information collection regarding financial, performance, psychological, security, social, and time-related consequences, which the authors demonstrate using the top five most valuable digital brands (Study 4).     

An Anticipating Calculus Approach to the Utility Maximization of an Insider

In this paper we consider a financial market with an insider that has, at time   t = 0  , some additional information of the whole developing of the market. We use the forward integral, which is an anticipating integral, and the techniques of the Malliavin calculus so that we can take advantage of the privileged information to maximize the expected logarithmic utility from terminal wealth.     

UTILITY MAXIMIZATION IN AN INSIDER INFLUENCED MARKET

We study a controlled stochastic system whose state is described by a stochastic differential equation with anticipating coefficients. This setting is used to model markets where insiders have some influence on the dynamics of prices. We give a characterization theorem for the optimal logarithmic portfolio of an investor with a different information flow from that of the insider. We provide explicit results in the partial information case that we extend in order to incorporate the enlargement of filtration techniques for markets with insiders. Finally, we consider a market with an insider who influences the drift of the underlying price asset process. This example gives a situation where it makes a difference for a small agent to acknowledge the existence of an insider in the market.     

Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options

In this paper, we consider the problem of the numerical computation of Greeks for a multidimensional barrier and lookback style options: the payoff function depends in a rather general way on the minima and maxima of the coordinates of the d -dimensional underlying asset process. Using Malliavin calculus techniques, we derive additional weights that enable computation of the Greeks using Monte Carlo simulations. Numerical experiments confirm the efficiency of the method. This work is a multidimensional extension of previous results (see Gobet and Kohatsu-Higa 2001 ).     

ARBITRAGE BOUNDS FOR PRICES OF WEIGHTED VARIANCE SWAPS

We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co‐maturing put options. We assume the put option prices do not admit arbitrage and deduce no‐arbitrage bounds on the weighted variance swap along with super‐ and sub‐replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model‐free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi‐infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model‐independent and probability‐free setup. In particular, we use and extend Föllmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take the form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk‐neutral expectations of discounted payoffs.     

OPTIMAL LIQUIDATION OF DERIVATIVE PORTFOLIOS

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.     

MULTIFRACTIONAL STOCHASTIC VOLATILITY MODELS

The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.     

时差型卫星定位系统目标位置标校算法的误差分析

针对时差型卫星定位系统位置标校算法中标校源选择的问题,首先建立了基于单个标校源的标校方法的误差分析模型,分析了参数的系统性误差和随机测量误差对标校效果的影响;其次,定义了两类定位误差改善因子,用来定量描述标校算法对定位精度的改善能力,并可以通过理论计算评价标校源的标校效果,从而为标校源的选择提供了依据。最后,仿真分析验证了误差分析模型及定位误差改善因子的有效性。     

The role of channel quality in customer equity management

Maximization of customer equity is a core objective of customer–company relationship management. We present an extended model of customer equity for determining the optimal allocation of marketing resources across acquisition and retention activities. Focusing on the negative relationship between acquisition and retention, we motivate channel quality as a relevant decision variable, explicate its role in the model, and demonstrate the existence of an optimal value. In addition, rather than making concavity assumptions about acquisition and retention rate response curves, we use the flexible ADBUDG model (Little, JDC, Models and Managers: the Concept of a Decision Calculus. Manag Sci 1970; 16(8): 466–484.), which allows for both S-shaped and strictly-concave relationships, and parameterize it using decision calculus. We show how to estimate and apply the model and then provide sensitivity analyses with respect to changes in the true values of model parameters as well as inaccuracy in managerial inputs. We conclude by comparing our model with extant models and discussing the implications of our research.     

When and how consumers are willing to exchange data with retailers: An exploratory segmentation

Consumer data is a crucial resource for retailers. Yet accessing this data increasingly requires consumers to willingly participate in data exchange. This paper draws on social exchange theory and privacy calculus to investigate differences in consumer willingness to exchange data with retailers. Consumers are also profiled on their perceptions of retailer's use and abuse of data, along with the antecedents and outcomes of these perceptions. We employ a cross-sectional quantitative survey and collect data from a sample of 463 US consumers. For statistical analysis, we employ a latent class segmentation and identify six consumer segments which differ in their perceptions of the consumer-retailer data exchange. The key drivers of these segment differences include privacy concerns, technology readiness, and general trust of, and engagement with, retail brands. The segments also differ in their subsequent views towards the use and abuse of their data by retailers, including willingness to exchange data. Hence, when accessing and utilizing consumer data, it is important that brands identify different segments, and adapt their approach accordingly.     

STOCK LIQUIDATION VIA STOCHASTIC APPROXIMATION USING NASDAQ DAILY AND INTRA-DAY DATA

By focusing on computational aspects, this work is concerned with numerical methods for stock selling decision using stochastic approximation methods. Concentrating on the class of decisions depending on threshold values, an optimal stopping problem is converted to a parametric stochastic optimization problem. The algorithms are model free and are easily implementable on-line. Convergence of the algorithms is established, second moment bound of estimation error is obtained, and escape probability from a neighborhood of the true parameter is also derived. Numerical examples using both daily closing prices and intra-day data are provided to demonstrate the performance of the algorithms.     

EXPLICIT IMPLIED VOLATILITIES FOR MULTIFACTOR LOCAL‐STOCHASTIC VOLATILITY MODELS

We consider an asset whose risk‐neutral dynamics are described by a general class of local‐stochastic volatility models and derive a family of asymptotic expansions for European‐style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three‐halves stochastic volatility, and SABR local‐stochastic volatility.     

GREEKS FORMULAS FOR AN ASSET PRICE MODEL WITH GAMMA PROCESSES

Greeks formulas of Delta, Rho, Vega, and Gamma are derived in closed form for asset price dynamics described by gamma processes and Brownian motions time‐changed by a gamma process. The model considered here includes many well‐known models of practical interest, such as the variance gamma model and the Black–Scholes model. Our approach is based upon the Malliavin calculus for jump processes by making full use of a scaling property of gamma processes with respect to the Girsanov transform. The existence of their variance is investigated. Numerical results are provided to illustrate that the derived Greeks formulas have faster rate of convergence relative to the finite difference method.