Cox-Ross-Rubinstein Model Research Articles

Calculation Option Price on Shares PT. Nippon Indosari Corpindo Tbk. with Fuzzy Binomial Tree Model

The Cox-Ross-Rubinstein binomial tree model is widely used in stock and derivative securities calculations, such as options calculations. The binomial CRR model assumes that the parameter increases in option prices and decreases in option prices so this model produces stock price movements up and down stock price movement. However, stock price movements show price fluctuations and cause the volatility of the value to be unsuitable. In this study, modeling stock and option price movements using a fuzzy binomial tree model. The data used was data on the movement of the stock price of Nippon Indosari Corpindo Ltd Plc from February 2021 to January 2022. The results showed that for February 2022, with a risk size of 90%, the selling price options with the greatest volatility of 51.6484081, medium volatility of 33.33154354, and the smallest volatility of 28.17155892.

Pricing and Optimization Model of European Options

Financial market forecasts are gradually gaining the attention of investors. The options market is an integral and important part. Commonly used analysis methods include traditional BS formula and CRR model. This article obtained the transaction data of SP&500 from February 2013 to February 2018, used the data of the first 4 years to train the model, and finally explored the actual price and predicted value of the European call option. To forecast the stock price, a hidden Markov model (HMM) is first applied. And on this basis, improve the parameter estimation in the traditional method BS-CRR, and compare each model with the actual value of the option. Compared with the traditional model, to some extent, the Hidden Markov Model can predict the state of the stock market., get rid of the limitations of the BS model assumptions, predict the direction of the price trend, and verify the feasibility of its application in the financial field.

Limiting Cases of the Black-Scholes Type Asymptotics of Call Option Pricing in the Generalised CRR Model

The article concerns the generalised Cox‑Ross‑Rubinstein (CRR) option pricing model with new formulas for changes in upper and lower stock prices. The formula for option pricing in this model, which is the Black‑Scholes type formula, and its asymptotics are presented. The aim of the paper is to analyse limiting cases of the obtained asymptotics using probability theory and later data from the Warsaw Stock Exchange. Empirical analyses of option pricing in the generalised CRR model confirm the calculated limits.

Biosignature line ratios of [P ii] in exoplanetary and nebular environments

ABSTRACT Being the backbone element of deoxyribonucleic acid, phosphorus is a key component in the search for life in the Universe. To aid in its detection, we present line emissivity ratios for the five lowest-lying forbidden [P ii] transitions, namely those among the levels 3s23p2(3P0, 3P1, 3P2, 1D2, and 1S0). The wavelengths range between 0.44 and 70 µm, and several lie within the spectroscopic domain observable with the JWST. These line ratios have been calculated using a new collisional-radiative-cascade (CRR) model combining calculated collision strengths and level-specific recombination rate coefficients, with both data sets computed using the accurate Breit–Pauli R-matrix method. The CRR model includes a new scheme for (e + ion) recombination to emission-line formation. We compare its effect to models incorporating only electron-impact excitation and spontaneous radiative decay. We find that (e + ion) recombination has a significant impact on all line ratios, and represents a major improvement in physical accuracy of emission-line models.

Numerical Approaches of Pricing European Options in the Cox-Ross-Rubinstein Models

The Cox-Ross-Rubinstein (CRR) market model is one of the simplest and easiest ways to analyze the option pricing model. CRR has been employed to evaluate a European Option Pricing (call options) model without complex elements, including dividends, stocks, and stock indexes. Instead, it considers only a continuous dividend yield, futures, and currency options. The CRR model is simple but strong enough to describe the general economic intuition behind option pricing and its principal techniques. Also, it gives us basic ideas on how to develop financial products based on current deviations and volatilities. The paper investigates the CRR model using numerical approaches with python code. It provides a practical event using the mathematical model to demonstrate the application of the model in the financial market. First, the paper provides a simple example to figure out the basic concept of the model. Only a two-period binomial model based on the introductory definitions of the call options makes us understand the concept more easily and quickly. Next, we used actual data on Tesla stock fluctuations from the Nasdaq website (See section 3). We developed the python code to make it easier to understand figures, including tables and graphs. The code allows us to visualize and simplify the model and output data. The code analyzes the stock data to evaluate the probability of the stock’s price increasing or decreasing. Then, it used the CRR model to estimate all possible cases for the stock’s prices and investigate the call and put option pricing. The code was based on the introductory code of binomial option pricing, but we improved it to get more information and provide more detailed results from the data. The detailed codes are provided in section 3 of the paper. As a result, we believe the CRR model is a fundamental formula, but the improved python code can suggest a new direction for evaluating the probability and investigating the value of the stocks. Also, we expect to develop the code to extend the Black Scholes Pricing model, increasing the number of periods.

