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With the development of mathematical applications for stock and energy markets, the subject of financial mathematics has attracted more attention from mathematicians in recent years. The main topics of my work have focused on the stock and energy markets, in particular, the electricity market. The first part of my thesis introduces stochastic modeling and derivatives pricing with pure and jump-diffusion models for energy markets, the second part develops a method to value an exotic option on stock markets and the third part demonstrates my financial toolbox for financial applications. Due to the lack of the analytic evaluation formulas for options in most applications, various numerical approaches have been applied in this thesis such as the binomial tree approach, and finite differences for solving partial differential equations.

This paper explores option-implied information measures for optimal portfolio allocation. We introduce two state variables constructed from option prices. The first state variable is the risk-premium on the risky asset and the second variable is the market price of risk. We also explore a lognormal distribution, a mixture of lognormal distributions, and a binomial tree for constructing the implied risk-neutral density function. Using a combination of statistical and economic measures applied to a portfolio given by the 1-month US Treasury bill and the S&P 500 Index we show the good performance of option-implied information measures for optimal portfolio allocation.

International audience; The function solution to the functional scheme derived from the Binomial tree financial model with local volatility converges to thesolution of a diffusion equation of type ht(t, x)+ x2σ2(t,x) hxx(t, x) = 0 as the number of discrete dates n → ∞. Contrarily to classical numerical methods, in particular finite difference methods, the principle is only based on a discretization in time. We establish the uniform convergence in time of the scheme and provide the rate of convergence when the payoff function is not necessarily smooth as in finance. We illustrate the convergence result and compare its performance to the finite difference and finite element methods by numerical examples.

The aerospace industry is a rich source of innovation and novel technologies. However, structural drivers, such as product complexity and stringent performance requirements, inhibit the uptake of new technologies and present challenges to portfolio management specific to technology-rich environments. Although project risk is often managed through staged funding processes associated with intermediate milestones, this process is typically decoupled from portfolio valuation. In this paper, we use a simple real options binomial tree model to value the National Aeronautics and Space Administration portfolio funded through the Small Business Innovation Research program, accounting quantitatively for the funding stages and future opportunities accompanying technology infusion. For the 2009 portfolio, the average net project investment is $166k, suggesting the important role that early-stage funding plays in launching innovation efforts. We study the staged architecture to offer insight to managers of commercial and public portfolios alike.

We develop a general framework for statically hedging European-style options with nonstandard terminal payoffs which can be applied to mixed static-dynamic and semi-static hedges for many path dependent exotic options. This framework provides a new model-free method of derivatives pricing that builds on recent advances in transform-based numerical approaches. The goal is achieved by separating the hedging and pricing problems to obtain model-free replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoffs functions is accomplished by computing a set of "hedge coefficients" for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.

We construct a discrete-time option valuation model capable of taking into consideration multiple exercises and stochastic interests rates under a generalized Brennan–Rubinstein framework, and further apply the Geske and Johnson (1984) method for the valuation of American options. For implementation, we use only the once- and twice-exercisable option values to approximate American option values. Our numerical results show that the effects of stochastic interest rates are very large, particularly for out-of-the-money American put options with a high-risk underlying asset price. For options with a high-risk return on asset as the underlying assets, the effects of stochastic interest rates are negligible, even for options with long-term maturity. Finally, the sign of correlation between asset prices and bond prices plays an important role in determining whether the values of American put options under stochastic interest rates are larger or smaller than those under constant interest rates.

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.

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.

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

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.

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.

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.

