Empirical Tests of Canonical Nonparametric American Option Pricing Methods

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

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

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

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.

A Jumping Smile

We develop a general method to infer martingale equivalent probability density functions (PDFs) for asset prices using American options prices. The early exercise feature of American options precludes expressing the option price in terms of the PDF of the price of the underlying asset. We derive tight bounds for the option price in terms of the PDF and demonstrate how these bounds, together with observed option prices, can be used to estimate the parameters of the PDF. We infer the distribution for the price of crude oil during the Persian Gulf crisis and find the distribution differs significantly from that recovered using standard techniques.

The valuation of mineral assets and equities is an extremely contentious issue. This thesis investigates the valuation of gold mining initial public offerings through the use of a binomial option pricing approach. The study provides mixed results. Although statistical results indicate the binomial option pricing model is an effective equity valuation technique, an analysis of the estimation errors that arise by comparing the binomial equity value to the market equity value indicates the opposite. This analysis does, however, provide interesting insights into the validity of the model and the circumstances in which the application of the model will be most appropriate. In addition to the contribution made to the academic literature, this thesis also makes a significant practical contribution. One investment professional estimates that real option pricing techniques will become a widely used valuation methodology within the next five years. This thesis is a significant step towards achieving this expectation.

<title>Abstract</title> This study explores the put-call duality under the Truncated Normal Distribution (TND) model in option pricing which assumes that the log-prices of the underlying assets are bounded above and below. The model is compared with famous Black \&amp; Scholes Model which has been consistently been considered as the benchmark model for option pricing. Using the model, we derived a model-consistent put-call duality identity and applied it in the construction of an objective function that improves the accuracy and consistency of option pricing. This is in conjunction with the various modifications that have been put forward to "improve" the option price. The purpose of this study was to extend Zhu \&amp; He's work by studying the the put-call duality relation of the proposed model and compare it with the results obtained using the classical Black \&amp; Scholes (BS) Model. Similarly, this study investigated the implied volatility of the proposed model and compared it with the BS model. From the duality constraint calibration, the proposed model performed significantly better than the traditional Black-Scholes model. A numerical analysis was done using real market data for S\&amp;P 500 stock price data retrieved on June 27, 2025. Model performance check such as AIC, BIC and pricing errors indicated that outstanding performance of the TND model. From the methodology and numerical analysis, there is evidence that the duality relation holds for the Truncated Normal Distribution model as is the case with the Black \&amp; Scholes model. The implied volatility indicated that the TND model was sensitive for in-the-money and out-of-the money options. Therefore, the enforcement of the duality constraint in the pricing of European option improved the market consistency ensures that no arbitrage opportunities exist in the market that cab provide enormous risk-free earnings in the market.

This paper investigates the approximated arbitrage bounds of option prices in an incomplete market setting and draws implications for option pricing and risk management. It gives consideration to periods of global financial crisis and European sovereign debt crisis. To this end, we employ the gain-loss ratio method combined with the market-implied risk-neutral distribution calculated by binomial tree to investigate the options price bounds. Our implied gain-loss bounds of option prices are preference-free and parametric-free to avoid the misspecification error of subjective choice on the benchmark model of gain-loss ratio, and consequently, greatly reduce model risk and market risk. The empirical results show that there are option prices breaking the gain-loss bounds, even after taking into account the market information. This means that a good risk management technique and good-deal investment opportunities exist if the implied binomial tree is used as a benchmark model in the gain-loss bounds.

Chapter 9 - Option pricing

The paper describes an alternative options pricing method which uses a binomial tree linked to an innovative stochastic volatility model. The volatility model is based on wavelets and artificial neural networks. Wavelets provide a convenient signal/noise decomposition of the volatility in the nonlinear feature space. Neural networks are used to infer future volatility from the wavelets feature space in an iterative manner. The bootstrap method provides the 95% confidence intervals for the options prices. Market options prices as quoted on the Chicago Board Options Exchange are used for performance comparison between the Black‐Scholes model and a new options pricing scheme. The proposed dynamic volatility model produces as good as and often better options prices than the conventional Black‐Scholes formulae.

We use additive processes to price options on the Standard and Poor's 500 index (SPX) for the sake of comparison of pricing performance across both model class and family of time-one distribution. Each of the additive processes in this study is defined using one of the following: subordination, Sato's (2002) construction of self-similar additive processes from self-decomposable distributions, or both. We find that during the year 2005: (1) for a given family of time-one distributions, four-parameter self-similar additive models consistently yielded lower pricing errors than those of four-parameter subordinated, and time-inhomogeneous additive models, (2) for a given class of additive models, the time-one marginal given by the normal inverse Gaussian distribution consistently yielded lower pricing errors than those of the variance gamma distribution. Market and model benchmarks for the additive models under consideration are obtained via the bid-ask spreads of the options and Lévy stochastic volatility model prices, respectively.

Factorization of European and American Option Prices under Complete and Incomplete Markets

We develop a general method for estimating the implied, martingale equivalent, probability density function (PDF) for futures prices from American options prices. The early exercise feature of American options precludes expressing the price of the option in terms of the PDF. There exist tight bounds for the price of American options in terms of the PDF. We demonstrate how these bounds, together with observed option prices, can be used to estimate the parameters of the PDF. We estimate the distribution for crude oil during the Persian Gulf crisis and find the distribution differs significantly from that recovered using standard techniques.

The calibration of volatility models from observable option prices is a fundamental problem in quantitative finance. The most common approach among industry practitioners is based on the celebrated Dupire's formula [6], which requires the knowledge of vanilla option prices for a continuum of strikes and maturities that can only be obtained via some form of price interpolation. In this paper, we propose a new local volatility calibration technique using the theory of optimal transport. We formulate a time continuous martingale optimal transport problem, which seeks a martingale diffusion process that matches the known densities of an asset price at two different dates, while minimizing a chosen cost function. Inspired by the seminal work of Benamou and Brenier [1], we formulate the problem as a convex optimization problem, derive its dual formulation, and solve it numerically via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm. The solution effectively reconstructs the dynamic of the asset price between the two dates by recovering the optimal local volatility function, without requiring any time interpolation of the option prices.