How to handle the COS method for option pricing

The Fourier Cosine Expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the Greeks. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results

Greeks' pitfalls for the COS method in the Laplace model

The Greeks Delta, Gamma and Speed are the first, second and third derivatives of a European option with respect to the current price of the underlying asset. The Fourier cosine series expansion method (COS method) is a numerical method for approximating the price and the Greeks of European options. We develop a closed-form expression of Speed for various European options in the Laplace model and we provide sufficient conditions for the COS method to approximate Speed. We show empirically that the COS method may produce numerically nonsensical results if theses sufficient conditions are not met

The multidimensional COS method for option pricing

The multidimensional COS method is a numerical tool to price financial options, which depend on several underlyings. The method makes use of the characteristic function $\varphi$ of the logarithmic returns of the underlyings and it is advantageous if the Fourier-cosine coefficients $v_{\boldsymbol{k}}$ of the payoff function are given in closed-form. However, in important cases, neither $\varphi$ nor $v_{\boldsymbol{k}}$ are given analytically but need to be recovered numerically. In this article, we prove the convergence of the multidimensional COS method including numerical uncertainty on $\varphi$ and $v_{\boldsymbol{k}}$. Our analysis helps to understand how the approximation errors on $\varphi$ and $v_{\boldsymbol{k}}$ propagate in the COS method

Validation of machine learning based scenario generators

Machine learning methods are getting more and more important in the development of internal models using scenario generation. As internal models under Solvency 2 have to be validated, an important question is in which aspects the validation of these data-driven models differs from a classical theory-based model. On the specific example of market risk, we discuss the necessity of two additional validation tasks: one to check the dependencies between the risk factors used and one to detect the unwanted memorizing effect. The first one is necessary because in this new method, the dependencies are not derived from a financial-mathematical theory. The latter one arises when the machine learning model only repeats empirical data instead of generating new scenarios. These measures are then applied for an machine learning based economic scenario generator. It is shown that those measures lead to reasonable results in this context and are able to be used for validation as well as for model optimization

Sequential decompositions at their limit

Sequential updating (SU) decompositions are a well-known technique for creating profit and loss (P&L) attributions, e.g., for a bond portfolio, by dividing the time horizon into subintervals and sequentially editing each risk factor in each subinterval, e.g., FX, IR or CS. We show that SU decompositions converge for increasing number of subintervals if the P&L attribution can be expressed by a smooth function of the risk factors. We further consider average SU decompositions, which are invariant with respect to the order or labeling of the risk factors. The averaging is numerically expensive, and we discuss several ways in which the computational complexity of average SU decompositions can be significantly reduced

American and exotic options in a market with frictions

In a market with frictions, bid and ask prices are described by sublinear pricing functionals, which can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model with transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of American options and compute an implied liquidity to describe the bidask spread. This method describes liquidity over time very well, compared to the classical approach of describing bid and ask prices by quoting bid and ask implied volatilities.Peer ReviewedPostprint (author's final draft

Advanced stock price models, concave distortion functions and liquidity risk in finance

Esta tesis consta de tres ensayos. En el primer ensayo, probamos empíricamente el desempeño de los precios de varios modelos financieros avanzados para opciones exóticas. Calibramos seis modelos avanzados para precios de acciones a una serie de datos de mercado reales de opciones europeas en el DAX, el índice de referencia de la Bolsa alemana. A través de una simulación de Monte Carlo, calculamos precios de opciones de barrera para todos los modelos y comparamos los precios modelados con los precios del mercado de las opciones de barrera. El modelo Bates reproduce bien los precios de las opciones de barrera. El modelo BNS sobrevalora y los modelos Lévy con cambio temporal estocástico y con efecto de palanca subestiman los precios de las opciones exóticas. Un análisis heurístico sugiere que el diferente grado de fluctuación de las trayectorias aleatorias de los modelos es el responsable de producir diferentes precios para las opciones de barrera. En el segundo ensayo de esta tesis se examinan medidas de riesgo coherentes y funciones de distorsión cóncavas. Una familia de funciones cóncavas de distorsión es un conjunto de funciones cóncavas y crecientes, con dominio e imagen igual al intervalo unitario. Se usan las funciones de distorsión para definir medidas de riesgo coherentes. Demostramos que cualquier familia de funciones de distorsión que cumpla una ecuación de traslación, puede ser representada por una función de distribución. Una aplicación se puede encontrar en la ciencia actuarial: los principios de primas basados en los momentos son fáciles de entender, pero en general no son monótonos y no se pueden utilizar para comparar los riesgos de diferentes contratos de seguros entre sí. Nuestro teorema de representación permite comparar dos riesgos de seguros entre sí de acuerdo con un principio de primas basado en un momento, definiendo adecuadamente una medida de riesgo coherente. En el último ensayo de esta tesis, investigamos los mercados financieros con fricciones, donde los precios de compra y venta de instrumentos financieros se describen mediante funciones de precios sublineares. Estas funciones pueden definirse recursivamente utilizando medidas de riesgo coherentes. En un modelo binomial y en la presencia de costes de transacción, demostramos la convergencia de los precios de compra y venta para varias opciones europeas y americanas, en particular opciones plain vanillas, asiáticas, lookback y de barrera. Realizamos varios experimentos numéricos para confirmar los hallazgos teóricos. Aplicamos los resultados a los datos de mercado reales de las opciones plain vanilla europeas y americanas y calculamos una liquidez implícita para describir la diferencia de precios de compra y venta. Este método describe muy bien la liquidez en comparación con el enfoque clásico de describir la diferencia entre los precios de compra y venta con las volatilidades implícitas de dichos precios.This thesis consists of three essays. In the first essay, we test empirically the pricing performance of several advanced financial models. We calibrate six advanced stock price models to a time series of real market data of European options on the DAX, a German blue chip index. Via a Monte Carlo simulation, we price barrier down-and-out call options for all models and compare the modelled prices to given real market data of the barrier options. The Bates model reproduces barrier option prices well. The BNS model overvalues and Lévy models with stochastic time-change and leverage undervalue the exotic options. A heuristic analysis suggests that the different degree of fluctuation of the random paths of the models are responsible of producing different prices for the barrier options. The second essay of this thesis discusses the relationship between coherent risk measures and concave distortion functions. A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment based premium principle by defining an appropriate coherent risk measure. In the last essay of this thesis, we investigate financial markets with frictions, where bid and ask prices of financial intruments are described by sublinear pricing functionals. Such functionals can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model in the presence of transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of European and American plain vanilla options and compute an implied liquidity to describe the bid-ask spread. This method describes liquidity over time very well, compared to the classical approach of describing the bid-ask spread by quoting bid and ask implied volatilities

