Trapped by the Tails of the Bivariate Normal Distribution

Electronic Journal of Applied Statistical Analysis, 2020

In this paper we propose a numerical scheme to estimate the price of a barrier option in a general framework. More precisely, we extend a classical Sequential Monte Carlo approach, developed under the hypothesis of deterministic volatility, to Stochastic Volatility models, in order to improve the efficiency of Standard Monte Carlo techniques in the case of barrier options whose underlying approaches the barriers. The paper concludes with the application of our procedure to two case studies in a SABR model.

SIAM Journal on Financial Mathematics, 2013

We develop a conditional sampling scheme for pricing knock-out barrier options under the Linear Transformations (LT) algorithm from Imai and Tan (2006), ref. . We compare our new method to an existing conditional Monte Carlo scheme from Glasserman and Staum , ref.

International Journal of Finance & Banking Studies (2147-4486)

The aim of this paper is to evaluate barrier options by considering volatility as stochastic following the CIR process used in Heston (1993). To solve this problem, we used Monte Carlo simulation. We studied the effects of stochastic volatility on the value of the barrier option by considering different values of the determinants of the option. We illustrated these effects in twelve graphs. We found that in general, regardless of the parameter under study, the stochastic volatility model significantly overvalues the in-the-money (ITM) barrier options, and slightly the deep-in-the money (DIP) options, while slightly undervaluing the near-out-the money (NTM) options.

2017

We obtain closed form expressions for the exact no-arbitrage prices, as well as estimates, of some types of multivariate options with barriers that are generated by hyperplanes placed on a collection (or vector) of stock prices (they are not placed individually on each stock of the collection). A novelty for the estimates is that we combine ideas of convex analysis with tools of stochastic theory.

Finance and Stochastics, 2009

We suggest two new fast and accurate methods, Fast Wiener-Hopf method (WHF-method) and Iterative Wiener-Hopf method (IWH-method), for pricing barrier options for a wide class of Lévy processes. Both methods use the Wiener-Hopf factorization and Fast Fourier Transform algorithm. Using an accurate albeit relatively slow finite-difference algorithm developed in Levendorskiǐ et al (2006) (FDS-method), we demonstrate the accuracy and fast convergence of the two methods for processes of finite variation. We explain that the convergence of the methods must be better for processes of infinite variation, and, as a certain supporting evidence, demonstrate with numerical examples that the results obtained by two methods are in extremely good agreement. Finally, we use FDS, WHF and IWH-methods to demonstrate that Cont and Volchkova method (CV-method), which is based on the approximation of small jumps by an additional diffusion, may lead to sizable relative errors, especially near the barrier and strike. The reason is that CV-method presumes that the option price is of class C 2 up to the barrier, whereas for processes without Gaussian component, it is typically not of class C 1 at the barrier.

Physica A: Statistical Mechanics and its Applications

We report call option pricing for up-and-out style barrier options through the use of a neural net model. A synthetic data set was constructed from the real LIFFE standard option price data by use of the Rubenstein and Reiner analytic model (Risk September (1991) 28). Unbiased estimates at the 95% confidence level were achieved for realistic barriers (barrier 4% or more above maxðS 0 ; X Þ).

Options involving two barriers, including the case of rebates, are analyzed and analytic valuation formulas given. The analysis is self-contained and intuitive, and relies only on de®ning properties of Brownian sample paths. The connection with results obtained by Laplace transforms is made explicit.

Review of Derivatives Research, 2013

Imposing a symmetry condition on returns, Carr and Lee [2009] show that (double) barrier derivatives can be replicated by a portfolio of European options and can thus be priced using fast Fourier techniques (FFT). We show that prices of barrier derivatives in stochastic volatility models can alternatively be represented by rapidly converging series, putting forward an idea by . This representation turns out to be faster and more accurate than FFT. Numerical examples and a toolbox of a large variety of stochastic volatility models illustrate the practical relevance of the results.

Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase the simulation efficiencies of the Monte-Carlo methods. Our approach in this research work will consider the use of antithetic variate techniques to estimate the fair prices of barrier options. Next, we use the quasi-Monte Carlo method, together with Sobol sequence to estimate the values of the same option. An extended version of the Black-Scholes model will serve as basis for the exact prices of these exotic options. The resulting simulated prices will be compared to the exact prices. The research concludes by showing some results which proves that when random numbers are generated via low discrepancy sequences in contrast to the normal pseudo-random numbers, a more efficient simulation method is ensued. This is further applicable in pricing complex derivatives without closed formsolutions.

Advances in Dynamic Games, 2010

We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black-Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11] and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.

Econometric Modeling: Capital Markets - Asset Pricing eJournal, 2018

We analyze model risk for the pricing of barrier options. In contrast to existing literature, this paper is based on an empirical data set of over 40,000 bonus certificates to analyze the real market extent of model risk for traded barrier options instead of purely synthetic options. For this purpose a local volatility model, the Heston model and the Bates model are applied. Furthermore, we add to the literature on the behavior of issuers of retail derivatives in terms of model choice. We find evidence that the majority of the issuers prefer stochastic volatility over local volatility models, while they do not use the even more realistic Bates model which incorporates jumps in the underlying.

2001

In this paper the problem of estimating the ratio of variances, σ, in a bivariate normal distribution with unknown mean is considered from a decisiontheoretic point of view. First, the UMVU estimator of σ is derived, and then it is shown to be inadmissible under two specific loss functions, namely, the squared error loss and the entropy loss. The derivation of the results is done by conditioning on an auxiliary negative binomial random variable.

Developing Country Studies, 2014

We present an original Probabilistic Monte Carlo (PMC) model for pricing European discrete barrier options. Based on Monte Carlo simulation, the PMC model computes the probability of not crossing the barrier for knock-out options and crossing the barrier for knock-in options. This probability is then multiplied by an average sample discounted payoff of a plain vanilla option that has the same inputs as the barrier option but without barrier and to which we have applied a filter. We test the consistency of our model with an analytical solution (Merton 1973 and Reiner & Rubinstein 1991) adjusted for discretization by Broadie et al. (1997) and a naive numerical model using Monte Carlo simulation presented by Clewlow & Strickland (2000). We show that the PMC model accurately price barrier equity options. Market participants in need of selecting a reliable and simple numerical method for pricing discrete barrier options will find our paper appealing. Moreover, the idea behind the method ...

Journal of Taibah University for Science, 2017

In this paper, we apply an improved version of Monte Carlo methods to pricing barrier options. This kind of options may match with risk hedging needs more closely than standard options. Barrier options behave like a plain vanilla option with one exception. A zero payoff may occur before expiry, if the option ceases to exist; accordingly, barrier options are cheaper than similar standard vanilla options. We apply a new Monte Carlo method to compute the prices of single and double barrier options written on stocks. The basic idea of the new method is to use uniformly distributed random numbers and an exit probability in order to perform a robust estimation of the first time the stock price hits the barrier. Using uniformly distributed random numbers decreases the estimation of first hitting time error in comparison with standard Monte Carlo or similar methods. It is numerically shown that the answer of our method is closer to the exact value and the first hitting time error is reduced.

International Journal of Theoretical and Applied Finance, 2009

This paper examines the pricing of barrier options when the price of the underlying asset is modeled by branching process in 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.