Correction to: Semi-analytical prices for lookback and barrier options under the Heston model

Decisions in Economics and Finance - In this note, we point out a mistake in Theorem 1 of De De Gennaro Aquino and Bernard (Decis Econ Finance 42(2):715–741, 2019) and provide some missing...     

The valuation of barrier options under a threshold rough Heston model

In this paper, we propose a novel model for pricing double barrier options, where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process, which is driven by the convolution of a fractional kernel with the CIR process. The new model both captures the leverage effect and produces rough paths for the volatility process. The model also nests the threshold diffusion, Heston and rough Heston models. We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method. We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.     

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.     

On pricing lookback options under the CEV process

Abstract We consider the problem of pricing European lookback options when the underlying asset price is driven by a constant elasticity of variance (CEV) process. The evaluation model is based on the binomial approximation developed by Nelson and Ramaswamy (1990) and we show how to apply it in the case of such options. We develop simple pricing algorithms that compute accurate estimates of the option prices.     

The pricing of lookback options and binomial approximation

Refining a discrete model of Cheuk and Vorst, we obtain a closed formula for the price of a European lookback option at any time between emission and maturity. We derive an asymptotic expansion of the price as the number of periods tends to infinity, thereby solving a problem posed by Lin and Palmer. We prove, in particular, that the price in the discrete model tends to the price in the continuous Black–Scholes model. Our results are based on an asymptotic expansion of the binomial cumulative distribution function that improves several recent results in the literature.     

Pricing external barrier options in a regime-switching model

External barrier options are two-asset options where the payoff is defined on one asset and the barrier is defined on another asset. In this paper, we derive the Laplace transforms of the prices and deltas for the external single and double barrier options where the underlying asset prices follow a regime-switching model with finite regimes. The derivation is made possible because we can obtain the joint Laplace transform of the first passage time of one asset value and the value of the other asset. Numerical inversion of the Laplace transforms is used to calculate the prices of external barrier options.     

Moment explosions in the rough Heston model

Decisions in Economics and Finance - We show that the moment explosion time in the rough Heston model, introduced by El Euch and Rosenbaum in 2016, is finite if and only if it is finite for the...     

Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff

Abstract We analyze the Galerkin infinite element method for pricing European barrier options and, more generally, options with discontinuous payoff. The infinite element method is a simple and efficient modification of the more common finite element method. It keeps the best features of finite elements, i.e., bandedness, ease of programming, accuracy. Three main aspects are considered: (i) the degeneracy of the pricing PDE models at hand; (ii) the presence of discontinuities at the barriers or in the payoff clause and their effects on the numerical approximation process; (iii) the need for resorting to suitable numerical methods for unbounded domains when appropriate asymptotic conditions are not specified. The numerical stability and convergence of the proposed method are proved. Mathematics Subject Classification (2000): 65N30, 65J10 Journal of Economic Literature Classification: G13, C63     

Valuation of options on the maximum of two prices with default risk under GARCH models

In this paper, we work under GARCH models to value options on the maximum or the minimum of two prices. In addition, we consider not only two underlying asset prices but also geometric average ones. Further, default risk is also incorporated in a reduced-form model. In the proposed framework, closed-form pricing formulae of options on the maximum with or without default risk are derived and then used to perform numerical examples.     

A discrete-time algorithm for pricing double barrier options

Decisions in Economics and Finance -     

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α-Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.     

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.     

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.

     

The values and incentive effects of options on the maximum or the minimum of the stock prices and market index

In this study, we employ the certainty equivalent principle to investigate cost efficiency and incentives of the options on the maximum or the minimum of the stock prices and market index levels. In addition, the options with averaging features are also considered. Numerical results show that options on the maximum are more cost efficient and incentive-efficient than traditional ones. As for options on the minimum, they are more cost efficient than traditional ones only when the weight in the options is not very large. However, options on the minimum also provide stronger incentives to increase stock prices than traditional ones.     

A spatio-temporal model of house prices in the USA

This paper provides an empirical analysis of changes in real house prices in the USA using State level data. It examines the extent to which real house prices at the State level are driven by fundamentals such as real per capita disposable income, as well as by common shocks, and determines the speed of adjustment of real house prices to macroeconomic and local disturbances. We take explicit account of both cross-sectional dependence and heterogeneity. This allows us to find a cointegrating relationship between real house prices and real per capita incomes with coefficients (1,−1)$(1,-1)$, as predicted by the theory. We are also able to identify a significant negative effect for a net borrowing cost variable, and a significant positive effect for the State level population growth on changes in real house prices. Using this model we then examine the role of spatial factors, in particular, the effect of contiguous states by use of a weighting matrix. We are able to identify a significant spatial effect, even after controlling for State specific real incomes, and allowing for a number of unobserved common factors. We do, however, find evidence of departures from long run equilibrium in the housing markets in a number of States notably California, New York, Massachusetts, and to a lesser extent Connecticut, Rhode Island, Oregon and Washington State.     

Prediction market prices under risk aversion and heterogeneous beliefs

In this paper, we examine the properties of prediction market prices when risk averse traders have heterogeneous beliefs in state probabilities. We show that the equilibrium state prices equal the mean beliefs of traders about that state if and only if the traders’ common utility function is logarithmic. We also provide a necessary and sufficient condition ensuring that the state prices are systematically below or above the mean beliefs of traders, thus providing a rational explanation to the favorite-longshot bias in prediction markets.     

A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge

Barrier options based upon the extremum of more than one underlying prices do not allow for closed-form pricing formulas, and thus require numerical methods to evaluate. One example is the autocallable structured product with knock-in feature, which has gained a great deal of popularity in the recent decades. In order to increase numerical efficiency for pricing such products, this paper develops a semi-analytic valuation algorithm which is free from the computational burden and the monitoring bias of the crude Monte Carlo simulation. The basic idea is to combine the simulation of the underlying prices at certain time points and the exit (or non-exit) probability of the Brownian bridge. In the literature, the algorithm was developed to deal with a single-asset barrier option under the Black–Scholes model. Now we extend the framework to cover two-asset barrier options and autocallable product. For the purpose, we explore the non-exit probability of the two-dimensional Brownian bridge, which has not been researched before. Meanwhile, we employ the actuarial method of Esscher transform to simplify our calculation and improve our algorithm via importance sampling. We illustrate our algorithm with numerical examples.     

2007年第一季度 城市地价动态监测报告

根据中国城市地价动态监测系统对全国重点地区和主要城市监测结果的分析，2007年第一季度我国主要城市地价状况如下——