Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options

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

The importance of design robustness and reliability is a well-established notion and practice in today’s industry. Manufacturing companies strive to achieve Six-Sigma quality measures. While Virtual Prototyping is a key-factor in accelerating the product development process while reducing development costs; it has not contributed in the quest for improved product reliability and robustness performance since it is based on deterministic approaches. This paper provides a systematic approach to design for sixsigma simultaneously addressing variability and uncertainty present in real life on most design parameters. The first step is the ability to flexibly integrate and federate best-of-class virtual prototyping simulation software programs -most based on Finite Element Modeling techniques- thus, capturing existing engineering processes. These engineering processes provide the deterministic calculation engines for assessing design performance outputs of interest, such as -but not limited to- stress, structural impact, eigenfrequencies, forced response functions, fatigue life. Second is the assessment of design parameter variability on the design performance outputs. Analytical methods for the efficient calculation of dynamic response variability examined in this work include reliability-based methods such as First and Second Order Reliability methods, the Importance Sampling Monte Carlo (ISMC) method and the Transformed Importance Latin Hypercube Sampling (TILHS) method. They have been assessed for their ability to accurately predict the failure probability in a reasonable amount of time. The latter criterion is paramount for successful application to large industrial FE models, where the computation time is the main bottleneck. The accuracy has been examined with the comparison of the aforementioned methods with classical and advanced Monte-Carlo methodologies. Uncertainty is treated with the use of Transformation Method based on Fuzzy arithmetic concepts. Fuzzy arithmetic is an attractive mathematical concept to solve problems with uncertain parameters. The Transformation Method (TM) follows a fuzzy approach based on the concept of α-cuts, i.e. discretization of the parameter’s magnitude into a number of intervals. Uncertainty is then assessed by solving the problem in a systematic way: for each combination of the values attained by the uncertain parameters a deterministic computation of the response is performed. In this paper, the TM is demonstrated on an airplane FE model, to assess the effect of material stiffness uncertainty on the FRF from wing tip to pilot seat. A second demonstration example on this model aims to ensure a certain frequency difference between two free-free modes that may result in flutter at certain air speed; several reliability methods are applied for this purpose, namely Monte Carlo simulations, FORM and Importance Sampling.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

In statically indeterminate timber structures, the stiffness of the members and the semi-rigid behaviour of joints influence the force distribution in the structure, affecting its performance and reliability. With the ambition to provide a reliable basis for the increasing interest in building more complex and statically indeterminate timber structures, this paper aims at identifying the impact of connections on the reliability of such structures by performing a study of a simple statically indeterminate system, i.e. a simple beam with rotational springs at its supports. Using UQlab, a Matlab-based toolbox, a parametric study varying the stiffness and the level of ductility of connections is performed. Different methods such as Monte Carlo Simulations, Subset, Importance Sampling and Adaptive Kriging with Subset simulation are applied and compared in terms of efficiency and computational time. Advanced modelling strategies such as importance sampling are effective means to reduce computational costs especially in complex models. The sensitivity analyses on input parameters reveal that the joint stiffness and ductility are amongst the most important input variables. Moreover, the results showed that an increment in reliability is gained increasing the level of ductility of joints of intermediate and high stiffness.

본 논문은 오류 마루 영역에서 우수한 성능을 가지는 비이진 LDPC (low-density parity-check) 부호의 설계 방법을 제안하고 성능을 검증한다. 제안된 설계 방법은 비이진 LDPC 부호의 이진 최소 거리(binary minimum distance)를 최대화하도록 패리티 검사 행렬의 비이진 원소 값들을 결정한다. BPSK (binary phase-shift keying) 변조 방식 하에서 제안된 방법으로 설계된 비이진 LDPC 부호가 오류 마루(error floor) 영역에서 우수한 성능을 가지는 것을 Monte Carlo 시뮬레이션과 중요도 표본 추출(importance sampling) 기법을 사용하여 검증한다. In this paper we propose a design algorithm for nonbinary LDPC (low-density parity-check) codes with low error-floors. The proposed algorithm determines the nonbinary values of the nonzero entries in the parity-check matrix in order to maximize the binary minimum distance of the designed nonbinary LDPC codes. We verify the performance of the designed nonbinary LDPC codes in the error-floor region by Monte Carlo simulation and importance sampling over BPSK (binary phase-shift keying) modulation.

