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Abstract:
These are my Lecture Notes for a course in Continuous Time Finance which I taught in the Summer term 2003 at the University of Kaiserslautern. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any comment on the notes. Please send your comments via e-mail to ce16@standrews.ac.uk.

Mathematical Finance, Financial Market Models, Stochastic Integration, Option Pricing

Abstract:
These are my Lecture Notes for a course in Discrete Time Finance which I taught in the Winter term 2005 at the University of Leeds. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any comment on the notes. Please send your comments via e-mail to ce16@st-andrews.ac.uk.

Discrete Time Finance, Mathematical Finance

Abstract:
These are my Lecture Notes for a course in Game Theory which I taught in the Winter term 2003/04 at the University of Kaiserslautern. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any comment on the notes. Please send your comments via e-mail to ce16@st-andrews.ac.uk.

Game Theory, Mathematical Economics, Set valued analysis, Fixed point theorems

Abstract:
In this paper the performance of locally risk-minimizing hedge strategies for European options in stochastic volatility models is studied from an experimental as well as from an empirical perspective. These hedge strategies are derived for a large class of diffusion-type stochastic volatility models, and they are as easy to implement as usual delta hedges. Our simulation results on model risk show that the locally risk-minimizing hedges are robust with respect to uncertainty and even misconceptions about the underlying data generating process. The empirical study indicates that locally risk-minimizing hedge strategies consistently produce lower standard deviations of profit-and-loss-ratios than delta hedges (over different time periods as well as in different markets). The more skewed the market and the more out-of-the-money the option, the higher the benefit.

Locally risk-minimizing hedge, delta hedge, stochastic volatility,

Abstract:
We discuss how implied volatilities for OTC traded Asian options can be computed by combining Monte Carlo techniques with the Newton method in order to solve nonlinear equations. The method relies on accurate and fast computation of the corresponding vegas of the option. In order to achieve this we propose the use of logarithmic derivatives instead of the classical approach. Our simulations document that the proposed method shows far better results than the classical approach. We also discuss the issue of variance reduction in order to optimize our method.

implied volatility, Monte Carlo simulation, Asian options, exotic options

Abstract:
We study the classical real option problem in which an agent faces the decision if and when to invest optimally into a project. The investment is assumed to be irreversible. This problem has been studied by Myers and Majd [18] for the case of a complete market, in which the risk can be perfectly hedged with an appropriate spanning asset, by McDonald and Siegel [16], who include the incomplete case but assume that the agent is risk neutral toward idiosyncratic risk and later by Henderson [12] who studies the incomplete case with risk aversion toward idiosyncratic risk under the assumption that the project value follows a geometric Brownian motion. We take up Henderson's utility based approach but assume as suggested by Dixit and Pindyck [4] as well as others, that the project value follows a mean reverting geometric Ornstein-Uhlenbeck process. The mean reverting structure of the project value process makes our model richer and economically more meaningful. By using techniques from optimal control theory we derive analytic expressions for the value and the optimal exercise time of the option to invest.

Real Options, Models of Mean-Reversion, Optimal Control, Incomplete Market Models

Abstract:
We study the classical geometric mean reversion process which has been used to model commodity prices by various authors in Economics and Finance. We obtain certain regularity results which guarantee positivity and the existence of a stationary distribution. More important we derive an analytical formula for the stationary distribution and all of its higher moments. Furthermore we derive a computationally simple but efficient recursive formula for the higher moments which we apply to moment matching.

Models of mean-reversion, equilibrium distributions

Abstract:
This paper derives an analytic expression for the distribution of the average volatility $\frac{1}{T-t} \int_t^T \sigma_s^2 ds$ in the stochastic volatility model of Hull and White. This result answers a longstanding question, posed by Hull and White (Journal of Finance 42, 1987), whether such an analytic form exists. Our findings are applied to obtain closed-form solutions for European and Digital call option prices. The paper also provides an explicit solution for the Delta hedge of a European call. Moreover, it is proved that the Delta hedge under the minimal martingale measure coincides with the locally $R$-minimizing hedge in the model considered here.

Stochastic volatility models, incomplete markets, Delta hedging, locally R-minimizing hedging strategies, Malliavin calculus

Abstract:
We show that the Heston volatility or equivalently the Cox-Ingersoll-Ross process is Malliavin differentiable and give an explicit expression for the derivative. This result assures the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model and the Cox-Ingersoll-Ross model for interest rates.

