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Cliquet-style options in different variants are basic building blocks in select products which are offered by German life insurance companies. We present both an analytical pricing approximation via the central limit theorem and a corresponding control variate Monte Carlo approach for their valuation. The control variate approach turns out to be a good alternative to the integral representation of Bernard and Li (SIAM J Financ Math, 4:353-371, 2013). Further, it can be modified to increase the efficiency of pricing cliquet-style options in the Heston price setting.

We apply the multilevel Monte Carlo method for option pricing problems using exponential Levy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Levy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and $\alpha$ -stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.

Investment decisions in adaptation are usually made under significant uncertainty due to climate change and socio-economic trends. In this study, we propose three ways to incorporate climate and socio-economic uncertainty into the assessment of an adaptation infrastructure designed to cope with flood-risk in the city of Bilbao. First, we use stochastic modelling to estimate the present value of expected damage over a time period, considering that extreme events may increase in the future. Second, we develop an additional calculation that incorporates two risk measures used in financial economics: Value-at-Risk and Expected Shortfall, the latter being a less common but better risk indicator. Third, we illustrate a case of Real Options Analysis (ROA) in which a binomial tree is used to study whether the best decision at present is to invest now or to delay the investment decision.

In this work, the Generalized Binomial Distribution (GBD) combined with some basic financial concepts is applied to generate a model for determining the prices of a European call and put options. To demonstrate the behavior of the option prices (call and put) with respect to variables, some numerical examples and graphical illustration have been given in a concrete setting to illustrate the application of the obtained result of the study. It was observed that when there is an increase in strike prices, it leads to decrease in calls option price C(0) and increase in puts option price P(0). Decrease in interest rate leads to decrease in calls option price P(0), and increase in puts option price P(0), and decrease in expiration date leads to decrease in calls option price C(0) and decrease in puts option price P(0). It was also found that the problem of option price can be approached using Generalized Binomial Distribution (GBD) associated with finance terms.

With concerns regarding global climate change increasing, recent studies on adapting to nonstationary climate change recommended a different planning strategy that could spread risks. Uncertainty in global climate change should be considered in any decision-making processes for flood mitigation strategies, especially in areas within a monsoon climate regime. This study applied a novel planning method called real option analysis (ROA) to an important water resources planning practice in Korea. The proposed method can easily be applied to other watersheds that are threatened by flood risk under climate change. ROA offers flexibility for decision-makers to reflect uncertainty at every stage during the project planning period. We successfully implemented ROA using a binomial tree model, including two real options—delay and abandon—to evaluate flood mitigation alternatives for the Yeongsan River Basin in Korea. The priority ranking of the four alternatives between the traditional discount cash flow (DCF) and ROA remained the same; however, two alternatives that were assessed as economically infeasible using DCF, were economically feasible using ROA. The binomial decision trees generated in this study are expected to be informative for decision-makers to conceptualize their adaptive planning procedure.

Commercial potential in developing countries has always received a great attention from international investors, but this is not the case in Waste-to-Energy sector. Waste-to-Energy is bound up with various uncertainties rooted in its long-term nature therefore incorporating risks regarding political matters in developing countries makes it more complex. The present study substantiates the incompatibility of classic valuation methods in risky projects. Consequently, to deal with the riskiness of Waste-to-Energy investment in less developed countries, the combination of binomial tree analysis and Decoupled NPV is proposed. The hybrid approach is deployed to value a Waste-to-Energy project in Iran, and all evidence attest to the robustness of the method. The contribution of this paper can open up new vistas for investing in Waste-to-Energy industry, thus abating the catastrophic effects of landfill gas emissions.

The pricing of options in the fast mean-reverting stochastic volatility model using the singular perturbation method has received a considerable amount of attention in the last two decades. However, it is not to easy to estimate the accuracy of the approximation if the payoff function is not smooth or bounded, as is the case for European call options. In this article, we introduce a new novel approach for pricing options in the fast mean-reverting stochastic volatility model. Combinations of Fourier analysis and singular perturbation methods enable us to estimate the accuracy easily. We also show that this method allows us to derive the price of European and Bermudan options in the fast mean-reverting stochastic volatility environment with jumps.

