Lecture:The first lecture is on Wed 28/9 then

Michaelmas term : Wed, weeks 1,2,3,4,5,7,8,9,10,11

Time: 10.00-12.00

Seminar:The first seminar is on Mon 3/10 then

Michaelmas term : Mon, weeks 2,3,4,5,7,8,9,10,11

Time: 12.00-13.00

Lecture: LSE, OLD.3.25

Seminar: LSE, CLM.2.04

Map and direction can be found here: http://www.lse.ac.uk/mapsAndDirections/findingYourWayAroundLSE.aspx

Analysis and algebra at the level of a BSc in pure or applied mathematics and basic statistics and probability theory with stochastic processes. Knowledge of measure theory is not required as the course gives a self-contained introduction to this branch of analysis.

The course covers core topics in measure theoretic probability and modern stochastic calculus, thus laying a rigorous foundation for studies in statistics, actuarial science, financial mathematics, economics, and other areas where uncertainty is essential and needs to be described with advanced probability models. Emphasis is on probability theory as such rather than on special models occurring in its applications. Brief review of basic probability concepts in a measure theoretic setting: probability spaces, random variables, expected value, conditional probability and expectation, independence, Borel-Cantelli lemmas Construction of probability spaces with emphasis on stochastic processes. Operator methods in probability: generating functions, moment generating functions, Laplace transforms, and characteristic functions. Notions of convergence: convergence in probability and weak laws of large numbers, convergence almost surely and strong laws of large numbers, convergence of probability measures and central limit theorems. If time permits and depending on the interest of the students topics from stochastic calculus might be covered as well.

Williams, D. (1991): Probability with Martingales. Cambridge University Press;

Kallenberg, O. (2002). Foundations of modern probability. Springer;

Billingsley, P(2008). Probability and measure. John Wiley& Sons;

Jacod, J., & Protter, P. E. (2003). Probability essentials. Springer;

Dudley, R. M. (2002). Real analysis and probability (Vol. 74). Cambridge University Press

Tuesday at 16.00 to 17.30 and Thursday at 15.30 to 17.00

Date: start October 11 2016, end December 8 2016

Lecture Room 3 (CDT space), Imperial College London

The purpose of this course is to provide an introduction to an important class of stochastic processes - continuous time Markov processes. Usually, a discrete time Markov process is deﬁned by specifying the law that leads from the state at one time to that at the next time. One cannot use this approach in continuous time. It is necessary to describe the transition law inﬁnitesimally in time, and then prove that this description leads to a well-deﬁned process for all time. Markov processes are some of the most widely used stochastic processes - they have applications in biology, ﬁnance, physics and other ﬁelds.

We start with a few examples - Brownian motion and continuous time Markov chains. Using these examples we can deﬁne continuous time Markov processes in greater generality. The key ingredient is the Hille-Yosida theorem, which links the generator (inﬁnitesimal description of the process) to the semigroup (evolution of the process over time). Usually only the generator is known explicity. We will explore how one can deduce properties of the process from information about the generator. As examples we will look at modiﬁcations of Brownian motion, in which there is special behavior at the boundary of the state space. We will also brieﬂy introduce stochastic diﬀerential equations and some of their properties.

The second part of the course will be about stationary and quasi-stationary distributions of Markov processes. If one has a deterministic system (ODE), then one is usually interested in whether there exists an equilibrium. For stochastic processes the natural analogue is whether there exists a stationary distribution. We will look at examples from stochastic diﬀerential equations (SDE) and study the long time behavior of the processes. We will give conditions under which the processes converge in some

1. Feller Markov semigroups: properties and examples

2. From Feller Markov processes to inﬁnitesimal description: How to get a generator from a semigroup/resolvent, examples

3. From inﬁnitesimal description to Feller Markov process: How to get a semigroup from a generator, quasi left continuity

4. Existence of a Feller Markov process with a given Feller Markov semigroup

5. Kolmogorov condition for path continuity

6. Examples and applications

7. A short introduction to SDE

8. Stationary distributions of Markov processes: theory and applications

Measure theoretic probability and real analysis. Some knowledge of functional analysis would also be helpful.

