Pricing

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)

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 de- signed 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 impor- tant 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 va- rious 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 between
the 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-Rho
relations 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. [1]
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 impor-
tant model implications, such as mean-reversion, non-stationarity of the
volatility term-structure etc. The calibration is performed with real mar-
ket 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 tech-
nique of Monte Carlo simulation in the tree. This combined method is
very attractive since it allows to price path-dependent options in a lat-
tice 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.