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.     

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.     

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.     

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.     

Options and earnings announcements: an empirical study for the European Options Exchange

In this paper we give an introduction in option pricing theory and explicitly specify the Black-Scholes model. Although market participants use this and similar models to price options, they violate one of the fundamental assumptions of the model. They do not set a constant value for the volatility of the underlying asset over time, but change the volatility even during a day. By means of event study methodology we investigate the volatility of the underlying asset and the volatility implicit in option prices around earnings announcements by firms. We find that the volatility in option prices increases before the announcement date and drops sharply afterwards. The volatility of the underlying stocks is higher only at the announcement dates and we do not observe a higher volatility around these dates. Hence, the constant volatility of the underlying asset, which is one of the assumptions in the Black-Scholes model, does not hold. However, the market seems to correctly anticipate the change in volatility, by correcting option prices.     

Exchange options for catastrophe risk management

In this paper, we investigate the pricing issue and catastrophe risk management of exchange options. Exchange options allow the holder to exchange its stocks for another at maturity and can be seen as an extended version of catastrophe equity put options with another traded asset price as strike prices. Since option holders have to issue new shares to exercise the option, we illustrate the differences between option prices calculated using pre-exercise and post-exercise share prices. The effects of default risk on option prices and risk management are also considered. Finally, risk management analysis shows that exchange options can effectively hedge catastrophe risk.     

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.     

Multi-step barrier products and static hedging

This paper examines multi-step barrier options with an arbitrary payoff function using extended static hedging methods. Although there have been studies using extended reflection principles to obtain joint distribution functions for barrier options with complex barrier conditions, and static hedging methods to evaluate limited barrier options with well-known payoff functions, we obtain an explicit expression of barrier option price which has a general payoff function under the Black–Scholes framework assumption. The explicit multi-step barrier options prices we discuss in this paper are not only useful in that they can handle different levels and time steps barrier and all types of payoff functions, but can also extend to pricing of barrier options under finite discrete jump–diffusion models with a simple barrier. In the last part, we supplement the theory with numerical examples of various multi-step barrier options under the Black–Scholes or discrete jump–diffusion model for comparison purposes.     

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.     

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.     

Dividend paying assets,the unit root property,and suboptimality

We consider the stationary equilibria of one good overlapping generations (OLG) economies with a sequence of possibly incomplete asset markets and prove two results. First, we show that if some asset always pays a nonnegative dividend, then its price changes sign across states if and only if the Perron root of every agent’s matrix of intertemporal rates of substitution exceeds one. Second, we provide sufficient conditions in terms of dividends and asset prices such that, keeping asset prices fixed, a conditionally Pareto improving allocation is induced by a stationary reassignment of a single asset. When taken together the results show that when for some agent the Perron root exceeds one, the existence of an asset that pays a strictly positive dividend in every state is sufficient to induce an improvement.     

Impact of volatility jumps in a mean-reverting model: Derivative pricing and empirical evidence

This paper investigates the critical role of volatility jumps under mean reversion models. Based on the empirical tests conducted on the historical prices of commodities, we demonstrate that allowing for the presence of jumps in volatility in addition to price jumps is a crucial factor when confronting non-Gaussian return distributions. By employing the particle filtering method, a comparison of results drawn among several mean-reverting models suggests that incorporating volatility jumps ensures an improved fit to the data. We infer further empirical evidence for the existence of volatility jumps from the possible paths of filtered state variables. Our numerical results indicate that volatility jumps significantly affect the level and shape of implied volatility smiles. Finally, we consider the pricing of options under the mean reversion model, where the underlying asset price and its volatility both have jump components.     

Monetary policy,asset prices and consumption in China

This paper studies the wealth channel in China. Although the wealth channel has been found to be functioning in many advanced countries, its existence is yet to be explored in most emerging economies, also in China. In order to illuminate dynamics between monetary policy, asset prices and consumption, we use the structural vector autoregression method. The findings support the view that a loosening of China's monetary policy does indeed lead to higher asset prices. Furthermore, a positive shock to residential prices increases household consumption, while the role of stock prices seems to be small from the households’ point of view. Finally, we test the existence of the wealth channel more formally to find out whether those changes in asset prices that are caused by monetary policy are significant enough to increase consumption. In summary, the wealth channel remains weak but there are some signs of it via residential prices. The results are not that different from those attained for the advanced economies, where the size of the wealth channel has been found to be limited.     