Malliavin calculus, stochastic volatility models, Heston model, Cox-Ingersoll-Ross process

Abstract:
We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of the first author in order to derive approximate option pricing formulas in the context of the Heston model. Numerical results are given.

Malliavin calculus, stochastic volatility models, Heston model, Cox-Ingersoll-Ross process, Hull and White formula, Option pricing

Abstract:
Due to the increasing risk of inflation and diminishing pension benefits, insurance companies have started selling inflation-linked products. Selling such products the insurance company takes over some or all of the inflation risk from their customers. On the other side financial derivatives which are linked to inflation such as inflation linked bonds are traded on financial markets and appear to be of increasing popularity. The insurance company can use these products to hedge its own inflation risk. In this article we study how to optimally manage a pension fund taking positions in a money market account, a stock and an inflation linked bond, while financing investments through a continuous stochastic income stream such as the plan member's contributions. We use the martingale method in order to compute an analytic expression for the optimal strategy and express it in terms of observable market variables.

Pension mathematics, inflation, long-term investment

Abstract:
We show that under the Black Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility. This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the Black-Scholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the Black-Scholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options.

Asian options, volatility, vega, duration, qualitative risk-management

Abstract:
We solve a Dixit and Pindyck type irreversible investment problem in continuous time under the assumption that the project value follows a Cox-Ingersoll-Ross process. We indicate how the solution qualitatively differs from the two classical cases geometric Brownian motion and geometric mean reversion. Furthermore we discuss analytical properties of the Cox-Ingersoll-Ross process and demonstrate potential advantageous of this process as a model of the project value with regards to the classical ones.

Irreversible investment, real options, models of mean-reversion

Abstract:
We consider a continuous time market model, in which agents influence asset prices. The agents are assumed to be rational and maximizing expected utility from terminal wealth. They share the same utility function but are allowed to possess different levels of information. Technically our model represents a stochastic differential game with anticipative strategy sets. We derive necessary and sufficient criteria for the existence of Nash-equilibria and characterize them for various levels of information asymmetry. Furthermore we study in how far the asymmetry in the level of information influences Nash-equilibria and general welfare. We show that under certain conditions in a competitive environment an increased level of information may in fact lower the level of general welfare. This effect can not be observed in representative agent based models, where information always increases welfare. Finally we extend our model in a way, that we add prior stages, in which agents are allowed to buy and sell information from each other, before engaging in trading with the market assets. We determine equilibrium prices for particular pieces of information in this setup.

information, financial markets, stochastic differential games

Abstract:
Geometric mean reversion plays a fundamental role in economic dynamic models. While it is known, at least since Merton (1975) [9], that the equilibrium distribution of geometric mean reversion is a Gamma distribution, an explicit expression for the non-equilibrium distribution has not been known. In this article we compute the probability density function of X(t) where X(·) represents a general geometric mean reversion process.

Geometric mean reversion, non-equilibrium analysis, economic dynamics, econometrics

Abstract:
We study a class of differential public good games and show how static public good games can naturally be embedded into this class. This allows us to compare the outcomes in the static and the dynamic case. In the dynamic case we study the feedback Nah-equilibria and compare these to the Nash equilibria of the corresponding static game. To solve for feedback the Nash equilibria in the dynamic case, we solve the Hamilton-Jacobi-Bellmann equation by using the method of characteristic functions. Analytical results are given.

Public good games, differential games, optimal control

Abstract:
We adapt the evolutionary stock market model from Evstigneev, Hens, Schenk-Hoppé (2006) to a continuous time framework, where uncertainty in dividends is produced by a single Wiener process. The setup is therefore significantly different from Yang and Ewald (2008), who also study continuous time, but remain within the framework of random dynamical systems of non-diffusive type. For the case of fix-mix strategies we derive the stochastic differential equation which determines the evolution of the wealth processes of the various market players. These stochastic differential equations are highly non-linear and we find that it is impossible to solve them analytically. Instead we simulate the wealth dynamic for various initial setups of the market. A detailed discussion of our observations from the simulations is given.

Behavioral Finance, Evolutionary Finance, Wealth dynamics

Abstract:
In this article we study a game theoretic model of the conflict which arises between a monetary authority and the private sector with regard to the inflation-rate. Building up on the simple but illustrative one period game theoretic model introduced by Barro and Gordon, we assume that rather than playing a one shot game, monetary authority and private sector react to each other repeatedly for an infinite number of times. Both, the monetary authorities's and the private sector's reactions are assumed to be stochastic in the form of fixed behavioral transition probabilities. These probabilities are interpreted as strategies in a new game. We study the set of Nash-equilibira of this new game and how these correspond to the classical discretionary Nash-equilibrium of Barro-Gordon as well as the classical Non-Nash low inflationary state. In contrast to Barro-Gordon we show that the low-inflationary state can be realized as a Nash-equilibrium in our model.

