Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

Pricing Bounds on Barrier Options

Author index of volume 144

This study aims to examine the effectiveness of the Monte Carlo antithetic variate and control variate methods in pricing knock-out barrier options compared to the standard Monte Carlo method. The main problem in barrier option pricing is the high variance of estimates, which can reduce the accuracy and efficiency of results. The standard Monte Carlo method often requires a very large number of simulations to achieve stable results, which is computationally inefficient. To address this issue, this study employs variance reduction techniques, antithetic variate, and control variate. The findings indicate that both methods offer higher accuracy in price estimation compared to the standard Monte Carlo method. Further analysis reveals that the control variate method is more effective for pricing up and out barrier call options and down and out barrier call options, while the antithetic variate method excels in pricing up and out barrier put options and down and out barrier put options. This study underscores the importance of selecting the appropriate method according to the type of option involved to achieve accurate and efficient estimations.

Exact static hedging results for barrier options are usually based on unrealistic assumptions such as the availability of an infinite number of standard options or zero cost of carry. If these assumptions are dropped, static hedges may lead to large losses close to maturity of the barrier option. To avoid this, we analyze the well-known concept of super-replication in the context of static hedging both theoretically and numerically and apply it to the case of hedging barrier options in a stochastic volatility context. Numerical results show that our approach leads to very attractive static hedging strategies which super-replicate the barrier option’s payoff, match the delta and vega of the target option and only consist of a handful of liquidly traded standard options. Surprisingly, the identified cost-optimal static super-replication portfolios are only marginally more expensive than the barrier option itself.

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e-α n2.

Mellin Transform Approach for Pricing of Barrier Option under Levy Process with Minimal Martingle Measure

Pricing Continuously Monitored Barrier Options Under the Sabr Model: A Closed-Form Approximation

Barrier option pricing under the 2-hypergeometric stochastic volatility model

The pricing and static hedging of multi-step double barrier options

In this paper, we present our research on pricing window barrier options under a hybrid stochastic-local volatility (SLV) model in the foreign exchange (FX) market. Due to the hybrid effect of the local volatility and stochastic volatility components of the model, the SLV model can reproduce the market implied volatility surface, and can improve the pricing accuracy for exotic options at the same time. In this paper, numerical techniques such as Monte Carlo and finite difference methods for standard exotic barrier options under the SLV model are extended to pricing window barrier options and numerical results produced by the SLV model are used to examine the performance and accuracy of the model for pricing window barrier options.

Many developing economies are heavily exposed to commodity markets, leaving them vulnerable to the vagaries of international commodity prices. This paper examines the use of commodity options, including plain vanilla, risk reversal, and barrier options, to hedge such risks. It then proposes the use of a new structured product, a sovereign Eurobond with an embedded option on a specific commodity price. By extracting commodity price risk out of the bond, such an instrument insulates the bond default risk from commodity price movements, allowing it to be marketed at a lower credit spread. The product is also designed to help developing countries establish a credit derivatives market, which would in turn enhance the marketability and liquidity of sovereign bonds.

We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.

In this article barrier options are analytically evaluated under the regime-switching model, with the volatility of the underlying price being allowed to jump between different states following a Markov chain. The target barrier option prices are expressed in a Fourier cosine series after a particular approximation formula is obtained. The accuracy and efficiency of the newly derived formula are demonstrated through numerical experiments, demonstrating the formula’s potential for practical applications. TOPICS:Analysis of individual factors/risk premia, factor-based models, options Key Findings • Barrier options are analytically evaluated under the regime-switching model. • This approximation formula is written in the form of a converged Fourier cosine series. • The formula is shown to be very accurate and efficient, and has a great potential to be applied in practice.

A barrier option is an exotic path-dependent option contract that, depending on terms, automatically expires or can be exercised only if the underlying asset ever reaches a predetermined barrier price. Using a partial differential equation approach, we provide an integral representation of the barrier option price via the Mellin transform. In the case of knock-out barrier options, we obtain a decomposition of the barrier option price into the corresponding European option value minus a barrier premium. The integral representation formula can be expressed in terms of the solution to a system of coupled Volterra integral equations of the first kind. Moreover, we suggest some possible numerical approaches to the problem of barrier option pricing.