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Department of Statistics

ST909 Applications of Stochastic Calculus for Finance


Availability:

Commitment: 30 hours of lectures 10 hours of tutorials

Content:

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

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

Assessment: Exam (80%), 2 class tests (20%)

Illustrative Bibliography:

  • 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