Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.     

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.     

Approximate arbitrage-free option pricing under the SABR model

The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. (2002) provides a popular vehicle to model the implied volatilities in the interest rate and foreign exchange markets. To exclude arbitrage opportunities, we need to specify an absorbing boundary at zero for this model, which the existing analytical approaches to pricing derivatives under the SABR model typically ignore. This paper develops closed-form approximations to the prices of vanilla options to incorporate the effect of such a boundary condition. Different from the traditional normal distribution-based approximations, our method stems from an expansion around a one-dimensional Bessel process. Extensive numerical experiments demonstrate its accuracy and efficiency. Furthermore, the explicit expression yielded from our method is appealing from the practical perspective because it can lead to fast calibration, pricing, and hedging.     

Option Contracts and Vertical Foreclosure

A model of vertical integration is studied. Upstream firms sell differentiated inputs; downstream firms bundle them to make final products. Downstream products are sold as option contracts, which allow consumers to choose from a set of commodities at predetermined prices. The model is illustrated by examples in telecommunication and health markets. Equilibria of the integration game must result in upstream input foreclosure and downstream monopolization. Consumers may or may not benefit from integration.     

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.     

The behaviour of betting and currency markets on the night of the EU referendum

We study the behaviours of the Betfair betting market and the sterling/dollar exchange rate (futures price) during 24 June 2016, the night of the EU referendum. We investigate how the two markets responded to the announcement of the voting results by employing a Bayesian updating methodology to update prior opinion about the likelihood of the final outcome of the vote. We then relate the voting model to the real-time evolution of the market-determined prices as the results were announced. We find that, although both markets appear to be inefficient in absorbing the new information contained in the vote outcomes, the betting market seems less inefficient than the FX market. The different rates of convergence to the fundamental value between the two markets lead to highly profitable arbitrage opportunities.     

Carbon option price forecasting based on modified fractional Brownian motion optimized by GARCH model in carbon emission trading

With the rapid growth of carbon trading, the development of carbon financial derivatives such as carbon options has become inevitable. This paper established a model based on GARCH and fractional Brownian motion (FBM), hoping to provide reference for China's upcoming carbon option trading through carbon option price forecasting research. The fractal characteristic of carbon option prices indicates that it is reasonable to use FBM to predict option prices. The GARCH model can make up for the lack of fixed FBM volatility. In this paper, the daily closing prices of EUA option contracts on the European Energy Exchange are selected as samples for price prediction. The GARCH model was used to determine the return volatility, and then the FBM was used to calculate the forecast price for the next 60 days. The results showed that the predicted price can better fit the actual price. This paper further compares the price prediction results of this model with the other three models through line graphs and error evaluation indicators such as MAPE, MAE and MSE. It is confirmed that the prediction results of the model in this paper is the closest to the actual price.     

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.     

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.     

存货购销中最优期权合约交易量研究

在分析利用期权合约规避价格波动风险的原理的基础上,分别给出存货购销两个环节中可以运用的期权策略,然后利用均值方差模型计算使投资组合达到效用最大化时所对应的最优期权合约交易量及其对经营利润的影响,研究发现：在存货采购环节,企业可以通过购入看涨期权、购入看涨期权同时售出看跌期权两种策略控制采购价格波动的风险,在存货销售环节,企业可以通过购入看跌期权、同时购入看跌期权并售出看涨期权两种策略来稳定销售利润;从最优期权合约交易量及其对企业经营利润的影响来看,期权工具在控制存货采购价格、稳定销售利润中可以发挥良好作用。     

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.     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

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.     

