# Pricing

### Simultaneous arrival of orders to a limit order book

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

In this dissertation, we analyse and model the order flow to a limit order book (LOB). In particular, we focus on simultaneously arriving orders, an aspect which although present in many real-world data sets, is typically not explicitly included in state of the art models like the (non-marked) Hawkes processes. We filled this gap by comparing two different modelling approaches. First, we considered Cox processes directed by Lévy subordinators which have been successfully harnessed to model simultaneous arrival of insurance claims. Here, concurrent order arrivals are driven by jumps in the subordinator. Second, we examined marked Hawkes processes, where the information that orders arrive together is encompassed within marks. These processes are built upon an infectious character of orders, i.e. the arrival of an order alters the likelihood of consecutive orders. Our analysis was conducted with the open source programming language R and the sample data from Limit Order Book – The Efficient Reconstructor (LOBSTER), freely available at https://lobster.wiwi.hu-berlin.de/. Here, about two to three percent of order book events occur together with other order book events. We found that Hawkes processes fit the high-frequency order book data more accurately than Cox processes directed by Lévy subordinators. Furthermore, there is a multitude of goodness of fit (GOF) measures available for Hawkes processes. Hence, we recommend modelling high-frequency order flows with simultaneous arrival of orders by employing a (multivariate) marked Hawkes process. Suitable marks can be the total number or the total size of orders arriving to a component at the same time.

### Conditional Expectations for Euler-Maruyama Scheme Using Malliavin Calculus with Application to American Option Pricing

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

For pricing American or Bermudan style options using Monte Carlo based approaches one is confronted with calculating conditional expectations. The most popular method in practice to accomplish this is based on least-square regression which was proposed by Longstaff and Schwartz. In this work, we follow a different Monte Carlo based approach using Malliavin calculus techniques to calculate the conditional expectations as a quotient of unconditional expectations applying a generalized integration by parts formula. The dynamics of the underlying (asset) prices are driven by a multi-dimensional diffusion process with a local volatility term.

### Maturity Randomization Approach to the Pricing of American Options under Constant and Stochastic Volatility

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

The largest part of listed options traded on the global financial markets are American-styled. Still, except for some special cases no analytic closed-form solution for the American option value exists. Therefore American options are usually valued by means of numerical algorithms. However, in particular if stochastic volatility models are considered, those numerical methods tend to be computationally expensive. In this dissertation we study an alternative analytic approach to the valuation of American options: maturity randomization. The core of this approach is to replace the constant maturity of an American option by a random variable with a suitable distribution in order to reduce the complexity of the resulting pricing equations. We apply this method to the valuation of American puts and calls in the constant elasticity of variance model and a fast mean-reverting stochastic volatility model. To this end, we derive analytic pricing functions for maturity-randomized American option contracts which approximate the equivalent fixed-maturity American option. Comparing our results with those obtained from popular numerical approaches we find that maturity randomization yields very good approximations for the American option value and exercise boundary. Besides, since the pricing function is derived in an analytic form, computing numerical results hardly requires any computation time.

### Valuation of Mid-Curve Options Using Copulas

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

Upon exercise of a plain-vanilla European swaption the resulting swap starts immediately. A Mid-Curve Option (MCO) is a variation on a plain-vanilla swaption: it introduces a delay between exercise and start of the resulting swap. This delay needs to be taken into account when valuing MCOs. For purposes of valuation, an MCO can always be represented as an option on a weighted sum of two spot swap rates. The market approach is to freeze weights and treat them as constants. But in fact these weights are stochastic and depend on the entire yield curve. The approach described in this thesis does not treat the weights as constants. Instead, we use a method from a paper by Saroka/Rebonato to predict the likely shape of the yield curve and derive the weights from this. Then we perform a valuation of an MCO.

### FX Option pricing using local stochastic volatility models

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

The exact knowledge of an option price is essential when trade options. In 1973, Black and Scholes first derived the price of an option as a solution of a mathematical equation. Since then this framework have been substantially extended. The most common extensions are the local and the stochastic volatility. Local stochastic volatility methods try to overcome most of the problems with these two approaches while combining them. This thesis summarizes the mathematical requirements needed for the approaches building the basis for the local stochastic volatility. Additional results needed to derive a local stochastic volatility model are also stated. Furthermore the techniques for an implementation and which steps need careful handling are discussed.