Application of a limit Function Of Negative Hyper-geometric Distribution in Option Pricing.

This work introduces limit function of a negative hyper geometric distribution of the form .where and, which is applied in option pricing using wealth equation and some martingale tools. This paper present a simple discrete time model in compares. This work conclude that limit of negative hyper geometric as can be associated with financial terms that can be used to evaluate option values (non dividend) which gives the same numerical with CRR model.

THE USE OF NEGATIVE PROBABILITIES IN ECONOMICS

The concept of „negative probability” has become one of interest in research. Initially developed in connection with physics (negative probabilities are important in the analysis of quantum phenomena), nowadays it has applications in many other scientific disciplines. Its main advantage is that it extends the classical probability theory, which involves values between 0 and 1, allowing negative or supraunitary values. From this point of view, negative probabilities are a useful tool that facilitates the calculation and makes analysis more flexible through models. This paper aims to present the theoretical framework, the principles in relation to which the use of negative probabilities in economics, in general, can be understood. The existing literature highlights that the negative probabilities are appropriate in the analysis of stochastic processes and, from this perspective, has developed applications especially in the field of finance. In this sense, negative probabilities are used in models such as the binomial CRR model (The Cox-Ross-Rubinstein Market Model), that evaluate the price of derivatives on the capital market, those that include hidden variable (non-observable) etc. In our analysis we describe these models and, based on them, we try to identify the features of the negative probabilities that can be used in the economic field. The research is a theoretical one, leading the way for new approaches in the direction of using negative probabilities in macroeconomic modelling.

On the convergence scheme in the CRR model

We investigate the convergence scheme from the discrete to the continuous time model in the binomial tree model of Cox-Ross-Rubinstein (CRR). We introduce the notion of $\bar\Sigma $-decomposition and we classify the financial payoffs according to their r

Red Cell Alloimmunisation Is Associated with Development of Autoantibodies and Increased Red Cell Transfusion Requirements in Myelodysplastic Syndromes (MDS)