Renewable energy sources have become a very important issue due to environmental and sustainability concerns. In addition, most renewable energy electricity generation (RES-E) projects are characterized by considerable uncertainty and sequential decision-making. These projects’ net present value (NPV) is very often close to zero, which renders them a good candidate for the application of real options valuation methodology of investment project appraisal. This paper examines the real options valuation of a potential onshore wind farm project in Serbia. The binomial tree model spans a 15 year period consisting of a 2 year investment period and a 13 year operating period, the first 12 years of which are protected by Feed-in-Tariffs according to local renewable energy regulations. The authors examine a multi-phased compound (nested) path-dependent real option consisting of mutually exclusive options - a sequential option to invest as well as expand, repower, contract and abandon options. Volatility, which is calculated by means of a Monte Carlo simulation according to the logarithmic-present-value-returns approach, shows predominant sensitivity to the capacity factor simulated with the rescaled Weibull's probability distribution. The final binomial tree results show that the proposed sequence of options increases project value by transforming higher risk and lower return in the initial discounted cash flow model, to lower risk and higher return in the RO model. This paper contributes to the existing literature in at least two ways: it presents in-depth analysis of the real option application to the RES-E project and provides decision-makers with sophisticated tool for improving strategic thinking, capital budgeting and decision-making processes.

We propose an integral equation approach for pricing American-style Parisian down-and-out call options under the BlackScholes framework. For this type of options, the knock-out feature is activated only if the underlying asset price continuously remains below a pre-determined barrier for a sufficiently long period of time. As such, the corresponding pricing problem becomes a three-dimensional (3-D) free boundary problem, instead of a two-dimensional (2-D) one as is the case of one-touch barrier options, and this poses a computational challenge. In our approach , we first reduce the 3-D problem to a 2-D one, and then, by applying the Fourier sine transform to the resulting 2-D problem, we can derive a pair of coupled integral equations governing the option price at any given time in terms of (i) the option price at the barrier and (ii) the optimal exercise boundary at that time. This pair of coupled integral equations can be solved using the NewtonRaphson iterative procedure, after which, the option price, the optimal exercise boundary, and the hedging parameters can be obtained in a straightforward manner. A complexity analysis of the method, together with numerical results, show that the proposed approach is robust and significantly more efficient than existing uniform finite difference methods with CrankNicolson timestepping, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight into several interesting properties of the option price and the optimal exercise boundary.

Many financial assets, such as currencies, commodities, and equity stocks, exhibit both jumps and stochastic volatility, which are especially prominent in the market after the financial crisis. Some strategic decision making problems also involve American-style options. In this paper, we develop a novel, fast and accurate method for pricing American and barrier options in regime switching jump diffusion models. By blending regime switching models and Markov chain approximation techniques in the Fourier domain, we provide a unified approach to price Bermudan, American options and barrier options under general stochastic volatility models with jumps. The models considered include Heston, Hull–White, Stein–Stein, Scott, the 3/2 model, and the recently proposed 4/2 model and the α -Hypergeometric model with general jump amplitude distributions in the return process. Applications include the valuation of discretely monitored contracts as well as continuously monitored contracts common in the foreign exchange markets. Numerical results are provided to demonstrate the accuracy and efficiency of the proposed method.

This paper studies the pricing of some multivariate European options, namely Exchange options and Quanto options, when the risky assets involved are modelled by Markov-Modulated Lvy Processes (MMLPs). Pricing formulae are based upon the characteristic exponents by using the well known FFT methodology. We study these pricing issues both under a risk neutral martingale measure and the historical measure. The dependence between the assets components is incorporated in the joint characteristic function of the MMLPs. As an example, we concentrate upon a regime-switching version of the model of Ballotta etal. (2015) in which the dependence structure is introduced in a flexible way. Several numerical examples are provided to illustrate our results.

