King's College London
Financial Mathematics
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Core MSc Lecture Courses

Applied Probability and Stochastics (CMFM01) S1 Compulsory
Probability spaces, random variables, distributions, independence, product spaces. Expectation and conditional expectation. Moments, generating functions, characteristic functions. Random processes, filtrations and stopping times. Martingales, Brownian motion and the Poisson process. Elements of Itô integration.

Risk Neutral Valuation: Pricing and Hedging Derivatives (CMFM02) S1 Compulsory
Forward prices, discounting, arbitrage-free pricing; binomial trees, derivatives pricing in discrete time by use of binomial lattices, geometric Brownian motion, volatility and drift, martingales and conditional expectation, Itô calculus, hedging portfolios, replication, Black-Scholes model, put-call parity. Option deltas, gammas, vegas, and other sensitivities. Risk premium, risk-neutral measure. Arbitrage-free multi-asset financial markets.

Financial Markets (CMFM03) S1 (afternoons)
This course presents an overview of the world's financial markets, and a concise mathematical formulation of the main characteristics of financial instruments and trading practice, with an emphasis on quantitative aspects of options, futures, and other derivatives. Topics covered include: Spot markets for stocks, bonds, currencies, commodities. Forward markets. Commodity futures, financial futures. Stock options, index options, currency options, commodity options, interest rate options, options on futures. Interest rate swaps, currency swaps. Corporate bonds, treasury bonds, inflation-linked products. Energy and credit markets. Structured products, OTC derivatives.

Stochastic Analysis (CMFM04) S2
Itô's isometry and the Itô integral. Levy's characterisation theorem and Girsanov's theorem. Random time changes, martingale representation theorems. Stochastic differential equations and diffusions. Ornstein-Uhlenbeck process, Brownian bridge, Bessel processes. Fokker-Planck equation and Feynman-Kac formula. Cauchy and Dirichlet problems.

Distribution Theory (CMFM05) S1
Theory of families of discrete and continuous probability distributions. Binomial, Poisson, geometric, negative binomial, hypergeometric distributions. Uniform, exponential, normal, gamma, beta, Student and Cauchy distributions. Compound distributions. Applications of order statistics. Scale and displacement families. Exponential families. Elements of information geometry. Properties of fat-tailed distributions. Stable families. Applications to mathematical finance. 

Numerical and Computational Methods in Finance (CMFM06) S1, material needed for FM12.
Direct solutions of partial differential equations arising in financial applications by finite difference methods. Binomial and trinomial tree methods. Monte Carlo simulation techniques in finance. Times series analysis and discrete time price volatility models. Practical implementation of numerical and computational methods in finance by use of spreadsheets.

Interest Rate and Foreign Exchange Dynamics (CMFM07) S2 compulsory
Discount bonds, interest rates, yield curves. Basic state variable models: Vasicek, Hull-White, Cox-Ingersoll-Ross, lognormal short rate, rational log-normal model and others. Heath-Jarrow-Morton framework; introduction to market models. Pricing formulae for caps, floors, swaptions. Multi-currency interest rate models.

Exotic Derivatives (CMFM08) S2
This course surveys pricing and risk management techniques for a number of important financial derivatives. Topics covered include: barrier, quanto, forward-starting, Asian, lookback, power and compound options; options to exchange one asset for another, dual-asset knock-out options and basket options. Energy and weather derivatives; insurance linked products.

Portfolio Risk Management (CMFM09) S2
Theory of risk aversion and utility, with applications. Trading strategies, consumption and portfolio value processes, expected utility from consumption and terminal wealth. Portfolio optimisation, solution of Merton problem. Transaction costs, market incompleteness, portfolio constraints.

Credit Risk Management (CMFM10) S2
Default events and survival indicator processes. Price processes for assets with jump risk. Overview of credit derivative structures. Lando's formula for risky discount bonds. Applications to credit default swaps, corporate and sovereign coupon bonds, and structured loan facilities. Duffie-Singleton formula for partial recovery on default. Introduction to structural models for default. Basic elements of copula theory and extremal events with applications to topics in credit risk.

Applied Computational Finance (CMFM12) S2
Computation of relevant special functions. Computation of standard and simple exotic Black-Scholes type results by the use of a formula. Implementation of Newton-Raphson iteration and other equation-solving methods. Implementation of binomial and trinomial trees. Numerical solution of SDEs and their implementation. Implementation of explicit and implicit finite-difference schemes. Course taught in two programming languages with assessed coursework.

Elective MSc Lecture Courses

Basic analysis (CMMS05)
A self-contained excursion through some fundamentals of functional analysis. Topics to be discussed: Banach spaces, open mapping and closed graph theorems, Hahn-Banach theorem, dual spaces, Banach-Alaoglu theorem.

Operator theory (CMMS08)
Hilbert and Banach spaces. Riesz representation theorem. The adjoint. Orthogonal projections. Spectral theory of bounded linear operators. Spectral theorem for bounded linear operators. Spectral theorem for compact operators. Applications to differential and integral equations.

Spectral theory of Markov chains (CMMS09)
The course will study the spectral properties of operators associated with Markov Chains and random walks on graphs. Topics to be discussed: Markov transition matrices, invariant states, ergodicity, continuous time Markov chains, the infinitesimal generator, etc.


Notes:
Subject to approval by the Programme Director, any course from the list of courses offered for the
MSc in Mathematics can be taken as an elective MSc lecture course, for example:

  • Advanced neural nets (CMNN15)
  • Information theory in neural networks (CMNN14)
  • Statistical mechanics of neural nets (CMNN13)
  • Measure theory (CMMS13)
  • Partial differential equations (CMMS12)
  • Statistical dynamics (CMMS35)

Not all MSc courses that are described here will necessarily run in a given year. Most, though not all, of the core financial mathematics lecture courses are likely to take place in the early evening. Some courses will run at other times.

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