Showing papers on "Binomial options pricing model published in 2022"

A Quantum Algorithm for Pricing Asian Options on Valuation Trees

Abstract: We develop a novel quantum algorithm for approximating the price of a discrete floating-strike Asian option based on an underlying valuation tree. The paths of the tree are encoded in bit-representation into a qubit register, where quantum state preparation is used to load the corresponding distribution onto the states. We implement the expectation value of the option pricing formula as a composition of the price probabilities, the payout and an indicator function, mapping their respective values to amplitudes of additional qubits. Thus, the underlying no longer has to be discretized into the same bit values for different times, resulting in smaller quantum circuits. The algorithm may be used with quantum amplitude estimation, enabling a quadratic speed-up over classical Monte Carlo methods.

Embedding a net present value analysis into a binomial tree with a real option analysis

Abstract: A net present value (NPV) analysis with associated financial statements is embedded into a binomial tree that allows the analysis to evolve through time and, at any given moment in time, to vary with different states of the world. The modeling of the cash flows within the binomial tree requires a variable component based on relationships with a specifically defined random variable and a component that is fixed at a given moment in time. Although after-tax operating cash flows are used in this treatment with a real options example, the binomial tree method can be applied to other definitions of project cash flows.

An introduction to Monte Carlo-Tree (MC-Tree) method

Abstract: The article aims to introduce concepts in option pricing and risk management. Pricing and risk management is one of the fundamental problems in financial mathematics. Then readers may explore further to understand how to use mathematical models in pricing and risk management. More specifically, our research introduces a new method called Monte Carlo-Tree (MC-Tree), for option pricing and risk management with high accuracy.

Asian Options in a Market with High Volatility: Perspective and Evidence from Zoom and Peloton

Abstract: Contemporarily, uncertainties strongly affect the stock market, especially for the options market that fluctuates dramatically. On this basis, Asian options have become a favorable option for investors due to their low-risk characteristic. In this paper, two recently highly volatile assets, Zoom (ZM) and Peloton (PTON), were selected as investigation targets underlying Asian options. Specifically, both the Black-Scholes method and Monte-Carlo method were applied to price the Asian options for ZM and PTON. Based on calculations with the Black-Scholes formula and simulation with the Monte-Carlo method, we demonstrated that the prices of Asian options are approximately half of the corresponding Vanilla options. Besides, the expected returns are also nearly half of the corresponding Vanilla options. Moreover, according to the comparison between the calculations and simulation, the Black-Scholes formula eventually goes inaccurate while the strike price goes large or less than the original price. However, if the Monte-Carlo method is applied with enough times of simulations, the price of the Asian option will be reflected accurately. Overall, the Asian option is indeed a better choice for prudent investors in a highly volatile market.

Discrete-Time Model Pricing of American Option Early Exercise Premia

Abstract: The characteristics of the early exercise of American options are full of research implications, and it is interesting to define the premia for the early exercise of different American options. In this paper, the option value is calculated via the risk-neutral measure, which define the payoff of an option at a certain time period through a discrete model called binomial tree and then discount the payoff by risk-neutral method to determine the option value. This paper first considered the conditions for early exercise of American options at the nodes, and then find out the law of premiums by taking the slope of the premium function.

Investigation on the Mechanism of American Put Option Pricing under Binomial Consideration

Abstract: American options are style of options which allows the holder to exercise their rights at any time before and including the expiration date. Different from European options, that only allows the holder to exercise their right at the expiration date, American options seems to bring the holders with more freedom. There are two common types of options: calls and puts. In this investigation, we will be focusing on American put options specifically. Puts give the buyer the right, but not the obligation for selling the underlying asset at the strike price in the contract. Moreover, options may have their own pricing theory, that is a model to estimate the value of options by assigning a price, which is known as premium. With the theory, American option holders could make the decision of holding the option or not. Therefore, the pricing model of an American option seems to be significant and important for the final decision.

Pricing exotic options in the incomplete market: An imprecise probability method

Abstract: This article considers a novel exotic option pricing method for incomplete markets. Nonparametric predictive inference (NPI) is applied to the option pricing procedure based on the binomial tree model allowing the method to evaluate exotic options with limited information and few assumptions. As the implementation of the NPI method is greatly simplified by the monotonicity of the option payoff in the tree, we categorize exotic options by their payoff monotonicity and study a typical type of exotic option in each category, the barrier option and the look-back option. By comparison with the classic binomial tree model, we investigate the performance of our method either with different moneyness or varying maturity. All outcomes show that our model offers a feasible approach to price the exotic options with limited information, which makes it can be utilized for both complete and incomplete markets.