Monetary Policy, Inflation, Game Theory, Stochastic Reaction Strategies

Abstract:
We combine and extend two existing lines of research in game theoretic studies of fisheries. The first line of research is the inclusion of the aspect of predation and the consideration of multi-species fisheries within classical game theoretic models of fisheries and goes back to Quirk and Smith (1977), Anderson (1975) and most recently Sumaila (1996). The models developed in this line are either static or discrete time and do not include ecological uncertainty. The second line of research includes continuous time and uncertainty, but focuses on single species models and does not capture any features of ecological interaction, see for example Jorgensen and Yeung (1996). In this article we develop a continuous time framework, where ecological interaction is described by a stochastic dynamics, including the cases of predator prey and competition. We obtain a stochastic differential game and derive feedback Nash-equilibrium strategies in semi-analytic form. Furthermore we compare the results with the case where fisheries regulations restrict each fishery as to only being allowed to fish one particular species and study the inefficiencies which arise from this. In addition to that, we also consider the case where fisheries cooperate. Here we observe quite different effects on the ecosystem, depending on whether the system is competitive or predator prey.

differential Games, Fisheries, Environmental and Resource Economics, Stochastic Optimal Control

Abstract:
Fershtman and Nitzan (1991) presented a continuous dynamic public good game model and solved the model for feedback Nash-equilibria. Wirl (1996) extended the model and considered nonlinear strategies. Both models do not include uncertainty and hence neglect an important factor in the theory of public goods. We extend the framework of Nitzan and Fershtman and include a diffusion term. We consider two cases. In the first case the volatility of the diffusion term is dependent on the current level of the public good. This setup will in principle lead to the same feedback strategies computed under certainty. In the second case the volatility is dependent on the current rate of public good provision by the agents. The result in this case is qualitatively different from the first one. We provide a detailed discussion of our results as well as numerical examples.

Stochastic differential games, Public goods, Hamilton-Jacobi-Bellman equations

Abstract:
We use Malliavin calculus and the Clark-Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic, that is closed form, expression. Numerical computations which are based on this expression are provided.

Asian options, option pricing, hedging, Malliavin calculus

Abstract:
We consider a model of a fishery in which the dynamic of the unharvested fish population is given by the stochastic logistic growth equation. Similar as in the classical deterministic analogon, we assume that the fishery harvests the fish population following a constant effort strategy. In a first step we derive the effort level that leads to maximum expected sustainable yield, which is understood as the expectation of the equilibrium distribution of the stochastic dynamics. This replaces the non-zero fixed point, in the classical deterministic setup. In a second step, we assume that the fishery is risk averse and that there is a trade off between expected sustainable yield and uncertainty measured in terms of the variance of the equilibrium distribution. We derive the optimal constant effort harvesting strategy for this problem. In a final step, we consider an approach which we call the mean-variance analysis to sustainable fisheries. Similar as in the now classical mean-variance analysis in Finance, going back to Markowitz (1957), we study the problem of maximizing expected sustainable yields under variance constraints, and dual to this, minimizing the variance, e.g. risk, under guaranteed minimum expected sustainable yields. We derive explicit formulas for the optimal fishing effort in all four problem considered and study the effects of uncertainty, risk aversion and mean reversion speed on fishing efforts.

Fisheries, Environmental and Resource Economics, Sustainability, Maximum Sustainable Yield, Stochastic dynamic fisheries models

Abstract:
In this paper we study the question what value an agent in a generalized Black-Scholes model with partial information attributes to the complementary information, i.e. the information which is needed to gain full information. To do this we study the utility maximization problems from terminal wealth for the two cases partial information and full information. We assume that the drift term of the risky asset is a dynamic process of general linear type and the two levels of observation correspond to whether this drift term is observable or not. To obtain analytical tractable results we use logarithmic utility in combination with methods from stochastic filtering theory.