Purpose– The purpose of this paper is to examine the risk neutral and non-risk neutral pricing of Singapore Real Estate Investment Trusts (S-REITs) via comparing the average of the individual ratios (of deviation between expected and observed closing price/observed closing price) with the ratio (of standard deviation/mean) for closing prices via the binomial options pricing tree model.Design/methodology/approach– If the ratio (of standard deviation/mean) ratio > the ratio (of deviation between expected and observed closing price/observed closing price), then the deviation of closing prices from the expected risk neutral prices is not significant and that the S-REIT is consistent with risk neutral pricing. If the ratio (of deviation between expected and observed closing price/observed closing price) is greater, then the S-REIT is not consistent with risk neutral pricing.Findings– Capitacommercial Trust (CCT), Capitamall Trust (CMT) and Keppel Real Estate Investment Trust (REIT) have large positive differences between the two ratios (39.86, 30.79 and 18.96 percent, respectively), implying that these S-REITs are not trading at risk neutral pricing. Suntec REIT has a small positive difference of 2.35 percent between both ratios, implying that it is trading at risk neutral pricing. Ascendas REIT has the largest negative difference between the two ratios at −4.24 percent, to be followed by Mapletree Logistics Trust at −0.44 percent. Both S-REITs are trading at risk neutral pricing. The analysis shows that CCT, CMT and Keppel REIT exhibit risk averse pricing.Research limitations/implications– Results are consistent with prudential asset allocation for viable S-REIT portfolio investing but that not all these S-REITs exhibit strong market efficiency in their pricing.Practical implications– Pricing may be risk neutral over a certain period but investor sentiments, fear of risks and speculative activities could affect an S-REIT’s risk neutrality.Social implications– With enhanced risk diversification activities, the S-REITs should attain risk neutral pricing.Originality/value– Virtually no research of this nature has been undertaken for S-REITS.

In the literature there appear various kinds of binomial trees for pricing options on stocks under geometric Brownian motions (GBMs) with known cash dividends. The aim of this paper is to compare the performance of the existing binomial trees in aspect of the convergence rates, which are usually used to measure precisely how fast the approximate values converge to the exact one, and to give a theoretical proof of the convergence rates for the interpolation binomial trees which are based on a model that excludes the arbitrage possibilities. Also the paper extends the studies to the regime-switching models with known cash dividend payment.

American exercise presents difficulties for option valuation. For an in-the-money option, it becomes necessary at each point in time to consider whether to exercise or to hold on, based on expectations about the stock price at option expiration and also about the optionality value of potential exercise on every date when exercise will be possible in the future. An “Asian” option payoff presents its own valuation problems because the payoff is based on the arithmetic average of correlated lognormal prices, which is not lognormal. Adding stochastic volatility makes both of these problems much harder. But in this article, Russo and Staino are able to develop a lattice technique that deals with all three of these difficulties. The stochastic variable that the tree is built around is the variance, while both the current asset price and the running average of past prices to be included in calculating the payoff at expiration are carried along as auxiliary variables, in the form of sets of values at each node that span the range of possible values along all of the paths that lead to that node. Although there are several approximations in the procedure, accuracy is excellent, and execution is relatively fast.

Purpose– The purpose of this paper is to highlight how the basic binomial option pricing model (BOPM) might be used by regulators to help formulate rules, prior to financial crisis, that help prevent loan overstatement by banks in emerging market economies undergoing financial crises.Design/methodology/approach– The paper draws on the theory of soft budget constraints (SBC) to construct a simple model in which banks overstate loans to minimize losses. The model is used to illustrate how guarantees of bailout assistance (BA) (to banks) by crisis stricken countries’ financial authorities may encourage banks to overstate loans and delay the implementation of IFRS for loan valuation. However, the model also illustrates how promises of BA may be depicted as binomial put options which provide banks with the option of either: reporting loan values on poor projects accurately and receiving the loans’ liquidation values; or, overstating loans and receiving the guaranteed BA. An illustration is also provided of how authorities may use this representation to help minimize bank loan overstatement in periods of financial crisis. In order to provide an illustration of how the option value of binomial assistance may evolve during a financial crisis, the model is generalized to the Mexican financial crisis of the late 1990s. During this period, Mexican authorities’ guarantees of BA to the nation’s largest banks encouraged those institutions to overstate loans and delay the implementation of (previously adopted) international “best practices” based loan valuation standards.Findings– Application of the model to the Mexican financial crisis provides evidence that, in spite of Mexico’s “official” 1997 adoption of international “best accounting practices” for banks, “iron clad” guarantees of BA by the country’s financial authorities to Mexico’s largest banks provided those institutions with an incentive to knowingly overstate loans in the late 1990s and early 2000s.Research limitations/implications– The model is compared against only one country in which the BA was directly infused into banks’ loan portfolios. Thus, as conceived, it is directly applicable to crisis countries in which the bailout took this form. However, the many quantitative variations of SBC models as well as recent studies which have applied the binomial model to other forms of bailout (e.g. direct purchases of bank shares by authorities) suggest that the model could be modified to accommodate different bailout scenarios.Practical implications– The model and application show that guaranteed BA can be viewed as a put option and thatex-anteregulatory policies based on the correct valuation of the BA as a binomial option might prevent banks from overstating loans.Social implications– Use of the binomial or similar approaches to valuing BA may help regulators to determine the level of BA that will not encourage banks to overstate the value of their loans.Originality/value– Recent research has used the BOPM to value, on anex-postbasis, the BA which appears on the balance sheet of institutions which have been rescued. However, little research has advocated the use of this type of model to help prevent, on anex-antebasis, the overstatement of loans on poor projects.

