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.     

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.     

基于实物期权法的风险投资项目估价方法——一个文献综述

本文理清了实物期权及其在风险投资项目估价中应用的发展脉络,并提出了实物期权在风险投资项目估价应用中需要发展的地方。     

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.     

Hedging and pricing early-exercise options with complex fourier series expansion

We introduce a new numerical method called the complex Fourier series (CFS) method proposed by Chan (2017) to price options with an early-exercise feature—American, Bermudan and discretely monitored barrier options—under exponential Lévy asset dynamics. This new method allows us to quickly and accurately compute the values of early-exercise options and their Greeks. We also provide an error analysis to demonstrate that, in many cases, we can achieve an exponential convergence rate in the pricing method as long as we choose the correct truncated computational interval. Our numerical analysis indicates that the CFS method is computationally more comparable or favourable than the methods currently available. Finally, the superiority of the CFS method is illustrated with real financial data by considering Standard & Poor’s depositary receipts (SPDR) exchange-traded fund (ETF) on the S&P 500® index options, which are American options traded from November 2017 to February 2018 and from 30 January 2019 to 21 June 2019.     

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.     

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.     

Pricing European continuous-installment currency options with mean-reversion

In this paper, we consider European continuous-installment currency option under the mean-reversion environment. Specifically, we provide efficient pricing formula of installment currency put option via a partial differential equation (PDE) approach when the exchange rate follows the mean reverting lognormal model. Using the Mellin transform techniques, we derive the integral equation representation for the optimal stopping boundary from the PDE for pricing of the option. To verify the efficiency and accuracy of our approach, we provide computational results with the least square Monte Carlo method proposed by Longstaff and Schwartz (2001). We also present some numerical examples to examine the characteristics of the optimal boundaries and prices.     

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.     

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 spread options with counterparty risk and jump risk

In this paper, we investigate spread options with counterparty risk in a jump-diffusion model. Due to the fact that there is no closed-form formula of spread options with counterparty risk, we obtain analytical expressions of lower and upper bounds by employing the measure-change technique. Finally, we numerically check the accuracy of the bounds and analyze the impacts of counterparty risk and jump risk on spread option prices.     

Pricing of vulnerable exchange options with early counterparty credit risk

The exchange option is one of the most popular options in the over-the-counter (OTC) market, which enables the holder of two underlying assets to exchange one with another. In OTC markets, with the increasing apprehension of credit default risk in the case of option pricing since the global financial crisis, it has become necessary to consider the counterparty credit risk while evaluating the option price. In this study, we combine the vulnerable exchange option and early counterparty default risk to obtain the closed-form formula for the vulnerable exchange option with early counterparty credit risk by using the method of dimension reduction, Mellin transform, and the method of images. Moreover, we examine the pricing accuracy of the option value by comparing our closed-form solution with the formula derived by the Monte-Carlo simulation.     

基金费用价值化模型的建立及比较分析

随着基金业的不断发展,投资则越来越关注基金的投资价值。因此,寻找一种能客观评价投资者投资价值的工具就显得尤为重要。本文在分析几种常见的价值评估模型的基础上,建立投资成本与投资收益间关系的模型——费用价值化模型。这将为投资者提供投资决策的模型框架,定量的评价其投资成本和投资价值。     

A closer look at Black–Scholes option thetas

This paper investigates Black–Scholes call and put option thetas, and derives upper and lower bounds for thetas as a function of underlying asset value. It is well known that the maximum time premium of an option occurs when the underlying asset value equals the exercise price. However, we show that the maximum option theta does not occur at that point, but instead occurs when the asset value is somewhat above the exercise price. We also show that option theta is not monotonic in any of the parameters in the Black–Scholes option-pricing model, including time to maturity. We further explain why the implications of these findings are important for trading and hedging strategies that are affected by the decay in an option’s time premium.

Exponential stock models driven by tempered stable processes

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous Lévy processes. With a view of option pricing, we provide a systematic analysis of the existence of equivalent martingale measures, under which the model remains analytically tractable. This includes the existence of Esscher martingale measures and martingale measures having minimal distance to the physical probability measure. Moreover, we provide pricing formulae for European call options and perform a case study.     

Approximate analytic solution for Asian options with stochastic volatility

The valuation of Asian options is complicated because the arithmetic average of lognormal random variables is no longer lognormal. Furthermore, the stochastic volatility inherent in financial asset prices is easily observed. However, few academic studies consider the pricing and hedging of Asian options with stochastic volatility, despite the popularity of such options. This study extends the work of Hull and White (1987) and integrates the Taylor series expansion technique to derive an approximate analytic solution for Asian options with stochastic volatility. Numerical experiments show that the proposed approximate analytic solution performs favorably and is computationally efficient compared with large-sample simulations. The approximate analytic solution provides a practical approach for pricing and hedging Asian options with stochastic volatility and is both easy to implement and desirable in terms of computing speed.     

带Poisson跳的Black-Scholes模型的期权定价

主要讨论欧式期权的定价公式。首先给出一个B-S期权定价公式的简化方法,使具有一般微积分知识的读者就能理解;并假定股票价格过程遵循带Poisson跳的扩散过程,在股票预期收益率、波动率和无风险利率均为时间函数的情况下,得到欧式期权定价公式和买权与卖权之间的平价关系。     

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.