Introduction: Up to 90% of MDS patients require red blood cell (RBC) transfusions and 30-45% become RBC transfusion dependent (RBC-TD). However, literature addressing RBC alloimmunisation in MDS is limited. This study evaluates the incidence and clinical impact of RBC alloimmunisation in 817 adult MDS (or related disorders) patients enrolled in the South Australian-MDS registry.Methods: Demographic, clinical, laboratory and treatment (including transfusions) details of patients with at least 6-month follow up were analysed. RBC-TD was defined as >1unit of RBC transfusion every 8 weeks over a period of 4 months. Azacitidine, intensive chemotherapy or allogeneic stem cell transplantation were denoted as disease modifying therapies (DMT).Results: Median age at diagnosis was 73 years and 66% were male. During the study, 695 (85%) patients received >1 RBC unit, and 98 developed 175 alloantibodies. Of these, 11 patients developed alloantibodies following MDS-unrelated RBC transfusion well before the MDS diagnosis, and seven developed antibodies before their first documented RBC transfusion. Remaining 80 patients developed alloantibodies following MDS-related RBC transfusion. The cumulative incidence of RBC alloimmunisation with death as competing risk was 11% at 50 months following the first MDS-related RBC transfusion (Fig. 1A). Importantly, 73% and 50% of allo-immunised patients developed alloantibodies during first 20 RBC transfusions and within 6 months of commencement of RBC transfusion, respectively. Of the 98 allo-immunised patients, 50 (51%) developed one alloantibody, while 48 (49%) developed multiple alloantibodies. Rh (93/175; 54%) and Kell (41; 24%) alloantibodies were most frequent followed by Kidd (13, 7.5%) and Lutheran (10, 5.7%). Within Rh system anti-E, anti-C and anti-D were the most frequent.Autoantibodies were detected in significantly more alloimmunised than in non-alloimmunised patients (65% vs. 18%, p<0.0001; Fig 1B) and >85% autoantibodies occurred within 5 months of alloimmunisation. Though 73% of alloimmunised patients developed alloantibody during their first 20 RBC units, total RBC units transfused were significantly higher in alloimmunised compared to non-alloimmunised patients (90±100 vs. 30±52; p<0.0001) (Fig. 1C). In alloimmunised patients, total RBC transfused after alloimmunisation were significantly higher than before alloimmunisation (p<0.0001, Fig. 1D). In MDS patients, multiple other factors such as disease progression, infection, bleeding and treatment can influence RBC-transfusion intensity. Hence, we evaluated RBC-transfusion intensity in 33 patients meeting strict inclusion criteria (Fig. 2A). In these patients, RBC-transfusion intensity substantially increased after alloimmunisation (p=0.002; Fig. 2B-C).We then evaluated risk factors for alloimmunisation using RSF and R-PART analysis. RBC units transfused before alloantibody formation was the most important predictor of alloimmunisation, followed by RBC-TD status, treatment type and age. A classification tree using RPART suggested that 46% patients who were RBC-TD developed an alloantibody within first 20 units of RBC transfused. Landmark analyses using CRR model both at baseline (using age, sex, WHO subtype, IPSS-R risk groups and type of treatment) and at six months (baseline factors plus number of RBCs transfused within 6 months and RBC-TD status at 6 months) following the start of RBC transfusions were performed. At both time-points, alloimmunisation risk was significantly lower in DMT treated patients (HR -3.04; p=0.0023) and appears unrelated to number of RBC transfusions as patients managed with DMT received significantly more RBC units than patients treated with supportive care (p<0.0001).Conclusion: This study characterises alloimmunisation in a large statewide cohort of MDS patients and demonstrates a significant increase in RBC transfusion requirements following alloimmunisation, most probably due to development of additional alloantibodies and autoantibodies, resulting in subclinical/clinical haemolysis. Use of DMT is associated with lower alloimmunisation risk. Strategies mitigating alloimmunisation risk are critical for optimising RBC-transfusion management. [Display omitted] DisclosuresNo relevant conflicts of interest to declare.

Option pricing: a yet simpler approach

We provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox–Ross–Rubinstein (CRR) pricing model. The main tool used in this paper for simplifying the reasoning is applying static hedging arguments. In applying the static hedging principle, we consider Arrow–Debreu securities and digital options, or backward random processes. In the last case, the CRR model is extended to an infinite state space which leads to an interesting new phenomenon not present in the classical CRR model. At the end, we discuss the paradox involving the drift parameter mu in the Black–Scholes–Merton model pricing. We provide sensitivity analysis and an approximation of the speed of convergence for the asymptotically vanishing effect of drift in prices.

Nonparametric predictive inference for American option pricing based on the binomial tree model

In this article, we present the American option pricing procedure based on the binomial tree from an imprecise statistical aspect. Nonparametric Predictive Inference (NPI) is implemented to infer imprecise probabilities of underlying asset movements, reflecting uncertainty while learning from data, which is superior to the constant risk-neutral probability. In a recent article, we applied the NPI method to the European option pricing procedure that gives good results when the investor has non-perfect information. We now investigate the NPI method for American option pricing, of which imprecise probabilities are considered and updated for every one-time-step path. Different from the classic models, this method is shown that it may be optimal to early exercise an American non-dividend call option because our method considers all information that occurs in the future steps. We also study the performance of the NPI pricing method for American options via simulations in two different scenarios compared to the Cox, Ross and Rubinstein binomial tree model (CRR), first where the CRR assumptions are right, and second where the CRR model uses wrong assumptions. Through the performance study, we conclude that the investor using the NPI method tends to achieve good results in the second scenario.

