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.     

Valuing lookback options with barrier

In this paper, we introduce a new class of exotic options, termed lookback-barrier options, which literally combine lookback and barrier options by incorporating an activating barrier condition into the European lookback payoff. A prototype of lookback-barrier option was first proposed by Bermin (1998), where he intended to reduce the expensive cost of lookback option by considering lookback options with barrier. However, despite his novel trial, it has not attracted much attention yet. Thus, in this paper, we revisit the idea and extend the horizon of lookback-barrier option in order to enhance the marketability and applicability to equity-linked investments. Devising a variety of payoffs, this paper develops a complete valuation framework which allows for closed-form pricing formulas under the Black–Scholes model. Our closed-form pricing formulas provide a substantial advantage over the method of Monte Carlo simulation, because the extrema appearing in both of the lookback payoff and barrier condition would require a large number of simulations for exact calculation. Complexities involved in the derivation process would be resolved by the Esscher transform and the reflection principle of the Brownian motion. We illustrate our results with numerical examples.     

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

In this paper, an analytical approximation formula for pricing European options is obtained under a newly proposed hybrid model with the volatility of volatility in the Heston model following a Markov chain, the adoption of which is motivated by the empirical evidence of the existence of regime-switching in real markets. We first derive the coupled PDE (partial differential equation) system that governs the European option price, which is solved with the perturbation method. It should be noted that the newly derived formula is fast and easy to implement with only normal distribution function involved, and numerical experiments confirm that our formula could provide quite accurate option prices, especially for relatively short-tenor ones. Finally, empirical studies are carried out to show the superiority of our model based on S&P 500 returns and options with the time to expiry less than one month.     

A study on the bursting point of Bitcoin based on the BSADF and LPPLS methods

We aim to reveal the characteristics and mechanism of the Bitcoin bubble in 2019. First, we identify the period during which two important Bitcoin bubbles occurred based on the generalized supremum augmented Dickey-Fuller (GSADF) method. There are two significant bubble cycles. The first bubble lasted approximately 26 days from November 25, 2017, to December 21, 2017, while the second bubble lasted approximately one week from June 22 to June 29, 2019. The occurrence of the first bubble was related to the considerable expansion of initial coin offerings (ICOs) in 2017, while the formation of the second bubble was affected by the release of Libra. Second, as the GSADF method cannot be used to accurately infer the time at which a bubble bursts, we employ the log-periodic power law singularity (LPPLS) model for this purpose. We verify that the LPPLS method can not only infer the timing of a bubble burst but also shows stable results. Finally, we demonstrate the implications of the 2019 bubble. During the 2019 bubble, due to the increased supervision of European and American governments and the impact of hedging assets, the bubble’s duration was shorter, and the positive feedback mechanism was not as strong as that of the 2017 bubble. In addition, the oscillating frequency of the bubble in 2019 was low and unstable, which means that it would be more beneficial for investors to hold the currency for a long time.     

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.     

Calibration of stochastic volatility models: A Tikhonov regularization approach

We aim to calibrate stochastic volatility models from option prices. We develop a Tikhonov regularization approach with an efficient numerical algorithm to recover the risk neutral drift term of the volatility (or variance) process. In contrast to most existing literature, we do not assume that the drift term has any special structure. As such, our algorithm applies to calibration of general stochastic volatility models. An extensive numerical analysis is presented to demonstrate the efficiency of our approach. Interestingly, our empirical study reveals that the risk neutral variance processes recovered from market prices of options on S&P 500 index and EUR/USD exchange rate are indeed linearly mean-reverting.     

Probability of multiple crossings and pricing of double barrier options

This paper derives pricing formulas of standard double barrier option, generalized window double barrier option and chained option. Our method is based on probabilitic approach. We derive the probability of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. The price of a standard double barrier option is presented as an infinite sum that converges very rapidly. Although the price formula of standard double barrier option is the same with Kunitomo and Ikeda (1992), our method gives an intuitive interpretation for each term in the infinite series. From the intuitive interpretation we present the way how to approximate the infinite sum in the pricing formula and an error bound for the given approximation. Guillaume (2003) and Jun and Ku (2013) assumed that barriers are constant to price barrier options. We extend constant barriers of window double barrier option and chained option to curved barriers. By employing multiple crossing probabilities and previous skills we derive closed formula for prices of 16 types of the generalized chained option. Based on our analytic formulas we compute Greeks of chained options directly.     