Increasing penetration of wind energy in Germany challenges the transmission and distribution grid operators to ensure the sustained feed-in of wind energy. Energy storage technologies have become a widely discussed solution to balance intermittent electricity supply and demand. This paper introduces an economic real options-based analysis of a hydrogen wind energy storage system (H2-WESS) deployed adjacent to a nearshore wind farm in northern Germany. The H2-WESS can be used to produce and store hydrogen when feed-in management occurs to avoid the shutdown of wind turbines and when the spot market electricity price falls below the estimated (efficiency-adjusted) market price of hydrogen. Moreover, an H2-WESS can provide negative minute reserve capacity. The modular design of the H2-WESS gives an investor the option to expand its capacity and react to new project developments. A comprehensive simulation model considers all relevant volatile inputs such as stochastic wind conditions, feed-in management events, prices, and minute reserve calls. By means of a Monte Carlo simulation, the annual revenues and their volatility are computed with respect to forecasted technology improvements till 2030. On the basis of the simulation results, a binomial options pricing model is used to design four interdependent binominal trees and to evaluate a Bermuda-type compound expansion option. The decision trees, in which the investor can choose the maximum of the option to either expand to a further H2-WESS stage or to keep the option alive, uses 390 time steps and 76,050 decision nodes each. Every compound decision takes the option of smaller stages into account. The compound expansion option to invest in a 5, 10, 15 or 20 MW H2-WESS has a 15-year expiration time and is calculated to be about €2 million, compared to the net present value of a 5 MW H-WESS of about €-2.45 million. We conclude from the real options analysis that it is crucial for a realistic valuation of energy projects to incorporate the value of managerial flexibility that is influenced by various uncertainties. Compared to conventional power plants, and due to the modular design, the flexibility of the H2-WESS comprises many specific options.

A class of finite volume methods is developed for pricing either European or American options under jump-diffusion models based on a linear finite element space. An easy to implement linear interpolation technique is derived to evaluate the integral term involved, and numerical analyses show that the full discrete system matrices are M-matrices. For European option pricing, the resulting dense linear systems are solved by the generalised minimal residual (GMRES) method; while for American options the resulting linear complementarity problems (LCP) are solved using the modulus-based successive overrelaxation (MSOR) method, where the H+-matrix property of the system matrix guarantees convergence. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of these methods.

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.

The objective of this paper is to detect the arbitrage opportunity and to manage the bubbles in the Korean real estate market based on a binomial American option pricing model with heteroscedasticity. The limitation of previous research where the real options framework was first introduced is its macro-economic implication based on the utilization of the general housing indices. Therefore, in this paper, we extensively apply the model to different residential types and regions. The results suggest that the model can detect the realistic and reasonable trend of housing bubbles and the arbitrage opportunities for different times, residential types, and regions. We also simulate two scenarios to encourage the trades of real estate assets: promotion of early exercise, and the shortened Contract period. Performing arbitrage trading based on these two methods, we discover that both approaches effectively reduce the housing bubbles in all residential types and regions. Specifically, the promotion of early exercise reduces the housing bubble more effectively than the shortened Contract period. Hence, we advocate the utilization of the information obtained from the model to boost the transactions in the Korean real estate market in order to reduce the bubble-related risks and to support sustainable economic growth.

Derivative pricing is about cash flow discounting at the riskfree rate. This teaching has lost its meaning post the financial crisis, due to the addition of extra value adjustments (XVA), which also made derivatives pricing and valuation a very difficult task for investors. This article recovers a properly defined discount rate that corresponds to different collateral and margin schemes. A binomial tree model is developed, enabling end-users to price in counterparty default and funding risk. Coherent XVAs, if needed, naturally result from decomposing the discount rate, and can be computed on the same tree., Comment: 38 pages, 3 figures, 1 table

Financial markets are often modeled using a random walk, for example in the binomial option pricing model which is a discrete version of the Black-Scholes formula. This paper presents an alternative approach to option pricing based on a quantum walk model. The quantum walk, which incorporates superposition states and allows for effects such as interference, was originally developed in physics, but has also seen application in areas such as cognitive psychology, where it is used to model dynamic decision-making processes. It is shown here that the quantum walk model captures key aspects of investor behavior, while the collapsed state captures the observed behavior of markets. The resulting option price model agrees quite closely with the classical random walk model, but helps to explain some observed anomalies. The method also has the advantage that it can be run directly on a quantum computer.

