Numerical and Theoretical Foundations

Analysis on Variance Reduction Approaches with Dependence
Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2012)

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 the
underlying 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
L´evy processes: the Black-Scholes model, the Merton model, the Variance-Gamma
model 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.

Each Smile is Unique: Analysis of an Inverse Problem in Option Pricing
Master Thesis in "Mathematical Finance", Oxford (2006)
In the classical Black-Scholes world, the volatility in the model for the evolution
of an asset price is assumed to be constant. In this thesis we deal with an extended
model that allows the volatility to change with time and asset price, the local volatility
model.
The main focus lies on the theory of how to infer the functional form of the local
volatility from market traded options. This so-called inverse problem of option pricing
is established as a parameter identification problem within the framework of partial
differential equations of parabolic type. We derive an adapted partial differential
equation that yields an explicit formula for the local volatility function, and analyse
the properties of the formula. Assuming a particular form for the local volatility
surface, we show some uniqueness and stability results for the problem. Finally, we
give an overview of methods used in practice to extract the volatility surface from
prices of traded options.

Nonlinear Black Scholes Modelling - FDM vs FEM
Master Thesis in "Mathematical Finance", Oxford (2004)
A non-linear Black-Scholes model in d dimensions is presented allowing both, the volatilities and the correlations of the underlying assets, to be uncertain up to some upper and lower bound. A finite element scheme for discretising the asset space is derived. In particular, conform finite elements for the non-linear uncertain parameter model are presented for the cases d=1 and d=2. The finite element method is compared to the classical finite difference method in several numerical examples.

Implied Volatility Modelling
Master Thesis in "Mathematical Finance", Oxford (2004)
In the classical Black-Scholes theory the volatility is assumed to be constant. Contrary to this assumption we investigate pricing models with a non-constant volatility parameter, in particular stochastic volatility models. The main focus lies in models specifying a stochastic process for the implied volatility.

Robustness of the Least Squares Monte Carlo Method
Master Thesis in "Mathematical Finance", Oxford (2004)
The valuation and optimal exercise of American options is one of the most important
and difficult problems in option pricing theory today. The Least Square Monte Carlo
algorithm, developed by Longstaff and Schwartz in 2001, solves the early exercise
problem by estimating the continuation value using least square regression. In the
limit that the number of simulated paths and the number of terms in the regression
function go to infinity the almost sure convergence of the LSM algorithm and an
asymptotically Gaussian error distribution was shown by Clement, Lamberton and
Protter.
In this thesis the convergence and robustness of the algorithm is investigated for a
finite number of paths and terms in the regression function. The result unveils small
systematic errors for plain vanilla American options: The distribution of option prices
deviates from a Gaussian curve for some combinations of valuation parameters. Especially
if the continuation value was approximated with more than five basis functions
the distribution of prices shows asymmetric fat tails. Furthermore, the algorithm
is biased if either the continuation value is approximated with a polynomial of only
second order or the simulation is done with only 104 paths. For American average
options the bias remains even for 105 paths.

Completion of Market Data
Master Thesis in "Mathematical Finance", Oxford (2003)
Incomplete data is a very common problem financial institutions face when they collect financial time series in IT data bases from commercial data vendors for regulatory, accounting, and benchmarking purposes. The thesis at hand presents several data completion techniques, some of which are productively used in practice, and others which are less established. Optimal completion methods are then recommended, based on an empirical study for the completion of swap and forward rate curves in the currencies Deutsche Mark (respectively Euro), Pound Sterling, and US Dollar. The source code of all completion routines, programmed in a standard professional market data base environment, is listed in an appendix (it is available in electronic form upon request).

Simulating the Dynamics of the Risk Neutral Distribution
Master Thesis in "Mathematical Finance", Oxford (2003)
Knowledge about the dynamics of the risk neutral distribution is necessary
for the valuation of complex options on financial assets. We present
and explore a formalism (the twin formalism) that allows to simulate how
the risk neutral probability density function evolves with time. The main
idea of the twin formalism is to describe the risk neutral distribution as
a mixture of possible future distributions, and to simulate its time development
in a simplified ("twin") model space. This approach is model
independent in the sense that we may apply it to a large variety of distributions
(or processes): The formalism will automatically take care of
the necessary no-arbitrage conditions between the future and today's risk
neutral distribution. As an illustration, we apply the twin formalism to
several models and to the valuation of different options, in particular in
connection with the problem of so-called smiles in implied volatility. The
modeling framework is formulated in terms of probability distributions,
rather than processes. This means that one does not have to specify
the dynamics (i.e. stochastic differential equations) of the underlying processes.
It suffices to determine the distributions only at those points of
time that are relevant for the option one wants to price.