Tradability,closeness to market prices,and expected profit: their measurement for a binomial model of options pricing in a heterogeneous market

A reliable method of options pricing in real time would help various players, including hedgers and speculators, to make informed decisions. In this study, we develop an extensive simulation with multiple business environments, which includes the use of real data from the S&P 500 Index between the years 2010–2017 for the 30 days prior to expiration of the options. Forecasted tradability is computed based on the SH model: a theoretical model of real-time options pricing that takes into account players’ heterogeneity with regard to their willingness to accept offers proposed by the opposing player. The quality of the model is examined for the scenario in which the model players are speculators who act against the real market prices. We show that the equilibrium prices predicted by the SH model are close to the market prices (a deviation of up to approx. 3%) in an In-The-Money environment. Additionally, the tougher the players (i.e., the greater their level of unwillingness to accept a bid from the opposing player), the higher the average tradability. We also find that the level of willingness of the players has a greater effect on tradability than does option moneyness or the market trend.

     

Price support and spreads in the IPO aftermarket: An empirical microstructure study

Using the inventory components of spreads as a measure of inventory holding-risk, we test the hypothesis of Hanley et al. [Hanley, K. W., Kumar, A., & Seguin, P. J. (1993). Price stabilization in the market for new issues. Journal of Financial Economics, 34, 177–197] that price supports reduce market makers’ inventory holding-risk in the aftermarket of initial public offerings (IPOs). We find that both spreads and their inventory components are significantly smaller in the earlier periods of the IPO aftermarket than those in the later periods. More importantly, the inventory components of spreads are significantly smaller for stocks without over-allotment options (OAOs) exercised, and for stocks with lower or negative initial returns which are more likely to have price supports. The results are consistent with the price support hypothesis.     

Pricing perpetual options with stochastic discount interest rates

The aim of this study is to investigate the prices and optimal exercise strategies of certain perpetual American options on the asset under stochastic discount interest model. This is different from Gerber and Shiu (N Am Actuar J 2(3):101–112, 1998), in which the interest force was a constant; here we suppose that the accumulated interest function is perturbed by the standard Brownian motion and Poisson process. We obtain an explicit expression of optimal option-exercise boundary in the case of perpetual put option. Moreover, we get a corresponding result when the individual claim is described by an exponential distribution. Finally, we analyze the influence of certain coefficients in stochastic interest model on the optimal option-exercise boundary.     

Volatility smiles when information is lagged in prices

This study explores volatility smiles when stock market information is lagged, specifically in the REIT industry. A usual requirement is that REITs can only disseminate information relating to their property valuations once per year; therefore, this leads to the lagging effect. Within the context of exchange options (i.e. mergers), it seems that no study has researched on this theme. This article uses the Black & Scholes model to calculate implied volatilities and their corresponding implied options to illustrate arbitrage opportunities when exchange options emerge. The results illustrate that implied volatilities are different from non-implied volatilities. Further, arbitrage is still higher among REITs as opposed to other capital market instruments. Finally, just like other capital market instruments, REIT acquisitions generate alpha.     

Extracting market information from equity options with exponential Lévy processes

Lévy processes have been successfully applied in the modeling of financial assets. Useful information such as implied volatility, skewness, and risk-preferences can be derived from market option prices. In this paper, we advocate using Esscher conjugate Lévy processes to estimate risk-neutral and empirical densities. More specifically, we employ the exponential Meixner and NIG processes to calculate in closed form the pricing kernel in the equity market and then study the evolution of equity market behavior between 2002 and 2010. Our empirical analysis using S&P 500 options shows that the risk preferences of equity investors were signalling an anomaly in the market well before the subprime prime mortgage crisis (August 2007) and the crisis of confidence that followed, anticipating the downfall in equity markets in 2008, but then returning to normal levels in 2009.     

