# Numerische & Theoretische Grundlagen

### Approximating MVA along Low-Dimensional State Spaces

*Master Thesis in "Mathematical Finance", Oxford (2018)*

Banks are currently facing the problem of computing MVA, which is the present value effect of the future costs entailed by future initial margin postings. In this thesis, I present a method for approximating MVA. This method, which I call the projection method, consists in approximating the future initial margin amount in a future model scenario using sensitivities with respect to model state variables and the pseudo-inverse of the model-to-market Jacobian. I assess the projection methods performance for interest rate swaps and European swaptions, in the case where the risk-neutral dynamics of interest rates is given by the Hull-White one factor model.

### A principal components-based dividend term structure model

*Master Thesis in "Mathematical Finance", Oxford (2017)*

We develop and calibrate a term structure model for dividend futures based on principal components of the dividend futures’ covariance matrix. Dividend futures are cash-settled contracts on the cumulative dividends paid by a reference entity over a fixed time. Our model is based on a mean-reverting stochastic process. An advantage of modelling principal components instead of individual futures is that their covariance is diagonal by construction. Diagonality of the covariance matrix imposes strict requirements on the model’s reversion speed.

We also derive the explicit transformations of the model parameters from the risk-neutral to the real-world measure. Here we assume an affine structure of the market price of risk. We link annual excess returns of a buy-and-hold strategy with this market price of risk. Calibration in the risk-neutral measure is based on the complete history of daily EUROSTOXX 50 dividend futures prices. We also assume a simple seasonality of dividend payments based on the averaged, historical seasonality. The calibrated model is able to match a fixed subset of dividend futures market prices exactly and interpolate between or extrapolate beyond vailable market prices. Additionally, the model matches a subset of covariances exactly, whereas other covariances are matched approximately with high accuracy. Real-world parameters are calibrated via linear regression of dividend futures returns against an affine function of the principal components.

### A Markovian Limit-Order Book Market Model with underlying mixed Erlang distribution

*Master Thesis in "Mathematical Finance", Oxford (2017)*

Markovian limit-order book models are a well-known object of study. They provide analytical formulae interlinking properties on local scale - arrival intensities for instance - global properties of the price process - most prominently the variance. The queueing systems involved are usually modelled through Poisson processes - the order arrival waiting times are assumed to be exponentially distributed. In this dissertation, we propose queueing systems whose underlying order arrival waiting times are supposed to be distributed according to a more general mixed Erlang distribution. Based on a concretely observed limit-order book, it is shown that this approach leads to better estimates for local properties and - in some cases - slight improvements in the predictions of global properties.

### SABR Extensions to Negative Rates: A Comparative Analysis

*Master Thesis in "Mathematical Finance", Oxford (2017)*

Negative interest rates have become a common market feature over the last years, demanding adjustments of models used for the valuation of interest rate derivatives. For the popular SABR model, several extensions to negative interest rates have been proposed. In this work, we consider two of these approaches, namely a finite-difference scheme by Hagan et al. for the shifted SABR model, and an analytic solution by Antonov et al. for the free-boundary SABR model. We describe the theoretical foundations, implement the models, and analyze how well they calibrate to current market data for swaptions and caps. The well-established analytic formula for the shifted SABR model serves as a benchmark.

### Using Piecewise Constant Policy Time stepping to solve Hamilton-Jacobi-Bellman-Isaacs equations

*Master Thesis in "Mathematical Finance", Oxford (2017)*

In mathematical finance, Hamilton-Jacobi-Bellman (HJB) equations are well known in problems of stochastic optimization. These equations include a control parameter that can be adjusted to maximize respectively minimize the solution of the PDE. An extension of these non-linear PDEs to two control parameters leads to Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations that have a close connection to game theory where two players playing against each other choose their strategies to maximize their outcome of a differential game.

For some problems, analytic solutions exist, but more complex PDEs require numerical methods. One approach is using finite differences with an extension to include the optimization into the solution process. A well-known algorithm for this task is "Howard's" algorithm also called policy iteration. Another method called piecewise constant policy time stepping (PCPT) is the main topic of this work. We will present the basic ideas behind these methods and apply them to several numerical examples. We will consider the case of unknown parameters by example of an unknown volatility and combine this case with the pricing of an American option and the problem of mean-variance asset allocation. All those problems lead to HJBI equations and will be the test cases for PCPT as well as policy iteration as a benchmark. For this purpose, the methods will be implemented in MATLAB. Extensive simulations will be run to determine the performance of the different methods, concerning run times, required numbers of grid points and time steps as well as the capability to produce the correct solution of the problem at all.