### Derivative pricing frameworks with funding effects: How to put the FVA piece into the valuation-adjustment puzzle

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

In this thesis we consider recursive and non-linear pricing models that take into account funding constraints to maintain a self-nancing replicating portfolio. We derive the Crepey Model based on a Forward-Backward SDE (FBSDE) and reduce the problem to a Backward SDE (BSDE). Through the BSDE framework of the Crepey Model we derive the Burgard/Kjaer Model, the Brigo/Pallavicini Model and the Albanese/Andersen Model. For the derived models we present and prove the Funding Invariance Principle that enables us to discount with an arbitrary rate. For this arbitrary rate we adapt term structure modelling in a multi-currency setup such that the related bond-price processes are tradeable assets. With this setup we are able to modify common analytical pricing formulas for interest derivatives. Finally we specify an algorithm that solves the recursive BSDE problem with dierent schemes. These schemes are used to evaluate the impact of the dierent models as well as the numerical accuracy of the BSDE solution schemes.

### Accelerated Share Repurchase Programs

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

### Arbitrage-Free Pricing from a Fixed Income Macroeconomic Perspective

*Master Thesis in "Mathematical Finance", Oxford (2013) *

The Black-Scholes model for option pricing was the first widely accepted model for pricing and hedging financial derivatives and has become a starting point for much work in financial modeling. The model assumes that the price of the underlying in the derivative contract follows geometric Brownian motion and that the market is perfect. The input parameters of the Black-Scholes formula for option pricing are, the stock price, volatility, strike price, risk-free interest rate and the time to expiration of the option. An important parameter of this model is the underlyings price volatility. Using market prices the volatility can be inferred, which when used in the Black-Scholes model reproduces the market price; this is referred to as the implied volatility. A plot of the implied volatility vs the strike price and time to expiry of the derivative is referred to as the volatility surface. Prior to the stock market crash of October 1987 the volatility surface was basically flat for all strike prices. Since the crash the volatility surface has become skewed, being referred to as the volatility smile. The Black-Scholes model however assumes a constant volatility and cannot reproduce the smile. Several approaches have been developed to capture the volatility smile by modeling the volatility as a stochastic process. These can be grouped into three main categories, namely local volatility models, stochastic volatility models and jump diffusion models. Each of these model types have different strengths and weaknesses, but often entail tedious formulae which must be calibrated to different parameters. In this work an economic approach is taken to capture the volatility smile. By tweaking the assumptions of the Black Scholes model to allow for credit risk, a simple extension of the Black Scholes formula is developed capable of reproducing a volatility smile. The parameters to be calibrated in this model have a direct economic interpretation, and can be linked to the appearance of the smile post October 1987. The Black-Schole model naturally emerges as a special case of this more general model. The performance of the here-developed model, with respect to fitting the volatility smile and hedging is compared to that of the Black-Scholes model. This comparison is done for equity options on the S&P 500 index.

Chapter 1 discusses the background on smile modeling and various volatility models, highlighting the strengths and weaknesses of the various approaches used. Chapter 2 starts of from the Black-Scholes model and modifies the model based of simple economic arguments to develop an alternative modeling approach, named here the Black-Scholes Jump-Diffusion (BSJD) model. The formulae for the hedging parameters (Greeks) are developed. Chapter 3 discusses the challenges involved in computing a volatility surface from market data. The difficulties in calibration are also illustrated. In chapter 4 numerical results on calibration of the BSJD model to SPX volatility smile are presented, as well as a framework for validating the model. The performance of the BSJD model in hedging SPX options is then compared to the Black-Scholes model. The fifth and last chapter summarizes the insights gained from this work.

### Uncertainty in Option-Pricing Models Quantified by Relative Entropy

*Master Thesis in "Mathematical Finance", Oxford (2013) *

We study the problem of deriving bounds for equilibrium prices of non-traded assets in incomplete markets under model uncertainty. The starting point is a recent study by Boyarchenko, Cerrato, Crosby and Hodges (BCCH) who generalize the no-good-deal pricing method of Cochrane and Saá-Requejo to account for uncertainty in the investor’s probability model of future asset returns. The approach of bears formal similarity to Hansen and Sargent’s general theory of robust control under model uncertainty. Following an account of the basic definitions and results from Hansen and Sargent’s theory, the classical theory of good-deal pricing, and the BCCH generalization, we make some critical observations on the BCCH form which lead to a proposal for its modification. We finally discuss the problem of calibrating an uncertainty-adjusted good-deal pricing model to the level of an investor’s statistical uncertainty.