Keeping a weather eye on prediction markets: The influence of environmental conditions on forecasting accuracy

Increasingly, prediction markets are being embraced as a mechanism for eliciting and aggregating dispersed information and providing a means of deriving probabilistic forecasts of future uncertain events. The efficient market hypothesis postulates that prediction market prices should incorporate all information that is relevant to the performances of the contracts traded. This paper shows that such may not be the case in relation to information regarding environmental factors such as the weather and atmospheric conditions. In the context of horserace betting markets, we demonstrate that even after the effects of these factors on the contestants (horses and jockeys) have been discounted, the accuracy of the probabilities derived from market prices is affected systematically by the prevailing weather and atmospheric conditions. We show that significantly better forecasts can be derived from prediction markets if we correct for this phenomenon, and that these improvements have substantial economic value.     

The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange,stock, and bond markets

We study the forecasting of future realized volatility in the foreign exchange, stock, and bond markets from variables in our information set, including implied volatility backed out from option prices. Realized volatility is separated into its continuous and jump components, and the heterogeneous autoregressive (HAR) model is applied with implied volatility as an additional forecasting variable. A vector HAR (VecHAR) model for the resulting simultaneous system is introduced, controlling for possible endogeneity issues. We find that implied volatility contains incremental information about future volatility in all three markets, relative to past continuous and jump components, and it is an unbiased forecast in the foreign exchange and stock markets. Out-of-sample forecasting experiments confirm that implied volatility is important in forecasting future realized volatility components in all three markets. Perhaps surprisingly, the jump component is, to some extent, predictable, and options appear calibrated to incorporate information about future jumps in all three markets.     

Convex incentives in financial markets: an agent-based analysis

We investigate whether convex incentive contracts are a source of instability of financial markets as indicated by the results of a continuous double-auction asset market experiment performed by Holmen et al. (J Econ Dyn Control 40:179–194, 2014). We develop a model to replicate the setting of the experiment and perform an agent-based simulation where agents have linear or convex incentives. Extending the simulation by varying features of actual asset markets that were not studied in the experiment, our main results show that increasing the number of convex incentive contracts increases prices and volatility and decreases market liquidity, measured both as bid–ask spreads and volumes. We also observe that the influence of risk aversion on traders’ decisions decreases when there are convex contracts and that increasing the differences in initial wealth among the traders has similar effects as increasing number of convex incentive contracts.     

The spatial impact of employment centres on housing markets

The spatial impact of employment centres on housing markets. Spatial Economic Analysis. Local economic growth tends to affect neighbourhood house prices unevenly. It has been observed that prime locations experience price hikes far in excess of the surrounding local area. Yet, this phenomenon is not well captured by existing economic models. This research provides a model of spatial and temporal interactions between housing and employment markets. The results show that rapid growth of employment centres increases house prices in neighbouring locations even after adjusting for fundamentals. It is concluded that spatial clustering of companies creates an option value for existing and potential employees that goes beyond ease of access for commuting purposes.     

Consistent pricing of VIX options with the Hawkes jump-diffusion model

This paper presents a valuation of VIX options employing a Hawkes jump-diffusion model that captures the clustering pattern of jumps observed extensively in the financial markets. In the consistent framework, the valuation problem of VIX options is solved efficiently via the Fourier cosine expansion (COS) method. The Monte Carlo (MC) simulations are carried out to demonstrate the reliability and efficiency of the COS method. Furthermore, a sensitivity analysis is performed to show how option prices response to different parameters associated with jump clustering. Finally, empirical studies are conducted to provide evidence to support our jump specification in matching the VIX option surface.     

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.     

Collateral amplification under complete markets

This paper examines the robustness of the Kiyotaki–Moore collateral amplification mechanism to the existence of complete markets for aggregate risk. We show that, when borrowers can hedge against aggregate shocks at fair prices, the volatility of endogenous variables becomes identical to the first best in the absence of credit constraints. The collateral amplification mechanism disappears.To motivate the limited use of contingent contracts, we introduce costs of issuing contingent debt and calibrate them to match the liquidity and safety premia the data. We find that realistic costs of state contingent market participation can rationalize the predominant use of uncontingent debt. Amplification is restored in such an environment.