Evolving Yield Curves in the Real-World Measure
Master Thesis in "Mathematical Finance", Oxford (2003)
This work describes an algorithm, proposed in a paper by Rebonato et al. [RMJB],
which can be used to evolve yield curves in the real world measure over time periods
of the order of years. This algorithm will be used to simulate the USD-Libor curve.
The quality of the simulated curves is determined by comparing the path statistics as
there are the Eigensystem, the distribution of curvatures, and the structure of manyday
unconditional variance and many-day auto-correlation of the real curves with
simulated curves. For this purpose, the historical data is thoroughly investigated and
the results for the historical path statistics as well as the implications for the model
are discussed in detail.
The algorithm is introduced step-by-step starting from a very simple approach, which
fails to reproduce some of the aforementioned statistics to the final algorithm which
is capable of satisfying all the statistics, at least, qualitatively. A simple financial
explanation will be provided for the applied methodologies.

Predictor-Corrector Techniques for Solving LMM
Master Thesis in "Mathematical Finance", Oxford (2003)

Market Models versus Markov Functional Interest Rate Models
Master Thesis in "Mathematical Finance", Oxford (2002)

Practical Issues of the LMM
Master Thesis in "Mathematical Finance", Oxford (2002)
This thesis discusses the practical use of a LIBOR Market Model (LMM). In the first part the model is introduced, with special emphasis on the derivation of the drift term. Also, pricing within the model is explained and pricing formulae for analytically tractable instruments are derived. In general, however, analytical pricing is not feasible, so the second part presents an efficient method to numerically implement the LMM. In the next part, calibration of the model is discussed. At first, market-conditions of calibration are derived, explaining why quoted cap-volatilities can be directly used in the LMM. Furthermore, an analytical expression to approximately calculate corresponding (Black) swaption volatilities is obtained. Then, second, correlation and volatility functions are introduced in order to reduce the number of parameters that have to be fitted to market data. Using these functions the model is finally calibrated to the EUR cap and swaption market. The last part presents two examples. The first in strument chosen to be priced is a Ratchet Cap,an exotic instrument. Its price is calculated for various values of the parameters and the findings are discussed.In particular, the analysis of a simple deterministic scenario permits us to understand the qualitative features of the price's dependence on the input parameters. The second instrument chosen to be priced is a Bermudan Swaption, an asset that involves an optimal exercise decision. Two methods to numerically solve the optimal exercise problem are presented. In a computer experiment the dependence of the price on early exercise strategies is analyzed. One numerical method is shown not to produce arbitrage-free prices. The appendices gather a historical analysis of yield curve movements justifying assumptions made in the calibration, also an introduction to the modeling of Skews and Smiles within the LMM is presented.

Firm- Value Models with Jumps
Master Thesis in "Mathematical Finance", Oxford (2002)
In this treatise the implications of a firm-value model where the asset value is driven
by a jump-diffusion process are discussed in comparison to a diffusion-driven model. A
comprehensive view on both the equity-related and the credit-related properties of the
model is provided by studying not only credit-default-swap spreads but the value of the
stock and options on the stock of the company as well.
The results for a diffusion-driven firm-value model of the Black-Cox-Longstaff-Schwartz
type are first produced for later reference. It is demonstrated that the diffusion model
is able to reproduce some stylized empirical facts such as the shape of the mediumto-
long-term spread curve and the skew in the implied volatility. However, it fails to
capture other aspects of market data such as the non-zero short-term credit spread and
the flattening of the skew with increasing maturity.
For the jump-diffusion approach subsequently discussed, a Monte Carlo simulation method
is suggested that can be used for arbitrary jump amplitude distributions. This procedure
is quite efficient because it does not rely on a discretization of the problem in time and
calculates the path between jumps in a single step.
When specializing the jump-diffusion approach to a model with double-exponential jump
amplitudes, the Laplace transform of the solutions of the pricing equation is presented
in closed form for credit default swaps and stock options. The solutions, which can be
determined by applying the Gaver-Stehfest inversion algorithm, were demonstrated to be
in agreement with the results of the Monte Carlo simulation. While the implied stockoption
volatilities still show the skew and the medium-to-long-term behavior of credit
spreads does not change qualitatively compared to the diffusion approach, the jumpdiffusion
model also produces a non-zero short-term credit spread and a skew diminishing
with increasing maturity. Finally, we fit the model to volatility and credit-spread market
data for an arbitrarily chosen example. From these results it is concluded that the jumpdiffusion
approach is more consistent with qualitative and quantitative features found in
market data than the diffusion approach.

Applications of Statistical Bootstrapping in Finance
Diploma Thesis in "Mathematical Finance", Oxford (2000)
This thesis presents applications of statistical resampling techniques, specifically bootstrapping, to problems in finance. In particular, solutions to a range of problems in the calculation of Monte Carlo Value-at-Risk (VaR) are presented. For example, we provide methods for improving backtesting stability, acceleration of Monte Carlo VaR convergence by orders of magnitude, and accounting for covariance matrix uncertainty in VaR estimates. Extensive numerical tests on large number of randomly generated portfolios show the effectiveness of the suggested solutions.