Recent empirical studies in Equity markets show evidence that, while asset log-returns are largely uncorrelated, it is possible to predict with some accuracy their future sign. Such prediction is made over a given forecast horizon based solely on the observed sign of the cumulative log-return over a lookback horizon. This manuscript proposes a methodology to study the impact of such findings on option pricing by embedding into the risk premium the effects of signed path dependence. This is achieved by devising a model-free empirical risk-neutral distribution based on Polynomial Chaos Expansions coupled with stochastic bridge interpolators that includes information from the entire set of observable European call option prices under all available strikes and maturities for a given underlying asset. Under the real-world measure we propose a price dynamics model that is compatible with an asset price process that is largely uncorrelated but still exhibits signed path dependence. The risk premium behaviour is subsequently inferred non-parametrically via a stochastic bridge interpolation that couples the risk neutral Polynomial Chaos Expansion result with the signed path dependence mixture binomial tree dynamic model to obtain a dynamic stochastic model for the implied risk premium process.

In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a &ldquo, flexible&rdquo, version of Joshi&rsquo, s tree, and develop the corresponding convergence theory in the European case: we find a closed form formula for the coefficients of 1/n and 1/n3/2 in the expansion of the error. Then we define several optimized versions of the tree, and find closed form formulae for the parameters of these optimal variants. In a numerical study, we found that in the American case, an optimized variant of the tree significantly improved the performance of Joshi&rsquo, s original split tree.

For the first time, a framework is introduced that allows for a real options analysis to be performed in an EVA/MVA-embedded binomial tree. This framework enhances traditional EVA/MVA analysis so that it can capture the additional value generated through strategic decision making during a project’s life. The EVA calculation is separated into three parts: a variable component, a fixed component in regard to the cash flow, and a fixed component in regard to the cost of capital. A reconciliation of methods shows that the EVA/MVA framework produces the same real option valuation as an equivalent NPV-embedded binomial tree.

The binomial option pricing is developed from first principles. The underlying security is assumed to evolve in discrete steps of time. An option is shown, using the principle of no arbitrage, to be equivalent to a dynamic portfolio composed of the underlying security and risk-free cash. The option price is shown to be equivalent to a random evolution of the option price provided the underlying security has a martingale evolution. In a simple and transparent discrete-time model, the under-pinning of option theory is seen to be based on the binomial probability distribution. The binomial option pricing model, which is based on the binomial random variable, shows the utility of probability theory in option pricing.

This chapter starts with an overview of the zoology of exotic options, i.e., with options that are more complex than plain vanilla ones. Some exotic options can be valued by a modification of the Black-Scholes formula, while for some there are more complicated formulas, developed in the context of the geometric Brownian motion, and the others can be valued only numerically using Monte Carlo simulations, binomial tree techniques, or partial differential equations. For most of the exotic derivatives, it turns out that the geometric Brownian motion model calibrated to value correctly the plain vanilla options might give quite imprecise results. The empirical phenomenon called the volatility smile (or surface) demonstrates that the market does not, in fact, believe in lognormal returns and the volatility constant over time. This fact has led to the development of various alternative stochastic models that try to capture better the behavior of market prices, especially the jumps and stochastic volatilities of the underlying asset returns. We will discuss some of the best-known models in the last section.

Automatic differentiation is now used extensively in finance to determine the greeks of financial securities. Unfortunately when using this technique with binomial trees the second order sensitivities, such as gamma, cannot be calculated due to the interaction of the structure of the lattice and singularities in the derivatives of payoff functions. In this paper we use recent results on the likelihood ratio method on binomial trees to overcome this issue. The algorithm for determining delta via the likelihood ratio method is extremely easy and efficient to implement, and when combined with automatic differentiation yields continuous gammas.

