Introductory description

This module is available for students on a course where it is a listed option and as an Unusual Option to students who have completed the prerequisite modules.

Pre-requisites

• ST401 Stochastic Methods in Finance; OR ST908 Stochastic Calculus for Finance (MSMF students)

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Option Pricing and Hedging in Continuous Time

• Pricing Europeans via equivalent martingale measures, numeraire, fundamental valuation formula, arbitrage and admissible strategies
• Pricing Europeans via PDEs (brief review)
• Completeness for the Black Scholes economy
• Implied volatility, market implied distributions, Dupire
• Stochastic volatility and incomplete markets
• Pricing a vanilla swaption, Black's formula for a PVBP-digital swaption
• Multicurrency Economy
• Black-Scholes economy with dividends
• Economy with possibility of default CVA, DVA of a vanilla swap

Applications across Asset classes

Interest Rates: Term Structure Models

• Short rate models. Introduction to main examples, implementation of Hull-White
• Market Models (Brace, Gaterek and Musiela approach), specification in terminal and spot measure
• Pricing callable interest rate derivatives with market models, drift approximation and
  separability, implementation via Longstaff-Schartz
• Greeks via Monte Carlo for market models, pathwise method, likelihood ratio method.
• Markov-functional models
• Practical issues in choice of model for various exotics, Bermudan swaptions
• Calibration: global versus local
• Stochastic volatility models, SABR

Credit

• Description of main credit derivative products: CDS, First-to-default swaps, CDOs
• Extension of integration by parts, Ito's formula, Doleans exponential to cover jumps
• Martingale characterization of single jump processes, Girsanov's Theorem
• State variable, default and enlarged filtrations
• Filtration switching formula
• Intensity-correlation versus default-events correlation
• Conditional Jump Diffusion approach to modelling of default correlation

FX

• Stochastic local volatility models, calibration,
• Gyongy's Theorem
• Barrier options

Time permitting:

• Equity
• Dividends
• Volatility as an asset class, variance swaps, volatility derivatives
• Heston model

Learning outcomes

By the end of the module, students should be able to:

• Demonstrate understanding of the conceptual framework for the pricing and hedging of derivatives using EMMs and be able to apply it in a range of models of financial markets.
• Demonstrate in-depth knowledge of some models, including for interest rate term structures, credit risk and stochastic volatility.
• Be able to use techniques and concepts from stochastic analysis, such as the Markov property, Ito’s formula and changes of measure to study models of financial markets. 
• Show some awareness of how models are used in practice: their interpretation and implementation. 
• Be able to communicate mathematical results about financial models clearly and precisely.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

Demonstrate understanding of the conceptual framework for the pricing and hedging of derivatives using EMMs and be able to apply it in a range of models of financial markets.
Demonstrate in-depth knowledge of some models, including for interest rate term structures, credit risk and stochastic volatility.
Be able to use techniques and concepts from stochastic analysis, such as the Markov property, Ito’s formula and changes of measure to study models of financial markets.
Show some awareness of how models are used in practice: their interpretation and implementation.
Be able to communicate mathematical results about financial models clearly and precisely.

Transferable skills

TBC