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.     

CVaR HEDGING USING QUANTIZATION‐BASED STOCHASTIC APPROXIMATION ALGORITHM

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value‐at‐risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.     

The valuation of reset options with multiple strike resets and reset dates

This article makes two contributions to the literature. The first contribution is to provide the closed‐form pricing formulas of reset options with strike resets and predecided reset dates. The exact closed‐form pricing formulas of reset options with strike resets and continuous reset period are also derived. The second contribution is the finding that the reset options not only have the phenomena of Delta jump and Gamma jump across reset dates, but also have the properties of Delta waviness and Gamma waviness, especially near the time before reset dates. Furthermore, Delta and Gamma can be negative when the stock price is near the strike resets at times close to the reset dates. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:87–107,2003     

PRICING SWAPTIONS UNDER MULTIFACTOR GAUSSIAN HJM MODELS

Several approximations have been proposed in the literature for the pricing of European‐style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin‐Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European‐style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by‐product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade‐off than all the approximations already proposed in the literature.     

Internal determinants of recycling behaviour by university students: a cross‐country comparative analysis

University students are a key group for research since they will be the consumers and the intellectual vanguard of the future and, therefore, a reference group for other consumers. Accordingly, data obtained from university students in Spain (n = 640) and the US (n = 597) were analysed to identify the main internal factors which lead them to participate in recycling activities. Given that morality‐based theories, and more specifically the Value–Belief–Norm (VBN) Theory, might be very useful in explaining pro‐environmental behaviour which requires personal sacrifices and whose benefits are mainly for the environment itself, that is the theoretical approach that we take for the selection and analysis of factors. Our findings support our thesis that environmental knowledge is a factor that should be (but is not currently) considered in the framework of VBN theory for predicting recycling behaviour. Although university students from Spain and the US have very different recycling rates, the internal factors that explain their recycling behaviour are very similar (motivations, perceived consumer effectiveness (PCE), environmental knowledge and gender). The elasticity of recycling behaviour to changes in internal factors is estimated with a view to making predictions, and altruistic motivations and PCE are found to be the factors that have the greatest effect in terms of improving recycling behaviour, followed by environmental knowledge. These predictions could help university policy makers take better decisions about the factors on which they need to act to increase recycling activities. Some guidelines for consideration in future intervention strategies to encourage this group to recycle are also provided.     

Unspanned stochastic volatility in the multifactor CIR model

Empirical evidence suggests that fixed‐income markets exhibit unspanned stochastic volatility (USV), that is, that one cannot fully hedge volatility risk solely using a portfolio of bonds. While Collin‐Dufresne and Goldstein (2002, Journal of Finance, 57, 1685–1730) showed that no two‐factor Cox–Ingersoll–Ross (CIR) model can exhibit USV, it has been unknown to date whether CIR models with more than two factors can exhibit USV or not. We formally review USV and relate it to bond market incompleteness. We provide necessary and sufficient conditions for a multifactor CIR model to exhibit USV. We then construct a class of three‐factor CIR models that exhibit USV. This answers in the affirmative the above previously open question. We also show that multifactor CIR models with diagonal drift matrix cannot exhibit USV.     

Portfolio Value-at-Risk with Heavy-Tailed Risk Factors

This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.     

Convergence of optimal expected utility for a sequence of discrete‐time markets

We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that .     

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond B^T with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.     

THE TRACKING ERROR RATE OF THE DELTA‐GAMMA HEDGING STRATEGY

We analyze the convergence rate of the quadratic tracking error, when a Delta‐Gamma hedging strategy is used at N discrete times. The fractional regularity of the payoff function plays a crucial role in the choice of the trading dates, in order to achieve optimal rates of convergence.     

Mean–field moral hazard for optimal energy demand response management

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a continuum of consumers with mean–field interaction, whose consumption is impacted by a common noise. We formulate the problem as a Principal–Agent problem with moral hazard in which the Principal—she—is an electricity producer who observes continuously the consumption of a continuum of risk‐averse consumers, and designs contracts in order to reduce her production costs. More precisely, the producer incentivizes each consumer to reduce the average and the volatility of his consumption in different usages, without observing the efforts he makes. We prove that the producer can benefit from considering the continuum of consumers by indexing contracts on the consumption of one Agent and aggregate consumption statistics from the distribution of the entire population of consumers. In the case of linear energy valuation, we provide closed‐form expression for this new type of optimal contracts that maximizes the utility of the producer. In most cases, we show that this new type of contracts allows the Principal to choose the risks she wants to bear, and to reduce the problem at hand to an uncorrelated one.     

MEAN–VARIANCE PORTFOLIO OPTIMIZATION WITH STATE‐DEPENDENT RISK AVERSION

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.     

Hedging nontradable risks with transaction costs and price impact

A risk‐averse agent hedges her exposure to a nontradable risk factor U using a correlated traded asset S and accounts for the impact of her trades on both factors. The effect of the agent's trades on U is referred to as cross‐impact. By solving the agent's stochastic control problem, we obtain a closed‐form expression for the optimal strategy when the agent holds a linear position in U. When the exposure to the nontradable risk factor is nonlinear, we provide an approximation to the optimal strategy in closed‐form, and prove that the value function is correctly approximated by this strategy when cross‐impact and risk‐aversion are small. We further prove that when is nonlinear, the approximate optimal strategy can be written in terms of the optimal strategy for a linear exposure with the size of the position changing dynamically according to the exposure's “Delta” under a particular probability measure.     

MODIFIED LELAND’S STRATEGY FOR A CONSTANT TRANSACTION COSTS RATE

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay‐off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper, we suggest a modification of Leland’s strategy ensuring that the approximation error vanishes in the limit.     

Existence of a calibrated regime switching local volatility model

By Gyöngy's theorem, a local and stochastic volatility model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility (LV) function is equal to the ratio of the Dupire LV function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a stochastic differential equation (SDE) nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in Guyon and Henry‐Labordère [Risk Magazine, 25, 92–97], provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. When the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level, we prove existence of a solution to the associated Fokker–Planck equation under the condition that the range of the squared stochastic volatility factor is not too large. We then deduce existence to the calibrated model by extending the results in Figalli [Journal of Functional Analysis, 254(1), 109–153].     

商业银行风险管理中的模糊分析方法

将商业银行面临的风险分解成若干个风险因素子系统,以金融理论为指导建立各子系统风险因素评价的隶属度矩阵,并在此基础上采用模糊间接识别模型及贴近度的概念,构造各子系统风险因素模糊识别模型;通过各子系统模糊识别模型的合成,建立整个商业银行系统的模糊识别模型,借以达到各商业银行风险排序、化解金融隐患之目的。     

Static and semistatic hedging as contrarian or conformist bets

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semistatic portfolios should more properly be thought of as separate classes of derivatives, with nontrivial, model‐dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from the Carr–Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance‐minimizing portfolios. We explain why the exact semistatic hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance‐minimizing semistatic portfolios. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener–Hopf factors and Laplace–Fourier inversion.     

CONVERGENCE OF BARRIER OPTION PRICES IN THE BINOMIAL MODEL

In this paper, we study the rate of convergence of the European barrier call option price given by the CRR binomial model to the Black–Scholes price as the number of periods n tends to infinity. In general the error is of order and we give explicit formulas for the coefficients of and 1/n in the asymptotic expansion of the error. These coefficients depend on the positions of the barrier and strike in the binomial lattice and enable us to give a rigorous explanation of the observed fact that the error is of order 1/n when n is chosen in an appropriate way.     

THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path‐dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first‐touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double‐barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally.     

Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.