Partial information, value of information, stochastic optimal

Abstract:
We consider Merton's version of the Solow model Merton (1975), where capital per labor is assumed to follow the diffusion process: dk(t)=[sf(k(t))-(n+lambda-sigma2)k(t)]dt + sigmak(t)dW(t), with constant per capital savings rate s. Merton defined a golden rule in this context as one for which expected utility from consumption c=(1-s)f(k) under the equilibrium distribution of capital/output becomes maximal. We discuss some of Merton's results and their limitations and then provide an alternative setup, in which we consider a mean-variance optimizer. We show then, that unless in Merton, risk-aversion and volatility do have an effect on Golden rule consumption, even if a Cobb-Douglas production function is assumed.

Economic Growth, Golden Rule, Solow model, Risk Aversion

Abstract:
In this article we consider a continuous time framework where the central bank has two objectives regarding inflation. On the one hand there is the usual trade off between output and inflation as seen in the original Barro Gordon model, evolving continuously (and discounted) over time. In addition the bank faces a terminal payoff which is linear in delivered inflation during that period due to physical payoff the bank has to deliver at maturity (the bank is short an inflation linked bond ILB). The private agents in this model, unlike as usual, are not supposed to have a rational expectation. Instead we assume a learning strategy for them given by some stochastic differential equation. Actual inflation can be controlled by the Central Bank up to the some uncertainty term. This simple and stochastic optimal control problem can be solved entirely and analytically as we will see.

Monetary Policy, Optimal Control, Inflation Indexed Bond

Abstract:
We study forward prices and prices of European call options, which are written on a renewable resource. The price of this resource is assumed to follow the inverse of a geometric mean reverting process. We assume that the resource is not tradable, until the option matures at time T and study the dynamics of the forward prices of the resource. In contrast to Black (1976) we show that forward prices do not evolve according to a geometric Brownian motion, but follow a more complex process. Even though, we are able to compute forward prices in closed form. For the case of an option we show that the Black (1976) formula needs to be adapted in such a way, that the normal distribution is replaced by a reciprocal T-distribution, to get at least a very good approximation of the true option price. We include numerical evidence to strengthen our result. We also include an analytic expression for the true option price in terms of an integral representation. Finally, we derive pricing formulas for options written on forward contracts, and show how forwards contracts can be hedged under the assumption that there is a spanning asset.

Options, Commodities, Renewable Resources, Risk management

Abstract:
We show that Australian options are in fact Asian options in the classical sense. By using this relationship and a relationship between the underlying of an Australian option and geometric mean reversion, we are able to give a quick proof of the Milevsky and Posner result for the approximation of Asian options by using the reciprocal (Γ)Gamma-distribution. This result then also carries over to Australian options, giving us an approximation in terms of the inverse (Γ)Gamma-distribution for Australian options. We point out some problems that are attached to this approach. Furtheron, we provide an analytical formula for the price of an Australian option. Finally we derive a PDE for the price of an Australian option, which unlike in the classical Asian case, has only one state variable, rather than two. This enables us to infer from Carr, Ewald and Xiao (2008), that Australian options are increasing in volatility, answering a question that was indicated by Moreno and Navas (2008).

Asian options, Australian options

Abstract:
In this paper we discuss the Malliavin differentiability of a particular class of Feller diffusions which we call $\delta$-diffusions. This class is given by \begin{equation*} d\nu_t=\kappa(\theta-\nu_t))dt \eta \nu_t^{\delta}d\mathbb W_t^2, \delta\in[\frac{1}{2},1] \end{equation*} and appears to be of relevance in Finance, in particular for interest and foreign-exchange models, as well as in the context of stochastic volatility models. We extend the result obtained in Alos and Ewald (2008) for $\delta=\frac{1}{2}$ and proof Malliavin differentiability for all $\delta \in [\frac{1}{2},1]$.

Malliavin calculus, Feller diffusions, Greeks, Option pricing

Abstract:
We show how infinite horizon stochastic optimal control problems can be solved via studying their finite horizon approximations. This often leads to analytical solutions for the infinite horizon problem, even when the complexity of the finite horizon approximation is to large, as in order to allow analytical solutions in the finite horizon case.

Stochastic optimal control, Hamilton-Jacobi-Bellman equation

Abstract:
We combine methods for portfolio optimization in incomplete markets which are due to Karatzas et al. [6] with methods proposed by Nualart based on Malliavin Calculus to model insider trading within a stochastic volatility model. We compute the optimal portfolio within a certain set of insider strategies for a general stochastic volatility model but also apply the methods to explicit examples. We further discuss how the Heston model fits into this context.

