Narrow Results

Since the ability to control the investor’s income or loss within a certain range, barrier option has been among the most popular path-dependent options where its payoff depends on whether or not the underlying asset’s price reaches a given “barrier”. First, assuming the underlying asset as an uncertain variable for the case that the Caputo fractional-order derivative is adopted instead of the ordinary derivative, the real financial market is better modeled by the uncertain fractional-order differential equation with Caputo type. Then, a first-hitting time model which can measure the exercise ability is innovatively presented. Second, based on the first-hitting time theorem of the uncertain fractional-order differential equation, the reliability index (including validity and survival index) for the proposed model is obtained, and four types of European barrier option (including up-and-in call, down-and-in put, up-and-out put, and down-and-out call options) pricing formulas are obtained accordingly. Lastly, applying the predictor–corrector method, numerical algorithms are provided for calculating European barrier and the reliability index, numerical experiments and corresponding sensitivity analysis are also illustrated concerning various conditions.

We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases:

• If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general.

• If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general.

We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

This paper examines the pricing of barrier options when the price of the underlying asset is modeled by a branching process in a random environment (BPRE). We derive an analytical formula for the price of an up-and-out call option, one form of a barrier option. Calibration of the model parameters is performed using market prices of standard call options. Our results show that the prices of barrier options that are priced with the BPRE model deviate significantly from those modeled assuming a lognormal process, despite the fact that for standard options, the corresponding differences between the two models are relatively small.

This paper investigates no arbitrage relationships between the prices of two one-touch options with the same maturity but with different barrier levels. We find the no-arbitrage range of prices for a no-touch option, given the price of a second no-touch option and the prices of co-maturing vanilla call options. The upper and lower bounds are the cost of a super-replicating portfolio and a sub-replicating portfolio respectively. These consist of call options, put options, digital options and a one-touch option. We assume that the underlying process is a continuous martingale but do not postulate a model.

Barrier options are one of the most popular exotic options. In this contribution, we propose a performance barrier option, which is a type of barrier option defined with the $N$th period logarithm return rate process on an underlying asset over the time interval $[0,T]$, $N\le T\le 2N$. We show that the price of this performance barrier option is determined by the joint distribution of a Slepian process and its maximum. Furthermore, we derive a tractable formula for this joint distribution and obtain explicit formulas for the up-out-call performance option and up-out-put performance option.

This paper proposes a new hybrid algorithm to price the arithmetic Asian options under the geometric Brownian motion (GBM). The proposed algorithm is based on the control variate technique, such that the control variable is a combination of the barrier arithmetic Asian option and the geometric Asian option, which each one will be estimated by the importance sampling and the control variate techniques, respectively. Besides, we drive a conditional expectation for the estimator that it can reduce variance of simulations. The merits of the proposed algorithm for pricing arithmetic Asian options are illustrated by several examples.

The presence of discrete dividends complicates the derivation and form of pricing formulas even for vanilla options. Existing analytic, numerical, and theoretical approximations provide results of varying quality and performance. Here, we compare the analytic approach, developed and effective for European puts and calls, of Buryak and Guo with the formulas, designed in the context of barrier option pricing, of Dai and Chiu.

In this chapter, we propose the structural model in terms of the Stair Tree model and barrier option to evaluate the fair deposit insurance premium in accordance with the constraints of the deposit insurance contracts and the consideration of bankruptcy costs. First, we show that the deposit insurance model in Brockman and Turle (2003) is a special case of our model. Second, the simulation results suggest that insurers should adopt a forbearance policy instead of a strict policy for closure regulation to avoid losses from bankruptcy costs. An appropriate deposit insurance premium can alleviate potential moral hazard problems caused by a forbearance policy. Our simulation results can be used as reference in risk management for individual banks and for the Federal Deposit Insurance Corporation (FDIC).