Continuous Time Markov Processes : An Introduction’ T. M. Liggett

Diﬀusions, Markov Processes and Martingales’ L.C.G. Rogers and R. Williams

Foundations of Modern Probability’ O. Kallenberg

Stochastic stability of diﬀerential equations’ R. Khasminskii

*********The lectures will be held at

1) October 12, 2016 - from 12.00 to 15.00

2) October 21, 2016 – from 13.00 to 16.00

3) November 3, 2016 - from 12.00 to 15.00 –

4) November 9, 2016 - from 12.00 to 15.00

5) December 1, 2016 - from 12.00 to 15.00 –

In this course a vast number of equity derivatives models is presented by implementing a standardized, uniform pricing approach. The first part is devoted to the development of a very general pricing representation that includes the major classes of equity derivatives models developed in literature (from Black-Scholes-Merton, CEV, local volatility to AJD and Pure Jump models). Generally these models are presented with heterogeneous techniques, notations and a plethora of various pricing representation; by starting from the simplest examples, the course bring back all the models in a uniform, formalized approach. In the second part the standardized analytical environment allows the regular deployment of powerful numerical techniques (i.e. adaptive quadrature schemes) that can solve in a flexible way the pricing and hedging problems regardless of the model’s theoretical complexity: the course aims to end up with the building of the code of an efficient, modular “all-purpose pricing and calibration engine.

Date: Fridays 4, 11, 18 and 25 November, and 4 December

Time: 9.00-12.00

Cass Bussiness School, 106 Bunhill Row, EC1Y 8TZ

Room: 3003 for the first four dates and 3002 for the last date

Important: pre-registration required to be allowed into the building. Each participant will be issued with an access card at the start of the course

1) High level view: Mean-variance (MV) preferences in the context of translation-invariant hulls of expected utility (Filipović & Kupper 2007); monotonization (ibid; Maccheroni et al. 2009)

2) The main applications: Optimal portfolios, good-deal price bounds and indifference prices (in general and then specifically for MV preferences)

3) Abstraction: Reduction of 2) to quadratic hedging = minimization of L^2 distance over subspace of stochastic integrals; CAPM in the context of quadratic hedging

4) Discrete time: dynamic programming vs sequential regressions: Li & Ng 2000, Bertsimas et al. 2001, Gugushvili 2003, Č. 2004; Foellmer & Schweizer 1989, Č. & Kallsen 2009

5) Key insights from discrete time: Foellmer-Schweizer decomposition; opportunity-neutral measure; minimal martingale measure and variance-optimal measure; \mathcal{E}-martingales (Choulli et al. 1998)

6) Primary examples: Hedging in a multinomial lattice for plain vanilla options and exotics; continuous-time limit; relation to Lévy processes

7) The same again but for semimartingales: Applications to Lévy processes, stochastic volatility and hidden Markov chain models; semi-explicit results using Fourier transform (Duffie & Richardson 2002, Hubalek et al. 2006, Č. & Kallsen 2007, 2008, Kallsen et al. 2009, Kallsen & Pauwels 2009, 2010, Kallsen & Vierthauer 2009, Denkl et al. 2009)

8) Lots of ‘interesting’ technical details: Schweizer’s set of admissible strategies; closedness of marketed subspace; (non)existence of F-S decomposition; sufficient conditions for admissibility; Girsanov theorem for signed measures (Choulli et al. 1998 and references therein)

9) Asymptotic theory motivated by 6): Tracking error as an approximation to hedging error (Bertsimas et al. 2000); results for smooth and non-smooth pay-offs (Gobet & Temam 2002); related results from literature with transaction costs; approximation of Lévy model results by Black-Scholes greeks.