### Valuation of Non-Deliverable Self-Quanto Forwards in an FX Options Market with Smile

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2013)*

The subject of this thesis is the pricing of non-deliverable self-quanto forward contracts by incorporating the volatility smile structure that can be observed in the FX options market. In general a quanto derivative is a cash-settled product where the payoff in one currency is converted into another currency. The pricing of several quanto FX derivatives can be approximated based on the analytic solutions provided by the Black-Scholes equation. However, due to intrinsic model assumptions the prices obtained within a Black-Scholes framework are often not consistent with the market quotations. Even though the quanto forward is a rather simple product, it has convexity and requires adjustments due to the volatility smile structure. We will present and analyze different approximations for the quanto forward payoff, based on vanilla FX call and/or put options. Special attention is paid to a market-consistent construction of the FX volatility surface as well as the simulation of the time evolution of the exchange rate and volatility surface within a Heston model approach.

### Asset Allocation under a Conditional Diversification Measure

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

In this thesis we consider the problem of diversifying investments in common market securities under certain restrictions, such as budget constraints, etc. Therefore we adapt the entropy diversification measure as well as the conditional principal component decomposition, both proposed by Meucci (2009). We derive a rigorous and powerful theoretical framework describing the geometry of the conditional principal components, which particularly allows us to prove the existence of such decompositions. Furthermore, we apply numerical tests to selected index securities, compare two approaches of portfolio selection (mean-variance vs. mean-diversification) and illustrate the differences that arise between the efficient frontiers. A propagated aim in this thesis is the asset allocation of mutually uncorrelated portfolios, which are naturally given by the principal components. Thus, finally, we back-test several rebalancing strategies based on a principal component decomposition and verify whether the resulting portfolios are closely uncorrelated.

### Approximate smiles in an extended SABR model

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

Hagan’s asymptotic expansion fo the SABR model is widely used by practitioners for fitting the smile of vanilla interest-rate options. However, it is well known to break down for very long dated options, large volatility of volatility or very small strikes. With the current very low short-term rates in the market, deficiencies in the underlying CEV model, namely an absorbing boundary at zero rates, become more acute. Thus, other choices of local volatility such as shifted log-normal or shifted CEV receive more attention as a basis for a stochastic-volatility extension. One recent empirical analysis of very long time-series data for interest rates suggests three regimes of interest rate dynamics depending on the level of rates: log-normal behaviour at very low rates, normal dynamics at intermediate rates and shifted log-normal behaviour at very large rates. In this thesis, we review two types of approximation schemes used in the literature for the standard, CEV-based SABR model: Asymptotic expansions for small time to maturity τ as well as a mixing approach for ρ = 0 suggested by Barjaktarevich and Rebonato. Both approaches are applied to an extended SABR model with a general local volatility function C(f). Approximate results are compared to Monte Carlo and a two-dimensional finite difference scheme for a few choices of C(f), including one that models the three regimes of.

### Fast Calculation of Greeks for Equity Basket Options by Monte Carlo using Adjoint Methods

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

### Analysis on Variance Reduction Approaches with Dependence

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

### On the Calibration of the SABR–Libor Market Model Correlations

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

This work is concerned with the SABR-LMM model. This is a term structure model of interest forward rates with stochastic volatility that is a natural extension of both, the LIBOR market model (Brace-Gatarek-Musiela [1997]) and the SABR stochastic volatility model of Hagan et al. [2002].

While the seminal approximation formula (developed by Hagan et al. [2002]) to implied Black volatility using the SABR model parameters allows for a successful calibration of each forward rate dynamics to the volatility smile of the respective caplets/floorlets, an adequate calibration of the rich correlation structure of SABR-LMM (correlations among the forward rates, the volatilities and the cross correlations) is a challenging topic and of great interest in practice. Although widely used for calibration, it is well known that swaptions’ volatilities carry only little information about correlations among the forward rates. As practically successful for the classical LMM, desirable would be to take the market swap rate correlations into account for the model calibration.

In this study we develop a new approach of calibrating the model correlations, aiming at incorporating the market information about the forward rate correlations implied from more correlation-sensitive products such as CMS spread derivatives, in which also swap rate correlations are involved. To this end we derive a displaced-diffusion model for the swap rate spreads with a SABR stochastic volatility. This we achieve by applying the Markovian projection technique which approximates the dynamics of the basket of forward rates, in terms of the terminal distribution, by a univariate displaced-diffusion. The CMS spread derivatives can then be priced using the SABR formulas for the implied volatility, taking the whole market smile of CMS spread options into consideration. For the ATM values in the payoff measure of the projected SDE we use a standard smile-consistent replication of the necessary convexity adjustment with swaptions.Numerical simulations conclude the work, giving a comparison between this method and the classical one of calibrating the model correlations to swaption volatilities. Furthermore, we study the performance of different parameterizations of the correlation (sub-)matrices.

