# COMPG012 - Financial Engineering

This database contains 2016-17 versions of the syllabuses. For current versions please see here.

Code COMPG012 MSc Basic probability and differential equations 1 Riaz Ahmad (100%) An introduction to the applied mathematical and computational aspects of Quantitative Finance. Successful application of the necessary probability and differential equation based approach to the pricing of financial derivatives; using both quantitative and numerical techniques.

# Content

1. Financial Products and Markets: Time value of money and applications. Equities, indices, foreign exchange and commodities. Futures, Forwards and Options. Payoff and P&L diagrams. Put-Call parity.

2. Stochastic Calculus: Brownian motion and properties, Itô’s lemma and Itô integral. Stochastic Differential Equations – drift and diffusion; Geometric Brownian Motion and Vasicek model.

3. Binomial model: Key assumptions. Delta hedging and no arbitrage. Risk-neutral probability and replicating portfolio. Single step and multi-period model. European and American options.

4. Black-Scholes Model: Assumptions, PDE and pricing formulae for European calls and puts. Extending to dividends, FX and commodities. The Greeks and risk management - theta, delta, gamma, vega & rho and their role in hedging. Two factor models and multi-asset options.

5. Mathematics of early exercise: Perpetual American calls and puts; optimal exercise strategy and the smooth pasting condition.

6. Computational Finance: Solving the pricing PDEs numerically using Explicit Finite Difference Scheme.

Stability criteria. Introduction to Monte Carlo technique for derivative pricing. Random number generation in Excel – RAND(), NORMSINV(), simulating random walks, correlations. Examining statistical properties of stock returns.

7. Stochastic interest rate models: Fixed income world – zero coupon bonds and coupon bearing bonds; yield curves, duration and convexity. Bond Pricing Equation (BPE). Popular models for the spot rate - Vasicek, CIR, Ho & Lee and Hull & White. Solutions of the BPE.

8. Introduction to Exotics: Basic features and classification of exotic options. Simple exotics – Binaries, one-touch, power options, compound and exchange options. Weak and strong path dependency - barriers, Asians and Lookbacks. Sampling - continuous and discrete. Pricing using the PDE framework.

# Method of Instruction

30 hours of lectures including 3 hours of computing sessions

# Assessment

The course has the following assessment components:

• Assessed assignments (30%)

• 2.5 hour written examination in the summer (70%)

To pass this course, students must:

• Obtain an overall pass mark of 50% for all sections combined.

# Resources

Recommended Book

• Paul Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley & Sons, 2007

Notes

• CFA Level 1 Quants (slide pack), Fitch Learning