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.     

Heterogeneous beliefs with herding behaviors and asset pricing in two goods world

We construct a incomplete information equilibrium model with heterogeneous beliefs and herding behaviors to identify their joint effects on the dynamics of asset prices. Herding behaviors make investors revise some of their estimations about expected growth rates of goods streams toward to the other one’s by a manner of weighted average of their own forecast and the other’s. As we expected, herding behaviors generate influences on the Radon Nikodym derivative, that is so-called “sentiment” as in Dumas et al. (2009), and in turn not only impact the dynamics of asset prices but also generate influences on investors’ survivals. We also show that introducing heterogeneous beliefs with herding behaviors permits to explain both the Backus–Smith puzzle and the mixed results about the influences of herding behaviors on asset prices. Moreover, we uncover that herding behaviors have positive influences on stocks’ risk premiums.     

Asset price,risk transfer and economic activities: Firm-level evidence from China

The paper investigates the extent of the impact from changes in asset price and risk on corporate investment behaviors as well as the real economy. The results support the unidirectional causality effects from asset price fluctuations on the macro-level. By applying quarterly data of Chinese listed companies, we further find the existence of balance sheet effect on the firm-level, which suggests that the changes in asset prices and risk affect the net asset value, and consequently influence corporate investment decisions. More importantly, the balance sheet effect appears to be much more significant after the implementation of new fair value accounting standards in 2007. The impact on the real economy from asset price risk is found to be more prominent as well.     

Evaluation of American option prices in a path integral framework using Fourier–Hermite series expansions

In this paper we review the path integral technique which has wide applications in statistical physics and relate it to the backward recursion technique which is widely used for the evaluation of derivative securities. We formulate the pricing of equity options, both European and American, using the path integral framework. Discretising in the time variable and using expansions in Fourier–Hermite series for the continuous representation of the underlying asset price, we show how these options can be evaluated in the path integral framework. For American options, the solution technique facilitates the accurate determination of the early exercise boundary as part of the solution. Additionally, the continuous representation of the state variable allows the relatively accurate and efficient evaluation of the option prices and the delta hedge ratio.     

Valuation and martingale properties of shadow prices: An exposition

Concepts of asset valuation based on the martingale properties of shadow (or marginal utility) prices in continuous-time, infinite-horizon stochastic models of optimal saving and portfolio choice are reviewed and compared with their antecedents in static or deterministic economic theory. Applications of shadow pricing to valuation are described, including a new derivation of the Black–Scholes formula and a generalised net present value formula for valuing an indivisible project yielding a random income. Some new results are presented concerning (i) the characterisation of an optimum in a model of saving with an exogenous random income and (ii) the use of random time transforms to replace local by true martingales in the martingale and transversality conditions for optimal saving and portfolio choice.     

基于多标度分形理论的金融资产收益非对称性测度方法研究

基于多标度分形理论,提出了一种新的更适用于实际金融资产收益数据的非对称性测度方法——两阶段非对称性检验法（Two-step asymmetry testing,TAT）,并运用Monte Carlo模拟考察了其与传统的偏度系数检验法的非对称性判定结论差异。实证结果表明：总体来讲,本文提出的两阶段非对称性检验法在常用检验水平下取得了较偏度系数法更为准确的金融资产收益非对称性判定结论,且两阶段非对称性检验法较偏度系数法更适用于具有非独立、非正态特性数据的非对称性检验。     

Forecasting the volatility of currency exchange rates

Currency volatility is defined to be the standard deviation of day-to-day changes in the logarithm of the exchange rate. After a discussion of statistical models for exchange rates, the paper describes methods for choosing and assessing volatility forecasts using open, high, low and close prices. Results for DM/$ futures prices at the IMM in Chicago from 1977 to 1983 show high and low prices are valuable when seeking accurate volatility forecasts. The best forecasts are a weighted average of present and past high, low and close prices, with adjustments for weekend and holiday effects. The forecasts can be used to value currency options.