On the Robustness of Conceptual Rainfall‐Runoff Models to Calibration and Evaluation Data Set Splits Selection: A Large Sample Investigation

AbstractConceptual rainfall‐runoff (CRR) models are widely used for runoff simulation and for prediction under a changing climate. The models are often calibrated with only a portion of all available data at a location and then evaluated independently with another part of the data for reliability assessment. Previous studies report a persistent decrease in CRR model performance when applying the calibrated model to the evaluation data. However, there remains a lack of comprehensive understanding about the nature of this “low transferability” problem and why it occurs. In this study we employ a large sample approach to investigate the robustness of CRR models across calibration/validation data splits. Specially, we investigate (1) how robust is CRR model performance across calibration/evaluation data splits, at catchments with a wide range of hydroclimatic conditions; and (2) is the robustness of model performance somehow related to the hydroclimatic characteristics of a catchment? We apply three widely used CRR models, GR4J, AWBM and IHACRE_CMD, to 163 Australian catchments having long‐term historical data. Each model was calibrated and evaluated at each catchment, using a large number of data splits, resulting in a total of 929,160 calibrated models. Results show that (1) model performance generally exhibits poor robustness across calibration/evaluation data splits and (2) lower model robustness is correlated with specific catchment characteristics, such as higher runoff skewness and aridity, highly variable baseflow contribution, and lower rainfall‐runoff ratio. These results provide a valuable benchmark for future model robustness assessments and useful guidance for model calibration and evaluation.

Nonparametric predictive inference for European option pricing based on the binomial tree model

In finance, option pricing is one of the main topics. A basic model for option pricing is the Binomial Tree Model, proposed by Cox, Ross, and Rubinstein in 1979 (CRR). This model assumes that the underlying asset price follows a binomial distribution with a constant upward probability, the so-called risk-neutral probability. In this article, we propose a novel method based on the binomial tree. Rather than using the risk-neutral probability, we apply Nonparametric Predictive Inference (NPI) to infer imprecise probabilities of movements, reflecting more uncertainty while learning from data. To study its performance, we price the same European options utilising both the NPI method and the CRR model and compare the results in two different scenarios, firstly where the CRR assumptions are right, and secondly where the CRR model assumptions deviate from the real market. It turns out that our NPI method, as expected, cannot perform better than the CRR in the first scenario, but can do better in the second scenario.

Computing Risk Measures of Life Insurance Policies through the Cox–Ross–Rubinstein Model

The problem of computing risk measures of life insurance policies is complicated by the fact that two different probability measures, the real-world probability measure along the risk horizon and the risk-neutral one along the remaining time interval, have to be used. This implies that a straightforward application of the Monte Carlo method is not available and the need arises to resort to time consuming nested simulations or to the least squares Monte Carlo approach. We propose to compute common risk measures by using the celebrated binomial model of Cox, Ross, and Rubinstein (1979) (CRR). The main advantage of this approach is that the usual construction of the CRR model is not influenced by the change of measure and a unique lattice can be used along the whole policy duration. Numerical results highlight that the proposed algorithm computes highly accurate values.

Survival outcomes of locally advanced prostate cancer in patients aged < 50years after local therapy in the contemporary US population.

To assess survival outcomes of locally advanced prostate cancer (LAPC) in patients aged < 50years after local therapy (LT), as compared to that in the older patients (≥ 50years). Moreover, effectiveness of postoperative radiation therapy (PRT) after radical prostatectomy (RP) was also assessed in patients aged < 50years. Within the Surveillance, Epidemiology, and End results database (2004-2014), non-metastatic cT3-4 LAPC patients treated with LT (RP, RT or RP+RT) were identified. After propensity score matching (PSM), cancer-specific mortality (CSM), overall survival (OS), and other-cause mortality (OCM) rates were assessed. Multivariable competing risk regression (MVA CRR) model was also used in our analysis. 1507 younger (< 50years) and 34833 older (≥ 50years) LAPC patients treated with LT were identified. Younger patients with LAPC had overall more aggressive disease features than their older counterparts. After PSM, younger patients yielded higher cumulative CSM rates than the older patients (P = 0.046). However, OS and cumulative OCM rates were significantly higher (P = 0.038 and P < 0.0001, respectively) in the older cohort. In the MVA CRR model, younger patients yielded higher CSM (P = 0.02). Specifically, younger patients resulted in higher CSM in Gleason score 8-10, cT3b/4 stage, cN1 stage, and patients treated with RP. No statistically significant differences were found in patients treated with RP versus RP+PRT in all parameters. LAPC patients aged < 50years yielded higher CSM after LT, specifically after RP, compared with the older counterparts (≥ 50years). No significant differences were observed in RP versus RP+PRT regarding survival outcomes in our analysis.