In this paper, we derive a pricing formula for arithmetic Asian options by using the Edgeworth series expansion. Our pricing formula consists of a Black-Scholes-Merton type formula and a finite sum with the estimation of the remainder term. Moreover, we present explicitly a method to compute each term in our pricing formula. The hedging formulas (greek letters) for the arithmetic Asian options are obtained as well. Our formulas for the long lasting question on pricing and hedging arithmetic Asian options are easy to implement with enough accuracy. Our numerical illustration shows that the arithmetic Asian options worths less than the European options under the standard Black-Scholes assumptions, verifies theoretically that the volatility of the arithmetic average is less than the one of the underlying assets, and also discovers an interesting phenomena that the arithmetic Asian option for large fixed strikes such as stocks has higher volatility (elasticity) than the plain European option. However, the elasticity of the arithmetic Asian options for small fixed strikes as trading in currencies and commodity products is much less than the elasticity of the plain European option. These findings are consistent with the ones from the hedgings with respect to the time to expiration, the strike, the present underlying asset price, the interest rate and the volatility.

We develop a numerical method based on the Cox et al. (1979) binomial tree option valuation approach that can accommodate arbitrarily complex capital structures with varying debt maturities and seniorities, as well as preferred stock and warrants. The method provides straightforward valuation for common bond market features such as convertibility and prepayment options, as well as sinking fund provisions, that have proven challenging to model analytically.

We introduce to the literature an intelligent lattice search algorithm to efficiently locate the optimal exercise boundary for American options. Lattice models can be accelerated by incorporating intelligent lattice search, truncation, and dynamic memory. We reduce computational runtime from over 18 minutes down to less than 3 seconds to estimate a 15,000-step binomial tree where the results obtained are consistent with a widely acclaimed literature. Delta and Implied Volatility can also be accelerated relative to standard models. Lattice estimation, in general, is considered to be slow and not practical for valuing large books of options or for promptly rebalancing a risk neutral portfolio. Our technique transforms standard binomial trees and renders them to be at least on par with commonly used analytical formulae. More importantly, intelligent lattice search can be tweaked to reach varying levels of accuracy with different step size, while conventional analytical formulae are less flexible. TOPICS:Options, derivatives

In this short paper, in order to price occupation-time options, such as (double-barrier) step options and quantile options, we derive various joint distributions of a mixed-exponential jump-diffusion process and its occupation times of intervals.

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows a local volatility model. The asymptotics for out-of-the-money, in-the-money, and at-the-money cases are derived, considering both fixed strike and floating strike Asian options. The asymptotics for the out-of-the-money case involves a non-trivial variational problem which is solved completely. We present an analytical approximation for Asian options prices, and demonstrate good numerical agreement of the asymptotic results with the results of Monte Carlo simulations and benchmark test cases in the Black-Scholes model for option parameters relevant in practical applications., Comment: 42 pages, 2 figures

This article provides a general accelerated recursive integration method for pricing American options based on stochastic volatility and double jump processes. Our proposed model is a generalization of the recursive integral representation method. American option prices can be evaluated by the sum of a corresponding European option price and an early exercise premium integral. Numerical results show that our proposed method is efficient and accuracy in pricing American options with stochastic volatility and double jump processes. © 2015 Wiley Periodicals, Inc. Jrl Fut Mark

The interest rate risk is an important factor in the valuation of timer options. Since the valuation of timer options with interest rate risk is a four-dimensional problem, the dimensionality curse causes tremendous difficulty in finding analytic solutions to the pricing of timer options. In this paper, a fast approximate analytic method is developed to price power style timer options with Vasicek interest rate model. The valuation of timer options with interest rate risk is formulated as a four-dimensional partial differential equation (PDE) using Δ-hedging approach. A dimension-reduction technique is then proposed to reduce the four-dimensional PDE into a two-dimensional nonlinear PDE. A perturbation approach is developed to solve the reduced two-dimensional nonlinear PDEs and then an explicit approximate analytic formula for the timer option is obtained. In particular, explicit approximate analytic formulas for timer options under both Heston and Hull–White models are further derived. Numerical examples of pricing timer options under the above two models are provided. Both the approximate analytic method and the crude Monte Carlo simulation method are used for the examples. The numerical results show that prices of timer options by both methods are close and the approximate analytic method is much faster than the crude Monte Carlo method.

