Binomial Tree Approach Research Articles

Carbon option pricing based on uncertain fractional differential equation: A binomial tree approach

In this paper, we use a Caputo–Hadamard uncertain fractional differential equation (UFDE) to describe the change of carbon emission rights price. Based on a binomial tree approach, a portfolio is constructed by buying options and shorting the underlying assets in a certain proportion. The pricing formulas for carbon finance European option are presented, and the algorithms are designed to formulate the price of carbon finance American option without explicit pricing formulas. Besides, a moment estimation method is used to obtain the parameter values in Caputo–Hadamard UFDE, and a method is provided to simulate the observation data of solution of Caputo–Hadamard UFDE. Finally, the specific Caputo–Hadamard UFDE with mean-reverting process is applied to simulating the change of carbon emission rights price in Chinese carbon market, and the feasibility of the proposed carbon option pricing method is illustrated by some numerical examples.

The impact of a convertible bond on the leverage of a firm

AbstractThe important issue of the impact of convertible bonds on the capital structure of firms has not attracted much attention. The authors use a binomial tree approach to price a convertible bond with convertibility throughout its life to calculate its debt and equity contribution to the capital structure. The authors will show that treatment of the convertible bonds by simply assigning its straight bond value portion to the debt, and the equity option portion to the owner's equity is not consistent with the foundations of finance theory and drastically overstates the debt component of these bonds and their contribution to the firm's leverage.

A Binomial Tree Approach to Pricing Vulnerable Option in a Vague World

The aim of this paper is pricing the vulnerable options in a vague world. Due to the vulnerability of financial markets and the economy environment in the real world, investors cannot always have precise information about firm value and default recovery rate in vulnerable option pricing. Therefore, following the framework of Klein in 1996, a fuzzy binomial tree pricing model is derived by modelling the firm value and default recovery rate as fuzzy numbers. The numerical results show that the precise information assumption about the firm value and recovery rate in Klein model may lead to underestimate the credit risk on the values of vulnerable options. This study aims to provide insights for future research on defaultable options pricing under imprecise market information.

Computation of Greeks in jump-diffusion models using discrete Malliavin calculus

In the last decade, many studies have investigated the computation of Greeks (sensitivity of options) for European options, American options, exotic options, and so on using Malliavin calculus. Moreover, many studies have derived Greeks using jump-diffusion models. In this paper, we investigate a new computation scheme to derive Greeks in a jump-diffusion model using discrete Malliavin calculus. This method enables us to obtain Greeks for European options using the binomial tree approach.

Computation of Greeks Using Binomial Tree

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.

Efficient willow tree method for European-style and American-style moving average barrier options pricing

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.

Discrete Malliavin calculus and computations of greeks in the binomial tree

This paper proposes new methods for computation of greeks using the binomial tree and the discrete Malliavin calculus. In the last decade, the Malliavin calculus has come to be considered as one of the main tools in financial mathematics. It is particularly important in the computation of greeks using Monte Carlo simulations. In previous studies, greeks were usually represented by expectation formulas that are derived from the Malliavin calculus and these expectations are computed using Monte Carlo simulations. On the other hand, the binomial tree approach can also be used to compute these expectations. In this article, we employ the discrete Malliavin calculus to obtain expectation formulas for greeks by the binomial tree method. All the results are obtained in an elementary manner.

Options based reserve procurement strategy for wind generators - Using binomial trees

Wind and solar PV are the most mature forms of renewable energy and are integral to our clean energy strategy. Their intermittency poses technical and economic challenges. Technical challenges are load balancing, frequency regulation, etc. Economic challenges include providing least costing load balancing (reserves) services to these intermittent generators. This paper considers a future electricity market situation wherein wind generators are required to forecast and bid to supply energy. The future electricity market treats wind generators similar to conventional generators penalizing for underproduction and pays poorly for overproduction. An intra-day ( 24 h) secondary market is proposed in this paper where a wind generator and a reserve provider can bilaterally trade in reserves. Reserves are traded in the market by purchasing options to buy reserves at predetermined strike prices by paying premiums. These reserves include call and put options to address underproduction and overproduction. A binomial tree approach for estimating possible deviation from the forecast value is used. A new optimization formulation is proposed that uses binomial tree option pricing technique to determine optimal values of strike prices and premiums for call and put options. Two examples illustrate the benefits of the proposed idea.

Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations

The recombining binomial tree approach, which has been initiated by Cox et al. (J Financ Econ 7: 229–263, 1979) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393–430, 1990) and Hull and White (J Financ Quant Anal 25: 87–100, 1990a), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65–85, 2010) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price.

An FFT-network for Lévy option pricing

This paper develops a simple network approach to American exotic option valuation under Lévy processes using the fast Fourier transform (FFT). The forward shooting grid (FSG) technique of the lattice approach is then generalized to expand the FFT-network to accommodate path-dependent variables. This network pricing approach is applicable to all Lévy processes for which the characteristic function is readily available. In other words, the log-value of the underlying asset can follow finite-activity or infinite-activity Lévy processes. With the powerful computation of FFT, the proposed network has a negligible additional computational burden compared to the binomial tree approach. The early exercise policy and option values in the continuation region are determined in a way very similar to that of the lattice approach. Numerical examples using American-style barrier, lookback, and Asian options demonstrate that the FFT-network is accurate and efficient.

