Binomial options pricing model

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting. The binomial model was first proposed by William Sharpe in the 1978 edition of Investments (), and formalized by Cox, Ross and Rubinstein in 1979 and by Rendleman and Bartter in that same year. For binomial trees as applied to fixed income and interest rate derivatives see . Use of the model The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American opt

Numerical methods for option pricing: polynomial approximation and high dimensionality

Francesco Statti

Options are some of the most traded financial instruments and computing their price is a central task in financial mathematics and in practice. Consequently, the development of numerical algorithms for pricing options is an active field of research. In general, evaluating the price of a specific option relies on the properties of the stochastic model used for the underlying asset price. In this thesis we develop efficient and accurate numerical methods for option pricing in a specific class of models: polynomial models. They are a versatile tool for financial modeling and have useful properties that can be exploited for option pricing. Significant challenges arise when developing option pricing techniques. For instance, the underlying model might have a high-dimensional parameter space. Furthermore, treating multi-asset options yields high-dimensional pricing problems. Therefore, the pricing method should be able to handle high dimensionality. Another important aspect is the efficiency of the algorithm: in real-world applications, option prices need to be delivered within short periods of time, making the algorithmic complexity a potential bottleneck. In this thesis, we address these challenges by developing option pricing techniques that are able to handle low and high-dimensional problems, and we propose complexity reduction techniques. The thesis consists of four parts: First, we present a methodology for European and American option pricing. The method uses the moments of the underlying price process to produce monotone sequences of lower and upper bounds of the option price. The bounds are obtained by solving a sequence of polynomial optimization problems. As the order of the moments increases, the bounds become sharper and eventually converge to the exact price under appropriate assumptions. Second, we develop a fast algorithm for the incremental computation of nested block triangular matrix exponentials. This algorithm allows for an efficient incremental computation of the moment sequence of polynomial jump-diffusions. In other words, moments of order 0, 1, 2, 3... are computed sequentially until a dynamically evaluated criterion tells us to stop. The algorithm is based on the scaling and squaring technique and reduces the complexity of the pricing algorithms that require such an incremental moment computation. Third, we develop a complexity reduction technique for high-dimensional option pricing. To this end, we first consider the option price as a function of model and payoff parameters. Then, the tensorized Chebyshev interpolation is used on the parameter space to increase the efficiency in computing option prices, while maintaining the required accuracy. The high dimensionality of the problem is treated by expressing the tensorized interpolation in the tensor train format and by deriving an efficient way, which is based on tensor completion, to approximate the interpolation coefficients. Lastly, we propose a methodology for pricing single and multi-asset European options. The approach is a combination of Monte Carlo simulation and function approximation. We address the memory limitations that arise when treating very high-dimensional applications by combining the method with optimal sampling strategies and using a randomized algorithm to reduce the storage complexity of the approach. The obtained numerical results show the effectiveness of the algorithms developed in this thesis.

EPFL2019

Polynomial models in finance

Damien Edouard Ackerer

This thesis presents new flexible dynamic stochastic models for the evolution of market prices and new methods for the valuation of derivatives. These models and methods build on the recently characterized class of polynomial jump-diffusion processes for which the conditional moments are analytic. The first half of this thesis is concerned with modelling the fluctuations in the volatility of stock prices, and with the valuation of options on the stock. A new stochastic volatility model for which the squared volatility follows a Jacobi process is presented in the first chapter. The stock price volatility is allowed to continuously fluctuate between a lower and an upper bound, and option prices have closed-form series representations when their payoff functions depend on the stock price at finitely many dates. Truncating these series at some finite order entails accurate option price approximations. This method builds on the series expansion of the ratio between the log price density and an auxiliary density, with respect to an orthonormal basis of polynomials in a weighted Lebesgue space. When the payoff functions can be similarly expanded, the method is particularly efficient computationally. In the second chapter, more flexible choices of weighted spaces are studied in order to obtain new series representations for option prices with faster convergence rates. The option price approximation method can then be applied to various stochastic volatility models. The second half of this thesis is concerned with modelling the default times of firms, and with the pricing of credit risk securities. A new class of credit risk models in which the firm default probability is linear in the factors is presented in the third chapter. The prices of defaultable bonds and credit default swaps have explicit linear-rational expressions in the factors. A polynomial model with compact support and bounded default intensities is developed. This property is exploited to approximate credit derivatives prices by interpolating their payoff functions with polynomials. In the fourth chapter, the joint term structure of default probabilities is flexibly modelled using factor copulas. A generic static framework is developed in which the prices of high dimensional and complex credit securities can be efficiently and exactly computed. Dynamic credit risk models with significant default dependence can in turn be constructed by combining polynomial factor copulas and linear credit risk models.

EPFL2017

Equilibrium Models for Derivatives Markets with Frictions

Yuan Zhang

This thesis develops equilibrium models, and studies the effects of market frictions on risk-sharing, derivatives pricing, and trading patterns. In the chapter titled "Imbalance-Based Option Pricing", I develop an equilibrium model of fragmented options markets in which option prices and bid-ask spreads are determined by the nonlinear risk imbalance between dealers and customers. In my model, dealers optimally exploit their market power and charge higher spreads for deep out-of-the-money (OTM) options, leading to an endogenous skew in both prices and spreads. In stark contrast to theories of price pressure in option markets, I show how wealth effects can make customers' net demand for options be negatively correlated with option prices. Under natural conditions, the skewness risk premium is positively correlated with the variance risk premium, consistent with the data. In the chapter titled "The Demand for Commodity Options", we develop a simple equilibrium model in which commercial hedgers, i.e., producers and consumers, use commodity options and futures to hedge price and quantity risk. We derive an explicit relationship between expected futures returns and the hedgers' demand for out-of-the-money options, and show that the demand for both calls and puts are positively related to expected returns, and the relationship is asymmetric, tilted towards puts. We test and confirm the model predictions empirically using the commitment of traders report from CFTC. In the chapter titled "Electronic Trading in OTC Markets vs. Centralized Exchange", we model a two-tiered market structure in which an investor can trade an asset on a trading platform with a set of dealers who in turn have access to an interdealer market. The investor's order is informative about the asset's payoff and dealers who were contacted by the investor use this information in the interdealer market. Increasing the number of contacted dealers lowers markups through competition but increases the dealers' costs of providing the asset through information leakage. We then compare a centralized market in which investors can trade among themselves in a central limit order book to a market in which investors have to use the electronic platform to trade the asset. With imperfect competition among dealers, investor welfare is higher in the centralized market if private values are strongly dispersed or if the mass of investors is large.

EPFL2018