This paper presents a novel Monte Carlo method for option pricing that is based on a general equilibrium model. The advantage of the method compared to the standard risk-neutral pricing approach is that it does not require the specification of a market price of risk, making the method particularly suitable for pricing in incomplete markets. The method produces a strongly consistent estimator for the option price which exhibits the same error convergence rate as the standard risk-neutral pricing Monte Carlo approach. For illustration, the procedure is applied to the pricing of options under stochastic volatility.

Office leases are generally agreed upon for extended terms, with possible options to leave or to renew in favor of the tenant. Tenants who have no options during the life of their lease expect to pay a lower rent than those who do. In this letter, we built up a conceptual framework based on binomial tree for the pricing of options embedded in a lease contract. Results show that lease options are dependent upon market rents volatility.

In recent years there has been a remarkable growth of multi-asset options. These options exhibit sensitivity to the volatility of the underlying assets, as well as to their correlations. The call versus call is a product commonly used to trade correlation within the inter-dealer broker markets. The buyer of correlation buys a European call on the equally weighted basket option and sells a weighted average of European calls on each asset. In this case, the following important question arises: Is the information provided by equally weighted basket options enough to price other European multi-asset exotic derivatives such as worst-of or outperformance options? This article investigates this issue under a stochastic correlation framework. Importantly, this article shows that, when pricing multi-asset exotic derivatives, matching the prices of European equally weighted basket options, quoted in the market, does not guaranty the absence of model risk even in the case where the exotic payoff is observed only at maturity.

Purpose – The purpose of this paper is to investigate empirically the forecasting performance of jump-diffusion option pricing models of (Merton and Bates) with the benchmark Black–Scholes (BS) model relative to market, for pricing Nifty index options of India. The specific period chosen for this study canvasses the extreme up and down limits (jumps) of the Indian capital market. In addition, equity markets keep on facing high and low tides of financial flux amid new economic and financial considerations. With this backdrop, the paper focuses on finding an impeccable option-pricing model which can meet the requirements of option traders and practitioners during tumultuous periods in the future. Design/methodology/approach – Envisioning the fact, the all option-pricing models normally does wrong valuation relative to market. For estimating the structural parameters that governs the underlying asset distribution purely from the underlying asset return data, we have used the nonlinear least-square method. As an approach, we analyzed model prices by dividing the option data into 15 moneyness-maturity groups – depending on the time to maturity and strike price. The prices are compared analytically by continuously updating the parameters of two models using cross-sectional option data on daily basis. Estimated parameters then used to figure out the forecasting performance of models with corresponding BS and market – for pricing day-ahead option prices and implied volatility. Findings – The outcomes of the paper reveal that the jump-diffusion models are a better substitute of classical BS, thus improving the pricing bias significantly. But compared to jump-diffusion model of Merton’s, the model of Bates’ can be applied more uniquely to find out the pricing of three popularly traded categories: deep-out-of-the-money, out-of-the-money and at-the-money of Nifty index options. Practical implications – The outcome of this research work reveals that the jumps are important components of pricing dynamics of Nifty index options. Incorporation of jump-diffusion process into option pricing of Nifty index options leads to a higher pricing effectiveness, reduces the pricing bias and gives values closer to the market. As the models have been tested in extreme conditions to determine the dominant effectuality, the outcome of this paper helps traders in keeping the investment protected under normal conditions. Originality/value – The specific period chosen for this study is very unique; it canvasses the extreme up and down limits (jumps) of the Indian capital market and provides the most apt situation for testifying the pricing competitiveness of the models in question. To testify the robustness of models, they have been put into a practical implication of complete cycle of financial frame.