The impact of negative interest rates on the pricing of options written on equity: a technical study for a suitable estimate of early termination

Abstract: This work aims to investigate the main problems that impact the pricing models and the sensitivity measures of American options written on shares without a pay-out, in the presence of negative interest rates with a specific focus on the Monte Carlo method. The first paragraph carries out a review of the anomalies caused by such an odd condition and focuses thereafter on the core topic of the research by treating a wide range of numerical models suitable for unbiased evaluation of the early exercise, thus expanding the existing literature. The two following paragraphs are dedicated to describing the models used for the correct estimation of fair value: binomial lattice models (Cox-Ross-Rubinstein - CRR Tree, Leisen Reimer - LR Tree, Jarrow-Rudd - JR Tree and Tian Tree), trinomial stochastic trees, Finite Difference Method (FDM) scheme and the Longstaff-Schwartz Monte Carlo. Particular attention is paid to this last approach which allows to combine the flexibility of traditional numerical integration schemes for stochastic processes on equity with the estimation of the convenience of exercising the American option ahead of time. After conducting quantitative tests both on pricing and on the estimation of sensitivity measures, the LR Tree was selected as the most performing deterministic algorithm to be compared with the Monte Carlo stochastic technique. The final part of the work focuses on quantifying the valuation gap introduced by negative interest rates in the valuation of American options written on an unprofitable underlying comparing the traditional valuation approach and the deterministic Leisen Reimer model and the Longstaff-Schwartz stochastic model.

Valuation of Index Options.

Abstract: This dissertation examines the valuation of index options. The first chapter analyzes the value of early exercise for an index option, specifically the Standard and Poor's 100 Index (OEX) option. The value is found by estimating European put-call parity and comparing it to the price difference between an American call and put. Zivney (1991) estimated this value for closing prices. The value of early exercise is re-estimated by using bid-ask prices, effectively mitigating the non-synchronous data problem, which may be severe in Zivney's study. The results from using intraday bid-ask prices are compared to last bid-ask and transaction prices. The value of early exercise is added to the Black-Scholes European option model value, and the combination is found to price index options better. The average estimate of early exercise is about 4.1 percent for calls in-the-money and 10.87 percent for puts in-the-money. The second chapter of the dissertation looks at various index proxies. This analysis seeks to discern how arbitragers are capturing arbitrage profits and thereby keeping the options near some equilibrium price. The index is proxied with other index options, the Standard and Poor's 500 Index futures, and small mimicking portfolios of stocks. Arbitrage profits are examined. The benchmark prices are generated with the Black-Scholes option pricing model, the Black-Scholes model plus the value of early exercise found above, the binomial option pricing model, and put-call parity. The mimicking portfolios of stocks produce the poorest profits. This is due to the deviation between the portfolio and the actual index increasing with time. The final section examines price discovery. Using a technique relying on vector error correction models moving averages, it is found that most price discovery is found in the cash OEX index, with a smaller portion occurring in the SPX futures. Virtually no price discovery is found in other options.

Pricing of bond options in India

Abstract: Bond options are widely used for managing interest rate risks. Bond options were introduced in India in December 2019. Various bond option pricing models are used by practitioners in the market. This study prices a European call option on 10-Year G-Sec using three models, the Jamshidian's option pricing formula based on Vasicek's model, the Black's option pricing model and Black, Derman and Toy's no arbitrage model. The results show that the option prices are different for different models. The study concludes that volatility of interest rates is an important consideration in each of the models and needs to be standardised to get comparable results.

Option Pricing: Examples and Open Problems

Abstract: The option pricing problem is equivalent with the hedging problem of the option, i.e. what the writer should do in order to hedge the risk that she undertakes selling a contract and moreover what is the probability of profit selling at a specific price. The probability of profit is also a useful information for the buyer. The hedging strategy should be practically possible for the writer otherwise has no meaning. In this note we will discuss the option pricing problem and in particular the effect of the volatility on the binomial model which is a way to hedge practically a specific option in contrast to every pricing model that assumes rebuilding of the replicating portfolio continuously in time. In order to use the binomial model we have to modified it accordingly as we have seen in a previous paper. We also point out three open problems regarding the binomial option pricing model.