### Long-term behavior of a zero-intelligence model for limit order books

*Master Thesis in "Mathematical Finance", Oxford (2017)*

Limit order books offer - among many other applications - a very popular method of automatized market making (i.e. turning order flow into prices) and are used in a wide variety of markets. The order flow is the process which drives price dynamics.

While many models for the order flow attempt to capture the strategies of the agents placing the order, a very popular class of models, termed `zero-intelligence models', seeks to determine which features of empirical price time series can be explained by random arrival of orders and in absence of any strategy.

In this thesis, we consider a typical representative of zero-intelligence models and study numerically the long-term (several hours to days) behavior of the resulting price trajectories. We find that the long-term behavior of the resulting prices is not compatible with empirical market time series, because the average volatility of simulated prices is found to decrease strongly with time. We analyze the reasons for this and find indications that this might actually be a general feature of zero-intelligence models rather than an individual shortcoming of the particular model studied.

While it is possible that some issue of the particular model studied here have

already been noted by practitioners, we believe the systematic study presented

here to be novel. The possible extension to the entire class of models and the

associated implications are also new results.

### Multilevel and multi-index Monte Carlo for portfolio Value-at-Risk

*Master Thesis in "Mathematical Finance", Oxford (2016)*

### A semigroup approach to regular asymptotic expansions with applications to Black-Scholes equations

*Master Thesis in "Mathematical Finance", Oxford (2016)*

This thesis is inspired by regular asymptotic expansions to the European put presented in a lecture by Dr. Jeff Dewynne on Asymptotic Methods in Module V as part of the part-time course in Mathematical Finance at the University of Oxford in autumn 2014. Throughout the course it was assumed that asymptotic expansions exist and converge to the respective European put. In this work we present a different approach to regular asymptotic expansions of the European put via strongly continuous semigroups. We first develop and prove semigroup theoretical results like existence and uniqueness of solutions, existence and convergence of asymptotic expansions and explicit formulae. These results are then applied to different Black-Scholes partial differential equations. It turns out that our approach is well calibrated, i.e., the results applied to the European put coincide with those presented in the Asymptotic Methods lecture. The theoretical results are enhanced by numerical examples.

### Computing Sensitivities of CVA Using Adjoint Algorithmic Differentiation

*Master Thesis in "Mathematical Finance", Oxford (2015)*

In this thesis it is shown how one can use adjoint algorithmic differentiation in order to compute sensitivities of a CVA computation for a portfolio of single currency plain vanilla swaps, where the underlying model is chosen such that it takes into account the phenomenon of wrong way risk. Besides giving a basic introduction to CVA in general and describing the concrete model used for this work, this work focuses on the computational techniques needed to implement the adjoint sensitivity computation for first order interest, credit, model volatility and correlation sensitivities. The implementation presented in this work is based on the library dco-c++, developed by NAG / RWTH Aachen.

Die Master Thesis als PDF-Dokument finden Sie hier.

### The trending Ornstein-Uhlenbeck Process and its Applications in Mathematical Finance

*Master Thesis in "Mathematical Finance", Oxford (2015)*

### Mixed Sequences and Application to Multilevel Algorithms

*Master Thesis in "Mathematical Finance", Oxford (2012)*

In this thesis we study mixed sequences and their application to multilevel approaches to price financial derivatives.

### Multilevel Monte Carlo Methods for American Options

*Master Thesis in "Mathematical Finance", Oxford (2012)*

The aim of this Thesis is to show how the ideas from multilevel Monte Carlo (MLMC) methods can be used in the valuation of American options. The Core idea is to use and adapt the multilevel Monte Carlo method introduced in (Gil08) and (Gil) to find the value of an American option under a given exercise rule and then optimize over all exercise rules from a feasible set.

### Simulating Yield Curves under the Real-World Measure

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2012)*

This work reviews a model proposed by Rebonato et al. to evolve a yield curve

under the real-world measure. The model is capable to reproduce several essential features observed for the real-world yield curve like the statistics of the curvature and the principle components. It includes a so-called spring mechanism preventing arbitrage opportunities. The mechanism causes an interaction between changes in yield of neighbouring maturities. Consequently, the correlations are not exactly reproduced by the model. This work analyses the impact of the spring mechanism on the correlation and describes a further enhancement of the model to recover the correlation observed in the real-world data.

### Evaluating Sensitivities of Bermudan Swaptions

*Master Thesis in "Mathematical Finance", Oxford (2011)*

### Utility-based pricing and hedging via functional differentiation

*Master Thesis in "Mathematical Finance", Oxford (2010)*

We present an approach to utility-based pricing and hedging in incomplete markets. For the limit of an infinitesimally small number of options, we derive a formula for the marginal optimal hedging strategy that is compatible with the marginal indifference price of Davis. For this, we use the concept of functional differentiation. The proposed framework is conceptually related and probably equivalent to a proposal of Kramkov and Sirbu. We apply it to the case of a jump diffusion process and compare it conceptually and numerically with Merton's and the minimal variance approach.