In this paper, we analyze offshore outsourcing contracts using the trinomial model, which is very useful in analyzing real options associated with offshoring projects. Earlier studies by Grenadier (1995) and Pashley, Krishnaswamy and Gilbert (1997) used the option pricing model to analyze lease contracts and debt contracts respectively. So far, however, no studies address offshore outsourcing contracts. Gopal, Mukhopadhyay, Krishnan, and Sivaramakrishnan (2003), Ethiraj, Kale, Krishnan, and Singh, (2004), Gopal and Koka (2010), and Gopal and Koka (2012) found through empirical studies a significant relationship between profits and type of contracts, and price and types of contracts respectively. But they do not provide a theoretical basis for their findings. In this paper we provide a theoretical basis for such a relationship employing the real options analysis using trinomial option pricing model.

In this article we propose a novel approach to reducing the computational complexity of various approximation methods for pricing discrete time American or Bermudan options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation, we propose a multilevel low biased estimate for the price of the option. It turns out that the resulting complexity gain can be of order $\varepsilon^{-1}$ with $\varepsilon$ denoting the desired precision. The performance of the proposed multilevel algorithms is illustrated by a numerical example.

Recently the entropy-based valuation of European options ( Stutzer, 1996 ) has been extended to American option pricing. In this paper, we improve the pricing accuracy by incorporating informative risk-neutral moments (RNMs), which are recovered from a set of market-available option data, as constraints into the entropy framework. With these RNMs, an appropriate risk-neutral measure close enough to the correct one is achieved. An adjusted least-squares algorithm is then utilized to determine the optimal exercising strategy. The results based on simulations and empirical analysis demonstrate that our method can price American options rather accurately and significantly outperforms the benchmark methods.

Using intraday data from the KOSPI 200 Index options, we examine the pricing performance of alternative option pricing models. For comparison, we consider the Black and Scholes (Journal of Political Economy, 81, 1973, 637) model, a simple traders’ rule known as the ad hoc Black-Scholes model, the deterministic volatility model, the stochastic volatility model, and the stochastic volatility with jumps model. Contrary to the findings of Jackwerth and Rubinstein (Recovering stochastic processes from option prices), Li and Pearson (A “horse race” among competing option pricing models using S&P 500 Index options), and Kim (Journal of Futures Markets, 29, 2009, 999) using daily data, we find that the most complicated model, namely the stochastic volatility with jumps model, shows the best performance for pricing the KOSPI 200 Index options. Overall, our evidence from intraday data indicates that the traders’ rules do not dominate mathematically more sophisticated models.

This paper studies the valuation and hedging problems of forward-start rainbow options (FSROs). By combining the characteristics of both multiple assets and forward-start feature, this new type of derivative has many potential applications, for instance, to incorporate the reset provision in rainbow options for investors or hedgers or design more effective executive compensation plans. The main contribution of this paper is a novel martingale pricing technique for options whose payoffs are associated with multiple assets and time points. Equipped with this technique, the analytic pricing formula and the formulae of the delta and gamma of the FSRO are first derived. We conduct numerical experiments to verify these formulae and examine the characteristics of the FSRO’s price and Greek letters. To demonstrate the importance and general applicability of the proposed technique, we also apply it to deriving the pricing formula for the discrete-sampling lookback rainbow options.

We consider the valuation of options with stressed-beta in a reduced form model. Under this two-state beta model, we provide the analytic pricing formulae for the European options and American options as the integral forms. Specifically, we provide the integral representation of the early exercise premium of an American put option. We use the quadrature method to evaluate the integral forms and we measure the performance of our pricing framework comparing the benchmarks set by the trinomial tree method. It turns out that our pricing framework with the quadrature methods are computationally efficient and accurate. We also calibrate the market data successfully.

Numerical solution methods for option pricing fall into two broad classes, either the lattice framework or Monte Carlo simulation. Monte Carlo simulation is problematic for American options, while the Binomial and Trinomial approaches have difficulty incorporating stochastic jumps. The probability density at expiration that is generated by a lattice typically has different skewness and kurtosis than is embedded in option market prices. The Binomial can be fitted imposing a terminal density with the right moments, using the Johnson system of distributions for example, but the problem still remains that the number of distinct nodes to calculate increases quadratically with the number of time steps. This article proposes the “Willow Tree” method, which combines a small number of large time steps that connect to many possible next period nodes, with the constraint that the terminal density, constructed from the Johnson system, has the same mean, volatility, skewness, and kurtosis as the risk neutral density extracted from prices in the options market. The result is a much faster and more accurate valuation than may be achieved through existing approaches.

