ST909 Applications of Stochastic Calculus for Finance
- This is a core module for the MSc in Mathematical Finance.
- PhD students interested in taking the module should consult the lecturer.
Commitment: 30 hours of lectures 10 hours of tutorials
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 the 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 the choice of model for various exotics, Bermudan swaptions
Calibration: global versus local
- Stochastic volatility models, SABR
- 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
- Stochastic local volatility models, calibration
- Gyongy's Theorem
- Barrier options
- Volatility as an asset class, variance swaps, volatility derivatives
- Heston model
Assessment: Exam (80%), 2 class tests (20%)
Bergomi L (2016) Stochastic volatility modelling, Chapman and Hall
Buehler H (2009) Volatility Markets: Consistent Modeling, Hedging and Practical
Implementation of Variance Swap Market Models VDM Verlag Dr. Müller
Elouerkhaoui, Y (2017), Credit Correlation: Theory and Practice, Macmillan.
Hunt PJ and Kennedy JE, (2004), Financial Derivatives in Theory and Practice, second edition, Wiley.
Homescu, C, Local Stochastic Volatility Models: Calibration and Pricing (2014)
Pelsser A, (2000), Efficient Methods for Valuing Interest Rate Derivatives, Springer.
Glasserman P, (2004), Monte Carlo Methods in Financial Engineering, Springer.
Gatheral J, (2006) The Volatility Surface: A Practitioners Guide, Wiley
Examination Period: April