### A Bayesian approach to option pricing

*Master Thesis in "Quantitative Finance", Frankfurt School of Finance & Management (2011)*

### Linear and Seasonal Trends in Temperature Derivatives

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

This work evaluates methods for pricing weather derivatives, in particular temperature swaps, using real and simulated data, from the perspectives of accuracy and potential accuracy given a model. While atmospheric modeling can be used for forecasting temperatures for contracts of up to a few weeks, actuarial or arbitrage methods are preferred for most intrayear contracts. We contribute to the discussion by comparing the performance of index and daily modeling, two actuarial methods addressing the temperature as the underlying variable of contracts. For this purpose, the accuracy of simulated forecasts was determined for both methods in a jack-knife analysis of historical data. We also address the issue of detrending, by comparing the performance of several procedures with respect to a linear stochastic temperature model of variable sample length. Our findings on the subject models of the underlying suggest a similar performance of both methods for maturities below one month and a better performance of index modeling for larger maturities up to half a year. In the analysis of detrending, we establish the regions of highest accuracy in the parameter space for a flat-line, linear and a damped linear detrending. We find that damped linear detrending outperforms the other methods in a thin parameter region which may exist in nature for sample lengths between ten and twenty years.

### Markov Functional interest rate models with stochastic volatility

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

With respect to modelling of the (forward) interest rate term structure under consideration of the market observed skew, stochastic volatility Libor Market Models (LMMs) have become predominant in recent years. A powerful representative of this class of models is Piterbarg's forward rate term structure of skew LMM (FL-TSS LMM). However, by construction market models are high- dimensional which is an impediment to their efficient implementation. The class of Markov functional models (MFMs) attempts to overcome this inconvenience by combining the strong points of market and short rate models, namely the exact replication of prices of calibration instruments and tractability. This is achieved by modelling the numeraire and terminal discount bond (and hence the entire term structure) as functions of a low-dimensional Markov process whose probability density is known.

This study deals with the incorporation of stochastic volatility into a MFM framework. For this sake an approximation of Piterbarg's FL-TSS LMM is devised and used as pre-model which serves as driver of the numeraire discount bond process. As a result the term structure is expressed as functional of this pre-model. The pre-model itself is modelled as function of a two-dimensional Markov process which is chosen to be a time-changed brownian motion. This approach ensures that the correlation structure of Piterbarg's FL-TSS is imposed onto the MFM, especially the stochastic volatility component is inherited. As part of this thesis an algorithm for the calibration of Piterbarg's FL-TSS LMM to the swaption market and the calibration of a two-dimensional Libor MFM to the (digital) caplet market was implemented. Results of the obtained skew and volatility term structure (Piterbarg parameters) and numeraire discount bond functional forms are presented.

### Iterative Drift Approximations for the LIBOR Market Model in the Spot Measure

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

### Semi-analytic Lattice Integration of a Markov Functional Term Structure Model

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

One common use of Markov functional models is to approximate LIBOR market models, and to avoid complications the terminal forward measure is typically used. If this method is applied to long term structures (ten or more years), the distribution of the early LIBORs in the term structure has a very large tail, which is normally not completely captured by common numerical techniques (either Monte Carlo or grid-based methods). A numerical method that is frequently applied to Markov functional models is known as the semi-analytic lattice integrator (Sali) tree. This thesis examines the implications of the long tails on the Sali tree. Adequate boundary conditions and grid sizes are derived in order to capture the effect of the long tails. It turns out that this method either exhibits stability problems or demands a relatively small lattice spacing. The reason for this is examined in detail and several variations of the Sali tree to avoid this effect are suggested and analysed. Furthermore the optimisation of the grid parameters is considered in order to reduce the necessary computation time.

### Pricing and Risk Measurement of Seasonal Commodities

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

### Spread Products in the Presence of Smiles and Dependencies

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

### 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.

### Pricing of path-dependent basket options using a copula approach

*Master Thesis in "Mathematical Finance", Oxford (2008)*

The pricing of basket options is usually a difficult task as assets of a basket usually show significant dependence structures which have to be incorporated appropriately in mathematical models. This becomes especially important if a derivative depends on the whole path of an option. In general pricing approaches linear correlation between the different assets are used to describe the dependence structure between them. This does not take into account that empirical multivariate distributions tend to show fat tails. One tool to construct multivariate distributions to impose a nonlinear dependence structure is the use of copula functions.

In the thesis the general framework of the use of copulas and pricing of basket options using Monte Carlo simulation is presented. On the base of the general framework an algorithm for the pricing of path-dependent basket options with copulas is developed and implemented. This algorithm conducts the calibration of the model to market data and performs a simulation and estimates the fair price of a basket option. In order to investigate the impact of the use of different copulas and marginals the algorithm is applied to a selection of basket options. It is analyzed how the proposed alternative approach affects the fair price of the option. In particular, a comparison to standard approaches assuming multivariate normal distributions is made. The results show that the use and the choice of copulas and especially the choice of alternative marginals can have a significant impact on the price of the options.

