Commitment:

• 3 hours of lectures per week.

Aims:

• To provide an introduction to three advanced topics in Mathematical Finance.
• Compute and explain key variables in the relevant models.
• Apply appropriate mathematical techniques.
• Analyse and compare different modelling approaches.

Content:

Three topics will be covered each year, motivated by current questions relevant to the financial industry. Example topics are algorithmic trading, introduction to market microstructures, limit order books.

Algorithmic trading would cover topics such as electronic markets - market participants, order types, and the limit order book. Stochastic optimal control and stopping; the dynamic programming principle and HUB equation. Optimal execution models with temporary and permanent price impact, linear and non-linear impact. Optimal execution models with limit orders and market orders; fill probabilities. Market Making. Targeting volume, for example, VWAP schedules.

Introduction to market microstructures would cover topics such as: order flow and liquidity; inventory risk, trade size and market depth; measuring liquidity and price discovery; static limit order markets; dynamic limit order markets; high-frequency trading; trading strategies.

Limit order books would cover topics such as main statistical characteristics of LOB: the distribution of the time of arrivals, volumes, placement and cancellation of orders, the shape of LOB and the intraday seasonality, modelling in physical time and event-driven times. Agent-based models, microstructure of the double auction, zero-intelligence, econophysics approaches via reaction-diffusion and decomposition-evaporation processes. Markov models of LOB, diffusive limits, large-scale limits, queueing theory modelling, Hawkes processes, SDEs and PDEs modelling, game-theoretic modelling.

Assessment:

• Three in-class tests: each worth 5% of the final module mark.
• Final 2 hour exam: worth 85% of the final module mark.

Illustrative Bibliography (for above topics):

Algorithmic trading

• Almgren (2003), optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Math. Finance, 10, 1- 18.
• Cartea, Jaimungal and Penalva, (2015), Algorithmic and high-frequency trading, CUP.

Introduction to market microstructures

• Foucault T., Pagano M. and A. Roell, (2013) Market liquidity: theory, evidence and policy. Oxford University Press.
• Hasbrouck J. (2007) Empirical market microstructure, Oxford University Press

Limit order books

• Abergel F. et al. (2016), Limit order books, CUP.
• Lu. X and Abergel, F. (2018) High-dimensional Hawkes processes for limit order books: modelling, empirical analysis and numerical calibration, Quant Finance, 18(2), 249 - 264.
• Cont, R., Stoikov, S. and Talreja, R. (2010), A stochastic model for order book dynamics, Operation Research, 58, 549 - 563.