Financial statements and an accompanying NPV calculation are embedded into a binomial tree. This generalization of traditional static NPV analysis allows the financial statements to both evolve through time and, at any given time, to vary with states of the world (similar to a Monte Carlo analysis). Modelling the component cash flows in a tree reveals dynamic detailed structure, leading to a more useful NPV analysis than if only the final cash flow value was modelled in a tree or if component cash flows were modelled without a tree. This dynamic detail provides credible cash flow forecasts that can improve hedging of adverse events and allow for leveraging of beneficial circumstances. The financial statements take the form of pro forma after-tax operating cash flows in this treatment. However, any cash flow model driven by the random variable in the tree and allowing for separate treatment of fixed costs, can be used. The benefits of this technique are illustrated via a real options example.

Electronic Health Records (EHRs) is a key form of healthcare records that attracts a great deal of attention. It is regarded that the sharing of healthcare records is a vital strategy for improving the quality of healthcare services and reducing medical expenses. Presently, the cloud computing paradigm supports many significant characteristics in Electronics Health Records (EHRs). However, in the cloud model, the patients have no longer control over their crucial healthcare records after outsourcing the records to cloud servers, which can lead to severe data integrity related concerns. Therefore, in this manuscript, we introduce a novel secure and efficient cloud based EHR data storage system by constructing Binary Binomial Tree like data structure. Furthermore, we extend the proposed model to attain significant cloud model based public auditing characteristics such as data dynamics, batch auditing, privacy preserving, blockless verification, data traceability, and recoverability. Extensive experiments and results demonstrate that the proposed cloud based EHR system is secure and efficient over other existing strategies.

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler-Maruyama type scheme. We show that our methods allow to obtain efficient and accurate European and American option prices. Numerical experiments are provided, and show the reliability and the efficiency of the algorithms., Comment: arXiv admin note: text overlap with arXiv:1503.03705

I document a sizeable bias that might arise when valuing out of the money American options via the Least Square Method proposed by Longstaff and Schwartz (2001). The key point of this algorithm is the regression-based estimate of the continuation value of an American option. If this regression is ill-posed, the procedure might deliver biased results. The price of the American option might even fall below the price of its European counterpart. For call options, this is likely to occur when the dividend yield of the underlying is high. This distortion is documented within the standard Black&ndash, Scholes&ndash, Merton model as well as within its most common extensions (the jump-diffusion, the stochastic volatility and the stochastic interest rates models). Finally, I propose two easy and effective workarounds that fix this distortion.

Highway infrastructure is critical to the sustainable development of a country. Nonetheless, these capital-intensive projects, especially in build-operate-transfer (BOT) contracts, are pron...

A real options (RO) formulation is proposed for decision-making on the timing to upgrade the seismic performance of existing seaports with increasing throughput demand in earthquake prone areas. The pay-off of the seismic upgrade investment option is estimated based on projected net earnings, repair cost, and downtime for a damaging reference seismic event having a pre-specified annual probability of occurrence. These projections inform a discrete-time RO binomial tree, following the American option valuation framework, which propagates the probability of the reference seismic event assuming Poisson temporal distribution of earthquake occurrence. The net present value of the expected annual payoff of the considered investment is used as an index supporting risk-informed decision-making discounted by the weighted average cost of capital (WACC). Numerical examples pertaining to decision makers with different capital cost, namely port authorities and terminal operators, operating in different economic environments typical of developed and developing countries are furnished to illustrate the applicability of the proposed RO formulation. It is found that high WACC and/or low throughput growth bring the optimal seismic upgrade timing forward, while earthquake consequences and upgrade cost have almost no influence on this timing.

To price options using replication in imperfect markets, both Knightian uncertainty and transaction cost have to be taken into account. In this paper, we put an uncertainty factor into volatility, assume investors minimize the root mean square error of replication when they choose hedging ratio, and derive European option price by a recursive procedure. To avoid high transaction cost caused by continuous hedging, we establish a discrete and binomial replication model considering both uncertainty and transaction cost. Numerical examples imply that option price contains both risk premium and uncertainty premium, and it is an approximately linearly increasing function of transaction cost but a nonlinearly increasing function of uncertainty. Additionally, both uncertainty and transaction cost have effects on the price of the at-the-money option, but they almost have no impact on the price of deeply in-the-money or out-of-the-money options. Empirical analysis of the Shanghai 50ETF options market indicates that the Black–Scholes model tended to underestimate the market price, whereas our model better estimates market prices.