### Pricing of Options on Index Credit Default Swaps

*Master Thesis in "Mathematical Finance", Oxford (2008)*

### An Analysis and a Numerical Framework for Calibration in Piterbarg's Stochastic Volatility LIBOR Market Model

*Master Thesis in "Mathematical Finance", Oxford (2008)*

This work elaborates a stochastic volatility model with time-dependent parameters for the pricing of interest rate derivatives. While pricing with this model can be done with a straight forward Monte Carlo simulation, its calibration constitutes a difficult high dimensional problem. It requires easy to evaluate formulas for plain vanilla prices or, alternatively, the direct derivation of parameters for the Monte Carlo method. This second approach is taken by Vladimir Piterbarg who developed efficient parameter averaging techniques that are used to solve the calibration problem in the parameter domain. While general concepts of this calibration method were presented by Piterbarg, the results that can be achieved by this calibration depend on the numerical techniques, modelling assumptions, and algorithms that are applied. This work enhances the analysis and reasoning for the approximations, adds a concrete computational framework and numerical techniques, and contributes an efficient calibration scheme. The results of this work show that by the use of the appropriate modelling tools and algorithms a fast calibration is obtained that allows recovering European Swaption prices in a Monte Carlo method with high precision.

### Correlation Hedge Performance of Quanto Options

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

Quanto options are financial derivatives with the payoff pattern of vanilla options but settle in a foreign currency using a contractually fixed FX rate. This master thesis summarizes how such products can be priced and provides an overview of various sensitivities for quanto call and put options. The performance of discrete hedging strategies is studied numerically taking different sets of sensitivities into account.

One common feature of quanto options is that their prices depend on correlations, which are usually hard to measure in real-world markets. Numeric results are presented which show that the lack of knowledge of the correlation deteriorates the hedge performance. Likewise, the performance is also adversely affected when using higher-order hedging strategies jointly with non-exact market models.

Using a special FX market consisting of three currencies it is possible to deduce implicit correlations and therefore to dynamically hedge this market parameter. Numerical results show that the hedge performance is considerably improved when taking the correlation sensitivities into account when constructing a hedging strategy.

### Evaluation of Variance Swaps in Crash Scenarios

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

### Efficient Volatility Calibration by Markovian Projection in Forward LIBOR Models

*Master Thesis in "Mathematical Finance", Oxford (2008)*

The Brace-Gątarek-Musiela (BGM) or LIBOR market model, which is based on the assump- tion that forward term rates follow lognormal processes under their corresponding forward measures, has established itself as a benchmark model for pricing and risk managing interest rate derivatives. One of the many virtues of the BGM model is that is provides a justification for the use of Black's formula for caplets.

European swaptions are in this framework priced with an approximation formula. While the BGM model has established a standard for incorporating all available at-the-money information, it is less successful in recovering other essential characteristics of interest rate markets, particularly volatility skews and smiles. Various extensions of the BGM model designed to incorporate skew and smile effects have been proposed. Valuation in any of these forward LIBOR models is typically done by Monte-Carlo simulation.

The usual calibration procedures in these models are complicated and computationally expensive. Another important fact to notice is that calibration algorithm is performed on a best-fit basis. Although very complicated, the problem of deriving efficient calibration techniques by finding accurate and fast approximations to prices of benchmark European swaptions and caplets is of great interest. The Markovian projection method is a novel approach to volatility calibration and represents a way of deriving efficient, analytical approximations to European-style option prices on various underlyings.

It is a powerful technique that seeks to optimally approximate a complex underlying process with a simpler one, keeping essential properties of the initial process and is, in principle, applicable to any diffusion model.

In this thesis, the method is developed for a Markovian projection onto displaced-diffusion and its application to single and cross-currency forward LIBOR models is investigated. Its application to the efficient calibration of European swaptions and FX options in the presence of market skews is shown and analytical approximations for both of these option types are derived. The analytical approximations are tested numerically and the results are compared to results obtained by alternative models and from the literature. Limitations of the projection onto displaced-diffusion are discussed and improvements and possible extensions of the method to capture implied volatility smiles are presented.

### A Two-Regime Markov-Chain Model for the Swaption Matrix

*Master Thesis in "Mathematical Finance", Oxford (2006)*

The most commonly used models for pricing complex interest rate products are

LIBOR Market Models. These models describe the evolution of a set of forward

LIBOR rates. For a given set of LIBOR rates such a model is defined by the term

structures of their volatilities and the correlations among these rates. It is therefore of vital importance to use a parameterization for these quantities that is able to capture the important features of the market the model is calibrated to. Since this is usually the swap market a good parameterization of the LIBOR Market Model should also be a good model for the volatility term structure of forward rates, i.e. the swaption matrix which contains all available information about the volatility term structure in the swap market.