Pricing American barrier options with discrete dividends by binomial trees

We are concerned with the problem of pricing plain-vanilla and barrier options with cash dividends in a piecewise lognormal model. In the plain-vanilla case, we offer a method with provides thin upper and lower bounds of the exact binomial price. In the barrier case, we provide an efficient algorithm based on suitable interpolation techniques. As by-product, we provide a new method for pricing American barrier options with continuous dividends.     

Leverage effect on stochastic volatility for option pricing in Hong Kong: A simulation and empirical study

This paper explores the importance of incorporating the financial leverage effect in the stochastic volatility models when pricing options. For the illustrative purpose, we first conduct the simulation experiment by using the Markov Chain Monte Carlo (MCMC) sampling method. We then make an empirical analysis by applying the volatility models to the real return data of the Hang Seng index during the period from January 1, 2013 to December 31, 2017. Our results highlight the accuracy of the stochastic volatility models with leverage in option pricing when leverage is high. In addition, the leverage effect becomes more significant as the maturity of options increases. Moreover, leverage affects the pricing of in-the-money options more than that of at-the-money and out-of-money options. Our study is therefore useful for both asset pricing and portfolio investment in the Hong Kong market where volatility is an inherent nature of the economy.     

A moments and strike matching binomial algorithm for pricing American Put options

This paper is dedicated to a new binomial lattice method called Moments and Strike Matching (MSM) consistent with the Black–Scholes model in the limit of an infinite step number and such that the Strike K is equal to one of the final nodes of the tree. The method is very easy to implement, since the parameters are explicitly given. Asymptotic expansions are obtained for the MSM European Put price and delta, which motivates the use of Richardson extrapolation. A numerical comparison with the best lattice based numerical methods known in literature, shows the efficiency of the proposed algorithm for pricing and hedging American Put options.      

分数布朗运动环境下的美式看涨期权的定价方法

在分数布朗运动模拟算法基础上,提出了分数布朗运动环境下标的资产价格过程的一种数值模拟方法。然后应用于欧式和美式看涨期权定价。结果表明,该方法具有很快的收敛速度,而且基于最小二乘方法和偏最小二乘方法的美式看涨期权价格,都与对应的欧式看涨期权价格几乎完全一样。这恰恰验证了不支付红利的条件下,美式看涨期权不应该提前执行的理论论断。     

Pricing VIX options with stochastic volatility and random jumps

This study presents an analytical exact solution for the price of VIX options under stochastic volatility model with simultaneous jumps in the asset price and volatility processes. We shall demonstrate that our new pricing formula can be used to efficiently compute the numerical values of a VIX option. While we also show that the numerical results obtained from our formula consistently match those obtained from Monte Carlo simulation perfectly as a verification of the correctness of our formula, numerical evidence is offered to illustrate that the correctness of the formula proposed in Lin and Chang (J Futur Markets 29(6), 523–543, 2009) is in serious doubt. Moreover, some important and distinct properties of VIX options (e.g., put-call parity, hedging ratios) are also examined and discussed.     

Environmental regulations in private and mixed duopolies: Taxes on emissions versus green R&D subsidies

In this article, we compare two kinds of environmental regulations—emissions taxes and green R&D subsidies—in private and mixed-duopoly markets in the presence of R&D spillovers. We show that a green R&D subsidy is better (worse) than an emissions tax when the green R&D is efficient (inefficient), irrespective of R&D spillovers, whereas the existence of a publicly owned firm encourages the government to adopt a subsidy policy. We also show that the optimal policy choice depends on R&D efficiency and spillovers. In particular, when green R&D is inefficient and the spillover rate is low (high), the government should choose an emissions tax and (not) privatize the state-owned firm. When green R&D is efficient, however, an R&D subsidy is better, but a privatization policy is not desirable for society, irrespective of spillovers.     

Valuation of piecewise linear barrier options

This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.     

期权供应合约下的分销商两阶段式最优订货决策

本文引入期权机制,建立具有一定柔性的两阶段式订购模型;讨论分销商第二阶段的决策,即分销商如何选择期权的执行数量;解决了期权供应合约中一个重要的决策问题。最后,分析了最优期权执行量与四个相关因素的关系。这四个因素为:分销商单位缺货成本,分销商出售产品的单价,期权的执行价格,以及期末分销商手中过剩成品的单位残值。