Water Saving Management Contract (WSMC) is believed as an efficient water-saving model and developed in recent years. However, because the high upfront investment cost and great uncertainty of WSMC, the static evaluation methods have defects in evaluating the return on investment of WSMC projects. Real option theory makes up for these defects and is applied in this paper with the expectation to evaluate the uncertainty of a Shared WSMC project and the economic feasibility of investing in the project. First, a new expression of volatility is presented based on the characteristics of Shared WSMC projects. The volatility contains uncertainties in water price, the amount of water-saving, waterborne fee, and operating & maintenance cost simultaneously. Then, it is assumed that the water saving service company is entitled options to abandon and expand the Shared WSMC project during the contract period. Based on the existing model, the exercise prices of the two options are improved. Furthermore, the real option value (ROV) of a Shared WSMC project is solved using the binomial tree option pricing model, and the economic feasibility of the project is comprehensively evaluated by combining with the static net present value and ROV organically. Finally, the Shared WSMC project of the Hebei University of Engineering in China is studied to verify the validity of the proposed method. The obtained results indicate that the economic value of the Shared WSMC project is expanded by the ROV, which means that additional value can be gained through flexible strategies and management in the context of uncertainty. Besides, the additional economic benefits of the two options vary significantly for different Shared WSMC projects. Expand option is particularly essential when Shared WSMC projects with a considerable return on investment. Conversely, the abandon option should be taken seriously. Especially when both options are considered together, the additional economic benefits are sub-additive.

The Black–Scholes model is the rare closed-form formula for pricing options, but its shortcomings are well known. Adding stochastic volatility, American or Bermudan early exercise, non-diffusive jumps in the returns process, and/or other “exotic” payoff features quickly takes one into the realm where approximate answers must by computed by numerical methods. Binomial and other lattice models, Monte Carlo simulation, and numerical solution of the valuation partial differential equation are the standard techniques. They are well known to eventually arrive at as close an approximation to the exact model value as one wants but can require a large amount of calculation and execution time to get there. In this article, the authors present what amounts to the culmination of a series of papers on using quadrature methods to solve option problems of increasing complexity. Quadrature entails modeling and approximating the transition densities between critical points in the option’s lifetime. Depending on the specifics of the option’s payoff, looking at only a few points in time can greatly reduce the amount of calculation needed to price it. For example, a Bermudan option that can be exercised at three possible early dates needs quadrature calculations for only those three dates plus expiration, while other numerical methods require calculations for every date and possible asset price. This article summarizes how to apply quadrature to standard option problems, from plain-vanilla Black–Scholes up to Bermudan and American options under a stochastic volatility jump-diffusion returns process. The procedure can be speeded up even more by use of Richardson extrapolation and caching techniques that allow reuse of results calculated in intermediate steps. The range of derivatives models that can be valued extremely efficiently using quadrature is very broad.

We develop a simple and efficient tree approach for pricing options under stochastic volatility. Our method encompasses the models of Heston, Hull-White, Stein-Stein, α -Hypergeometric, 3/2 and 4/2 models. Numerical results are provided to illustrate the effectiveness of the proposed method.

The implied volatility is the unique volatility value that makes the celebrated Black–Scholes formula yields a traded option’s price. Implied volatilities at varying strike prices and maturities form the implied volatility surface. Empirically, this surface is never flat. The local-volatility (LV) model is an option model that attempts to fit the implied volatility surface. It is popular because the preference freedom of the Black–Scholes model is retained. A tree consistent with the implied volatility surface generated by an LV model is known as implied tree. Past attempts to construct the implied tree, however, are prone to having invalid transition probabilities. In fact, this problem occurs even when the volatility surface is flat as in the Black–Scholes model. This paper unearths a potentially fundamental reason for that failure: the trees contain repelling fixed points. As efficient and valid trees for general LV models remain elusive despite decades of research, this paper turns to separable LV models. An efficient and valid binomial tree is then built for such models. Our novel tree is named the waterline tree because its upper part (the part that is above the water, so to speak) matches the moments of the price, whereas the lower part matches the moments of its logarithmic price. This break from traditional trees ensures that only attracting fixed points remain. Numerical results confirm the excellent performance of the waterline tree.