It is a well known feature of the interest rate markets that in times of market

turmoil the volatility term structure undergoes sudden changes in shape when the market enters these excited regimes. After a short period the market is normal again which also switches the shape of the volatility term structure back to normal. Rebonato and Kainth proposed an approach to capture this market feature in a LIBOR Market Model. They use the most common parameterization of the volatility term structure to describe the normal market situation and the excited regime each with one set of stochastic parameters. In the present work a more simplistic approach is taken by keeping the parameters constant. The model captures the switches betweenthe two regimes by transition probabilities.

### Weather Derivatives Valuation

*Diploma Thesis in "Mathematical Finance", Oxford (2006) *

In 1997 weather became a commodity, quantifiable in currency amounts and tradable on public exchanges and over the counter: weather derivatives, financial instruments based on an underlying weather index, were traded for the first time between US energy companies. Weather derivatives provide hedges against weather related risks, such as the risk of reduced consumption of electricity for air conditions in cool summers or the risk of cancellations at a ski resort in a rainy winter.Today, the weather derivatives market has emerged, but most contracts are still traded over the counter. The main deterrent for many potential investors is probably the fact that up-to-date there is no commonly accepted pricing method for weather derivatives.

This paper reviews different pricing approches for weather derivatives published during the past years, including actuarial pricing methods - such as burn analyis as well as index and daily modeling, arbitrage pricing, equilibrium pricing, benchmark pricing, underlying market analysis pricing and a general valuation equation for weather derivatives. The advantages and limitations of each method are discussed, in particular with respect to its applicability in the market.

It is concluded that daily modeling of a weather variable using a highly sophisticated autoregressive model that considers trends and seasonality (e.g., the SAROMA model) and potentially heteroskedasticy of temperature fluctuations (e.g., ARFIMA-FIGARCH) could be established as pricing standard for weather derivatives. Further research is necessary in improving the model algorithms covering seasonality and atmospheric oscillations and to include other contracts besides standard temperature derivatives.

### Valuation of Basket Options

*Master Thesis in "Mathematical Finance", Oxford (2005)*

Accurate valuation of basket options requires the use of numerical methods. This survey describes and compares the widely used Monte Carlo method, deterministic integration rules, the semi-analytical moment matching method and an approximation via Taylor expansion. Significant improvement in the accuracy of Monte Carlo methods can be achieved by using a suitable control variate. Two such approaches, the first based on the geometric average, the second on a partitioning algorithm, are discussed in detail. In situations where most of the variance of the terminal basket value can be explained by a small number of risk factors, a quasi-Monte Carlo method based on the Halton sequence turns out to be superior to standard Monte Carlo. Deterministic integration proves to be very competitive if the number of underlying assets is small. An approach based on the Gauss-Hermite formula is described. Moment matching methods for up to the third moment are easy to implement and provide a fast approximation of the option value, largely independent of the number of assets. Approaches using the lognormal, the reciprocal Gamma and a shifted lognormal distribution are compared. Ju's method of expanding the characteristic function of the distribution of the terminal basket value in a Taylor series does also perform very well with an achieved accuracy similar to a method matching the third moment.

### Monte Carlo evaluation of Delta and Gamma of barrier options using Bismut type formulae

*Master Thesis in "Mathematical Finance", Oxford (2005)*

In this thesis we obtain Bismut type stochastic representations of first and second order derivatives of diffusion semigroups. The results are comparable to the likelihood ratio formulae by Glasserman and other publications within the framework of Malliavin calculus by Fournié et al. In contrast to those references, our presentation is completely based on Itô diffusion theory and martingale methods.

The main idea is that the composition of the price process put into the corresponding semigroup with inverse time scale yields a martingale which smoothly depends on the initial spot price, and the differential of this martingale family is again a martingale containing the gradient of the semigroup. By taking expectations and iterating the procedure, we end up with flexible and general theorems on the gradient and Hessian of the semigroups both in the unbounded case and the bounded situation with Dirichlet boundary conditions.

The Hessian representation theorem is inspired by an earlier work on diffusions on Riemannian manifolds by Elworthy and Li, but generalizes their result and transfers it to the mathematical finance literature where, to our knowledge, it is unknown so far. For applications to finance, we discuss formulae for Delta and Gamma of standard European as well as for (also European) barrier options, both for the Black-Scholes model and a very general class of payoff functions. We obtain numerical data by transferring the theory to Monte Carlo algorithms. For options with a single underlying we compare the simulation results to the analytic formulae for barrier options with both vanilla and digital payoffs. In order to take account of the continuous monitoring of the barrier, we include a Brownian bridge type bias correction proposed by Beaglehole et al.