We present an original Probabilistic Monte Carlo (PMC) model for pricing European discrete barrier options and compound real options. On the basis of Monte Carlo (MC) simulation, for barrier options the PMC model computes the probability of not crossing the barrier for knock-out options and crossing the barrier for knock-in options. This probability is then multiplied by an average sample discounted payoff of a plain vanilla option that has the same inputs as the barrier option but barrier-free, and to which we have applied a filter. We test the consistency of our model with an analytical solution (Merton, 1973; Reiner and Rubinstein, 1991) adjusted for discretization by Broadie et al (1997) and a naive numerical model using MC simulation presented by Clewlow and Strickland (2000). Our study shows that the PMC model accurately prices barrier options. Moreover, the idea behind the method is simple and can be applied to the pricing of complex derivatives, easing the valuation step significantly: we illustrate the versatility of the PMC model in pricing sequential compound real options. Market participants needing to select a reliable, versatile and simple numerical method for pricing options with embedded features will find our article appealing.

Public rental housing (PRH) in China is mainly invested by the government at present. The huge capital demand brings it great pressure and a series of problems appear meanwhile. Public&ndash, private partnership (PPP) has been regarded as a way to solve the funding dilemma of PRH. However, the PRH project is not attractive for the private sector since the expected profit seems unsatisfactory based on traditional valuation methods. To improve this situation, this paper proposed an investment valuation model from a real option perspective. For the private sector, three types of options, including deferral option, abandonment option, and expansion option, were identified during the concession period of a PRH PPP project. On this basis, a two-stage binomial tree model was constructed for estimating the investment value. Then, the proposed model was tested in a hypothetical example of a typical PRH PPP project in Chongqing, China. The result shows that great potential value can be excavated through flexible strategies and adaption to uncertainties. This paper provides a deep analysis on the gaps of the real option application in public housing investment assessment, which is meaningful for improving the supply efficiency and financial sustainability of PRH.

Single and double barrier options on more than one underlying with stochastic volatility are usually priced via Monte Carlo simulation due to the non-existence of closed-form solutions for their value. In this paper, for a special dependence structure, the prices of some two-asset barrier derivatives, like double-digital options and correlation options can be derived analytically using generalized Fourier transforms and some conditions on the characteristic functions. We study the influence of the various parameters on these prices and show that these formulas can be easily and quickly computed. We also extend our approach to further allow for a random correlation structure.

Purpose – This paper aims to demonstrate the practical application of real options analysis to the evaluation of multistage projects, using an example involving a commercial real estate development. Design/methodology/approach – The approach demonstrated builds on static discounted cash flow (DCF) analysis and requires knowledge of only the binomial option pricing model. Findings – Real options analysis can be implemented in a spreadsheet and only one parameter – the volatility of the price of the completed project – needs to be estimated in addition to those required for static DCF analysis. The approach described can be used to evaluate a project at any stage of development, which is especially useful when the suspension of partly completed projects is under consideration. Originality/value – The paper shows how to carry out real options analysis of complex multistage development projects using straightforward valuation tools, making an important project evaluation technique more readily available to practitioners.

In this article, we consider the problem of pricing lookback options in certain exponential Levy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Levy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Levy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A tra...

We propose a stochastic dynamic program (SDP) for valuing options on stock-index futures. SDP accommodates European- as well as American-style options, and price limits on the underlying futures contracts. Our numerical investigation shows convergence, robustness, and efficiency. SDP presents some advantages and disadvantages with respect to the binomial tree and finite differences, and stands as a viable alternative to these classic numerical methodologies for option valuation. Our empirical investigation, which focuses on American options on the S&P 500 futures contract, is almost perfect for implicit volatilities, but somewhat mitigated for historical volatilities. In volatile markets, we recommend short time windows for the volatility estimation step. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:1185–1201, 2014

