存在交易成本的权证避险策略实证研究

权证发行人在存在交易成本时对冲风险,若按照B—S理论进行动态连续避险操作,将造成巨大的交易成本,致使B-S动态连续避险不可行。因此存在交易成本时,对避险的操作都采用间断性避险。本文在统一均值方差框架下,系统全面的比较了存在交易成本的五种避险策略。在比例交易成本情形下,Whalley—Wilmott避险策略优于其他所有策略,当避险误差的标准差相同时该策略的交易成本最小;其次分别是delta固定避险带避险策略,基于标的资产价格变化的避险策略,Leland避险模型和间断的B—S避险策略。随着波动率σ上升,无风险利率γ下降,基于变动的避险策略相对于基于时间的策略优势更大。     

Option pricing with an illiquid underlying asset market

We examine how price impact in the underlying asset market affects the replication of a European contingent claim. We obtain a generalized Black–Scholes pricing PDE and establish the existence and uniqueness of a classical solution to this PDE. Unlike the case with transaction costs, we prove that replication with price impact is always cheaper than superreplication. Compared to the Black–Scholes case, a trader generally buys more stock and borrows more (shorts and lends more) to replicate a call (put). Furthermore, price impact implies endogenous stochastic volatility and an out-of-money option has lower implied volatility than an in-the-money option. This finding has important implications for empirical analysis on volatility smile.     

期权“隐含波动率微笑”成因分析

Black-Scholes期权定价模型低估深实值和深虚值期权的现象称为“波动率微笑”。其主要原因是资产价格过程假设和市场机制因素给期权卖方的△套期保值带来了额外风险和成本。确定波动率和随机波动率研究都对BS模型做出了修正。     

The valuation of American barrier options using the decomposition technique

In this paper, we propose an alternative approach for pricing and hedging American barrier options. Specifically, we obtain an analytic representation for the value and hedge parameters of barrier options, using the decomposition technique of separating the European option value from the early exercise premium. This allows us to identify some new put-call ‘symmetry’ relations and the homogeneity in price parameters of the optimal exercise boundary. These properties can be utilized to increase the computational efficiency of our method in pricing and hedging American options. Our implementation of the obtained solution indicates that the proposed approach is both efficient and accurate in computing option values and option hedge parameters. Our numerical results also demonstrate that the approach dominates the existing lattice methods in both accuracy and efficiency. In particular, the method is free of the difficulty that existing numerical methods have in dealing with spot prices in the proximity of the barrier, the case where the barrier options are most problematic.     

American contingent claims under small proportional transaction costs

American options are considered in the binary tree model under small proportional transaction costs. Dynamic programming type algorithms, which extend the Snell envelope construction, are developed for computing the ask and bid prices (also known as the upper and lower hedging prices) of such options together with the corresponding optimal hedging strategies for the writer and for the seller of the option. Representations of the ask and bid prices of American options in terms risk-neutral expectations of stopped option payoffs are also established in this setting.     

Utility indifference valuation for jump risky assets

We discuss utility maximization problems with exponential preferences in an incomplete market where the risky asset dynamics is described by a pure jump process driven by two independent Poisson processes. This includes results on portfolio optimization under an additional European claim. Value processes of the optimal investment problems, optimal hedging strategies and the indifference price are represented in terms of solutions to backward stochastic equations driven by the Poisson martingales. Via a duality result, the solution to the dual problems is derived. In particular, an explicit expression for the density of the minimal martingale measure is provided. The Markovian case is also discussed. This includes either asset dynamics dependent on a pure jump stochastic factor or claims written on a correlated non tradable asset.     

Option pricing and replication with transaction costs and dividends

This paper derives optimal perfect hedging portfolios in the presence of transaction costs within the binomial model of stock returns, for a market maker that establishes bid and ask prices for American call options on stocks paying dividends prior to expiration. It is shown that, while the option holder's optimal exercise policy at the ex-dividend date varies according to the stock price, there are intervals of values for such a price where the optimal policy would depend on the holder's preferences. Nonetheless, the perfect hedging assumption still allows the derivation of optimal hedging portfolios for both long and short positions of a market maker on the option.     

American contingent claims under small proportional transaction costs

American options are considered in the binary tree model under small proportional transaction costs. Dynamic programming type algorithms, which extend the Snell envelope construction, are developed for computing the ask and bid prices (also known as the upper and lower hedging prices) of such options together with the corresponding optimal hedging strategies for the writer and for the seller of the option. Representations of the ask and bid prices of American options in terms risk-neutral expectations of stopped option payoffs are also established in this setting.     