To show that the theory is applicable in higher dimensions as well, we treat the particular example of a two-asset correlation option including a linear barrier related to the weighted average of the share prices. Due to the lack of an analytic expression, we compute both the price and the Delta by Monte Carlo simulation, where the algorithm for Delta is based on Bismut representation.

### Pricing Discrete Barrier Options with Time-dependent Coefficients

*Master Thesis in "Mathematical Finance", Oxford (2004)*

An extension of the Broadie-Glasserman expansion for discrete barrier options is developed for discrete barrier options with time-dependent coefficients. This expansion can be used to efficiently price discrete barrier option with time-dependent coefficients approximately avoiding heavy numerical calculations.

### Investigation of the Stochastic-Volatility Libor Market Model

*Master Thesis in "Mathematical Finance", Oxford (2004) *

In this thesis, the influence of stochastic volatility on the pricing of vanilla and correlation-dependent exotic interest-rate instruments in the Libor market model is investigated by comparison of prices in the stochastic volatility model to those obtained in a standard Libor market model.

### Relation between Greeks for American Options

*Master Thesis in "Mathematical Finance", Oxford (2004)*

The Greeks or sensitivites of option values with respect to parameters such as spot price, risk free rate of interest, or volatility of the underlying - to name but a few - play an important role for hedging, valuing and managing the risk of options. In the present work we address the derivation of the so called Gamma-Vega and Delta-Rhorelations for American options.

American options are valued numerically. In this context, calculating sensitivities

consumes additional computer power. Furthermore, the accuracy is usually worse than for the option value itself. The relations therefore save computing power and improve accuracy of results.

### HJM-Type Models with Discontinuities

*Master Thesis in "Mathematical Finance", Oxford (2004) *

The present work considers generalisations of the HJM model to semimartingale

driving terms. The bond pricing formula is recovered in the general case, and a bond option is priced in a HJM model driven by a jump diffusion.

### Valuing American-Asian Options with the Longstaff-Schwartz Algorithm

*Master Thesis in "Mathematical Finance", Oxford (2003)*

The Least-Squares Monte Carlo (LSM) algorithm of Longstaff & Schwartz is a

new and powerful approach for the valuation of the price of American options. This approach can also be applied to exotic, path-dependent options where the payoff and the value of the option depends on the value of the underlying, averaged over a given time-window (Asian options). So far, only American-Asian options have been considered where the starting point of the time window used for averaging is fixed.

American-Asian options with rolling time-window, i.e. a time window of constant width are particularly complex since they constitute a non-Markovian problem, that can not be transformed to a problem with a finite number of state variables.

In this work, the LSM algorithm is applied to American-Asian options with rolling

time window. The value of the option is determined. The convergence of the algorithm is studied in dependence of the maximum degree of the polynomials used in the regression and the number of base variables.

### A Finite Element Implementation of Generalized Passport Options

*Master Thesis in "Mathematical Finance", Oxford (2003) *

Except for special cases passport options and general options on trading accounts do not have closed-form solutions. Here we show how to derive approximate solutions using finite element methods. We also show that finite elements offer advantages in computing the hedge parameters. The results are applied to several examples.

### Pricing of Discrete Barrier Options

*Master Thesis in "Mathematical Finance", Oxford (2003)*

This study addresses the pricing of discrete barrier options using analytical methods and numerical simulations. For discrete barrier options, the asset price is only monitored at instants ti = iT/m, where T is the expiration date and m − 1 is the number of monitoring points (i = 1, 2, ...,m−1). The analytical solution for discrete barrier options involves mdimensional integrals, which are not analytically tractable for options with a high number of monitoring points (m > 5). The use of numerical procedures for pricing discrete barrier options becomes increasingly difficult at even higher numbers of monitoring points, e.g. m 100. In this case, it is convenient to perform an asymptotic expansion that becomes exact in the limit as the number of monitoring points goes to infinity. Broadie et al. have derived a continuity correction for discrete barrier options that satisfies this condition.

We show that the continuity correction for single barrier options with discrete monitoring can also be derived within the framework of matched asymptotic expansions. This method can be extended to derive an asymptotic expression for double barrier options with discrete monitoring.