No mercado de gás natural (GN), muitos contratos incorporam flexibilidades no volume, conhecidas como opções de swing, as quais concedem ao titular a opção de exercer o direito de receber volumes maiores ou menores, de acordo com o mercado. Neste artigo, o preço do GN é a principal fonte de incerteza e foi modelado com um processo estocástico seguindo o modelo de dois fatores de Schwartz e Smith (2000) ao qual foi incorporada a sazonalidade trimestral. Para estimar os preços à vista do GN, usando os preços dos contratos futuros do Henry Hub, implementou-se o filtro de Kalman. O apreçamento da opção foi realizado através do modelo de árvore binomial bivariável desenvolvido por Hahn e Dyer (2011). O valor da opção de swing foi positivo nos dois casos analisados, mostrando que essa opção tem valor e deve ser analisada para inclusão nos contratos de GN. As características do contrato analisado foram as mesmas especificadas em Jaillet, Ronn e Tompaidis (2004). In the natural gas (NG) market, contracts incorporate flexibility in the volume of the product. These contracts are known as swing options. Such contracts allow the option holder to exercise the right to receive greater or smaller amounts of NG contracted in accordance with market. Variation in the price of NG, which is the main source of uncertainty, was modeled in this study as a stochastic process using the two-factor model of Schwartz and Smith (2000), which was incorporated into assessment of the quarterly seasonality. To estimate NG spot prices using the Henry Hub prices of futures contracts traded on the NYMEX, it was necessary to implement the Kalman filter. The pricing of the option was conducted using the binomial tree model bi-variable developed by Hahn and Dyer (2011). The value of the swing option was positive in both cases analyzed, indicating that this option should be considered for inclusion in NG contracts. The characteristics of the analysis were the same as those specified in Jaillet, Ronn and Tompaidis (2004).

The binomial model is a workhorse for valuing options with early exercise possibilities, including options with American or Bermudan exercise and barrier options. The procedure is straightforward when the underlying asset does not pay dividends, and it can be readily adapted to a payout at a fixed proportional rate. But discrete dividends cause trouble because they make the tree “splinter” rather than recombining at each time step. If the tree recombines, an up move followed by a down move produces the same asset price as a down move followed by an up move. But if the intermediate date corresponds to an ex-dividend date and the same fixed amount is subtracted from the up and down nodes, the tree no longer recombines at the subsequent step. The number of nodes in a non-recombining tree increases at a geometric rate, quickly leading to unreasonably long execution times. A number of alternative procedures have been developed to deal with this problem. In this article, Areal and Rodrigues adopt a new approach, accepting the splintering of the binomial lattice but then applying several techniques to accelerate the calculations. They show that their procedures are faster and more accurate than the current methods that force the tree to recombine despite discrete dividend payouts.

PurposeThe purpose of this paper is to construct option pricing models for real estate development by considering and incorporating institutional arrangements, direct interactions and financial constraints in the model. It extends the application of real option theory from the framework borrowed from financial option pricing, and considers the case where a development company has restrictions from outside environment and financial constraint. It explores the effects of these additional practical factors on real asset project value and development timing. This paper makes contributions to bridge the theoretical models and practical applications.Design/methodology/approachReal estate development is modelled in the binomial option pricing framework with the considerations of time‐to‐build, foregone rent if delaying, institutional environment and capital budgeting. The investment timings are derived from the models and sensitivity analysis is conducted to explore the effects of these factors.FindingsApart from the factors in traditional option pricing theory, this paper confirms that the contractual covenants, positive synergies between properties and financial status of the firm, which enhance or restrict real flexibility embedded in the development land, influence project value and investment timing. Numerical examples illustrate the effects of these factors. It is argued that the valuation of real options should place emphasis on industry‐specific characteristics and start from the perspective of the firm rather than individual options.Practical implicationsThe models constructed in this paper and the results can be directly used in the practical real estate development.Originality/valueThis paper incorporates many practical factors in real estate development which are not investigated in previous studies. It values the option project from the firm perspective rather than project perspective as previous studies. It also shows the effects of institutional arrangement and firm factors on project value and development timing.

The Wang transform allows for a simple, yet intuitive approach to pricing options with underlying based on geometric Brownian motion. This paper shows how the approach by Hamada and Sherris can be used to price some exotic options. Examples showing the convergence of the Wang price to the Black–Scholes price for a Margrabe option, a geometric basket option and an asset-or-nothing option are given. We also take a look at the range of prices achievable using the Wang transform for these options.

The willow tree algorithm, first developed by Curran in 1998, provides an efficient option pricing procedure. However, it leads to a large bias through Curran’s sampling strategy when the number of points at each time step is not large. Thus, in this paper, a new sampling strategy is proposed. Compared with Curran’s sampling strategy, the new strategy gives a much better estimation of the standard normal distribution with a small number of sampling points. We then apply the willow tree algorithm with the new sampling strategy to price path-dependent options such as American, Asian and American moving-average options. The numerical results illustrate that the willow tree algorithm is much more efficient than the least-squares Monte Carlo method and binomial tree method with higher precision.

