# ST909: Continuous Time Finance for Interest Rate Models

###### Lecturer(s): Dr Jo Kennedy

* Prerequisite(s):** *ST401 Stochastic Methods in Finance or ST403 Brownian Motion or ST908 Probability and Stochastic Processes (non Statistics students)

** Commitment: **3 hours of lectures per week and 1 hour tutorial. This module runs in Term 2.

**Restrictions:**

Students who are not enrolled in the MSc in Financial Mathematics may take at most two of the following modules:

• ST906 Financial Time Series.

• ST909 Continuous Time Finance for Interest Rate Models.

• ST958 Topics in Mathematical Finance.

*Important: This module is Core for MSc Financial Mathematics students.*

** Timetable: **Seminars will take place from 12:00-13:00 in WBS3.006, this will not show on student timetables.

*Content:*

**Mathematical Foundations**

1. Monotone convergence, dominated convergence and Fatou’s lemma for conditional expectation. Optional sampling theorem. Finite variation processes as integrators, quadratic variation for continous martingales, Meyer’s Theorem.

2. Continuous Local Martingales, properties of the stochastic integral with respect to continuous local martingales. Continuous semimartingales as integrators, integration by parts and multidimensional Ito’s formula for continuous semimartingales, Levy’s Theorem.

3. Radon-Nikodym derivative, Girsanov’s Theorem for semimartingales, Novikov’s condition Martingale Representation Theorem

**Option Pricing in Continuous Time**

4. Pricing via PDEs (brief review)

Pricing via equivalent martingale measures, fundamental valuation formula, arbitrage and admissible strategies. Completeness for the Black Scholes economy. Pricing kernels

5. Implied volatility, market implied distributions. Stochastic volatility and incomplete markets. Multicurrency Economy.

**Term Structure Models**

5. ctd. Short rate models. Introduction to main examples, implementation of Hull-White

6. Review of main types of term structure models including Pure discount Bond, Heath-Jarrow-Morton, Flesaker-Hughston.

7. Market Models (Brace, Gaterek and Musiela approach), specification in terminal and spot measure.

8. 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.

9. Markov-functional models

10. Practical issues in choice of model for various exotics, Bermudan swaptions, TARNS

Calibration: global versus local

** Examination:** April Exam (80%) 2 class tests (20%)

* Deadline:* Assignment 1: week 8 and Assignment 2: week 10.

* Feedback:* Feedback on class tests will be returned after 4 weeks, following each test.