基于无套利对冲原理的信用风险期权定价

文章拓展了Klein假设中关于固定违约门槛的假设,构造可变违约门槛,根据无套利对冲原理,通过偏微分方程这种数学工具,推导出含信用风险的欧式脆弱期权价格波动的偏微分方程组和期权定价模型,进而求其显示解,得到类似于Black-Scholes公式的定价公式,该公式的推导过程比使用鞅理论推导更加浅显易懂。     

The market for crash risk

This paper examines the equilibrium when stock market crashes can occur and investors have heterogeneous attitudes towards crash risk. The less crash averse insure the more crash averse through options markets that dynamically complete the economy. The resulting equilibrium is compared with various option pricing anomalies: the tendency of stock index options to overpredict volatility and jump risk, the Jackwerth [Recovering risk aversion from option prices and realized returns. Review of Financial Studies 13, 433–451] implicit pricing kernel puzzle, and the stochastic evolution of option prices. Crash aversion is compatible with some static option pricing puzzles, while heterogeneity partially explains dynamic puzzles. Heterogeneity also magnifies substantially the stock market impact of adverse news about fundamentals.     

Risk aversion and the elasticity of substitution in general dynamic portfolio theory: Consistent planning by forward looking, expected utility maximizing investors

Using the measure of risk aversion suggested by Kihlstrom and Mirman [Kihlstrom, R., Mirman, L., 1974. Risk aversion with many commodities. Journal of Economic Theory 8, 361–388; Kihlstrom, R., Mirman, L., 1981. Constant, increasing and decreasing risk aversion with many commodities. Review of Economic Studies 48, 271–280], we propose a dynamic consumption-savings–portfolio choice model in which the consumer-investor maximizes the expected value of a non-additively separable utility function of current and future consumption. Preferences for consumption streams are CES and the elasticity of substitution can be chosen independently of the risk aversion measure. The additively separable case is a special case. Because choices are not dynamically consistent, we follow the “consistent planning” approach of Strotz [Strotz, R., 1956. Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies 23, 165–180] and also interpret our analysis from the game theoretic perspective taken by Peleg and Yaari [Peleg, B., Yaari, M., 1973. On the existence of a consistent course of action when tastes are changing. Review of Economic Studies 40, 391–401]. The equilibrium of the Lucas asset pricing model with i.i.d. consumption growth is obtained and the equity premium is shown to depend on the elasticity of substitution as well as the risk aversion measure. The nature of the dependence is examined. Our results are contrasted with those of the non-expected utility recursive approach of Epstein–Zin and Weil.     

Quadratic hedging schemes for non-Gaussian GARCH models

We propose different schemes for option hedging when asset returns are modeled using a general class of GARCH models. More specifically, we implement local risk minimization and a minimum variance hedge approximation based on an extended Girsanov principle that generalizes Duan׳s (1995) delta hedge. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization hedging strategies with respect to a pricing kernel. These approaches are investigated in the context of non-Gaussian driven models. Furthermore, we analyze these methods for non-Gaussian GARCH diffusion limit processes and link them to the corresponding discrete time counterparts. A detailed numerical analysis based on S&P 500 European call options is provided to assess the empirical performance of the proposed schemes. We also test the sensitivity of the hedging strategies with respect to the risk neutral measure used by recomputing some of our results with an exponential affine pricing kernel.     

Derivative trading by utility firms

This paper examines the use of derivatives by a utility company. The hedging problem for utilities is atypical; the goal is not strictly to minimize average costs. Rather, the objectives are to minimize the upside risk associated with extreme bills, volatility of bills, and average expected bills for consumers. We characterize the optimal positions on futures contracts and options on futures that a utility company should assume. The results indicate that the use of derivatives (both futures and options on futures) is an efficient means of optimizing the objective functions without exposing consumers to speculative risk.     

Mean-variance hedging for interest rate models¶with stochastic volatility

The mean-variance hedging approach for pricing and hedging claims in incomplete markets was originally introduced for risky assets. The aim of this paper is to apply this approach to interest rate models in the presence of stochastic volatility, seen as a consequence of incomplete information. We fix a finite number of bonds such that the volatility matrix is invertible and provide an explicit formula for the density of the variance-optimal measure which is independent of the chosen times of maturity. Finally, we compute the mean-variance hedging strategy for a caplet and compare it with the optimal stategy according to the local risk minimizing approach. Received: 14 July 2000 / Accepted: 10 April 2001     

Generalized affine transform on pricing quanto range accrual note

This paper was to price and hedge a quanto floating range accrual note (QFRAN) by an affine term structure model with affine-jump processes. We first generalized the affine transform proposed by Duffie et al. (2000) under both the domestic and foreign risk-neutral measures with a change of measure, which provides a flexible structure to value quanto derivatives. Then, we provided semi-analytic pricing and hedging solutions for QFRAN under a four-factor affine-jump model with the stochastic mean, stochastic volatility, and jumps. The numerical results demonstrated that both the common and local factors significantly affect the value and hedging strategy of QFRAN. Notably,  the factor of stochastic mean plays the most important role in either valuation or hedging. This study suggested that ignorance of these factors in a term-structure model will result in significant pricing and hedging errors in QFRAN. In summary, this study provided flexible and easily implementable solutions in valuing quanto derivatives.     

Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products

Guaranteed Minimum Withdrawal Benefits (GMWB) are popular riders in variable annuities with withdrawal guarantees. With withdrawals spread over the life of the annuities contract, the benefit promises to return the entire initial annuitization amount irrespective of the market performance of the underlying fund portfolio. Treating the dynamic withdrawal rate as the control variable, the earlier works on GMWB have considered the construction of a continuous singular stochastic control model and the numerical solution of the resulting pricing model. This paper presents a more detailed characterization of the pricing properties of the GMWB and performs a full mathematical analysis of the optimal dynamic withdrawal policies under the competing factors of time value of fund, optionality value provided by the guarantee and penalty charge on excessive withdrawal. When a proportional penalty charge is applied on any withdrawal amount, we can reduce the pricing formulation to an optimal stopping problem with lower and upper obstacles. We then derive the integral equations for the determination of a pair of optimal withdrawal boundaries. When a proportional penalty charge is applied on the amount that is above the contractual withdrawal rate, we manage to characterize the behavior of the optimal withdrawal boundaries that separate the domain of the pricing models into three regions: no withdrawal, continuous withdrawal at the contractual rate and an immediate withdrawal of a finite amount. Under certain limiting scenarios such as a high policy fund value, the time close to expiry, or a low value of guarantee account, we manage to obtain analytical approximate solution to the singular stochastic control model of dynamic withdrawals.     

No arbitrage conditions and liquidity

We extend the fundamental theorem of asset pricing to the case of markets with liquidity risk. Our results generalize, when the probability space is finite, those obtained by Kabanov et al. [Kabanov, Y., Stricker, C., 2001. The Harrison-Pliska arbitrage pricing theorem under transaction costs. Journal of Mathematical Economics 35, 185–196; Kabanov, Y., Rásonyi, M., Stricker, C., 2002. No-arbitrage criteria for financial markets with efficient friction. Finance and Stochastics 6, 371–382; Kabanov, Y., Rásonyi, M., Stricker, C., 2003. On the closedness of sums of convex cones in L⁰$L^0$ and the robust no-arbitrage property. Finance and Stochastics] and by Schachermayer [Schachermayer, W., 2004. The fundamental theorem of asset pricing under poportional transaction costs in finite discrete time. Mathematical Finance 14 (1), 19–48] for markets with proportional transaction costs. More precisely, we restate the notions of consistent and strictly consistent price systems and prove their equivalence to corresponding no arbitrage conditions. We express these results in an analytical form in terms of the subdifferential of the so-called liquidation function. We conclude the paper with a hedging theorem.     

Multi-player dynamic game model for Bitcoin transaction bidding prediction

With the rapid rise of cryptocurrencies, it has become an urgent problem to realize the flat use of digital currency, with making it really put into use, and giving full play to its utility in the current economic market. This paper innovatively takes the maximization of user benefit as the key point to predict transaction bidding price combining dynamic game theory. The bidding price of user transaction not only refers to historical transactions, but also considers the impact on future subsequences, and the result describes the interaction between transactions in detail. Also this paper proposes a method to express user satisfaction and establishes a user benefit model accordingly, so as to ensure the transaction is packaged successfully to the greatest extent within the acceptable range of transaction pricing. Finally this paper compares the proposed model with conventional machine learning prediction algorithms, finding that when user does not participate in the trading for the first time, the prediction effect of this proposal is better than that of machine learning over small data sets, moreover superior to machine learning methods in prediction accuracy and sensitivity, with a lower time complexity.     

Some Recent Developments in Futures Hedging

The use of futures contracts as a hedging instrument has been the focus of much research. At the theoretical level, an optimal hedge strategy is traditionally based on the expected–utility maximization paradigm. A simplification of this paradigm leads to the minimum–variance criterion. Although this paradigm is quite well accepted, alternative approaches have been sought. At the empirical level, research on futures hedging has benefited from the recent developments in the econometrics literature. Much research has been done on improving the estimation of the optimal hedge ratio. As more is known about the statistical properties of financial time series, more sophisticated estimation methods are proposed. In this survey we review some recent developments in futures hedging. We delineate the theoretical underpinning of various methods and discuss the econometric implementation of the methods.     

Discussion of dynamic programming and linear programming approaches to stochastic control and optimal stopping in continuous time

This paper seeks to highlight two approaches to the solution of stochastic control and optimal stopping problems in continuous time. Each approach transforms the stochastic problem into a deterministic problem. Dynamic programming is a well-established technique that obtains a partial/ordinary differential equation, variational or quasi-variational inequality depending on the type of problem; the solution provides the value of the problem as a function of the initial position (the value function). The other method recasts the problems as linear programs over a space of feasible measures. Both approaches use Dynkin’s formula in essential but different ways. The aim of this paper is to present the main ideas underlying these approaches with only passing attention paid to the important and necessary technical details.