### Non-Gaussian Distributions in Option Pricing

*Diploma Thesis in "Mathematical Finance", Oxford (2003) *

This project report presents alternative approaches for the pricing of financial derivatives which are based on more flexible and realistic models of asset prices. This is achieved by replacing the Gaussian distribution frequently used in models of financial instruments with various types of non-Gaussian distributions. One of these alternatives, the Gaussian mixture is investigated in more detail. The Expection Maximization (EM) algorithm is suggested as an efficient tool to estimate the parameters of such models. Furthermore it is shown that the Black-Scholes pricing formulas for plain vanilla European options can be used with little modifications under the assumption of mixed Gaussian returns. Monte-Carlo methods can also be efficiently applied. Based on observed market data the modified Black-Scholes formulas have been used to value European options on stock indices. The resulting values were compared to the corresponding Black-Scholes values. The use of mixed lognormal diffusions in modelling volatility smiles is discussed.

### A Monte Carlo pricing tool for American basket options on two assets

*Diploma Thesis in "Mathematical Finance", Oxford (2002)*

### Arbitrage-free Pricing of Exotic Interest Rate Derivatives in a Short-Rate Framework - Theory and Implementation

*Master Thesis in "Mathematical Finance", Oxford (2002) *

In this thesis we present the calibration of the one-factor short-rate model of Black, Derman and Toy and its application to the pricing of path-dependent options. First, we provide an overview about one-factor models in general before we investigate the speci c model of Black, Derman and Toy (BDT) in detail. We then show how to calibrate the BDT model to market prices of discount bonds and caps and examine the di erent model assumptions and implications. Moreover, we critically review all important model implications, such as mean-reversion, non-stationarity of the volatility term-structure etc. The calibration is performed with real market data and with a proprietary model which has been fully implemented in Visual Basic and Excel.

Finally, we apply the calibrated BDT short-rate tree to the pricing of non-standard caps, i.e. ratchet and sticky caps. For this, we use the technique of Monte Carlo simulation in the tree. This combined method is very attractive since it allows to price path-dependent options in a lattice which has been previously calibrated to market data. The reason for this combination of two numerical methods is that calibration of short-rate trees is practically very elegant and time-e/-cient but cannot handle path-dependent features. However, the path-dependent feature can be accomplished by a Monte Carlo simulation in the tree.

This thesis shows that the BDT model can be e/-ciently calibrated and applied to the pricing of exotic fixed income derivatives. However, it is important to understand all of the subtle model implications in order to correctly use the model in practice.

### Option Pricing for Discrete Hedging and Non-Gaussian Processes

*Master Thesis in "Mathematical Finance", Oxford (2002)*

The Black-Scholes option pricing method is correct under certain assumptions, among others continuous hedging and a log-normal underlying process. If any of these two assumptions is not fulfilled, a risk-less replication of an option is in general not possible.

To handle this case, a pricing method was proposed by Bouchaud and Sornette.

Similar to Black-Scholes, a hedging portfolio is considered. The hedging strategy is such that the risk of the hedging portfolio is minimized. An option price is then

deduced from this hedging strategy. Since a risk remains, the price includes a risk premium.

In this thesis, a new alternative method is presented, which instead of minimizing the portfolio risk minimizes the option price. This makes the option most competitive on the market. For the option writer, the ratio of return to risk is, by definition of the method, the same as for the Bouchaud-Sornette approach.

Both methods were compared with each other. For typical options, differences of up to 10 % of the price were found. The risk premium as well as fat tails in the underlying process give rise to volatility smiles for both methods. Furthermore, it was found that both methods are consistent with Black-Scholes pricing. The results converge towards the Black-Scholes result in the continuous time limit for a log-normal process.

### Valuation of American Options

*Diploma Thesis in "Mathematical Finance", Oxford (2001) *

The valuation of American options is a difficult problem. The basic reason is that

the asset price at which early exercise is optimal isn't known in advance and has to be found as part of the solution of the problem. In mathematical terms, a partial differential equation known as Black-Scholes equation has to be solved with a moving boundary condition. This is known in general as a moving boundary problem. Analytic solutions of this kind of problems can be found only in very special cases (e.g. for the American call on an asset paying a single discrete dividend during the lifetime of the option). However, because of the practical importance of American options, their efficient and accurate pricing is vital for option market participants. Finite difference methods can be used to solve the differential equation numerically, but in order to obtain an accurate solution, a considerable computational effort is necessary. Therefore other, more efficient methods have been developed.

In this thesis a comparison of numerical and approximative methods to solve this problem for equity (representing options on assets with discrete known payments) and FX options (representing options on assets with a continuous dividend yield or holding costs) is presented. The numerical methods are based on a binomial tree. A finite difference method the solution of which is considered exact is used as a benchmark all the other methods are compared to. The result is an assessment when these methods can be successfully applied.