Option Pricing and CVA Calculations using the Monte Carlo-Tree (MC-Tree) Method

Abstract: The binomial tree method and the Monte Carlo (MC) method are popular methods for solving option pricing problems. However in both methods there is a trade-off between accuracy and speed of computation, both of which are important in applications. We introduce a new method, the MC-Tree method, that combines the MC method with the binomial tree method. It employs a mixing distribution on the tree parameters, which are restricted to give prescribed mean and variance. For the family of mixing densities proposed here, the corresponding compound densities of the tree outcomes at final time are obtained. Ideally the compound density would be (after a logarithmic transformation of the asset prices) Gaussian. Using the fact that in general, when mean and variance are prescribed, the maximum entropy distribution is Gaussian, we look for mixing densities for which the corresponding compound density has high entropy level. The compound densities that we obtain are not exactly Gaussian, but have entropy values close to the maximum possible Gaussian entropy. Furthermore we introduce techniques to correct for the deviation from the ideal Gaussian pricing measure. One of these (distribution correction technique) ensures that expectations calculated with the method are taken with respect to the desired Gaussian measure. The other one (bias-correction technique) ensures that the probability distributions used are risk-neutral in each of the trees. Apart from option pricing, we apply our techniques to develop an algorithm for calculation of the Credit Valuation Adjustment (CVA) to the price of an American option. Numerical examples of the workings of the MC-Tree approach are provided, which show good performance in terms of accuracy and computational speed.

Option Pricing: Examples and Open Problems

Abstract: The option pricing problem is equivalent with the hedging problem of the option, i.e. what the writer should do in order to hedge the risk that she undertakes selling a contract and moreover what is the probability of profit selling at a specific price. The probability of profit is also a useful information for the buyer. The hedging strategy should be practically possible for the writer otherwise has no meaning. In this note we will discuss the option pricing problem and in particular the effect of the volatility on the binomial model which is a way to hedge practically a specific option in contrast to every pricing model that assumes rebuilding of the replicating portfolio continuously in time. In order to use the binomial model we have to modified it accordingly as we have seen in a previous paper. We also point out three open problems regarding the binomial option pricing model.

ESG-valued discrete option pricing in complete markets

Abstract: We consider option pricing using replicating binomial trees, with a two fold purpose. The first is to introduce ESG valuation into option pricing. We explore this in a number of scenarios, including enhancement of yield due to trader information and the impact of the past history of a market driver. The second is to emphasize the use of discrete dynamic pricing, rather than continuum models, as the natural model that governs actual market practice. We further emphasize that discrete option pricing models must use discrete compounding (such as risk-free rate compounding of $1+r_f \Delta t$) rather than continuous compounding (such as $e^{r_f \Delta t})$.

Options Pricing Comparison between the Black-Scholes Model and the Binomial Tree Model: A Case Study of American Equity Option and European-style Index Option

Abstract: In recent years, quantitative researchers used a wide range of models to price options, from the Black-Scholes model to more complex models such as the Heston model. This paper aims to analyze the effectiveness of the Black-Scholes model and the Binomial Tree model by using them to price Berkshire Hathaway’s equity options and European-style S&P 100 index options. The method used in this paper is gathering the market data of the options first. Second, using the data gathered to price the options by applying the Black-Scholes and Binomial Tree models. Third, comparing the derived theoretical price with the market price by getting the Sum of Square Errors. Lastly, determining the best model for each type of option. Through this research, the author found that comparing the two models, the Binomial Tree model derives a smaller Sum of Square Errors when pricing European-style index options, and the Black-Scholes model derives a smaller Sum of Square Errors when pricing American equity options. Thus, the Binomial Tree model is a better model to price European-style index options, and the Black-Scholes model is more effective when pricing American equity options.

Asian Chooser Option Pricing Based on Binominal Tree and Monte Carlo Simulation

Abstract: To deal with the increasingly complex financial market transaction scenarios, more and more sophisticated derivatives have been developed. One of them is the Asian chooser option. It is the combination of the Asian option and the chooser option. Firstly, this article introduces the basic idea of Asian options and Chooser options. Combining their traits, this article divides the pricing model of the Asian Chooser Option into 5 steps and figure out its analytic solution based on Binomial Tree Model. Then this article uses Monte Carlo Simulation to validate my analytic solutions using a simple example. The prices of the two methods are similar which indicates my analytic result is reliable.

Basics of Derivative Valuation

Abstract: We define frictionless markets and clarify what we mean by a “fair value” of a financial product in a frictionless market. Then we state and discuss the fundamental axiom in quantitative finance, the “no-arbitrage principle”. We give first applications of this NA principle and thereby derive the put-call-parity-equation and the formula for the fair price of futures. Finally, we provide the first steps towards the valuation of options: We define binomial stock-models, we give the formulas for the fair value of derivatives in such binomial models, and we show how hedging of derivatives in a binomial model is carried out.