Assessing the Potential Robustness of Conceptual Rainfall‐Runoff Models Under a Changing Climate

AbstractConceptual rainfall‐runoff (CRR) models are commonly used to assess the potential impact of climate change on water resources systems. However, they are often characterized by poorer performance when used to simulate a different climate compared to that of the calibration period. This is generally referred to as low model robustness, and these issues have been thoroughly explored using historical data. However, the implications of robustness are unknown for a changing climate where models may have to operate under conditions that lie beyond existing observations. This study extends these ideas to evaluate the “potential robustness” of different CRR models in the context of a changing climate. To achieve this aim, we combine a generalized split‐sample test framework with a stochastic weather generator. This allows us to assess the variabilities in runoff predictions obtained from using different calibration periods within each CRR model. We tested the potential robustness on three catchments with contrasting hydroclimatic conditions. We observed a consistent higher potential robustness in all models under drier conditions at all catchments. The three catchments illustrate contrasting patterns in the relative potential robustness of the three CRR models, which are related to both the structures of the CRR models and the unique catchment characteristics, highlighting the need of case‐specific assessment. This study illustrates a transferable empirical testing strategy to understanding variabilities in CRR model predictions. This approach can improve our knowledge of model behavior and thus informs the suitability of alternative models to simulate catchments hydrology under a changing climate.

Optimal hedging strategies in equity-linked products

Equity-linked insurance policies are one of the most widespread insurance products. In many cases such contracts have guarantees like a minimum return over the lifetime of the policy. Liabilities arising from such guarantees must be hedged by suitable investments. There are restrictions on hedging strategies in many jurisdictions but with the more flexible regulatory framework of Solvency 2 there are alternative ways to hedge certain guaranteed products using derivative securities. In this paper we investigate when it is optimal to switch to hedging liabilities with derivative securities in the framework of the Cox–Ross–Rubinstein model. This leads to optimal stopping problems that can be solved explicitly. Mortality is also incorporated in the model. The results may indicate the level of reserves necessary to meet obligations with the desired level of confidence. In particular the strategy may be applicable in adverse market conditions.

On the arbitrage price of European call options

ABSTRACTWe show that in a discrete price and discrete time model for option pricing, specifically that given by the Cox–Ross–Rubinstein model, the arbitrage price of a European call option can depend on parameters other than volatility (the standard deviation of the log asset price). We provide two theorems to illustrate this phenomenon. Our first theorem considers two securities with the same volatility so that at a specified time n0, with probability near 1, the two securities are equal. If their call options differ, both the discounted securities will be martingales. Our second theorem considers two securities with the same volatility so that at times n = 0, ..., N − 1 the securities are equal with probability near 1. If their call options differ, one of the discounted securities will be a martingale and the other discounted security will be a supermartingale.

On a recursive algorithm for pricing discrete barrier options

An alternative and simple algorithm for valuating the price of discrete barrier options is presented. This algorithm computes the price just exactly the same as the Cox–Ross–Rubinstein (CRR) model. As opposed to other pricing methodologies, this recursive algorithm utilizes only the terminal nodes of the binomial tree and it captures the intrinsic property, the knock-in or knock-out feature, of barrier options. In this paper, we apply the algorithm to compute the price of an Up and Out Put (UOP) barrier option and compare the results obtained from the CRR model. We then determine the time complexity of the algorithm and show that it is [Formula: see text].

Calibration of a CRR model based on an expanded SPT-based database for assessing soil liquefaction potential

A new calibrated empirical liquefaction triggering correlation equation with five parameters determined by trial and error is derived from the expanded SPT-based database of the Idriss and Boulanger (2010) and Xie (1984) case histories. The new calibrated correlation equation is reasonably conservative for the expanded SPT-based database and is insensitive to reasonable variations in the components of the liquefaction analysis framework adopted in this study. In addition, a probabilistic version of the new calibrated correlation equation expressed in the form of a mapping function that relates the liquefaction probability to a nominal safety factor obtained using the weighted maximum likelihood technique is verified using a weighted empirical probability approach and Monte Carlo simulations. Liquefaction, transition and non-liquefaction zones for clean sands are defined based on different probability contours, i.e., empirical correlations with different liquefaction probabilities. Liquefaction triggering correlations for liquefaction probabilities of 4%, 15% and 25% represent the very low, low, and moderate probability events, respectively, of a liquefiable soil being mistaken for a non-liquefiable soil in the liquefaction potential evaluation.