Polynomial models in finance | EPFL Graph Search

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.

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

Pricing interest rate, dividend, and equity risk

Sander Félix M Willems

This thesis studies the valuation and hedging of financial derivatives, which is fundamental for trading and risk-management operations in financial institutions. The three chapters in this thesis deal with derivatives whose payoffs are linked to interest rates, equity prices, and dividend payments. The first chapter introduces a flexible framework based on polynomial jump-diffusions (PJD) to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. Option prices are approximated efficiently using a moment matching technique based on the principle of maximum entropy. An extensive calibration exercise shows that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 index dividend futures and dividend options, and Euro Stoxx 50 index options. The second chapter revisits the problem of pricing a continuously sampled arithmetic Asian option in the classical Black-Scholes setting. An identity in law links the integrated stock price to a one-dimensional polynomial diffusion, a particular instance of the PJD encountered in the first chapter. The Asian option price is approximated by a series expansion based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however numerical experiments show that the bias can safely be ignored in practice. The last chapter presents a non-parametric method to construct a maximally smooth discount curve from observed market prices of linear interest rate products such as swaps, forward rate agreements, or coupon bonds. The discount curve is given in closed form and only requires basic linear algebra operations. The method is illustrated with several practical examples.

EPFL2019

Three Essays on Asset Pricing

Emmanuel Leclercq

In the first chapter,which is a joint work with Mathieu Cambou and Philippe H.A. Charmoy, we study the distribution of the hedging errors of a European call option for the delta and variance-minimizing strategies. Considering the setting proposed by Heston (1993), we assess the error distribution by computing its moments under the real-world probability measure. It turns out that one is better off implementing either a delta hedging or a variance-minimizing strategy, depending on the strike and maturity of the option under consideration. In the second paper, which is a joint work with Damir Filipovic and Loriano Mancini, we develop a practicable continuous-time dynamic arbitrage-free model for the pricing of European contingent claims. Using the framework introduced by Carmona and Nadtochiy (2011, 2012), the stock price is modeled as a semi-martingale process and, at each time t , the marginal distribution of the European option prices is coded by an auxiliary process that starts at t and follows an exponential additive process. The jump intensity that characterizes these auxiliary processes is then set in motion by means of stochastic dynamics of Itô's type. The model is a modification of the one proposed by Carmona and Nadtochiy, as only finitely many jump sizes are assumed. This crucial assumption implies that the jump intensities are taken values in only a finitedimensional space. In this setup, explicit necessary and sufficient consistency conditions that guarantee the absence of arbitrage are provided. A practicable dynamic model verifying them is proposed and estimated, using options on the S&P 500. Finally, the hedging of variance swap contracts is considered. It is shown that under certain conditions, a variance-minimizing hedging portfolio gives lower hedging errors on average, compared to a model-free hedging strategy. In the third and last chapter, which is a joint work with Rémy Praz, we concentrate on the commodity markets and try to understand the impact of financiers on the hedging decisions. We look at the changes in the spot price, variance, production and hedging choices of both producers and financiers, when the mass of financiers in the economy increases. We develop an equilibrium model of commodity spot and futures markets in which commodity production, consumption, and speculation are endogenously determined. Financiers facilitate hedging by the commodity suppliers. The entry of new financiers thus increases the supply of the commodity and decreases the expected spot prices, to the benefits of the end-users. However, this entry may be detrimental to the producers, as they do not internalize the price reduction due to greater aggregate supply. In the presence of asymmetric information, speculation on the futures market serves as a learning device. The futures price and open interest reveal different pieces of private information regarding the supply and demand side of the spot market, respectively. When the accuracy of private information is low, the entry of new financiers makes both production and spot prices more volatile. The entry of new financiers typically increases the correlation between financial and commodity markets.

EPFL2014

Pricing interest rate, dividend, and equity risk

Sander Félix M Willems

EPFL2019

Three Essays on Asset Pricing

Emmanuel Leclercq

EPFL2014

Numerical methods for option pricing: polynomial approximation and high dimensionality

Francesco Statti

EPFL2019