Trinomial tree model is excelled than binomial tree model in precision and calculation from an example. Triple Tree pricing error is much smaller, the result is simpler. By careful analysis, we conclude the real option value under trinomial tree model.

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

This paper is concerned with the pricing problem of the discrete arithmetic average Asian call option while the discrete dividends follow geometric Brownian motion. The volatility of the dividends model depends on the Markov-Modulated process. The binomial tree method, in which a more accurate factor has been used, is applied to solve the corresponding pricing problem. Finally, a numerical example with simulations is presented to demonstrate the effectiveness of the proposed method.

This paper sets out to provide a general framework for the pricing of average-type options via lower and upper bounds. This class of options includes Asian, basket, and options on the volume-weighted average price. The use of lower and upper bounds is proposed in response to the inherent difficulty in finding analytical representations for the true price of these options and the requirement for numerical procedures to be fast and efficient. We demonstrate that in some cases lower bounds allow for the dimensionality of the problem to be reduced and that these methods provide reasonable approximations to the price of the option.

This paper presents an extension of the double Heston stochastic volatility model by combining Hull-White stochastic interest rates. By the change of numeraire and quadratic exponential scheme, this paper develops a new simulation scheme for the extended model. By combining control variates and antithetic variates, this paper provides an efficient Monte Carlo simulation algorithm for pricing barrier options. Based on the differential evolution algorithm the extended model is calibrated to S&P 500 index options to obtain the model parameter values. Numerical results show that the proposed simulation scheme outperforms the Euler scheme, the proposed simulation algorithm is efficient for pricing barrier options, and the extended model is flexible to fit the implied volatility surface.

As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model. This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option; numerical calculation method is used to approximate the continuous convolution by calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy. By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable. Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.

Traditional methods of investment appraisal, like the Net Present Value, are not able to include the value of the operational flexibility of the project. In this paper, real options, and more specifically the option to expand, are assumed to be included in the project information in addition to the expected cash flows. Thus, to calculate the total value of the project, we are going to apply the methodology of the Net Present Value to the different scenarios derived from the existence of the real option to expand. Taking into account the analogy between real and financial options, the value of including an option to expand is explored by using the binomial options pricing model. In this way, estimating the value of the option to expand is a tool which facilitates the control of the uncertainty element implicit in the project. Keywords: Real options, Option to expand, Binomial options pricing model, Investment project appraisal

Binary Option Pricing Model (BOPM) is one approach that can be utilized to calculate the value of either call or put option. BOPM generally works by building a binomial tree diagram, also known as lattice diagram to explore all possible option values that occur based on the intrinsic price of the underlying asset in a range of specific time period. Due to its basic characteristics BOPM is renowned for its longer lead times in calculating option values while also acknowledged as a technique that can provide an assessment of the most fair option value. Therefore, this study aims to shorten the BOPM completion time of calculating the value of an option by building a computer cluster and presently implement the BOPM algorithm into a form that can be run in parallel processing. As an outcome, a Linux-based computer cluster has been built and conversion of BOPM algorithm so that it can be executed in parallel has been performed and also implemented using Python in this research. Conclusively, results of conducted experiments confirm that the implementation of BOPM algorithm in a form that can be executed in parallel and its application on a computer cluster was able to limit the growth of the completion time whilst at the same time maintain quality of calculated option values.

Moving average options are widely traded in financial markets, but exiting methods for pricing this type of option are too slow. This paper proposes two efficient willow tree methods for pricing European-style and American-style moving average barrier options (MABOs). We first solve the finite-dimensional partial differential equation model for discretely monitored MABOs by willow tree methods, and then compute the value of continuously monitored MABOs by Richardson’s two-point extrapolation. Our new willow tree method employs the interpolation error minimization technique to reduce complexity. The corresponding convergence rate and error bounds are also analyzed. It shows that our proposed methods can provide the same accuracy as the binomial tree approach and Monte Carlo simulation, but require much less computing time. The numerical experiments support our claims.