This paper proposes a new simulation method for pricing Bermudan derivatives that is applicable to problems where the transition density of the underlying asset price process is known analytically. We assume that the owner can exercise the option at a finite, although possibly large, number of exercise dates. The method is computationally efficient for high-dimensional problems and is easy to apply. Its efficiency stems from our use of quasi-Monte Carlo techniques, which have proven effective in the case of European derivatives. The valuation of a Bermudan derivative hinges on the optimal exercise strategy. The optimal exercise decision can be reduced to the evaluation of a series of conditional expectations with respect to different distributions. These expectations can be approximated by sampling from just a single distribution at each exercise point. We provide a theoretical basis for the selection of this distribution and develop a simple approximation that has good convergence properties. We...

It is acknowledged by the leading researchers that the fundamental force behind the emergence of advanced option pricing models was the result of abundant empirical research analysis, leading to the fact that the asset return distribution is non-log-normal. It obviously resulted in strong moneyness and maturity pricing biases of Black–Scholes (BS). The concept defined the assertive distributional assumptions for Nifty (benchmark index of India), because the stated view supported the idea and its implications majorly focused on establishing the well-protected alternative model. The core concept of almost all the option pricing models was substantially indicating towards the flexible distributional structure. Such behaviour not only correlated underlying stock returns and its volatility but also subdued the level of skewness and kurtosis. Therefore, this article endeavours to investigate empirically the out-of-sample moneyness–maturity forecasting performance of deterministic volatility option pricing models during the recent waves of economic imbalance. It forecasts S&P CNX Nifty 50 index options of India against significant BS option pricing model and also attempts to find out forecasting performances of a family of deterministic volatility option pricing models, namely, Practitioner Black-Scholes (PBS), Gram–Charlier (GC) and Constant Elasticity of Variance (CEV). The accuracy of models’ prices testify to the relative market price of Nifty index options using error metrics. For the purpose of this research, the prices are compared analytically through continuously updating the parameters of the models using cross-sectional option data, almost on a daily basis. The underlying focus of the article emphasizes on how to investigate the parameters of various models, pertaining specifically to a period of turbulence in the Indian economy. Eventually, while giving verdict to the results deduced, we measure out that the PBS model has smaller out-of-sample valuation errors in pricing Nifty Index options than the CEV and GC. However, it remains challenging to root out price bias completely out of all models we have framed out and applied.

We present a numerical method for pricing Bermudan options on a large number of underlyings. The asset prices are modeled with exponential time-inhomogeneous jump-diffusion processes. We improve the least-squares Monte Carlo method proposed by Longstaff and Schwartz introducing an efficient variance reduction scheme. A control variable is obtained from a low-dimensional approximation of the multivariate Bermudan option. To this end, we adapt a model reduction method called proper orthogonal decomposition (POD), which is closely related to principal component analysis, to the case of Bermudan options. Our goal is to make use of the correlation structure of the assets in an optimal way. We compute the expectation of the control variable by either solving a low-dimensional partial integro-differential equation or by applying Fourier methods. The POD approximation can also be used as a candidate for the minimizing martingale in the dual pricing approach suggested by Rogers. We evaluate both approaches in numerical experiments.

The Binomial model and similar lattice methods are workhorses of practical derivatives valuation. But returns processes more realistic than lognormal diffusions with constant parameters easily create difficulties for them. One of the most important extensions of the Black-Scholes paradigm is to allow stochastic volatility, but even nonstochastic timevarying volatility destroys the important property that the tree recombines, which limits the growth in the number of nodes as time advances. Stochastic volatility introduces a second random variable, which then requires adding another dimension to the tree, under the constraint that the return and volatility changes must maintain the same degree of correlation as in the data. The Heston model features correlation in return and volatility shocks, but building it into a lattice is tricky. In this article, Ruckdeschel, Sayer, and Szimayer develop a lattice method that begins with a binomial tree for the volatility and a trinomial tree for stock price, and then connects them in such a way that the empirical degree of correlation between return and volatility is maintained. Efficiency relative to existing methods is increased, and in some cases it is possible to improve performance further by matching higher moments as well.