Malliavin calculus, insider trading, portfolio optimization, assymetric information

Abstract:
An interpretation of the conflict between male and female parents during the process of caring for their common offspring by means of Game Theory was given in Houston and Davies. [A.I. Houston, N.B. Davies, The evolution of cooperation and life history in the dunnock Prunella modularis, in: R.M. Sibly, R.H. Smith (Eds.), Behavioral Ecology, Blackwell Scientific Publications, 1985, pp. 471-487]. Mathematically, this model represents a static game with continuous strategy sets. Recently, this model was reconsidered in a dynamic discrete time framework which also included state dependencies [J.M. McNamara et al., A dynamic game-theoretic model of parental care, J. Theor. Biol. 205 (2000) 605-623]. In this article, we give an interpretation of the parental care conflict in continuous time by means of a differential game with state dependent strategies.

Models of parental care, Behavioral ecology, Differential games, Game theory

Abstract:
We consider a diffusion (Xt) satisfying the stochastic differential equation dXt = à3B2(Xt, u)dt à3C3(Xt, v)dWt where u and v are parameters and consider the problem of minimizing certain functionals of the form View the MathML source in u and v where ti set membership, variant [0, T] are not necessarily distinct time points. For this we combine classical gradient methods with techniques from Malliavin calculus. The proposed technique has a particular advantage to classical techniques in the case when the functions hi are not continuous or have singularities. This is the case when the functions hi represent certain quantiles, i.e. hi(x)colon, equals1{xless-than-or-equals, slantpi} and the problem is to choose the parameters u, v in a way that the stochastic model fits the quantiles best.

Malliavin calculus, Monte-Carlo simulation, Calibration, Gradient methods, Diffusion-models, Optimization

Abstract:
We develop a continuous-time evolutionary market model where prices are endogenously generated by supply and demand. Investment strategies are assumed to be fix-mix, which means that the relative budget shares are constant in time. The model is therefore a hybrid. While given portfolio rules remain constant over time, assets, market-clearing and in particular market shares of the individual portfolio strategies evolve in continuous time. Our main goal is to understand the wealth dynamics which describes the evolution of market shares. We study its asymptotic properties and identify evolutionary stable investment strategies. These strategies prevent entrants to the financial market from gaining wealth in the long run and furthermore, in the existence of a small diversified number of mutant strategies, drive the invading strategy out of the market. Our definition of evolutionary stability is therefore a close adaptation of Maynard-Smith and Price's (1973) original definition of an ESS [8].

Evolutionary Economics, Evolutionary Finance, continuous-time portfolio theory, endogenously determined asset prices, evolutionary stability of trading strategies

Abstract:
We discuss the application of gradient methods to calibrate mean reverting stochastic volatility models. For this we use formulas based on Girsanov transformations as well as a modification of the Bismut-Elworthy formula to compute the derivatives of certain option prices with respect to the parameters of the model by applying Monte Carlo methods. The article presents an extension of the ideas to apply Malliavin calculus methods in the computation of Greek's.

Malliavin calculus, Monte Carlo simulation, Stochastic volatility models, Calibration, Gradient methods, Value at risk

Abstract:
In this article we show how the classical probabilistic technique of Malliavin calculus can be applied to study interesting aspects in the theory of stochastic differential games. These include in particular the aspect of information asymmetry. We identify the limitations of the classical setup and show how Malliavin calculus can overcome these.

Stochastic differential games, Malliavin calculus, dynamic games, information asymmetry

Abstract:
A stochastic differential game with the anticipative strategy sets is used to model a competition of two heterogeneously informed agents in a financial market. We interpret Nash-equilibria by a preference-suppressed measure where the agents of using a general utility function act as if they were logarithmic utility user. We derive necessary and sufficient criteria for the existence of Nash-equilibria and characterize them for various levels of information asymmetry. Furthermore we study how far the asymmetry in the level of information influences Nash-equilibria and general welfare in the case of logarithmic utility in which the closed form Nash-equilibria are obtainable.

function, information, financial markets, stochastic differential games

Abstract:
We implement the Heston stochastic volatility model by using multidimensional Ornstein-Uhlenbeck processes and a special Girsanov transformation, and consider the Malliavin calculus of this model. We derive explicit formulas for the Malliavin derivatives of the Heston volatility and the log-price, and give a formula for the local volatility which is approachable by Monte-Carlo methods.

stochastic volatility models, local volatility, Malliavin calculus, Monte Carlo methods