dataAmericanOption.csv

For a time-series of two days, February, 2nd and February, 5th, 2018, the data set contains end-of-day bid and ask prices of 80 plain vanilla, at-the-money American put and call options on the S&P500, or rather the SPDR S&P 500 ETF Trust, an exchange traded fund replicating the S&P500, with maturities ranging from about 3 to 8 month. The option prices were obtained from the Chicago Board Options Exchange via a Bloomberg Terminal

Advanced stock price models, concave distortion functions and liquidity risk in finance

Esta tesis consta de tres ensayos. En el primer ensayo, probamos empíricamente el desempeño de los precios de varios modelos financieros avanzados para opciones exóticas. Calibramos seis modelos avanzados para precios de acciones a una serie de datos de mercado reales de opciones europeas en el DAX, el índice de referencia de la Bolsa alemana. A través de una simulación de Monte Carlo, calculamos precios de opciones de barrera para todos los modelos y comparamos los precios modelados con los precios del mercado de las opciones de barrera. El modelo Bates reproduce bien los precios de las opciones de barrera. El modelo BNS sobrevalora y los modelos Lévy con cambio temporal estocástico y con efecto de palanca subestiman los precios de las opciones exóticas. Un análisis heurístico sugiere que el diferente grado de fluctuación de las trayectorias aleatorias de los modelos es el responsable de producir diferentes precios para las opciones de barrera. En el segundo ensayo de esta tesis se examinan medidas de riesgo coherentes y funciones de distorsión cóncavas. Una familia de funciones cóncavas de distorsión es un conjunto de funciones cóncavas y crecientes, con dominio e imagen igual al intervalo unitario. Se usan las funciones de distorsión para definir medidas de riesgo coherentes. Demostramos que cualquier familia de funciones de distorsión que cumpla una ecuación de traslación, puede ser representada por una función de distribución. Una aplicación se puede encontrar en la ciencia actuarial: los principios de primas basados en los momentos son fáciles de entender, pero en general no son monótonos y no se pueden utilizar para comparar los riesgos de diferentes contratos de seguros entre sí. Nuestro teorema de representación permite comparar dos riesgos de seguros entre sí de acuerdo con un principio de primas basado en un momento, definiendo adecuadamente una medida de riesgo coherente. En el último ensayo de esta tesis, investigamos los mercados financieros con fricciones, donde los precios de compra y venta de instrumentos financieros se describen mediante funciones de precios sublineares. Estas funciones pueden definirse recursivamente utilizando medidas de riesgo coherentes. En un modelo binomial y en la presencia de costes de transacción, demostramos la convergencia de los precios de compra y venta para varias opciones europeas y americanas, en particular opciones plain vanillas, asiáticas, lookback y de barrera. Realizamos varios experimentos numéricos para confirmar los hallazgos teóricos. Aplicamos los resultados a los datos de mercado reales de las opciones plain vanilla europeas y americanas y calculamos una liquidez implícita para describir la diferencia de precios de compra y venta. Este método describe muy bien la liquidez en comparación con el enfoque clásico de describir la diferencia entre los precios de compra y venta con las volatilidades implícitas de dichos precios.This thesis consists of three essays. In the first essay, we test empirically the pricing performance of several advanced financial models. We calibrate six advanced stock price models to a time series of real market data of European options on the DAX, a German blue chip index. Via a Monte Carlo simulation, we price barrier down-and-out call options for all models and compare the modelled prices to given real market data of the barrier options. The Bates model reproduces barrier option prices well. The BNS model overvalues and Lévy models with stochastic time-change and leverage undervalue the exotic options. A heuristic analysis suggests that the different degree of fluctuation of the random paths of the models are responsible of producing different prices for the barrier options. The second essay of this thesis discusses the relationship between coherent risk measures and concave distortion functions. A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment based premium principle by defining an appropriate coherent risk measure. In the last essay of this thesis, we investigate financial markets with frictions, where bid and ask prices of financial intruments are described by sublinear pricing functionals. Such functionals can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model in the presence of transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of European and American plain vanilla options and compute an implied liquidity to describe the bid-ask spread. This method describes liquidity over time very well, compared to the classical approach of describing the bid-ask spread by quoting bid and ask implied volatilities

Scenario generation for market risk models using generative neural networks

In this research, we show how to expand existing approaches of using generative adversarial networks (GANs) as economic scenario generators (ESG) to a whole internal market risk model - with enough risk factors to model the full band-width of investments for an insurance company and for a one year time horizon as required in Solvency 2. We demonstrate that the results of a GAN-based internal model are similar to regulatory approved internal models in Europe. Therefore, GAN-based models can be seen as a data-driven alternative way of market risk modeling