### Valuation of American Basket Options using Quasi-Monte Carlo Methods

*Master Thesis in "Mathematical Finance", Oxford ((2009)*

The valuation of American basket options is normally done by using the Monte Carlo approach. This approach can easily deal with multiple random factors which are necessary due to the high number of state variables to describe the paths of the underlyings of basket options (e.g. the German

Dax consists of 30 single stocks).

In low-dimensional problems the convergence of the Monte Carlo valuation can be speed up by using low-discrepancy sequences instead of pseudorandom numbers. In high-dimensional problems, which is definitely the case for American basket options, this benefit is expected to diminish.This expectation was rebutted for different financial pricing problems in recent studies. In this thesis we investigate the effect of using different quasi random sequences (Sobol, Niederreiter, Halton) for path generation and compare the results to the path generation based on pseudo-random numbers, which is used as benchmark.

American basket options incorporate two sources of high dimensionality, the underlying stocks and time to maturity. Consequently, different techniques can be used to reduce the effective dimension of the valuation problem. For the underlying stock dimension the principal component analysis (PCA) can be applied to reduce the effective dimension whereas for the time dimension the Brownian Bridge method can be used. We analyze the effect of using these techniques for effective dimension reduction on convergence behavior.

To handle the early exercise feature of American (basket) options within the Monte Carlo framework we consider two common approaches: The Threshold approach proposed by Andersen (1999) and the Least-Squares Monte Carlo (LSM) approach suggested by Longstaff and Schwartz (2001). We investigate both pricing methods for the valuation of American (basket) options in the equity market.

### Calibration of Interest Rate Models with Stochastic Volatility

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2009)*

Stochastic volatility models became more and more important in the last years. This thesis aims at finding practically relevant hints for a stable calibration of stochastic volatility models based on computational experiments.

### Valuation of Exotic Options under Jump Diffusion, A comparison of Monte Carlo Methods and Numerical Solution of Partial Integro-Differential Equation

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2008)*

In this thesis, two numerical methods are compared in their usability of calculating the values of exotic options under jump diffusion. As methods we consider both the Monte Carlo and the Finite Difference method. The exotic options that are evaluated with these two methods are Window Barrier Options in general, and Up-and-Out Window Barrier Call options in more detail. For both methods it is shown how the basic schemes can be extended from a Black-Scholes model to a jump diffusion model, and how the particularities of Window Barrier Options can be incorporated into the models. Both methods are then implemented and it is shown that the Monte Carlo method is a more flexible tool that allows for a quick implementation of diferent exotic options, but that the Finite Difference offers higher precision in exchange for a higher complexity in implementation.

Accompanying this thesis is a C# program in with which Window Barrier Options can be evaluated with the Monte Carlo and the Finite Difference method.

### The Continuous-Time Lattice Method

*Master Thesis in "Mathematical Finance", Oxford (2007)*

The Continuous-Time Lattice Method for the pricing of derivatives is presented in this thesis. This method can be applied if the underlying random process is a combination of a diffusion and a jump process. Instead of approximating the underlying process directly, the Markov generator of the process is approximated on a lattice. Therefore the state space of the approximating process is a finite set. Although the method can be applied to exotic derivatives, this thesis is concerned with the application to European vanilla equity options. The lattice is thus a discretization of the price of the underlying instrument. The time variable stays continuous. Using matrix diagonalization, the probability kernel of theunderlying random process is obtained and used for pricing.

A C++ program for the computation of the implied volatility surface using the Continuous-Time Lattice Method is implemented. Results are shown for European vanilla options if the underlying process is a combination of a CEV and a variance-gamma process. This model is shown to exhibit the overall features of the implied volatility smile exhibited by market-traded option prices. These features are an asymmetric smile and a flattening of the smile for longer times to maturity.

### Numerical solution of jump diffusion problems with early exercise

*Master Thesis in "Mathematical Finance", Oxford (2007)*

This thesis deals with the numerical valuation of options with early exercise. Pricing is done with several jump-diffusion models which all belong to the rich class of exponential Léevy processes: the Black-Scholes model, the Merton model, the Variance-Gammamodel and the Tempered Stable model.Two simulation approaches, a Monte Carlo and a finite difference method, will be presented. These methods are well adapted to jump diffusion problems and they are able to handle European as well as American options. Both simulation techniques will be applied to all of the pricing models mentioned above. Finally, the results and the computational complexity of both approaches will be compared with each other.

### Analysis on Variance Reduction Approaches with Dependence

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2012)*