Binomial tree is a method that can be used to determine price option contracts. In this method, the stock price movement is presented in the form of a tree with each branch representing the probability of the stock price to move up or move down. The purpose of this paper was to determine the price of the options contracts with the American type on Binomial Tree method and compare the three methods that is variance matching, proportional , and risk neutral of determining the value of price option contracts used in Binomial Tree method with Black-Schole method. The result of this research was the value of the options contract using the variance matching more similar with the value of the Black-Scholes contract.

Asset selection and timing decisions are major investment concerns. To resolve these issues simultaneously, a new class of rainbow trend options is proposed. The diversification effect of rainbow options can reduce the importance of asset selection decisions and trend options can mitigate unfavorable effects on market entry and exit decisions. We consider a general framework to facilitate the derivation of analytic pricing formulas for simple, pure, and Asian rainbow trend options using the martingale pricing method. The properties of these options and their Greeks are analyzed. We also investigate the performance of the dynamic delta hedging strategy for issuers of rainbow trend options. Last, this paper explores the applications of rainbow trend options for hedging price risks, designing executive stock options, modifying countercyclical capital buffer proposed by Basel Committee, and acting as control variates of the Monte Carlo simulation.

This paper studies the pricing of Asian options whose payoffs depend on the average value of an underlying asset during the period to a maturity. Since the Asian option is not so sensitive to the value of underlying asset, the possibility of manipulation is relatively small than the other options such as European vanilla and barrier options. We derive the pricing formula of geometric Asian options under the constant elasticity of variance (CEV) model that is one of local volatility models, and investigate the implication of the CEV model for geometric Asian options.

The study focusses on the price convergences, implied volatility structures, and hedging competencies of the discrete tree options pricing models. In addition, the paper also studies the efficiency of optimization techniques involved in the calibration of discrete time models. The process involves matching of quoted market option prices with the model prices – relied on the assumption that there are sufficiently large numbers of liquidly traded options. To minimize the price bias between the model and market, this paper uses Differential Evolution and Simulated Annealing methods of optimization. To benchmark, price statistics of European-style Nifty index options and American-style stock options traded on the National Stock Exchange (the world’s third largest Futures and Options exchange) of India have been used. The concepts of optimization and relative pricing are used to demonstrate the price effectiveness of the tree models with respect to continuous-time classical options pricing model of Black–Scholes–Merton. The outcome of the research reveals that the pricing and hedging errors of forward-inducted implied that binomial and trinomial trees are stable and lower with Simulated Annealing methods of optimization than their backward-inducted counterparts.

In this paper, a general binomial lattice framework, which is both computationally simple and numerically accurate, is developed for pricing real estate derivatives with stochastic interest rate. To obtain a computationally simple binomial tree with constant volatility, the transformation method and the probability density matching approach are introduced. A tilt parameter is then added to the jump movements to obtain smooth convergence. Therefore, the Richardson extrapolation (RE) can be used to enhance the convergence of the discrete binomial lattice models to continuous models when pricing European options. In addition, our smooth convergent models can also be applied to pricing American options.

Pricing options is an important problem in financial engineering. In many scenarios of practical interest, financial option prices associated to an underlying asset reduces to computing an expectation w.r.t.~a diffusion process. In general, these expectations cannot be calculated analytically, and one way to approximate these quantities is via the Monte Carlo method; Monte Carlo methods have been used to price options since at least the 1970's. It has been seen in Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier options using Sequential Monte Carlo' and Jasra, A. \& Del Moral, P. (2011) `Sequential Monte Carlo for option pricing' that Sequential Monte Carlo (SMC) methods are a natural tool to apply in this context and can vastly improve over standard Monte Carlo. In this article, in a similar spirit to Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier options using sequential Monte Carlo' and Jasra, A. \& Del Moral, P. (2011) `Sequential Monte Carlo for option pricing' we show that one can achieve significant gains by using SMC methods by constructing a sequence of artificial target densities over time. In particular, we approximate the optimal importance sampling distribution in the SMC algorithm by using a sequence of weighting functions. This is demonstrated on two examples, barrier options and target accrual redemption notes (TARN's). We also provide a proof of unbiasedness of our SMC estimate.