This paper shows two examples of how the analysis of option pricing problems can lead to computational methods efficiently implemented in parallel. These computational methods outperform “general purpose” methods (i.e., for example, Monte Carlo, finite differences methods). The GPU implementation of two numerical algorithms to price two specific derivatives (continuous barrier options and realized variance options) is presented. These algorithms are implemented in CUDA subroutines ready to run on Graphics Processing Units (GPUs) and their performance is studied. The realization of these subroutines is motivated by the extensive use of the derivatives considered in the financial markets to hedge or to take risk and by the interest of financial institutions in the use of state of the art hardware and software to speed up the decision process. The performance of these algorithms is measured using the (CPU/GPU) speed up factor, that is using the ratio between the (wall clock) times required to execute the code on a CPU and on a GPU. The choice of the reference CPU and GPU used to evaluate the speed up factors presented is stated. The outstanding performance of the algorithms developed is due to the mathematical properties of the pricing formulae used and to the ad hoc software implementation. In the case of realized variance options when the computation is done in single precision the comparisons between CPU and GPU execution times gives speed up factors of the order of a few hundreds. For barrier options, the corresponding speed up factors are of about fifteen, twenty. The CUDA subroutines to price barrier options and realized variance options can be downloaded from the website http://www.econ.univpm.it/recchioni/finance/w13. A more general reference to the work in mathematical finance of some of the authors and of their coauthors is the website http://www.econ.univpm.it/recchioni/finance/.

Option pricing is essential for market participants as it facilitates risk management and price discovery. This study provides a comprehensive elucidation of diverse option pricing models. The author delineates the development of option pricing models from the fundamental Black-Scholes-Merton model (BSM) to more intricate methodologies. The importance of comprehending option pricing for hedging strategies and risk management is crucial in financial mathematics. The study analyses several models, including the BS model, Jump Diffusion model (JDM), Binomial Tree model (BTM), Monte Carlo simulation (MCS), and stochastic volatility models (SVM), while addressing their applications, limitations, advantages, and disadvantages. This work delineates the history and distinctions between other models and the Black-Scholes-Merton model. The Black-Scholes-Merton model exerts the most substantial influence on financial markets. The alternative models exhibit greater flexibility and enhanced accuracy when confronting complicated and uncertain market trends. The trend of future development is bound to be the continuous updating of models to adapt to more complex situations.

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

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

In this paper, Crude Monte Carlo method and importance sampling are utilized to determine the reliability of long-term changes in groundwater level. Furthermore, different data analysis methods are used to determine the abnormal patterns and to investigate the cause of spatial variations of failure probability. For this purpose, three methods including robust covariance, one-class SVM, and Isolation Forest are applied to define the decision function. In the preliminary detection of the outliers, DFFITS and COOK measures are used to confirm the existence of abnormal plains in a two-dimensional space. The validity of prediction results is verified through the developed method of uncertain monitoring by selecting the most significant outlier points. In addition, the abnormal pattern detection methods are compared using non-random pattern discovery decision functions. The reliability analysis of groundwater is conducted during the two periods from 1994 to 2007 and 2008 to 2021. In the second period, parts of the eastern part of the northwest, central parts of the desert of Iran, and areas from west-southwest and east-south-east to other regions exposed to a lower probability of passing through the critical conditions. In contrast, the outcomes confirm the occurrence of drought with probability more than 80% for most of the plains. Eventually, the importance sampling method showed the closest relation in the correct distribution of the decision function. In contrast, due to the cluster shape and density of the outliers, the upper part of the decision function was determined with high certainty in the discovery of abnormal plains.

We present a very fast and accurate algorithm for calculating prices of finite lived double barrier options with arbitrary terminal payoff functions under regime-switching hyper-exponential jump-diffusion (HEJD) models, which generalize the double-exponential jump-diffusion model pioneered by Kou and Lipton. Numerical tests demonstrate an excellent agreement of our results with those obtained using other methods, as well as a significant increase in computation speed (sometimes by a factor of 5). The first step of our approach is Carr's randomization, whose convergence we prove for barrier and double barrier options under strong Markov processes of a wide class. The resulting sequence of perpetual option pricing problems is solved using an efficient iteration algorithm and the Wiener-Hopf factorization.

At the multiplexer of an ATM network, the admissible cell loss probability is less than 10−12 for some service categories. Ordinary Monte Carlo (MC) simulation requires considerable computation time to obtain such a small probability. The importance sampling (IS) technique is effective in accclerating the MC simulation. In an IS simulation, the underlying distribution is biased so as to generate more samples in the target event. In order to achieve a fast simulation, it is important to have a simulation distribution that yields an estimate with minimum variance. This is called the optimal simulation distribution. In this paper, we investigate the survivor function P(Q > q) of a stationary queue length Q in ATM queuing and obtain an IS estimate of P(Q > q) at about 10−2 by means of the optimal simulation distribution. Conventional methods [1, 2] for determining the optimal distribution require a lot of computation time. We apply the large deviation theory to this problem to obtain the optimal simulation distribution analytically and carry out an IS simulation. We present some numerical results for M/D/1 and 2-state MMPP/D/1 which demonstrate that our estimates are obtained in a very small computation time and are accurate when compared to MC simulation. © Scripta Technica, Inc. Electron Comm Jpn Pt 1, 80(12): 28–37, 1997

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.

Barrier option pricing under the 2-hypergeometric stochastic volatility model

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