Option-based valuation of mortgage-backed securities

Pure econometric approaches to pricing mortgage-backed securities (MBSs) - principal pricing vehicles used by financial practitioners - fail to capture their true risks. This point was powerfully driven home by the global financial crisis. Since prior to the crisis default rates of MBSs were quite modest, econometric pricing models systematically underestimated the possibility of default. As a result, MBSs were severely overvalued. It is widely believed that the global crisis was largely triggered by incorrect valuation of mortgage-backed securities. In the aftermath, it is important to revisit the foundations for pricing MBSs and to pay much closer attention to default risk. This paper introduces a comprehensive model for valuation of fixed-rate pass-through mortgagebacked securities in a simple option-based framework. In the model, we use bivariate binomial tree approach to simultaneously model prepayment and default options. Our simulation results demonstrate that the proposed model has sufficient flexibility to capture the two principal risks.

Empirical tests of canonical nonparametric American option‐pricing methods

AbstractAlcock and Carmichael (2008, The Journal of Futures Markets, 28, 717–748) introduce a nonparametric method for pricing American‐style options, that is derived from the canonical valuation developed by Stutzer (1996, The Journal of Finance, 51, 1633–1652). Although the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method. We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. Although the Alcock and Carmichael method fails to outperform a traditional implied‐volatility‐based Black–Scholes valuation or a binomial tree approach, our extension generates significantly lower pricing errors and performs comparably well to the implied‐volatility Black–Scholes pricing, in particular for out‐of‐the‐money American put options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:509–532, 2010

Empirical Tests of Canonical Nonparametric American Option Pricing Methods

Alcock and Carmichael (2008) introduce a nonparametric method for pricing American style options that is derived from the canonical valuation developed by Stutzer (1996). While the statistical properties of this nonparametric pricing methodology have been studied in a controlled simulation environment, no study has yet examined the empirical validity of this method.We introduce an extension to this method that incorporates information contained in a small number of observed option prices. We explore the applicability of both the original method and our extension using a large sample of OEX American index options traded on the S&P100 index. While the Alcock and Carmichael (2008) method fails to outperform a traditional implied volatility based Black-Scholes valuation or a Binomial Tree approach, our extension generates signi cantly lower pricing errors and performs comparably well to the implied-volatility Black-Scholes pricing, in particular for out-of-the-money American put options.

Option Implied Moments: An Application to Nikkei 225 Futures Options

This paper proposes an integrated process to recover the moments of the risk-neutral distribution using a Gram-Charlier expansion series and Rubinstein's implied binomial tree approach. The advantage of using the implied tree approach is that it accounts for the possibility of early exercise of American options. We apply the method to American-style options on Nikkei 225 futures. We then demonstrate how to use the implied moments for trading options.

Is Volatility Risk Priced in the Option Market?

Rubinstein (1994) shows evidence of a significant time pattern in the shape of the volatility smile after the crash of 1987 and proposes an implied binomial tree approach to overcome the empirical limitations of the Black and Scholes model. This approach, and more generally the class of generalized deterministic volatility models, is based on the assumption that the local volatility of the underlying asset is a known function of time and of the path and level of the underlying asset price. In these economies, options are redundant assets. We use this observation as a testable restriction and ask three questions. First, is the observed dynamics of the smile consistent with deterministic volatility models? Second, if volatility is stochastic, so that two assets cannot dynamically complete the market, is volatility also priced and if so how important is to model explicitly the price of volatility in the design of risk management strategies? We address this question by testing if the returns on the underlying and on at-the-money options span the asset prices in the economy or if we need additional information from returns on other options or the riskfree rate. Third, are there any differences in the spanning properties of the option market before and after the 1987 market crash? We cast these questions in terms of martingale restrictions on the pricing kernel and conduct tests based on daily S&P500 index options data from April 1986--December 1995. All our tests suggest that both in- and out-of-the-money options are needed for spanning purposes. This finding is even stronger in the postcrash period and suggests that returns on away-from-the-money options are driven by at least one additional economic factor compared to returns on at-the-money options. This finding is inconsistent with the implications of deterministic volatility models based on generalized deterministic volatility. The finding is consistent with explanations of the smile in which volatility is stochastic and priced in equilibrium and with models in which away-from-the-money options are used in equilibrium by a different specialized clientele, such as portfolio insurers, subject to different budget constraints and/or portfolio objectives than the typical investor in at-the-money options.

A Pricing Model for American Options with Stochastic Interest Rates

textabstractIn this paper we introduce a new methodology to price American put options under stochastic interest rates. The method is a combination of an analytic approach and a binomial tree approach. We construct a binomial tree for the forward risk adjusted tree and calculate analytically the expected early exercise value in each point. For American puts with stochastic interest rates the correlation between the stock price process has different influences on the European option values and the early exercise premiums. This results in a nonmonotonic relation between this correlation and the American put option value. Furthermore, there is evidence that the early exercise premium due to stochastic interest rates is much larger than established before by other researchers.